Reduced homography for recovery of pose parameters of an optical apparatus producing image data with structural uncertainty

[0001] The present invention relates generally to determining pose parameters (position and orientation parameters) of an optical apparatus in a stable frame, the pose parameters of the optical apparatus being recovered from image data collected by the optical apparatus and being imbued with a structural uncertainty that necessitates deployment of a reduced homography.

[0002] When an item moves without any constraints (freely) in a three-dimensional environment with respect to stationary objects, knowledge of the item's distance and inclination to one or more of such stationary objects can be used to derive a variety of the item's parameters of motion, as well as its complete pose. The latter includes the item's three position parameters, usually expressed by three coordinates (x, y, z), and its three orientation parameters, usually expressed by three angles (α, β, γ) in any suitably chosen rotation convention (e.g., Euler angles (ψ, θ, φ) or quaternions). Particularly useful stationary objects for pose recovery purposes include ground planes, fixed points, lines, reference surfaces and other known features.

[0003] Many mobile electronics items are now equipped with advanced optical apparatus such as on-board cameras with photo-sensors, including high-resolution CMOS arrays. These devices typically also possess significant on-board processing resources (e.g., CPUs and GPUs) as well as network connectivity (e.g., connection to the Internet, Cloud services and/or a link to a Local Area Network (LAN)). These resources enable many techniques from the fields of robotics and computer vision to be practiced with the optical apparatus on-board such virtually ubiquitous devices. Most importantly, vision algorithms for recovering the camera's extrinsic parameters, namely its position and orientation, also frequently referred to as its pose, can now be applied in many practical situations.

[0004] An on-board camera's extrinsic parameters in the three dimensional environment are typically recovered by viewing a sufficient number of non-collinear optical features belonging to the known stationary object or objects. In other words, the on-board camera first records on its photo-sensor (which may be a pixelated device or even a position sensing device (PSD) having one or just a few “pixels”) the images of space points, space lines and space planes belonging to one or more of these known stationary objects. A computer vision algorithm to recover the camera's extrinsic parameters is then applied to the imaged features of the actual stationary object(s). The imaged features usually include points, lines and planes of the actual stationary object(s) that yield a good optical signal. In other words, the features are chosen such that their images exhibit a high degree of contrast and are easy to isolate in the image taken by the photo-sensor. Of course, the imaged features are recorded in a two-dimensional (2D) projective plane associated with the camera's photo-sensor, while the real or space features of the one or more stationary objects are found in the three-dimensional (3D) environment.

[0005] Certain 3D information is necessarily lost when projecting an image of actual 3D stationary objects onto the 2D image plane. The mapping between the 3D Euclidean space of the three-dimensional environment and the 2D projective plane of the camera is not one-to-one. Many assumptions of Euclidean geometry are lost during such mapping (sometimes also referred to as projectivity). Notably, lengths, angles and parallelism are not preserved. Euclidean geometry is therefore insufficient to describe the imaging process. Instead, projective geometry, and specifically perspective projection is deployed to recover the camera's pose from images collected by the photo-sensor residing in the camera's 2D image plane.

[0006] Fortunately, projective transformations do preserve certain properties. These properties include type (that is, points remain points and lines remain lines), incidence (that is, when a point lies on a line it remains on the line), as well as an invariant measure known as the cross ratio. For a review of projective geometry the reader is referred to H. X. M. Coexter, Projective Geometry, Toronto: University of Toronto, 2^nd Edition, 1974; O. Faugeras, Three-Dimensional Computer Vision, Cambridge, Mass.: MIT Press, 1993; L. Guibas, “Lecture Notes for CSS4Sa: Computer Graphics—Mathematical Foundations”, Stanford University, Autumn 1996; Q.-T. Luong and O. D. Faugeras, “Fundamental Matrix: Theory, algorithms and stability analysis”, International Journal of Computer Vision, 17(1): 43-75, 1996; J. L. Mundy and A. Zisserman, Geometric Invariance in Computer Vision, Cambridge, Mass.: MIT Press, 1992 as well as Z. Zhang and G. Xu, Epipolar Geometry in Stereo, Motion and Object Recognition: A Unified Approach. Kluwer Academic Publishers, 1996.

[0007] At first, many practitioners deployed concepts from perspective geometry directly to pose recovery. In other words, they would compute vanishing points, horizon lines, cross ratios and apply Desargues theorem directly. Although mathematically simple on their face, in many practical situations such approaches end up in tedious trigonometric computations. Furthermore, experience teaches that such computations are not sufficiently compact and robust in practice. This is due to many real-life factors including, among other, limited computation resources, restricted bandwidth and various sources of noise.

[0008] Modern computer vision has thus turned to more computationally efficient and robust approaches to camera pose recovery. An excellent overall review of this subject is found in Kenichi Kanatani, Geometric Computation for Machine Vision, Clarendon Press, Oxford University Press, New York, 1993. A number of important foundational aspects of computational geometry relevant to pose recovery via machine vision are reviewed below to the benefit of those skilled in the art and in order to better contextualize the present invention.

[0009] To this end, we will now review several relevant concepts in reference to FIGS. 1-3. FIG. 1 shows a stable three-dimensional environment 10 that is embodied by a room with a wall 12 in this example. A stationary object 14, in this case a television, is mounted on wall 12. Television 14 has certain non-collinear optical features 16A, 16B, 16C and 16D that in this example are the corners of its screen 18. Corners 16A, 16B, 16C and 16D are used by a camera 20 for recovery of extrinsic parameters (up to complete pose recovery when given a sufficient number and type of non-collinear features). Note that the edges of screen 18 or even the entire screen 18 and/or anything displayed on it (i.e., its pixels) are suitable non-collinear optical features for these purposes. Of course, other stationary objects in room 10 besides television 14 can be used as well.

[0010] Camera 20 has an imaging lens 22 and a photo-sensor 24 with a number of photosensitive pixels 26 arranged in an array. A common choice for photo-sensor 24 in today's consumer electronics devices are CMOS arrays, although other technologies can also be used depending on application (e.g., CCD, PIN photodiode, position sensing device (PSD) or still other photo-sensing technology). Imaging lens 22 has a viewpoint O and a certain focal length f. Viewpoint O lies on an optical axis OA. Photo-sensor 24 is situated in an image plane at focal length f behind viewpoint O along optical axis OA.

[0011] Camera 20 typically works with electromagnetic (EM) radiation 30 that is in the optical or infrared (IR) wavelength range (note that deeper sensor wells are required in cameras working with IR and far-IR wavelengths). Radiation 30 emanates or is reflected (e.g., reflected ambient EM radiation) from non-collinear optical features such as screen corners 16A, 16B, 16C and 16D. Lens 22 images EM radiation 30 on photo-sensor 24. Imaged points or corner images 16A′, 16B′, 16C′, 16D′ thus imaged on photo-sensor 24 by lens 22 are usually inverted when using a simple refractive lens. Meanwhile, certain more compound lens designs, including designs with refractive and reflective elements (catadioptrics) can yield non-inverted images.

[0012] A projective plane 28 conventionally used in computational geometry is located at focal length f away from viewpoint O along optical axis OA but in front of viewpoint O rather than behind it. Note that a virtual image of corners 16A, 16B, 16C and 16D is also present in projective plane 28 through which the rays of electromagnetic radiation 30 pass. Because any rays in projective plane 28 have not yet passed through lens 22, the points representing corners 16A, 16B, 16C and 16D are not inverted. The methods of modern machine vision are normally applied to points in projective plane 28, while taking into account the properties of lens 22.

[0013] An ideal lens is a pinhole and the most basic approaches of machine vision make that an assumption. Practical lens 22, however, introduces distortions and aberrations (including barrel distortion, pincushion distortion, spherical aberration, coma, astigmatism, chromatic aberration, etc.). Such distortions and aberrations, as well as methods for their correction or removal are understood by those skilled in the art.

[0014] In the simple case shown in FIG. 1, image inversion between projective plane 28 and image plane on the surface of photo-sensor 24 is rectified by a corresponding matrix (e.g., a reflection and/or rotation matrix). Furthermore, any offset between a center CC of camera 20 where optical axis OA passes through the image plane on the surface of photo-sensor 24 and the origin of the 2D array of pixels 26, which is usually parameterized by orthogonal sensor axes (X_s, Y_s), involves a shift.

[0015] Persons skilled in the art are familiar with camera calibration techniques. These include finding offsets, computing the effective focal length f_eff (or the related parameter k) and ascertaining distortion parameters (usually denoted by α's). Collectively, these parameters are called intrinsic and they can be calibrated in accordance with any suitable method. For teachings on camera calibration the reader is referred to the textbook entitled “Multiple View Geometry in Computer Vision” (Second Edition) by R. Hartley and Andrew Zisserman. Another useful reference is provided by Robert Haralick, “Using Perspective Transformations in Scene Analysis”, Computer Graphics and Image Processing 13, pp. 191-221 (1980). For still further information the reader is referred to Carlo Tomasi and John Zhang, “How to Rotate a Camera”, Computer Science Department Publication, Stanford University and Berthold K. P. Horn, “Tsai's Camera Calibration Method Revisited”, which are herein incorporated by reference.

[0016] Additionally, image processing is required to discover corner images 16A′, 16B′, 16C′, 16D′ on sensor 24 of camera 20. Briefly, image processing includes image filtering, smoothing, segmentation and feature extraction (e.g., edge/line or corner detection). Corresponding steps are usually performed by segmentation and the application of mask filters such as Guassian/Laplacian/Laplacian-of-Gaussian (LoG)/Marr and/or other convolutions with suitable kernels to achieve desired effects (averaging, sharpening, blurring, etc.). Most common feature extraction image processing libraries include Canny edge detectors as well as Hough/Radon transforms and many others. Once again, all the relevant techniques are well known to those skilled in the art. A good review of image processing is afforded by “Digital Image Processing”, Rafael C. Gonzalez and Richard E. Woods, Prentice Hall, 3^rd Edition, Aug. 31, 2007; “Computer Vision: Algorithms and Applications”, Richard Szeliski, Springer, Edition 2011, Nov. 24, 2010; Tinne Tuytelaars and Krystian Mikolajczyk, “Local Invariant Feature Detectors: A Survey”, Journal of Foundations and Trends in Computer Graphics and Vision, Vol. 3, Issue 3, January 2008, pp. 177-280. Furthermore, a person skilled in the art will find all the required modules in standard image processing libraries such as OpenCV (Open Source Computer Vision), a library of programming functions for real time computer vision. For more information on OpenCV the reader is referred to G. R. Bradski and A. Kaehler, “Learning OpenCV: Computer Vision with the OpenCV Library”, O'Reilly, 2008.

[0017] In FIG. 1 camera 20 is shown in a canonical pose. World coordinate axes (X_w,Y_w,Z_w) define the stable 3D environment with the aid of stationary object 14 (the television) and more precisely its screen 18. World coordinates are right-handed with their origin in the middle of screen 18 and Z_w-axis pointing away from camera 20. Meanwhile, projective plane 28 is parameterized by camera coordinates with axes (X_c,Y_c,Z_c). Camera coordinates are also right-handed with their origin at viewpoint O. In the canonical pose Z_c-axis extends along optical axis OA away from the image plane found on the surface of image sensor 24. Note that camera Z_c-axis intersects projective plane 28 at a distance equal to focal length f away from viewpoint O at point o′, which is the center (origin) of projective plane 28. In the canonical pose, the axes of camera coordinates and world coordinates are thus aligned. Hence, optical axis OA that always extends along the camera Z_c-axis is also along the world Z_w-axis and intersects screen 18 of television 14 at its center (which is also the origin of world coordinates). In the application shown in FIG. 1, a marker or pointer 32 is positioned at the intersection of optical axis OA of camera 20 and screen 18.

[0018] In the canonical pose, the rectangle defined by space points representing screen corners 16A, 16B, 16C and 16D maps to an inverted rectangle of corner images 16A′, 16B′, 16C′, 16D′ in the image plane on the surface of image sensor 24. Also, space points defined by screen corners 16A, 16B, 16C and 16D map to a non-inverted rectangle in projective plane 28. Therefore, in the canonical pose, the only apparent transformation performed by lens 22 of camera 20 is a scaling (de-magnification) of the image with respect to the actual object. Of course, mostly correctable distortions and aberrations are also present in the case of practical lens 22, as remarked above.

[0019] Recovery of poses (positions and orientations) assumed by camera 20 in environment 10 from a sequence of corresponding projections of space points representing screen corners 16A, 16B, 16C and 16D is possible because the absolute geometry of television 14 and in particular of its screen 18 and possibly other 3D structures providing optical features in environment 10 are known and can be used as reference. In other words, after calibrating lens 22 and observing the image of screen corners 16A, 16B, 16C, 16D and any other optical features from the canonical pose, the challenge of recovering parameters of absolute pose of camera 20 in three-dimensional environment 10 is solvable. Still more precisely put, as camera 20 changes its position and orientation and its viewpoint O travels along a trajectory 34 (a.k.a. extrinsic parameters) in world coordinates parameterized by axes (X_w,Y_w,Z_w), only the knowledge of corner images 16A′, 16B′, 16C′, 16D′ in camera coordinates parameterized by axes (X_c,Y_c,Z_c) can be used to recover the changes in pose or extrinsic parameters of camera 20. This exciting problem in computer and robotic vision has been explored for decades.

[0020] Referring to FIG. 2, we now review a typical prior art approach to camera pose recovery in world coordinates (a.k.a. absolute pose, since world coordinates defined by television 14 sitting in room 10 are presumed stable for the purposes of this task). In this example, camera 20 is mounted on-board item 36, which is a mobile device and more specifically a tablet computer with a display screen 38. The individual parts of camera 20 are not shown explicitly in FIG. 2, but non-inverted image 18′ of screen 18 as found in projective plane 28 is illustrated on display screen 38 of tablet computer 36 to aid in the explanation. The practitioner is cautioned here, that although the same reference numbers refer to image points in the image plane on sensor 24 (see FIG. 1) and in projective plane 28 to limit notational complexity, a coordinate transformation exists between image points in the actual image plane and projective plane 28. As remarked above, this transformation typically involves a reflection/rotation matrix and an offset between camera center CC and the actual center of sensor 24 discovered during the camera calibration procedure (also see FIG. 1).

[0021] A prior location of camera viewpoint O along trajectory 34 and an orientation of camera 20 at time t=t_−i are indicated by camera coordinates using camera axes (X_c,Y_c,Z_c) whose origin coincides with viewpoint O. Clearly, at time t=t_−i camera 20 on-board tablet 36 is not in the canonical pose. The canonical pose, as shown in FIG. 1, obtains at time t=t_o. Given unconstrained motion of viewpoint O along trajectory 34 and including rotations in three-dimensional environment 10, all extrinsic parameters of camera 20 and correspondingly the position and orientation (pose) of tablet 36 change between time t=t_−i and t=t_o. Still differently put, all six degrees of freedom (6 DOFs or the three translational and the three rotational degrees of freedom inherently available to rigid bodies in three-dimensional environment 10) change along trajectory 34.

[0022] Now, at time t=t₁ tablet 36 has moved further along trajectory 34 from its canonical pose at time t=t_o to an unknown pose where camera 20 records corner images 16A′, 16B′, 16C′, 16D′ at the locations displayed on screen 38 in projective plane 28. Of course, camera 20 actually records corner images 16A′, 16B′, 16C′, 16D′ with pixels 26 of its sensor 24 located in the image plane defined by lens 22 (see FIG. 1). As indicated above, a known transformation exists (based on camera calibration of intrinsic parameters, as mentioned above) between the image plane of sensor 24 and projective plane 28 that is being shown in FIG. 2.

[0023] In the unknown camera pose at time t=t₁ a television image 14′ and, more precisely screen image 18′ based on corner images 16A′, 16B′, 16C′, 16D′ exhibits a certain perspective distortion. By comparing this perspective distortion of the image at time t=t₁ to the image obtained in the canonical pose (at time t=t_o or during camera calibration procedure) one finds the extrinsic parameters of camera 20 and, by extension, the pose of tablet 36. By performing this operation with a sufficient frequency, the entire rigid body motion of tablet 36 along trajectory 34 of viewpoint O can be digitized.

[0024] The corresponding computation is traditionally performed in projective plane 28 by using homogeneous coordinates and the rules of perspective projection as taught in the references cited above. For a representative prior art approach to pose recovery with respect to rectangles, such as presented by screen 18 and its corners 16A, 16B, 16C and 16D the reader is referred to T. N. Tan et al., “Recovery of Intrinsic and Extrinsic Camera Parameters Using Perspective Views of Rectangles”, Dept. of Computer Science, The University of Reading, Berkshire RG6 6AY, UK, 1996, pp. 177-186 and the references cited by that paper. Before proceeding, it should be stressed that although in the example chosen we are looking at rectangular screen 18 that can be analyzed by defining vanishing points and/or angle constraints on corners formed by its edges, pose recovery does not need to be based on corners of rectangles or structures that have parallel and orthogonal edges. In fact, the use of vanishing points is just the elementary way to recover pose. There are more robust and practical prior art methods that can be deployed in the presence of noise and when tracking more than four reference features (sometimes also referred to as fiducials) that do not need to form a rectangle or even a planar shape in real space. Indeed, the general approach applies to any set of fiducials defining an arbitrary 3D shape, as long as that shape is known.

[0025] For ease of explanation, however, FIG. 3 highlights the main steps of an elementary prior art approach to the recovery of extrinsic parameters of camera 20 based on the rectangle defined by screen 18 in world coordinates parameterizing room 10 (also see FIG. 2). Recovery is performed with respect to the canonical pose shown in FIG. 1. The solution is a rotation expressed by a rotation matrix R and a translation expressed by a translation vector h, or {R, h}.

[0026] In other words, the application of inverse rotation matrix R⁻¹ and subtraction of translation vector h return camera 20 from the unknown recovered pose to its canonical pose. The canonical pose at t=t_o is marked and the unknown pose at t=t₁ is to be recovered from image 18′ found in projective plane 28 (see FIG. 2), as shown on display screen 38. In solving the problem we need to find vectors p_A, p_B, p_C and p_D from viewpoint O to space points 16A, 16B, 16C and 16D through corner images 16A′, 16B′, 16C′ and 16D′. Then, information contained in computed conjugate vanishing points 40A, 40B can be used for the recovery. In cases where the projection is almost orthographic (little or no perspective distortion in screen image 18′) and vanishing points 40A, 40B become unreliable, angle constraints demanding that the angles between adjoining edges of candidate recovered screen 18 be 90° can be used, as taught by T. N. Tan et al., op. cit.

[0027] FIG. 3 shows that without explicit information about the size of screen 18, the length of one of its edges (or other scale information) only relative lengths of vectors p_A, p_B, p_C and p_D can be found. In other words, when vectors p_A, p_B, p_C and p_D are expressed by corresponding unit vectors {circumflex over (n)}_A, {circumflex over (n)}_B, {circumflex over (n)}_C, {circumflex over (n)}_D times scale constants λ_A, λ_B, λ_C, λ_D such that p_A={circumflex over (n)}_Aλ_A, p_B={circumflex over (n)}_Bλ_B, p_C={circumflex over (n)}_C, and p_D={circumflex over (n)}_Dλ_D, then only relative values of scale constants λ_A, λ_B, λ_C, λ_D can be obtained. This is clear from looking at a small dashed candidate for screen 18* with corner points 16A*, 16B*, 16C*, 16D*. These present the correct shape for screen 18* and lie along vectors p_A, p_B, p_C and p_D, but they are not the correctly scaled solution.

[0028] Also, if space points 16A, 16B, 16C and 16D are not identified with image points 16A′, 16B′, 16C′ and 16D′ then the in-plane orientation of screen 18 cannot be determined. This labeling or correspondence problem is clear from examining a candidate for recovered screen 18*.

[0029] Its recovered corner points 16A*, 16B*, 16C* and 16D* do not correspond to the correct ones of actual screen 18 that we want to find. The correspondence problem can be solved by providing information that uniquely identifies at least some of points 16A, 16B, 16C and 16D. Alternatively, additional space points that provide more optical features at known locations in room 10 can be used to break the symmetry of the problem. Otherwise, the space points can be encoded by any suitable methods and/or means. Of course, space points that present intrinsically asymmetric space patterns could be used as well.

[0030] Another problem is illustrated by candidate for recovered screen 18**, where candidate points 16A**, 16B**, 16C**, 16D** do lie along vectors p_A, p_B, p_C and p_D but are not coplanar. This structural defect is typically resolved by realizing from algebraic geometry that dot products of vectors that are used to represent the edges of candidate screen 18** not only need to be zero (to ensure orthogonal corners) but also that the triple product of these vectors needs to be zero. That is true, since the triple product of the edge vectors is zero for a rectangle. Still another way to remove the structural defect involves the use of cross ratios.

[0031] In addition to the above problems, there is noise. Thus, the practical challenge is not only in finding the right candidate based on structural constraints, but also distinguishing between possible candidates and choosing the best one in the presence of noise. In other words, the real-life problem of pose recovery is a problem of finding the best estimate for the transformation encoded by {R, h} from the available measurements. To tackle this problem, it is customary to work with the homography or collineation matrix A that expresses {R, h}. In this form, the well-known methods of linear algebra can be brought to bear on the problem of estimating A. Once again, the reader should remember that these tools can be applied for any set of optical features (fiducials) and not just rectangles as formed by screen 18 used for explanatory purposes in this case. In fact, any set of fiducials defining any 3D shape in room 10 can be used, as long as that 3D shape is known. Additionally, such 3D shape should have a geometry that produces a sufficiently large image from all vantage points (see definition of convex hull).

[0032] FIGS. 4A & 4B illustrate realistic situations in which estimates of collineation matrices A are computed in the presence of noise for our simple example. FIG. 4A shows on the left a full field of view 42 (F.O.V.) of lens 22 centered on camera center CC while camera 20 is in the canonical pose (also see FIG. 1). Field of view 42 is parameterized by sensor coordinates of photo-sensor 24 using sensor axes (X_s,Y_s) Note that pixelated sensors like sensor 24 usually take the origin of array of pixels 26 to be in the upper corner. Also note that camera center CC has an offset (x_sc,y_sc) from the origin. In fact, (x_sc,Y_sc) is the location of viewpoint O and origin o′ of projective plane 28 in sensor coordinates (previously shown in camera coordinates (X_c,Y_c,Z_c)—see FIG. 1). Working in sensor coordinates is initially convenient because screen image 18′ is first recorded along with noise by pixels 26 of sensor 24 in the image plane that is parameterized by sensor coordinates. Note the inversion of real screen image 18′ on sensor 24 in comparison to virtual screen image 18′ in projective plane 28 (again see FIG. 1).

[0033] On the right, FIG. 4A illustrates screen image 18′ after viewpoint O has moved along trajectory 34 and camera 20 assumed a pose corresponding to an unknown collineation A₁ with respect to the canonical pose shown on the left. Collineation A₁ consists of an unknown rotation and an unknown translation {R, h}. Due to noise, there are a number of measured image points {circumflex over (p)}_i=({circumflex over (x)}_i,ŷ_i), indicated by crosses, for corner images 16A′, 16B′, 16C′ and 16D′. (Here the “hat” denotes measured values not unit vectors.) The best estimate of collineation A₁, referred to as Θ (estimation matrix), yields the best estimate of the locations of corner images 16A′, 16B′, 16C′ and 16D′ in the image plane. The value of estimation matrix Θ is usually found by minimizing a performance criterion through mathematical optimization. Suitable methods include the application of least squares, weighted average or other suitable techniques to process measured image points {circumflex over (p)}_i({circumflex over (x)}_i,ŷ_i). Note that many prior art methods also include outlier rejection of certain measured image points {circumflex over (p)}_i=({circumflex over (x)}_i,ŷ_i) that could “skew” the average. Various voting algorithms including RANSAC can be deployed to solve the outlier problem prior to averaging.

[0034] FIG. 4B shows screen image 18′ as recorded in another pose of camera 20. This one corresponds to a different collineation A₂ with respect to the canonical pose. Notice that the composition of collineations behaves as follows: collineation A₁ followed by collineation A₂ is equivalent to composition A₁A₂. Once again, measured image points {circumflex over (p)}_i=({circumflex over (x)}_i,ŷ_i) for the estimate computation are indicated.

[0035] The distribution of measured image points {circumflex over (p)}_i=({circumflex over (x)}_i,ŷ_i) normally obeys a standard noise statistic dictated by environmental conditions. When using high-quality camera 20, that distribution is thermalized based mostly on the illumination conditions in room 10, the brightness of screen 18 and edge/corner contrast (see FIG. 2). This is indicated in FIG. 4B by a dashed outline indicating a normal error region or typical deviation 44 that contains most possible measured image points {circumflex over (p)}_i=({circumflex over (x)}_i,ŷ_i) excluding outliers. An example outlier 46 is indicated well outside typical deviation 44.

[0036] In some situations, however, the distribution of points {circumflex over (p)}_i=({circumflex over (x)}_i,ŷ_i) does not fall within typical error region 44 accompanied by a few outliers 46. In fact, some cameras introduce persistent or even inherent structural uncertainty into the distribution of points {circumflex over (p)}_i=({circumflex over (x)}_i,ŷ_i) found in the image plane on top of typical deviation 44 and outliers 46.

[0037] One typical example of such a situation occurs when the optical system of a camera introduces multiple reflections of bright light sources (which are prime candidates for space points to track) onto the sensor. This may be due to the many optical surfaces that are typically used in the imaging lenses of camera systems. In many cases, these multiple reflections can cause a number of ghost images along radial lines extending from the center of the sensor or camera center CC as shown in FIG. 1 to the point where the optical axis OA of the lens intersects with the sensor. This condition results in a large inaccuracy when using the image to measure the radial distance of the primary image of a light source. The prior art teaches no suitable formulation of the homography or collineation to nonetheless recover parameters of camera pose under such conditions.

[0038] In view of the shortcomings of the prior art, it is an object of the present invention to provide for recovering parameters of pose or extrinsic parameters of an optical apparatus up to and including complete pose recovery (all six parameters or degrees of freedom) in the presence of structural uncertainty that is introduced into the image data. The optical apparatus may itself be responsible for introducing the structural uncertainty and it can be embodied by a CMOS camera, a CCD sensor, a PIN diode sensor, a position sensing device (PSD), or still some other optical apparatus. In fact, the optical apparatus should be able to deploy any suitable optical sensor and associated imaging optics.

[0039] It is another object of the invention to support estimation of a homography representing the pose of an item that has the optical apparatus installed on-board. The approach should enable selection of an appropriate reduced representation of the image data (e.g., measured image points) based on the specific structural uncertainty. The reduced representation should support deployment of a reduced homography that permits the use of low quality cameras, including low-quality sensors and/or low-quality optics, to recover desired parameters of pose or even full pose of the item with the on-board optical apparatus despite the presence of structural uncertainty.

[0040] Yet another object of the invention is to provide for complementary data fusion with on-board inertial apparatus to allow for further reduction in quality or acquisition rate of optical data necessary to recover the pose of the optical apparatus or of the item with the on-board optical apparatus.

[0041] Still other objects and advantages of the invention will become apparent upon reading the detailed specification and reviewing the accompanying drawing figures.

[0042] The objects and advantages of the invention are provided for by a method and an optical apparatus for recovering pose parameters from imaged space points P_i using an optical sensor. The electromagnetic radiation from the space points P_i is recorded on the optical sensor at measured image coordinates {circumflex over (x)}_i,ŷ_i that define the locations of measured image points {circumflex over (p)}_i=({circumflex over (x)}_i,ŷ_i) in the image plane. A structural uncertainty introduced in the measured image points {circumflex over (p)}_i=({circumflex over (x)}_i,ŷ_i) is determined. A reduced representation of the measured image points {circumflex over (p)}_i=({circumflex over (x)}_i,ŷ_i) is selected based on the type of structural uncertainty. The reduced representation includes rays {circumflex over (r)}_i defined in homogeneous coordinates and contained in a projective plane of the optical apparatus. At least one pose parameter of the optical apparatus is then estimated with respect to a canonical pose of the optical apparatus by applying a reduced homography H that uses the rays {circumflex over (r)}_i of the reduced representation.

[0043] When using the reduced representation resulting in reduced homography H it is important to set a condition on the motion of the optical apparatus based on the reduced representation. For example, the condition can be strict and enforced by a mechanism constraining the motion, including a mechanical constraint. In particular, the condition is satisfied by substantially bounding the motion to a reference plane. In practice, the condition does not have to be kept the same at all times. In fact, the condition can be adjusted based on one or more of the pose parameters of the optical apparatus. In most cases, the most useful pose parameters involve a linear pose parameter, i.e., a distance from a known point or plane in the environment.

[0044] The pose parameter or parameters used in adjusting the condition on the motion of the optical apparatus, or of an item that has such optical apparatus installed on-board, can be recovered independently of the pose estimation step that deploys the reduced homography H. In some embodiments an auxiliary measurement can be performed to obtain the one or more pose parameters used for adjusting the condition. More precisely, an independent optical, acoustic, inertial or even RF measurement can be performed for this purpose. In the case of the optical measurement, the same optical apparatus can be deployed and the measurement can be a depth-from-defocus or a time-of-flight based measurement.

[0045] Depending on the embodiment, the type of optical apparatus and on the condition placed on the motion of the optical apparatus, the structural uncertainty will differ. In some embodiments, the structural uncertainty will be substantially radial, meaning that the uncertainty of measured image points {circumflex over (p)}_i=({circumflex over (x)}_i,ŷ_i) is uncertain along a radial direction from the center of the optical sensor or from the point of view O established by the optics of the optical apparatus. In other cases, the structural uncertainty will be substantially linear (e.g., along vertical or horizontal lines). Structural uncertainty differs from normal noise, which is mostly due to thermal noise, 1/f noise and shot noise, in that it exhibits a substantially larger spread than normal noise.

[0046] The present invention, including the preferred embodiment, will now be described in detail in the below detailed description with reference to the attached drawing figures.

[0077] The drawing figures and the following description relate to preferred embodiments of the present invention by way of illustration only. It should be noted that from the following discussion, alternative embodiments of the methods and systems disclosed herein will be readily recognized as viable options that may be employed without departing from the principles of the claimed invention. Likewise, the figures depict embodiments of the present invention for purposes of illustration only. One skilled in the art will readily recognize from the following description that alternative embodiments of the methods and systems illustrated herein may be employed without departing from the principles of the invention described herein.

[0078] The present invention will be best understood by initially referring to FIG. 5A. This drawing figure illustrates in a perspective view a stable three-dimensional environment 100 in which an item 102 equipped with an on-board optical apparatus 104 is deployed in accordance with the invention. It should be noted, that the present invention relates to the recovery of pose by optical apparatus 104 itself. It is thus not limited to any item that has optical apparatus 104 installed on-board. However, for clarity of explanation and a better understanding of the fields of use, it is convenient to base the teachings on concrete examples. In this vein, a cell phone or a smart phone embodies item 102 and a CMOS camera embodies on-board optical apparatus 104.

[0079] CMOS camera 104 has a viewpoint O from which it views environment 100. In general, item 102 is understood herein to be any object that is equipped with an on-board optical unit and is manipulated by a user or even worn by the user. For some additional examples of suitable items the reader is referred to U.S. Published Application 2012/0038549 to Mandella et al.

[0080] Environment 100 is not only stable, but it is also known. This means that the locations of exemplary stationary objects 106, 108, 110, 112, 114, 116 present in environment 100 and embodied by a refrigerator, a corner between two walls and a ceiling, a table, a microwave oven, a toaster and a kitchen stove, respectively, are known prior to practicing a reduced homography H according to the invention. More precisely still, the locations of non-collinear optical features designated here by space points P₁, P₂, . . . , P_i and belonging to refrigerator 106, corner 108, table 110, microwave oven 112, toaster 114 and kitchen stove 116 are known prior to practicing reduced homography H of the invention.

[0081] A person skilled in the art will recognize that working in known environment 100 is a fundamentally different problem from working in an unknown environment. In the latter case, optical features are also available, but their locations in the environment are not known a priori. Thus, a major part of the challenge is to construct a model of the unknown environment before being able to recover any of the camera's extrinsic parameters (position and orientation in the environment, together defining the pose). The present invention applies to known environment 100 in which the positions of objects 106, 108, 110, 112, 114, 116 and hence of the non-collinear optical features P₁, P₂, . . . , P₉ are known a priori, e.g., either from prior measurements, surveys or calibration procedures that may include non-optical measurements, as discussed in more detail below.

[0082] The actual non-collinear optical features designated by space points P_l, P₂, . . . , P₉ can be any suitable, preferably high optical contrast parts, markings or aspects of objects 106, 108, 110, 112, 114, 116. The optical features can be passive, active (i.e., emitting electromagnetic radiation) or reflective (even retro-reflective if illumination from on-board item 102 is deployed, e.g., in the form of a flash). In the present embodiment, optical feature designated by space point P₁ is a corner of refrigerator 106 that offers inherently high optical contrast because of its location against the walls. Corner 108 designated by space point P₂ is also high optical contrast. Table 110 has two optical features designated by space points P₃ and P₆, which correspond to its back corner and the highly reflective metal support on its front leg. Microwave oven 112 offers high contrast feature denoted by space point P₄ representing its top reflective identification plate. Space point P₅ corresponds to the optical feature represented by a shiny handle of toaster 114. Finally, space points P₇, P₈ and P₉ are optical features belonging to kitchen stove 116 and they correspond to a marking in the middle of the baking griddle, an LED display and a lighted turn knob, respectively.

[0083] It should be noted that any physical features, as long as their optical image is easy to discern, can serve the role of optical features. Preferably, more than just four optical features are selected in order to ensure better performance in pose recovery and to ensure that a sufficient number of them, preferably at least four, remain in the field of view of CMOS camera 104, even when some are obstructed, occluded or unusable for any other reasons. In the subsequent description, we will refer simply to space points P₁, P₂, . . . , P₉ as space points P_i or non-collinear optical features interchangeably. It will also be understood by those skilled in the art that the choice of space points P_i can be changed at any time, e.g., when image analysis reveals space points that offer higher optical contrast than those used at the time or when other space points offer optically advantageous characteristics. For example, the distribution of the space points along with additional new space points presents a better geometrical distribution (e.g., a larger convex hull) and is hence preferable for pose recovery.

[0084] As already indicated, camera 104 of smart phone 102 sees environment 100 from point of view O. Point of view O is defined by the design of camera 104 and, in particular, by the type of optics camera 104 deploys. In FIG. 5A, phone 102 is shown in three different poses at times t=t_−i, t=t_o and t=t₁ with the corresponding locations of point of view O being labeled. For purposes of better understanding, at time t=t_o phone 102 is held by an unseen user such that viewpoint O of camera 104 is in a canonical pose. The canonical pose is used as a reference for computing a reduced homography H according to the invention.

[0085] In deploying reduced homography H a certain condition has to be placed on the motion of phone 102 and hence of camera 104. The condition depends on the type of reduced homography H. The condition is satisfied in the present embodiment by bounding the motion of phone 102 to a reference plane 118. This confinement does not need to be exact and it can be periodically reevaluated or changed, as will be explained further below. Additionally, a certain forward displacement ε_f and a certain back displacement ε_b away from reference plane 118 are permitted. Note that the magnitudes of displacements ε_f, ε_b do not have to be equal.

[0086] The condition is thus indicated by the general volume 120, which is the volume bounded by parallel planes at ε_f and ε_b and containing reference plane 118. This condition means that a trajectory 122 executed by viewpoint O of camera 104 belonging to phone 102 is confined to volume 120. Indeed, this condition is obeyed by trajectory 122 as shown in FIG. 5A.

[0087] Phone 102 has a display screen 124. To aid in the explanation of the invention, screen 124 shows what the optical sensor (not shown in the present drawing) of camera 104 sees or records. Thus, display screen 124 at time t=t_o, as shown in the lower enlarged portion of FIG. 5A, depicts an image 100′ of environment 100 obtained by camera 104 when phone 102 is in the canonical pose. Similarly, display screen 124 at time t=t₁, as shown in the upper enlarged portion of FIG. 5A, depicts image 100′ of environment 100 taken by camera 104 at time t=t₁. (We note that image 100′ on display screen 124 is not inverted. This is done for ease of explanation. A person skilled in the art will realize, however, that image 100′ as seen by the optical sensor can be inverted depending on the types of optics used by camera 104).

[0088] FIG. 5B is another perspective view of environment 100 in which phone 102 is shown in the pose assumed at time t=t₁, as previously shown in FIG. 5A. In FIG. 5B we see electromagnetic radiation 126 generally indicated by photons propagating from space points P_i to on-board CMOS camera 104 of phone 102. Radiation 126 is reflected or scattered ambient radiation and/or radiation produced by the optical feature itself. For example, optical features corresponding to space points P₈ and P₉ are LED display and lighted turn knob belonging to stove 116. Both of these optical features are active (illuminated) and thus produce their own radiation 126.

[0089] Radiation 126 should be contained in a wavelength range that camera 104 is capable of detecting. Visible as well as IR wavelengths are suitable for this purpose. Camera 104 thus images all unobstructed space points P_i using its optics and optical sensor (shown and discussed in more detail below) to produce image 100′ of environment 100. Image 100′ is shown in detail on the enlarged view of screen 124 in the lower portion of FIG. 5B.

[0090] For the purposes of computing reduced homography H of the invention, we rely on images of space points P_i projected to correspondent image points p_i. Since there are no occlusions or obstructions in the present example and phone 102 is held in a suitable pose, camera 104 sees all nine space points P₁, . . . , P₉ and images them to produce correspondent image points p₁, . . . , p₉ in image 100′.

[0091] FIG. 5C is a diagram showing the image plane 128 of camera 104. Optical sensor 130 of camera 104 resides in image plane 128 and lies inscribed within a field of view (F.O.V.) 132. Sensor 130 is a pixelated CMOS sensor with an array of pixels 134. Only a few pixels 134 are shown in FIG. 5C for reasons of clarity. A center CC of sensor 130 (also referred to as camera center) is shown with an offset (x_sc,y_sc) from the origin of sensor or image coordinates (X_s,Y_s). In fact, (x_sc,y_sc) is also the location of viewpoint O and origin o′ of the projective plane in sensor coordinates (obviously, though, viewpoint O and origin o′ of the projective plane have different values along the z-axis).

[0092] All but imaged optical features corresponding to image points p₁, . . . , p₉ are left out of image 100′ for reasons of clarity. Note that the image is not shown inverted in this example. Of course, whether the image is or is not inverted will depend on the types of optics deployed by camera 104.

[0093] The projections of space points P_i to image points p_i are parameterized in sensor coordinates (X_s,Y_s). Each image point p_i that is imaged by the optics of camera 104 onto sensor 130 is thus measured in sensor or image coordinates along the X_s and Y_s axes. Image points p_i are indicated with open circles (same as in FIG. 5B) at locations that presume perfect or ideal imaging of camera 104 with no noise or structural uncertainties, such as aberrations, distortions, ghost images, stray light scattering or motion blur.

[0094] In practice, ideal image points p_i are almost never observed. Instead, a number of measured image points {circumflex over (p)}_i indicated by crosses are recorded on pixels 134 of sensor 130 at measured image coordinates {circumflex over (x)}_i,ŷ_i. (In the convention commonly adopted in the art and also herein, the “hat” on any parameter or variable is used to indicate a measured value as opposed to an ideal value or a model value.) Each measured image point {circumflex over (p)}_i is thus parameterized in image plane 128 as: {circumflex over (p)}_i=({circumflex over (x)}_i,ŷ_i) while ideal image point p_i is at: p_i=(x_i,y_i).

[0095] Sensor 130 records electromagnetic radiation 126 from space points P_i at various locations in image plane 128. A number of measured image points {circumflex over (p)}_i are shown for each ideal image point p_i to aid in visualizing the nature of the error. In fact, FIG. 5C illustrates that for ideal image point p₁ corresponding to space point P₁ there are ten measured image points {circumflex over (p)}_i. All ten of these measured image points {circumflex over (p)}_i are collected while camera 104 remains in the pose shown at time t=t₁. Similarly, at time t=t₁, rather than ideal image points p₂, p₆, p₉, sensor 130 of camera 104 records ten measured image points {circumflex over (p)}₂, {circumflex over (p)}₆, {circumflex over (p)}₉, respectively, also indicated by crosses. In addition, sensor 130 records three outliers 136 at time t=t₁. As is known to those skilled in the art, outliers 136 are not normally problematic, as they are considerably outside any reasonable error range and can be discarded. Indeed, the same approach is adopted with respect to outliers 136 in the present invention.

[0096] With the exception of outliers 136, measured image points {circumflex over (p)}_i are expected to lie within typical or normal error regions more or less centered about corresponding ideal image points p_i. To illustrate, FIG. 5C shows a normal error region 138 indicated around ideal image point p₆ within which measured image points {circumflex over (p)}₆ are expected to be found. Error region 138 is bounded by a normal error spread that is due to thermal noise, 1/f noise and shot noise. Unfortunately, measured image points {circumflex over (p)}₆ obtained for ideal image point p₆ lie within a much larger error region 140. The same is true for the other measured image points {circumflex over (p)}₁, {circumflex over (p)}₂ and {circumflex over (p)}₉—these also fall within larger error regions 140.

[0097] The present invention targets situations as shown in FIG. 5C, where measured image points {circumflex over (p)}_i are not contained within normal error regions, but rather fall into larger error regions 140. Furthermore, the invention addresses situations where larger error regions 140 are not random, but exhibit some systematic pattern. For the purpose of the present invention larger error region 140 exhibiting a requisite pattern for applying reduced homography H will be called a structural uncertainty.

[0098] We now turn to FIG. 5D for an enlarged view of structural uncertainty 140 about ideal image point p₉. Here, normal error region 138 surrounding ideal image point p₉ is small and generally symmetric. Meanwhile, structural uncertainty 140, which extends beyond error region 138 is large but extends generally along a radial line 142 extending from center CC of sensor 130. Note that line 142 is merely a mathematical construct used here (and in FIG. 5C) as an aid in visualizing the character of structural uncertainties 140. In fact, referring back to FIG. 5C, we see that all structural uncertainties 140 share the characteristic that they extend along corresponding radial lines 142. For this reason, structural uncertainties 140 in the present embodiment will be called substantially radial structural uncertainties.

[0099] Returning to FIG. 5D, we note that the radial extent of structural uncertainty 140 is so large, that information along that dimension may be completely unreliable. However, structural uncertainty 140 is also such that measured image points {circumflex over (p)}₉ are all within an angular or azimuthal range 144 that is barely larger and sometimes no larger than the normal error region 138. Thus, the azimuthal information in measured image points {circumflex over (p)}₉ is reliable.

[0100] For any particular measured image point {circumflex over (p)}₉ corresponding to space point P₉ that is recorded by sensor 130 at time t, one can state the following mapping relation:
A^T(t)P₉→p₉+δ_t→{circumflex over (p)}₉(t).  (Rel. 1)

[0101] Here A^T(t) is the transpose of the homography matrix A(t) at time t, δ_t is the total error at time t, and {circumflex over (p)}₉(t) is the measured image point {circumflex over (p)}₉ captured at time t. It should be noted here that total error δ_t contains both a normal error defined by error region 138 and the larger error due to radial structural uncertainty 140. Of course, although applied specifically to image point p₉, Rel. 1 holds for any other image point p_i.

[0102] To gain a better appreciation of when structural uncertainty 140 is sufficiently large in practice to warrant application of a reduced homography H of the invention and some of its potential sources we turn to FIG. 5E. This drawing shows space points P_i in environment 100 and their projections into a projective plane 146 of camera 104 and into image plane 128 where sensor 130 resides. Ideal image points p_i are shown here in projective plane 146 and designated by open circles, as before. Measured image points {circumflex over (p)}_i are shown in image plane 128 on sensor 130 and designated by crosses, as before. In addition, radial structural uncertainties 140 associated with measured image points {circumflex over (p)}_i are also shown in image plane 128.

[0103] An optic 148 belonging to camera 104 and defining viewpoint O is also explicitly shown in FIG. 5E. It is understood that optic 148 can consist of one or more lenses and/or any other suitable optical elements for imaging environment 100 to produce its image 100′ as seen from viewpoint O. Item 102 embodied by the smart phone is left out in FIG. 5E. Also, projective plane 146, image plane 128 and optic 148 are shown greatly enlarged for purposes of better visualization.

[0104] Recall now, that recovering the pose of camera 104 traditionally involves finding the best estimate Θ for the collineation or homography A from the available measured image points {circumflex over (p)}_i. Homography A is a matrix that encodes in it {R, h}. R is the complete rotation matrix expressing the unknown rotation of camera 104 with respect to world coordinates (X_w,Y_w,Z_w), and h is the unknown translation vector, which in the present case is defined as the distance between the location of viewpoint O when camera 104 (or smart phone 102) is in the canonical pose (e.g., at time t=t_o; see FIG. 5A) and in the unknown pose that is to be recovered. An offset d between viewpoint O in the canonical pose and the origin of world coordinates (X_w,Y_w,Z_w) parameterizing environment 100 is also indicated. As defined herein, offset d is a vector from world coordinate origin to viewpoint O along the Z_w axis of world coordinates (X_w,Y_w,Z_w). Thus, offset d is also the vector between viewpoint O and reference plane 118 to which the motion of camera 104 is constrained (see FIG. 5A). When referring to the distance between the world origin and reference plane 118 we will sometimes refer to the scalar value d of offset d as the offset or offset distance. Strictly speaking, that scalar value is the norm of the vector, i.e., d=| d|.

[0105] Note that viewpoint O is placed at the origin of camera coordinates (X_c,Y_c,Z_c). In the unknown pose shown in FIG. 5E, a distance between viewpoint O and the origin of world coordinates (X_w,Y_w,Z_w) is thus equal to d+ h. This distance is shown by a dashed and dotted line connecting viewpoint O at the origin of camera coordinates (X_c,Y_c,Z_c) with the origin of world coordinates (X_w,Y_w,Z_w).

[0106] In comparing ideal points p_i in projective plane 146 with actually measured image points {circumflex over (p)}_i and their radial structural uncertainties 140 it is clear that any pose recovery that relies on the radial portion of measured data will be unreliable. In many practical situations, radial structural uncertainty 140 in measured image data is introduced by the on-board optical apparatus, which is embodied by camera 104. The structural uncertainty can be persistent (inherent) or transitory. Persistent uncertainty can be due to radial defects in lens 148 of camera 104. Such lens defects can be encountered in molded lenses or mirrors when the molding process is poor or in diamond turned lenses or mirrors when the turning parameters are incorrectly varied during the turning process. Transitory uncertainty can be due to ghosting effects produced by internal reflections or stray light scattering within lens 148 (particularly acute in a compound or multi-component lens) or due to otherwise insufficiently optimized lens 148. It should be noted that ghosting can be further exacerbated when space points P_i being imaged are all illuminated at high intensities (e.g., high brightness point sources, such as markers embodied by LEDs or IR LEDs).

[0107] Optical sensor 130 of camera 104 can also introduce radial structural uncertainty due to its design (intentional or unintentional), poor quality, thermal effects (non-uniform heating), motion blur and motion artifacts created by a rolling shutter, pixel bleed-through and other influences that will be apparent to those skilled in the art. These effects can be particularly acute when sensor 130 is embodied by a poor quality CMOS sensor or a position sensing device (PSD) with hard to determine radial characteristics. Still other cases may include a sensor such as a 1-D PSD shaped into a circular ring to only measure the azimuthal distances between features in angular units (e.g., radians or degrees). Once again, these effects can be persistent or transitory. Furthermore, the uncertainties introduced by lens 148 and sensor 130 can add to produce a joint uncertainty that is large and difficult to characterize, even if the individual contributions are modest.

[0108] The challenge is to provide the best estimate Θ of homography A from measured image points {circumflex over (p)}_i=({circumflex over (x)}_i,ŷ_i) despite radial structural uncertainties 140. According to the invention, adopting a reduced representation of measured image points {circumflex over (p)}_i=({circumflex over (x)}_i,ŷ_i) and deploying a correspondingly reduced homography H meets this challenge. The measured data is then used to obtain an estimation matrix Θ of the reduced homography H rather than an estimate Θ of the regular homography A. To better understand reduced homography H and its matrix, it is important to first review 3D rotations in detail. We begin with rotation matrices that compose the full or complete rotation matrix R, which expresses the orientation of camera 104. Orientation is expressed in reference to world coordinates (X_w,Y_w,Z_w) with the aid of camera coordinates (X_c,Y_c,Z_c).

[0109] FIGS. 6A-D illustrate a general orthogonal rotation convention. Specifically, this convention describes the absolute orientation of a rigid body embodied by an exemplary phone 202 in terms of three rotation angles α_c, β_c and γ_c. Here, the rotations are taken around the three camera axes X_c, Y_c, Z_c, of a centrally mounted camera 204 with viewpoint O at the center of phone 202. This choice of rotation convention ensures that viewpoint O of camera 204 does not move during any of the three rotations. The camera axes are initially aligned with the axes of world coordinates (X_w,Y_w,Z_w) when phone 202 is in the canonical pose.

[0110] FIG. 6A shows phone 202 in an initial, pre-rotated condition centered in a gimbal mechanism 206 that will mechanically constrain the rotations defined by angles α_c, β_c and γ_c. Mechanism 206 has three progressively smaller concentric rings or hoops 210, 212, 214. Rotating joints 211, 213 and 215 permit hoops 210, 212, 214 to be respectively rotated in an independent manner. For purposes of visualization of the present 3D rotation convention, phone 202 is rigidly fixed to the inside of third hoop 214 either by an extension of joint 215 or by any other suitable mechanical means (not shown).

[0111] In the pre-rotated state, the axes of camera coordinates (X_c,Y_c,Z_c) parameterizing the moving reference frame of phone 202 are triple primed (X_c′″,Y_c′″,Z_c′″) to better keep track of camera coordinate axes after each of the three rotations. In addition, pre-rotated axes (X_c′″,Y_c′″,Z_c′″) of camera coordinates (X_c,Y_c,Z_c) are aligned with axes X_w, Y_w and Z_w of world coordinates (X_s,Y_s,Z_s) that parameterize the environment. However, pre-rotated axes (X_c′″,Y_c′″,Z_c′″) are displaced from the origin of world coordinates (X_s,Y_s,Z_s) by offset d (not shown in the present figure, but see FIG. 5E & FIG. 8). Viewpoint O is at the origin of camera coordinates (X_c,Y_c,Z_c) and at the center of gimbal mechanism 206.

[0112] The first rotation by angle α_c is executed by rotating joint 211 and thus turning hoop 210, as shown in FIG. 6B. Note that since camera axis Z_c′″ of phone 202 (see FIG. 6A) is co-axial with rotating joint 211 the physical turning of hoop 210 is equivalent to this first rotation in camera coordinates (X_c,Y_c,Z_c) of phone 202 around camera axis. In the present convention, all rotations are taken to be positive in the counter-clockwise direction as defined with the aid of the right hand rule (with the thumb pointed in the positive direction of the coordinate axis around which the rotation is being performed). Hence, angle α_c is positive and in this visualization it is equal to 30°.

[0113] After each of the three rotations is completed, camera coordinates (X_c,Y_c,Z_c) are progressively unprimed to denote how many rotations have already been executed. Thus, after this first rotation by angle α_c, the axes of camera coordinates (X_c,Y_c,Z_c) are unprimed once and designated (X_c″,Y_c″,Z_c″) as indicated in FIG. 6B.

[0114] FIG. 6C depicts the second rotation by angle β_c. This rotation is performed by rotating joint 213 and thus turning hoop 212. Since joint 213 is co-axial with once rotated camera axis X_c″ (see FIG. 6B) such rotation is equivalent to second rotation in camera coordinates (X_c,Y_c,Z_c) of phone 202 by angle β_c around camera axis X_c″. In the counter-clockwise rotation convention we have adopted angle β_c is positive and equal to 45°. After completion of this second rotation, camera coordinates (X_c,Y_c,Z_c) are unprimed again to yield twice rotated camera axes (X_c′,Y_c′,Z_c′).

[0115] The result of the third and last rotation by angle γ_c is shown in FIG. 6D. This rotation is performed by rotating joint 215, which turns innermost hoop 214 of gimbal mechanism 206. The construction of mechanism 206 used for this visualization has ensured that throughout the prior rotations, twice rotated camera axis (see FIG. 6C) has remained co-axial with joint 215. Therefore, rotation by angle γ_c′ is a rotation in camera coordinates (X_c,Y_c,Z_c) parameterizing the moving reference frame of camera 202 by angle γ_c about camera axis Y_c′.

[0116] This final rotation yields the fully rotated and now unprimed camera coordinates (X_c,Y_c,Z_c). In this example angle γ_c is chosen to be 40°, representing a rotation by 40° in the counter-clockwise direction. Note that in order to return fully rotated camera coordinates (X_c,Y_c,Z_c) into initial alignment with world coordinates (X_w,Y_w,Z_w) the rotations by angles α_c, β_c and γ_c need to be taken in exactly the reverse order (this is due to the order-dependence or non-commuting nature of rotations in 3D space).

[0117] It should be understood that mechanism 206 was employed for illustrative purposes to show how any 3D orientation of phone 202 consists of three rotational degrees of freedom. These non-commuting rotations are described or parameterized by rotation angles α_c, β_c and γ_c around camera axes Z_c′″, X_c″ and finally Y_c′. What is important is that this 3D rotation convention employing angles α_c, β_c, γ_c is capable of describing any possible orientation that phone 202 may assume in any 3D environment.

[0118] We now turn back to FIG. 5E and note that the orientation of phone 102 indeed requires a description that includes all three rotation angles. That is because the motion of phone 102 in environment 100 is unconstrained other than by the condition that trajectory 122 of viewpoint O be approximately confined to reference plane 118 (see FIG. 5A). More precisely, certain forward displacement ε_f and a certain back displacement ε_b away from reference plane 118 are permitted. However, as far as the misalignment of camera coordinates (X_c,Y_c, Z_c) with world coordinates (X_w,Y_w,Z_w) is concerned, all three rotations are permitted. Thus, we have to consider any total rotation represented by a full or complete rotation matrix R that accommodates changes in one, two or all three of the rotation angles. For completeness, a person skilled in the art should notice that all possible camera rotations, or, more precisely the rotation matrices representing them, are a special class of collineations.

[0119] Each one of the three rotations described by the rotation angles α_c, β_c, γ_c has an associated rotation matrix, namely: R(α), R(β) and R(γ). A number of conventions for the order of the individual rotations, other than the order shown in FIGS. 6A-D, are routinely used by those skilled in the art. All of them are ultimately equivalent, but once a choice is made it needs to be observed throughout because of the non-commuting nature of rotation matrices.

[0120] The full or complete rotation matrix R is a composition of individual rotation matrices R(α), R(β), R(γ) that account for all three rotations (α_c,β_c,γ_c) previously introduced in FIGS. 6A-D. These individual rotation matrices are expressed as follows:

[0121] $\begin{array}{cc}R\left(\alpha \right)=\left(\begin{array}{ccc}\cos \alpha & \sin \alpha & 0\\ -\sin \alpha & \cos \alpha & 0\\ 0& 0& 1\end{array}\right)& \left(\mathrm{Eq}.2A\right)\\ R\left(\beta \right)=\left(\begin{array}{ccc}1& 0& 0\\ 0& \cos \beta & \sin \beta \\ 0& -\sin \beta & \cos \beta \end{array}\right)& \left(\mathrm{Eq}.2B\right)\\ R\left(\gamma \right)=\left(\begin{array}{ccc}\cos \gamma & 0& -\sin \gamma \\ 0& 1& 0\\ \sin \gamma & 0& \cos \gamma \end{array}\right)& \left(\mathrm{Eq}.2C\right)\end{array}$

[0122] The complete rotation matrix R is obtained by multiplying the above individual rotation matrices in the order of the chosen rotation convention. For the rotations performed in the order shown in FIGS. 6A-D the complete rotation matrix is thus: R=R(γ_c)·R(β_c)·R(α_c).

[0123] It should be noted that rotation matrices are always square and have real-valued elements. Algebraically, a rotation matrix in 3-dimensions is a 3×3 special orthogonal matrix (SO(3)) whose determinant is 1 and whose transpose is equal to its inverse:
R^T=R⁻¹;Det(R)=1,  Eq. 3
where superscript “T” indicates the transpose, superscript “−1” indicates the inverse and “Det” designates the determinant.

[0124] For reasons that will become apparent later, in pose recovery with reduced homography H according to the invention we will use rotations defined by the Euler rotation convention. The convention illustrating the rotation of the body or camera 104 as seen by an observer in world coordinates is shown in FIG. 6E. This isometric diagram illustrates each of the three rotation angles applied to on-board optical unit 104.

[0125] In pose recovery we are describing what camera 104 sees as a result of the rotations. We are thus not interested in the rotations of camera 104, but rather the transformation of coordinates that camera 104 experiences due to the rotations. As is well known, the rotation matrix R that describes the coordinate transformation corresponds to the transpose of the composition of rotation matrices introduced above (Eq. 2A-C). From now on, when we refer to the rotation matrix R we will thus be referring to the rotation matrix that describes the coordinate transformation experienced by camera 104. (It is important to recall here, that the transpose of a composition or product of matrices A and B inverts the order of that composition, such that (AB)^T=B^TA^T.)

[0126] In accordance with the Euler composition we will use, the first rotation angle designated by ψ is the same as angle α defined above. Thus, the first rotation matrix R(ψ) in the Euler convention is:

[0127] $R\left(\psi \right)=\left(\begin{array}{ccc}\cos \psi & -\sin \psi & 0\\ \sin \psi & \cos \psi & 0\\ 0& 0& 1\end{array}\right).$

[0128] The second rotation by angle θ produces rotation matrix R(θ):

[0129] $R\left(\theta \right)=\left(\begin{array}{ccc}1& 0& 0\\ 0& \cos \theta & -\sin \theta \\ 0& \sin \theta & \cos \theta \end{array}\right).$

[0130] Now, the third rotation by angle ψ corresponds to rotation matrix R(ψ) and is described by:

[0131] $R\left(\varphi \right)=\left(\begin{array}{ccc}\cos \varphi & -\sin \varphi & 0\\ \sin \varphi & \cos \varphi & 0\\ 0& 0& 0\end{array}\right).$

[0132] The result is that in the Euler convention using Euler rotation angles φ,θ,ψ we obtain a complete rotation matrix R=R(φ)·R(θ)·R(ψ). Note the ordering of rotation matrices to ensure that angles φ,θ,ψ are applied in that order. (Note that in some textbooks the definition of rotation angles φ and ψ is sometimes reversed.)

[0133] Having defined the complete rotation matrix R in the Euler convention, we turn to FIG. 7 and review the reduced representation of measured image points {circumflex over (p)}_i according to the present invention. The representation deploys N-vectors defined in homogeneous coordinates using projective plane 146 and viewpoint O as the origin. By definition, an N-vector in normalized homogeneous coordinates is a unit vector that is computed by dividing that vector by its norm using the normalization operator N as follows: N[ū]=ū/∥ū∥.

[0134] Before applying the reduced representation to measured image points {circumflex over (p)}_i, we note that any point (a,b) in projective plane 146 is represented in normalized homogeneous coordinates by applying the normalization operator N to the triple (a,b,f), where f is the focal length of lens 148. Similarly, a line Ax+By+C=0, sometimes also represented as [A,B,C] (square brackets are often used to differentiate points from lines), is expressed in normalized homogeneous coordinates by applying normalization operator N to the triple [A,B,C/f]. The resulting point and line representations are insensitive to sign, i.e., they can be taken with a positive or negative sign.

[0135] We further note, that a collineation is a one-to-one mapping from the set of image points p_i′ seen by camera 104 in an unknown pose to the set of image points p_i as seen by camera 104 in the canonical pose shown in FIG. 5A at time t=t_o. The prime notation “′” will henceforth be used to denote all quantities observed in the unknown pose. As previously mentioned, a collineation preserves certain properties, namely: collinear image points remain collinear, concurrent image lines remain concurrent, and an image point on a line remains on the line. Moreover, a traditional collineation A is a linear mapping of N-vectors such that:
m_i′=±N[A^T m_i]; n_i′=±N[A⁻¹ n_i].  (Eq. 4)

[0136] In Eq. 4 m_i′ is the homogeneous representation of an image point p_i′ as it should be seen by camera 104 in the unknown pose, and n_i′ is the homogeneous representation of an image line as should be seen in the unknown pose.

[0137] Eq. 4 states that these homogenous representations are obtained by applying the transposed collineation A^T to image point p_i represented by m_i in the canonical pose, and by applying the collineation inverse A⁻¹ to line represented by n_i in the canonical pose. The application of the normalization operator N ensures that the collineations are normalized and insensitive to sign. In addition, collineations are unique up to a scale and, as a matter of convention, their determinant is usually set to 1, i.e.: Det∥A∥=1 (the scaling in practice is typically recovered/applied after computing the collineation). Also, due to the non-commuting nature of collineations inherited from the non-commuting nature of rotation matrices R, as already explained above, a collineation A₁ followed by collineation A₂ results in the total composition A=A₁·A₂.

[0138] Returning to the challenge posed by structural uncertainties 140, we now consider FIG. 7. This drawing shows radial structural uncertainty 140 for a number of correspondent measured image points {circumflex over (p)}_i associated with ideal image point {circumflex over (p)}_i′ that should be measured in the absence of noise and structural uncertainty 140. All points are depicted in projective plane 146. Showing measured image points {circumflex over (p)}_i in projective plane 146, rather than in image plane 128 where they are actually recorded on sensor 130 (see FIG. 5E), will help us to appreciate the choice of a reduced representation r_i′ associated to ideal image point p_i′ and extended to measured points {circumflex over (p)}_i. We also adopt the standard convention reviewed above, and show ideal image point p_i observed in the canonical pose in projective plane 146 as well. This ideal image point p_i is represented in normalized homogeneous coordinates by its normalized vector m_i.

[0139] Now, in departure from the standard approach, we take the ideal reduced representation r_i′ of point p_i′ to be a ray in projective plane 146 passing through p_i′ and the origin o′ of plane 146. Effectively, reducing the representation of image point p_i′ to just ray r_i′ passing through it and origin o′ eliminates all radial but not azimuthal (polar) information contained in point p_i′. The deliberate removal of radial information from ray r_i′ is undertaken because the radial information of a measurement is highly unreliable. This is confirmed by the radial structural uncertainty 140 in measured image points {circumflex over (p)}_i that under ideal conditions (without noise or structural uncertainty 140) would project to ideal image point {circumflex over (p)}_i′ in the unknown pose we are trying to recover.

[0140] Indeed, it is a very surprising finding of the present invention, that in reducing the representation of measured image points {circumflex over (p)}_i by discarding their radial information and representing them with rays {circumflex over (r)}_i (note the “hat”, since the rays are the reduced representations of measured rather than model or ideal points) the resultant reduced homography H nonetheless supports the recovery of all extrinsic parameters (full pose) of camera 104. In FIG. 7 only a few segments of rays {circumflex over (r)}_i corresponding to reduced representations of measured image points {circumflex over (p)}_i are shown for reasons of clarity. A reader will readily see, however, that they would all be nearly collinear with ideal reduced representation r_i′ of ideal image point p_i′ that should be measured in the unknown pose when no noise or structural uncertainty is present.

[0141] Due to well-known duality between lines and points in projective geometry (each line has a dual point and vice versa; also known as pole and polar or as “perps” in universal hyperbolic geometry) any homogeneous representation can be translated into its mathematically dual representation. In fact, a person skilled in the art will appreciate that the below approach developed to teach a person skilled in the art about the practice of reduced homography H can be recast into mathematically equivalent formulations by making various choices permitted by this duality.

[0142] In order to simplify the representation of ideal and measured rays r_i′, {circumflex over (r)}_i for reduced homography H, we invoke the rules of duality to represent them by their duels or poles. Thus, reduced representation of point {circumflex over (p)}_i′ by ray r_i′ can be translated to its pole by constructing the join between origin o′ and point {circumflex over (p)}_i′. (The join is closely related to the vector cross product of standard Euclidean geometry.) A pole or n-vector n_i′ is defined in normalized homogeneous coordinates as the cross product between unit vector ô=(0,0,1)^T (note that in this case the “hat” stands for unit vector rather than a measured value) from the origin of camera coordinates (X_c,Y_c,Z_c) at viewpoint O towards origin o′ of projective plane 146 and normalized vector m_i′ representing point p_i′.

[0143] Notice that the pole of any line through origin o′ will not intersect projective plane 146 and will instead represent a “point at infinity”. This means that in the present embodiment where all reduced representations r_i′ pass through origin o′ we expect all n-vectors n_i′ to be contained in a plane through viewpoint O and parallel to projective plane 146 (i.e., the X_c-Y_c plane). Indeed, we see that this is so from the formal definition for the pole of p_i′:
n_i′=±N(ô× m_i′),  (Eq. 5)
where the normalization operator N is deployed again to ensure that n-vector n_i′ is expressed in normalized homogeneous coordinates. Because of the cross-product with unit vector ô=(0,0,1)^T, the value of any z-component of normalized n-vector m_i′ is discarded and drops out from any calculations involving the n-vector n_i′.

[0144] In the ideal or model case, reduced homography H acts on vector m_i representing point p_i in the canonical pose to transform it to a reduced representation by m_i′ (without the z-component) for point p_i′ in the unknown pose (again, primes “′” denote ideal or measured quantities in unknown pose). In other words, reduced homography H is a 2×3 mapping instead of the traditional 3×3 mapping. The action of reduced homography H is visualized in FIG. 7.

[0145] In practice we do not know ideal image points p_i′ nor their rays r_i′. Instead, we only know measured image points {circumflex over (p)}_i and their reduced representations as rays {circumflex over (r)}_i. This means that our task is to find an estimation matrix Θ for reduced homography H based entirely on measured values {circumflex over (p)}_i in the unknown pose and on known vectors m_i representing the known points P_i in canonical pose (the latter also sometimes being referred to as ground truth). As an additional aid, we have the condition that the motion of smart phone 102 and thus of its on-board camera 104 is substantially bound to reference plane 118 and is therefore confined to volume 120, as illustrated in FIG. 5A.

[0146] We now refer to FIG. 8, which once again presents a perspective view of environment 100, but with all stationary objects removed. Furthermore, smart phone 102 equipped with the on-board camera 104 is shown at time t=t_o (canonical pose) and at time t=t₁ (unknown pose). World coordinates (X_w,Y_w,Z_w) parameterizing environment 100 are chosen such that wall 150 is coplanar with the (X_w-Y_w) plane. Of course, any other parameterization choices of environment 100 can be made, but the one chosen herein is particularly well-suited for explanatory purposes. That is because wall 150 is defined to be coplanar with reference surface 118 and separated from it by offset distance d (to within d−ε_f and d+ε_b, and recall that d=| d|).

[0147] From the prior art teachings it is known that a motion of camera 104 defined by a succession of sets {R, h} relative to a planar surface defined by a p-vector p={circumflex over (n)}_p/d induces the collineation or homography A expressed as:

[0148] $\begin{array}{cc}A=\frac{1}{k}\left(I-\stackrel{\_}{p}·{\stackrel{\_}{h}}^T\right)R\mathrm{with}k=\sqrt[3]{1-\left(\stackrel{\_}{p}·\stackrel{\_}{h}\right)},& \left(\mathrm{Eq}.6\right)\end{array}$
where I is the 3×3 identity matrix and h^T is the transpose (i.e., row vector) of h. In our case, the planar surface used in the explanation is wall 150 due to the convenient parameterization choice made above. In normalized homogeneous coordinates wall 150 can be expressed by its corresponding p-vector p, where {circumflex over (n)}_p is the unit surface normal to wall 150 and pointing away from viewpoint O, and d is the offset, here shown between reference plane 118 and wall 150 (or the (X_w-Y_w) plane of the world coordinates). (Note that the “hat” on the unit surface normal does note stand for a measured value, but is used instead to express the unit vector just as in the case of the ôunit vector introduced above in FIG. 7).

[0149] To recover the unknown pose of smart phone 102 at time t=t₁ we need to find the matrix that sends the known points P_i as seen by camera 104 in canonical pose (shown at time t=t_o) to points p_i′ as seen by camera 104 in the unknown pose. In the prior art, that matrix is the transpose, A^T, of homography A. The matrix that maps points p_i′ from the unknown pose back to canonical pose is the transpose of the inverse A⁻¹ of homography A. Based on the definition that any homography matrix multiplied by its inverse has to yield the identity matrix I, we find from Eq. 6 that A⁻¹ is expressed as:

[0150] $\begin{array}{cc}{A}^{-1}={\mathrm{kT}}^T\left(I+\frac{\stackrel{\_}{p}·{\stackrel{\_}{h}}^T}{1-\left(\stackrel{\_}{p}·\stackrel{\_}{h}\right)}\right).& \left(\mathrm{Eq}.7\right)\end{array}$

[0151] Before taking into account rotations, let's examine the behavior of homography A in a simple and ideal model case. Take parallel translation of camera 104 in plane 118 at offset distance d to world coordinate origin while keeping phone 102 such that optical axis OA remains perpendicular to plane 118 (no rotation—i.e., full rotation matrix R is expressed by the 3×3 identity matrix I). We thus have p=(0,0,1/d) and h=(δx,δy,0). Therefore, from Eq. 6 we see that homography A in such a simple case is just:

[0152] $A=\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ \frac{-\delta x}{d}& \frac{-\delta y}{d}& 1\end{array}\right).$

[0153] When z is allowed to vary slightly, i.e., between ε_f and ε_b or within volume 120 about reference plane 118 as previously defined (see FIG. 5A), we obtain a slightly more complicated homography A by applying Eq. 6 as follows:

[0154] $A=\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ \frac{-\delta x}{d}& \frac{-\delta y}{d}& 1-\frac{\delta z}{d}\end{array}\right)/k.$

[0155] The inverse homography A⁻¹ for either one of these simple cases can be computed by using Eq. 7.

[0156] Now, when rotation of camera 104 is added, the prior art approach produces homography A that contains the full rotation matrix R and displacement h. To appreciate the rotation matrix R in traditional homography A we show traditional pose recovery just with respect to wall 150 defined by known corners P₂, P₁₀, P₁₁ and P₁₂ (room 100 is empty in FIG. 8 so that all the corners are clearly visible). (By stating that corners P₂, P₁₀, P₁₁ and P₁₂ are known, we mean that the correspondence is known. In addition, note that the traditional recovery is not limited to requiring co-planar points used in this visualization.)

[0157] In the canonical pose at time t=t_o an enlarged view of display screen 124 showing image 100′ captured by camera 104 of smart phone 102 contains image 150′ of wall 150. In this pose, wall image 150′ shows no perspective distortion. It is a rectangle with its conjugate vanishing points v1, v2 (not shown) both at infinity. The unit vectors {circumflex over (n)}_v1, {circumflex over (n)}_v2 pointing to these conjugate vanishing points are shown with their designations in the further enlarged inset labeled CPV (Canonical Pose View). Unit surface normal {circumflex over (n)}_p, which is obtained from the cross-product of vectors {circumflex over (n)}_v1,{circumflex over (n)}_v2 points into the page in inset CPV. In the real three-dimensional space of environment 100, this corresponds to pointing from viewpoint O straight at the origin of world coordinates (X_w,Y_w,Z_w) along optical axis OA. Of course, {circumflex over (n)}_p is also the normal to wall 150 based on our parameterization and definitions.

[0158] In the unknown pose at time t=t₁ another enlarged view of display screen 124 shows image 100′. This time image 150′ of wall 150 is distorted by the perspective of camera 104. Now conjugate vanishing points v1, v2 associated with the quadrilateral of wall image 150′ are no longer at infinity, but at the locations shown. Of course, vanishing points v1, v2 are not real points but are defined by mathematical construction, as shown by the long-dashed lines. The unit vectors {circumflex over (n)}_v1, {circumflex over (n)}_v2 pointing to conjugate vanishing points v1, v2 are shown in the further enlarged inset labeled UPV (Unknown Pose View). Unit surface normal {circumflex over (n)}_p, again obtained from the cross-product of vectors {circumflex over (n)}_v1, {circumflex over (n)}_v2, no longer points into the page in inset UVP. In the real three-dimensional space of environment 100, {circumflex over (n)}_p still points from viewpoint O at the origin of world coordinates (X_w,Y_w,Z_w), but this is no longer a direction along optical axis OA of camera 104 due to the unknown rotation of phone 102.

[0159] The traditional homography A will recover the unknown rotation in terms of rotation matrix R composed of vectors {circumflex over (n)}_v1, {circumflex over (n)}_v2, {circumflex over (n)}_p in their transposed form {circumflex over (n)}_v1^T, {circumflex over (n)}_v2^T, {circumflex over (n)}_p^T. In fact, the transposed vectors {circumflex over (n)}_v1^T, {circumflex over (n)}_v2^T, {circumflex over (n)}_p^T simply form the column space of rotation matrix R. Of course, the complete traditional homography A also contains displacement h. Finally, to recover the pose of phone 102 we again need to find homography A, which is easily done by the rules of linear algebra.

[0160] In accordance with the invention, we start with traditional homography A that includes rotation matrix R and reduce it to homography H by using the fact that the z-component of normalized n-vector m_i′ does not contribute to n-vector n_i′ (the pole into which r_i′ is translated). From Eq. 5, the pole n_i′ representing model ray r_i′ in the unknown pose is given by:

[0161] $\begin{array}{cc}{\stackrel{\_}{n}}_i^{\prime }={\hat{o}}^{\prime }\times {\stackrel{\_}{m}}_i^{\prime }=\left(\begin{array}{ccc}0& -1& 0\\ 1& 0& 0\\ 0& 0& 0\end{array}\right){\stackrel{\_}{m}}_i^{\prime }=\left(\begin{array}{c}-y_i^{\prime }\\ x_i^{\prime }\\ 0\end{array}\right),& \left(\mathrm{Eq}.8\right)\end{array}$
where the components of vector m_i′ are called (x_i′,y_i′,z_i′). Homography A representing the collineation from canonical pose to unknown pose, in which we represent points p_i′; with n-vectors m_i′ can then be written with a scaling constant κ as:
m_i′=κA^T m_i′,  (Eq. 9)

[0162] Note that the transpose of A, or A^T, is applied here because of the “passive” convention as defined by Eq. 4. In other words, when camera 104 motion is described by matrix A, what happens to the features in the environment from the camera's point of view is just the opposite. Hence, the transpose of A is used to describe what the camera is seeing as a result of its motion.

[0163] Now, in the reduced representation chosen according to the invention, the z-component of n-vector m_i′ does not matter (since it will go to zero as we saw in Eq. 8). Hence, the final z-contribution from the transpose of the Euler rotation matrix that is part of the homography does not matter. Thus, by using reduced transposes of Eqs. 2A & 2B representing the Euler rotation matrices and setting their z-contributions to zero except for R^T(φ), we obtain a reduced transpose R_r^T of a modified rotation matrix R_r:
R_r^T=R_r^T(ψ)·R_r^T(θ)·R^T(φ).  (Eq. 10A)

[0164] Expanded to its full form, this transposed rotation matrix R_r^T is:

[0165] $\begin{array}{cc}{R}_r^T=\left(\begin{array}{ccc}\cos \psi & \sin \psi & 0\\ -\sin \psi & \cos \psi & 0\\ 0& 0& 0\end{array}\right)\left(\begin{array}{ccc}1& 0& 0\\ 0& \cos \theta & \sin \theta \\ 0& 0& 0\end{array}\right)\left(\begin{array}{ccc}\cos \varphi & \sin \varphi & 0\\ -\sin \varphi & \cos \varphi & 0\\ 0& 0& 1\end{array}\right),& \left(\mathrm{Eq}.10B\right)\end{array}$
and it multiplies out to:

[0166] $\left(\mathrm{Eq}.10C\right)$ $R_r^T=\left(\begin{array}{ccc}\cos \mathrm{\varphi cos}\psi -\cos \mathrm{\theta sin\varphi }\sin \psi & \cos \mathrm{\psi sin\varphi }+\cos \mathrm{\theta cos\varphi sin}\psi & \sin \mathrm{\theta sin}\psi \\ -\cos \mathrm{\theta cos}\mathrm{\psi sin\varphi }-\cos \mathrm{\varphi sin}\psi & \cos \mathrm{\theta cos}\mathrm{\varphi cos\psi }-\sin \mathrm{\varphi sin}\psi & \cos \mathrm{\psi sin}\theta \\ 0& 0& 0\end{array}\right).$

[0167] Using trigonometric identities on entries with multiplication of three rotation angles in the transpose of the modified rotation matrix R_r^T we convert expressions involving sums and differences of rotation angles in the upper left 2×2 sub-matrix of R_r^T into a 2×2 sub-matrix C as follows:

[0168] $\left(\mathrm{Eq}.11\right)$ $C=\frac{1}{2}\left(\begin{array}{cc}\begin{array}{c}-\cos \mathrm{\theta cos}\left(\varphi -\psi \right)+\cos \left(\varphi -\psi \right)\\ +\cos \mathrm{\theta cos}\left(\varphi +\psi \right)+\cos \left(\varphi +\psi \right)\end{array}& \begin{array}{c}-\cos \mathrm{\theta sin}\left(\varphi -\psi \right)+\sin \left(\varphi -\psi \right)\\ +\cos \mathrm{\theta sin}\left(\varphi +\psi \right)+\sin \left(\varphi +\psi \right)\end{array}\\ \begin{array}{c}-\cos \mathrm{\theta sin}\left(\varphi -\psi \right)+\sin \left(\varphi -\psi \right)\\ -\cos \mathrm{\theta sin}\left(\varphi +\psi \right)-\sin \left(\varphi +\psi \right)\end{array}& \begin{array}{c}\cos \mathrm{\theta cos}\left(\varphi -\psi \right)-\cos \left(\varphi -\psi \right)\\ +\cos \mathrm{\theta cos}\left(\varphi +\psi \right)+\cos \left(\varphi +\psi \right)\end{array}\end{array}\right)$

[0169] It should be noted that sub-matrix C can be decomposed into a 2×2 improper rotation (reflection along y, followed by rotation) and a proper 2×2 rotation.

[0170] Using sub-matrix C from Eq. 11, we can now rewrite Eq. 9 as follows:

[0171] $\begin{array}{cc}{\stackrel{\_}{m}}_i^{\prime }=\kappa \left(\begin{array}{cc}C& \sin \mathrm{\psi sin}\theta \\ & \cos \mathrm{\psi sin}\theta \\ 0& 0\end{array}\right)\left(\begin{array}{ccc}1& 0& -\delta x/d\\ 0& 1& -\delta y/d\\ 0& 0& \left(d-\delta z\right)/d\end{array}\right){\stackrel{\_}{m}}_i=\kappa H^T{\stackrel{\_}{m}}_i& \left(\mathrm{Eq}.12\right)\end{array}$

[0172] At this point we remark again, that because of the reduced representation of the invention the z-component of n-vector m_i′ does not matter. We can therefore further simplify Eq. 12 as follows:

[0173] $\begin{array}{cc}{\stackrel{\_}{m}}_i^{\prime }=\kappa \left(\begin{array}{cc}C& \stackrel{\_}{b}\\ 0& 0\end{array}\right){\stackrel{\_}{m}}_i,& \left(\mathrm{Eq}.13\right)\end{array}$
where the newly introduced column vector b follows from Eq. 12:

[0174] $\stackrel{\_}{b}=-C\left(\begin{array}{c}\delta x/d\\ \delta y/d\end{array}\right)+\frac{d-\delta z}{d}\left(\begin{array}{c}\sin \psi \\ \cos \psi \end{array}\right)\sin \theta .$

[0175] Thus we have now derived a reduced homography H, or rather its transpose H^T=[C, b].

[0176] We now deploy our reduced representation as the basis for performing actual pose recovery. In this process, the transpose of reduced homography H^T has to be estimated with a 2×3 estimation matrix Θ from measured points {circumflex over (p)}_i. Specifically, we set Θ to match sub-matrix C and two-dimensional column vector b as follows:

[0177] $\begin{array}{cc}\Theta =\left(\begin{array}{ccc}{\theta }_1& {\theta }_2& {\theta }_3\\ {\theta }_4& {\theta }_5& {\theta }_6\end{array}\right)=\left(\begin{array}{cc}C& \stackrel{\_}{b}\end{array}\right).& \left(\mathrm{Eq}.14\right)\end{array}$

[0178] Note that the thetas used in Eq. 14 are not angles, but rather the estimation values of the reduced homography.

[0179] When Θ is estimated, we need to extract the values for the in-plane displacements δx/d and δy/d. Meanwhile δz, rather than being zero when strictly constrained to reference plane 118, is allowed to vary between −ε_f and +ε_b. From Eq. 14 we find that under these conditions displacements δx/d, δy/d are given by:

[0180] $\begin{array}{cc}\left(\begin{array}{c}\delta x/d\\ \delta y/d\end{array}\right)\approx -C^{-1}\left(\begin{array}{c}{\theta }_3\\ {\theta }_6\end{array}\right)+C^{-1}\left(\begin{array}{c}\sin \psi \\ \cos \psi \end{array}\right)\sin \theta .& \left(\mathrm{Eq}.15\right)\end{array}$

[0181] Note that δz should be kept small (i.e., (d−δz)/d should be close to one) to ensure that this approach yields good results.

[0182] Now we are in a position to put everything into our reduced representation framework. For any given space point P_i, its ideal image point p_i in canonical pose is represented by m_i=(x_i,y_i,z_i)^T. In the unknown pose, the ideal image point p_i′ has a reduced ray representation r_i′ and translates to an n-vector n_i′. The latter can be written as follows:

[0183] $\begin{array}{cc}{\stackrel{\_}{n}}_i^{\prime }=\kappa \left[\begin{array}{c}-y_i^{\prime }\\ x_i^{\prime }\\ 0\end{array}\right].& \left(\mathrm{Eq}.16\right)\end{array}$

[0184] The primed values in the unknown pose, i.e., point p_i′ expressed by its x_i′ and y_i′ values recorded on sensor 130, can be restated in terms of estimation values θ₁, . . . , θ₆ and canonical point p_i known by its x_i and y_i values. This is accomplished by referring back to Eq. 14 to see that:
x_i′=θ₁x_i+θ₂y_i+θ₃,and
y_i′=θ₄x_i+θ₅y_i+θ₆.

[0185] In this process, we have scaled the homogeneous representation of space points P_i by offset d through multiplication by 1/d. In other words, the corresponding m-vector m_i for each point P_i is taken to be:

[0186] $\begin{array}{cc}{\stackrel{\_}{m}}_i=\left[\begin{array}{c}{x}_i\\ y_i\\ d\end{array}\right]\stackrel{1/d}{->}\left[\begin{array}{c}{x}_i\\ y_i\\ 1\end{array}\right].& \left(\mathrm{Eq}.17\right)\end{array}$

[0187] With our reduced homography framework in place, we turn our attention from ideal or model values ({circumflex over (p)}_i′=({circumflex over (x)}_i′,ŷ_i′)) to the actual measured values {circumflex over (x)}_i and ŷ_i that describe the location of measured points {circumflex over (p)}_i ({circumflex over (p)}_i=({circumflex over (x)}_i,ŷ_i)) produced by the projection of space points P_i onto sensor 130. Instead of looking at measured values {circumflex over (x)}_i and ŷ_i in image plane 128 where sensor 130 is positioned, however, we will look at them in projective plane 146 for reasons of clarity and ease of explanation.

[0188] FIG. 9A is a plan view diagram of projective plane 146 showing three measured values {circumflex over (x)}_i and ŷ_i corresponding to repeated measurements of image point {circumflex over (p)}_i taken while camera 104 is in the same unknown pose. Remember that, in accordance with our initial assumptions, we know which actual space point P_i is producing measurements {circumflex over (x)}_i and ŷ_i (the correspondence is known). To distinguish between the individual measurements, we use an additional index to label the three measured points {circumflex over (p)}_i along with their x and y coordinates in projective plane 146 as: {circumflex over (p)}_i1=({circumflex over (x)}_i1,ŷ_i1), {circumflex over (p)}_i2=({circumflex over (x)}_i2,ŷ_i2), {circumflex over (p)}_i3=({circumflex over (x)}_i3,ŷ_i3). The reduced representations of these measured points {circumflex over (p)}_i1, {circumflex over (p)}_i2, {circumflex over (p)}_i3 are the corresponding rays {circumflex over (r)}_i1, {circumflex over (r)}_i2, {circumflex over (r)}_i3 derived in accordance with the invention, as described above. The model or ideal image point p_i′, which is unknown and not measurable in practice due to noise and structural uncertainty 140, is also shown along with its representation as model or ideal ray r_i′ to aid in the explanation.

[0189] Since rays {circumflex over (r)}_i1, {circumflex over (r)}_i2, {circumflex over (r)}_i3 remove all radial information on where along their extent measured points {circumflex over (p)}_i1, {circumflex over (p)}_i2, {circumflex over (p)}_i3 are located, we can introduce a useful computational simplification. Namely, we take measured points {circumflex over (p)}_i1, {circumflex over (p)}_i2, {circumflex over (p)}_i3 to lie where their respective rays {circumflex over (r)}_i1, {circumflex over (r)}_i2, {circumflex over (r)}_i3 intersect a unit circle UC that is centered on origin o′ of projective plane 146. By definition, a radius rc of unit circle UC is equal to 1.

[0190] Under the simplification the sum of squares for each pair of coordinates of points {circumflex over (p)}_i1, {circumflex over (p)}_i2, {circumflex over (p)}_i3, i.e., ({circumflex over (x)}_i1,ŷ_i1), ({circumflex over (x)}_i2,ŷ_i2), ({circumflex over (x)}_i3,ŷ_i3), has to equal 1. Differently put, we have artificially required that {circumflex over (x)}_i²+ŷ_i²=1 for all measured points. Furthermore, we can use Eq. 5 to compute the corresponding n-vector translations for each measured point as follows:

[0191] ${\hat{n}}_i=\left[\begin{array}{c}-{\hat{y}}_i\\ {\hat{x}}_i\\ 0\end{array}\right].$

[0192] Under the simplification, the translation of each ray {circumflex over (r)}_i1, {circumflex over (r)}_i2, {circumflex over (r)}_i3 into its corresponding n-vector {circumflex over (n)}_i1, {circumflex over (n)}_i2, {circumflex over (n)}_i3 ensures that the latter is normalized. Since the n-vectors do not reside in projective plane 146 (see FIG. 7) their correspondence to rays {circumflex over (r)}_i1, {circumflex over (r)}_i2, {circumflex over (r)}_i3 is only indicated with arrows in FIG. 9A.

[0193] Now, space point P_i represented by vector m_i=(x_i,y_i,z_i)^T (which is not necessarily normalized) is mapped by the transposed reduced homography H^T. The result of the mapping is vector m_i′=(x_i′,y_i′,z_i′). The latter, because of its reduced representation as seen above in Eq. 8, is translated into just a two-dimensional pole n_i′=(−y_i′,x_i′). Clearly, when working with just the two-dimensional pole n_i′ we expect that the 2×3 transposed reduced homography H^T of the invention will offer certain advantages over the prior art full 3×3 homography A.

[0194] Of course, camera 104 does not measure ideal data while phone 102 is held in the unknown pose. Instead, we get three measured points {circumflex over (p)}_i1, {circumflex over (p)}_i2, {circumflex over (p)}_i3, their rays {circumflex over (r)}_i1, {circumflex over (r)}_i2, {circumflex over (r)}_i3 and the normalized n-vectors representing these rays, namely {circumflex over (n)}_i1, {circumflex over (n)}_i2, {circumflex over (n)}_i3. We want to obtain an estimate of transposed reduced homography H^T in the form of estimation matrix Θ that best explains n-vectors {circumflex over (n)}_i1, {circumflex over (n)}_i2, {circumflex over (n)}_i3 we have derived from measured points {circumflex over (p)}_i1, {circumflex over (p)}_i2, {circumflex over (p)}_i3 to ground truth expressed for that space point P_i by vector n_i′. This problem can be solved using several known numerical methods, including iterative techniques. The technique taught herein converts the problem into an eigenvector problem in linear algebra, as discussed in the next section.

[0195] We start by noting that the mapped ground truth vector n_i′ (i.e., the ground truth vector after the application of the homography) and measured n-vectors {circumflex over (n)}_i1, {circumflex over (n)}_i2, {circumflex over (n)}_i3 should align under a correct mapping. Let us call their lack of alignment with mapped ground truth vector n_i′ a disparity h. We define disparity h as the magnitude of the cross product between n_i′ and measured unit vectors or n-vectors {circumflex over (n)}_i1, {circumflex over (n)}_i2, {circumflex over (n)}_i3. FIG. 9B shows the disparity h_i1 between n_i′, which corresponds to space point P_i, and {circumflex over (n)}_i1 derived from first measurement point {circumflex over (p)}_i1=({circumflex over (x)}_i1,ŷ_i1). From the drawing figure, and by recalling the Pythagorean theorem, we can write a vector equation that holds individually for each disparity h_i as follows:
h_i²+( n_i′,{circumflex over (n)}_i)²= n_i′, n_i′  (Eq. 18)

[0196] Substituting with the actual x and y components of the vectors in Eq. 18, collecting terms and solving for h_i², we obtain:
h_i²=(y_i′)²+(x_i′)²−(y_i′)²(ŷ_i)²−(x_i′)²({circumflex over (x)}_i)²−2(x_i′y_i′)({circumflex over (x)}_iŷ_i)  (Eq. 19)

[0197] Since we have three measurements, we will have three such equations, one for each disparity h_i1, h_i2, h_i3.

[0198] We can aggregate the disparity from the three measured points we have, or indeed from any number of measured points, by taking the sum of all disparities squared. In the present case, the approach produces the following performance criterion and associated optimization problem:

[0199] $\begin{array}{cc}\begin{array}{c}\min \\ \mathrm{over}{\theta }_1,\dots ,{\theta }_6\end{array}J=\frac{1}{2}\sum h_i^2\mathrm{such}\mathrm{that}\mathrm{Det}{\mathrm{\Theta \Theta }}^T=1.& \left(\mathrm{Eq}.20\right)\end{array}$

[0200] Note that the condition of the determinant of the square symmetric matrix ΘΘ^T is required to select one member out of the infinite family of possible solutions. To recall, any homography is always valid up to a scale. In other words, other than the scale factor, the homography remains the same for any magnification (de-magnification) of the image or the stationary objects in the environment.

[0201] In a first step, we expand Eq. 19 over all estimation values θ₁, . . . , θ₆ of our estimation matrix Θ. To do this, we first construct vectors θ=(θ₁, θ₂, θ₃, θ₄, θ₅, θ₆) containing all estimation values. Note that θ vectors are six-dimensional.

[0202] Now we notice that all the squared terms in Eq. 19 can be factored and substituted using our computational simplification in which {circumflex over (x)}_i²+ŷ_i²=1 for all measured points. To apply the simplification, we first factor the square terms as follows:
(y_i′)²+(x_i′)²−(y_i′)²(ŷ_i)²−(x_i′)²({circumflex over (x)}_i)²=(x_i′)²(1−{circumflex over (x)}_i²)+(y_i′)²(1−ŷ_i²)

[0203] We now substitute (1−{circumflex over (x)}_i²)=ŷ_i² and (1−ŷ_i²)={circumflex over (x)}_i² from the condition {circumflex over (x)}_i²+ŷ_i²=1 and rewrite entire Eq. 19 as:
h_i²=(x_i′)²(ŷ_i)²+(y_i′)²({circumflex over (x)}_i)²−2(x_i′y_i′)({circumflex over (x)}_iŷ_i).

[0204] From elementary algebra we see that in this form the above is just the square of a difference. Namely, the right hand side is really (a−b)²=a²−2ab+b² in which a=(x_i′)²(ŷ_i)² and b=(y_i′)²({circumflex over (x)}_i)². We can express this square of a difference in matrix form to obtain:

[0205] $\begin{array}{cc}{h}_i^2=\left(x_i^{\prime }{\hat{y}}_i,{\hat{y}}_i^{\prime }{\hat{x}}_i\right)·\left[\left(\begin{array}{c}{x}_i^{\prime }{\hat{y}}_i\\ y_i^{\prime }{\hat{x}}_i\end{array}\right)-\left(\begin{array}{c}{y}_i^{\prime }{\hat{x}}_i\\ x_i^{\prime }{\hat{y}}_i\end{array}\right)\right].& \left(\mathrm{Eq}.21\right)\end{array}$

[0206] Returning now to our purpose of expanding over vectors θ, we note that from Eq. 14 we have already obtained expressions for the expansion of x_i′ and y_i′ over estimation values θ₁, θ₂, θ₃, θ₄, θ₅, θ₆. To recall, x_i′=θ₁x_i+θ₂y_i+θ₃ and y_i′=θ₄x_i+θ₅y_i+θ₆. This allows us to reformulate the column vector [x_i′,ŷ_i,y_i′,{circumflex over (x)}_i] and expand it over our estimation values as follows:

[0207] $\begin{array}{cc}\left[\begin{array}{c}{x}_i^{\prime }{\hat{y}}_i\\ y_i^{\prime }{\hat{x}}_i\end{array}\right]=\left[\begin{array}{cc}{\hat{y}}_i\left[x_i,y_i,1\right]& \begin{array}{ccc}0& 0& 0\end{array}\\ \begin{array}{ccc}0& 0& 0\end{array}& {\hat{x}}_i\left[x_i,y_i,1\right]\end{array}\right]·\stackrel{\_}{\theta }.& \left(\mathrm{Eq}.22A\right)\end{array}$

[0208] Now we have a 2×6 matrix acting on our 6-dimensional column vector θ of estimation values.

[0209] Vector [x_i,y_i,1] in its row or column form represents corresponding space point P_i in canonical pose and scaled coordinates. In other words, it is the homogeneous representation of space points P_i scaled by offset distance d through multiplication by 1/d.

[0210] By using the row and column versions of the vector m_i we can rewrite Eq. 22A as:

[0211] $\begin{array}{cc}\left[\begin{array}{c}{x}_i^{\prime }{\hat{y}}_i\\ y_i^{\prime }{\hat{x}}_i\end{array}\right]=\left[\begin{array}{cc}{\hat{y}}_i{\stackrel{\_}{m}}_i^T& 0\\ 0& {\hat{x}}_i{\stackrel{\_}{m}}_i^T\end{array}\right]·\stackrel{\_}{\theta },& \left(\mathrm{Eq}.22B\right)\end{array}$
where the transpose of the vector is taken to place it in its row form. Additionally, the off-diagonal zeroes now represent 3-dimensional zero row vectors (0,0,0), since the matrix is still 2×6.

[0212] From Eq. 22B we can express [y_i′,{circumflex over (x)}_i,x_i′,ŷ_i]^T as follows:

[0213] $\left[\begin{array}{c}{y}_i^{\prime }{\hat{x}}_i\\ x_i^{\prime }{\hat{y}}_i\end{array}\right]=\left[\begin{array}{cc}0& {\hat{x}}_i{\stackrel{\_}{m}}_i^T\\ {\hat{y}}_i{\stackrel{\_}{m}}_i^T& 0\end{array}\right]·\stackrel{\_}{\theta }.$

[0214] Based on the matrix expression of vector [x_i′,ŷ_i,y_i′,{circumflex over (x)}_i]^T of Eq. 22B we can now rewrite Eq. 21, which is the square of the difference of these two vector entries in matrix form expanded over the 6-dimensions of our vector of estimation values θ as follows:

[0215] $\begin{array}{cc}{h}_i^2={\stackrel{\_}{\theta }}^T·\left[\begin{array}{cc}{\hat{y}}_i{\stackrel{\_}{m}}_i& 0\\ 0& {\hat{x}}_i{\stackrel{\_}{m}}_i\end{array}\right]\left[\begin{array}{cc}{\hat{y}}_i{\stackrel{\_}{m}}_i^T& -{\hat{x}}_i{\stackrel{\_}{m}}_i^T\\ -{\hat{y}}_i{\stackrel{\_}{m}}_i^T& {\hat{x}}_i{\stackrel{\_}{m}}_i^T\end{array}\right]·\stackrel{\_}{\theta }.& \left(\mathrm{Eq}.23\right)\end{array}$

[0216] It is important to note that the first matrix is 6×2 while the second is 2×6 (recall from linear algebra that matrices that are n by m and j by k can be multiplied, as long as m=j).

[0217] Multiplication of the two matrices in Eq. 23 thus yields a 6×6 matrix that we shall call M. The M matrix is multiplied on the left by row vector θ^T of estimation values and on the right by column vector θ of estimation values. This formulation accomplishes our goal of expanding the expression for the square of the difference over all estimation values as we had intended. Moreover, it contains only known quantities, namely the measurements from sensor 130 (quantities with “hats”) and the coordinates of space points P_i in the known canonical pose of camera 104.

[0218] Furthermore, the 6×6 M matrix obtained in Eq. 23 has several useful properties that can be immediately deduced from the rules of linear algebra. The first has to do with the fact that it involves compositions of 3-dimensional m-vectors in column form m_i and row form m_i^T. A composition taken in that order is very useful because it expands into a 3×3 matrix that is guaranteed to be symmetric and positive definite, as is clear upon inspection:

[0219] ${\stackrel{\_}{m}}_i·{\stackrel{\_}{m}}_i^T=\left[\begin{array}{ccc}{x}_i^2& x_i{y}_i& x_i\\ x_i{y}_i& y_i^2& y_i\\ x_i& y_i& 1\end{array}\right].$

[0220] In fact, the 6×6 M matrix has four 3×3 blocks that include this useful composition, as is confirmed by performing the matrix multiplication in Eq. 23 to obtain the 6×6 M matrix in its explicit form:

[0221] $M=\left[\begin{array}{cc}{\hat{y}}_i^2{\stackrel{\_}{m}}_i{\stackrel{\_}{m}}_i^T& -{\hat{x}}_i{\hat{y}}_i{\stackrel{\_}{m}}_i{\stackrel{\_}{m}}_i^T\\ -{\hat{x}}_i{\hat{y}}_i{\stackrel{\_}{m}}_i{\stackrel{\_}{m}}_i^T& {\hat{x}}_i^2{\stackrel{\_}{m}}_i{\stackrel{\_}{m}}_i^T\end{array}\right]=\left[\begin{array}{cc}{S}_{02}& -S_{11}\\ -S_{11}& S_{22}\end{array}\right].$

[0222] The congenial properties of the m_i· m_i^T 3×3 block matrices bestow a number of useful properties on correspondent block matrices S that make up the M matrix, and on the M matrix itself. In particular, we note the following symmetries:
S₀₂^T=S₀₂;S₂₀^T=S₂₀;S₁₁^T=S₁₁;M^T=M.

[0223] These properties guarantee that the M matrix is positive definite, symmetrical and that its eigenvalues are real and positive.

[0224] Of course, the M matrix only corresponds to a single measurement. Meanwhile, we will typically accumulate many measurements for each space point P_i. In addition, the same homography applies to all space points P_i in any given unknown pose. Hence, what we really need is a sum of M matrices. The sum has to include measurements {circumflex over (p)}_ij=({circumflex over (x)}_ij,ŷ_ij) for each space point P_i and all of its measurements further indexed by j. The sum of all M matrices thus produced is called the Σ-matrix and is expressed as:
Σ=Σ_i,jM.

[0225] The Σ-matrix should not be confused with the summation sign used to sum all of the M matrices.

[0226] Now we are in a position to revise the optimization problem originally posed in Eq. 20 using the Σ-matrix we just introduced above to obtain:

[0227] $\begin{array}{cc}\begin{array}{c}\min \\ \stackrel{\_}{\theta }\end{array}J=\frac{1}{2}\left({\stackrel{\_}{\theta }}^T\Sigma \stackrel{\_}{\theta }\right)\mathrm{such}\mathrm{that}\stackrel{\_}{\theta }=1.& \left(\mathrm{Eq}.24\right)\end{array}$

[0228] Note that the prescribed optimization requires that the minimum of the Σ-matrix be found by varying estimation values θ₁, θ₂, θ₃, θ₄, θ₅, θ₆ succinctly expressed by vector θ under the condition that the norm of θ be equal to one. This last requirement is not the same as the original constraint that Det∥ΘΘ^T∥=1, but is a robust approximation that in the absence of noise produces the same solution and makes the problem solvable with linear methods.

[0229] There are a number of ways to solve the optimization posed by Eq. 24. A convenient procedure that we choose herein involves well-known Lagrange multipliers method that provides a strategy for finding the local minimum (or maximum) of a function subject to an equality constraint. In our case, the equality constraint is placed on the norm of vector θ. Specifically, the constraint is that ∥ θ∥=1, or otherwise put: θ^T· θ=1. (Note that this last expression does not produce a matrix, since it is not an expansion, but rather an inner product that is a number, in our case 1. The reader may also review various types of matrix and vector norms, including the Forbenius norm for additional prior art teachings on this subject).

[0230] To obtain the solution we introduce the Lagrange multiplier λ as an additional parameter and translate Eq. 24 into a Lagrangian under the above constraint as follows:

[0231] $\begin{array}{cc}\begin{array}{c}\min \\ \stackrel{\_}{\theta },\lambda \end{array}J=\frac{1}{2}\left({\stackrel{\_}{\theta }}^T\Sigma \stackrel{\_}{\theta }\right)+\frac{\lambda }{2}\left(1-{\stackrel{\_}{\theta }}^T\stackrel{\_}{\theta }\right).& \left(\mathrm{Eq}.25\right)\end{array}$

[0232] To find the minimum we need to take the derivative of the Lagrangian of Eq. 25 with respect to our parameters of interest, namely those expressed in vector θ. A person skilled in the art will recognize that we have introduced the factor of ½ into our Lagrangian because the derivative of the squared terms of which it is composed will yield a factor of 2 when the derivative of the Lagrangian is taken. Thus, the factor of ½ that we introduced above will conveniently cancel the factor of 2 due to differentiation.

[0233] The stationary point or the minimum that we are looking for occurs when the derivative of the Lagrangian with respect to θ is zero. We are thus looking for the specific vector θ* when the derivative is zero, as follows:

[0234] $\begin{array}{cc}\frac{dJ}{d{\stackrel{\_}{\theta }}_{\stackrel{\_}{\theta }={\stackrel{\_}{\theta }}^{*}}}=\Sigma {\stackrel{\_}{\theta }}^{*}-\lambda {\stackrel{\_}{\theta }}^{*}=0.& \left(\mathrm{Eq}.26\right)\end{array}$

[0235] (Notice the convenient disappearance of the ½ factor in Eq. 26.) We immediately recognize that Eq. 26 is a characteristic equation that admits of solutions by an eigenvector of the Σ matrix with the eigenvalue λ. In other words, we just have to solve the eigenvalue equation:
Σ θ*=λ θ*,  (Eq. 27)
where θ*is the eigenvector and λ the corresponding eigenvalue. As we noted above, the Σ matrix is positive definite, symmetrical and has real and positive eigenvalues. Thus, we are guaranteed a solution. The one we are looking for is the eigenvector θ* with the smallest eigenvalue, i.e., λ=λ_min.

[0236] The eigenvector θ* contains all the information about the rotation angles. In other words, once the best fit of measured data to unknown pose is determined by the present optimization approach, or another optimization approach, the eigenvector θ* provides the actual best estimates for the six parameters that compose the reduced homography H, and which are functions of the rotation angles φ,θ,ψ we seek to find (see Eq. 12 and components of reduced or modified rotation matrix R_r^T in Eq. 11). A person skilled in the art will understand that using this solution will allow one to recover pose parameters of camera 104 by applying standard rules of trigonometry and linear algebra.

[0237] We now turn to FIG. 10A for a practical example of camera pose recovery that uses the reduced homography H of the invention. FIG. 10A is an isometric view of a real, stable, three-dimensional environment 300 in which the main stationary object is a television 302 with a display screen 304. World coordinates (X_w,Y_w,Z_w) that parameterize environment 300 have their origin in the plane of screen 304 and are oriented such that screen 304 coincides with the X_w-Y_w plane. Moreover, world coordinates (X_w,Y_w,Z_w) are right-handed with the Z_w-axis pointing into screen 304.

[0238] Item 102 equipped with on-board optical apparatus 104 is the smart phone with the CMOS camera already introduced above. For reference, viewpoint O of camera 104 in the canonical pose at time t=t_o is shown. Recall that in the canonical pose camera 104 is aligned such that camera coordinates (X_c,Y_c,Z_c) are oriented the same way as world coordinates (X_w,Y_w,Z_w). In other words, in the canonical pose camera coordinates (X_c,Y_c,Z_c) are aligned with world coordinates (X_w,Y_w,Z_w) and thus the rotation matrix R is the identity matrix I.

[0239] The condition that the motion of camera 104 be essentially confined to a reference plane holds as well. Instead of showing the reference plane explicitly in FIG. 10A, viewpoint O is shown with a vector offset d from the X_w-Y_w plane in the canonical position. The offset distance from the X_w-Y_w plane that viewpoint O needs to maintain under the condition on the motion of camera 104 from the plane of screen 304 is just equal to that vector's norm, namely d. As already explained above, offset distance d to X_w-Y_w plane may vary slightly during the motion of camera 104 (see FIG. 5A and corresponding description). Alternatively, the accuracy up to which offset distance d is known can exhibit a corresponding tolerance.

[0240] In an unknown pose at time t=t₂, the total displacement between viewpoint O and the origin of world coordinates (X_w,Y_w,Z_w) is equal to d+ h. The scalar distance between viewpoint O and the origin of world coordinates (X_w,Y_w,Z_w) is just the norm of this vector sum. Under the condition imposed on the motion of camera 104 the z-component (in world coordinates) of the vector sum should always be approximately equal to offset distance d set in the canonical pose. More precisely put, offset distance d, which is the z-component of vector sum d+ h should preferably only vary between d−ε_f and d+ε_b, as explained above in reference to FIG. 5A.

[0241] In the present embodiment, the condition on the motion of smart phone 102, and thus on camera 104, can be enforced from knowledge that allows us to place bounds on that motion. In the present case, the knowledge is that smart phone 102 is operated by a human. A hand 306 of that human is shown holding smart phone 102 in the unknown pose at time t=t₂.

[0242] In a typical usage case, the human user will stay seated a certain distance from screen 304 for reasons of comfort and ease of operation. For example, the human may be reclined in a chair or standing at a comfortable viewing distance from screen 304. In that condition, a gesture or a motion 308 of his or her hand 306 along the z-direction (in world coordinates) is necessarily limited. Knowledge of the human anatomy allows us to place the corresponding bound on motion 308 in z. This is tantamount to bounding the variation in offset distance d from the X_w-Y_w plane or to knowing that the z-distance between viewpoint O and the X_w-Y_w plane, as required for setting our condition on the motion of camera 104. If desired, the possible forward and back movements that human hand 306 is likely to execute, i.e., the values of d−ε_f and d+ε_b, can be determined by human user interface specialists. Such accurate knowledge ensures that the condition on the motion of camera 104 consonant with the reduced homography H that we are practicing is met.

[0243] Alternatively, the condition can be enforced by a mechanism that physically constrains motion 308. For example, a pane of glass 310 serving as that mechanism may be placed at distance d from screen 304. It is duly noted that this condition is frequently found in shopping malls and at storefronts. Other mechanisms are also suitable, especially when the optical apparatus is not being manipulated by a human user, but instead by a robot or machine with intrinsic mechanical constraints on its motion.

[0244] In the present embodiment non-collinear optical features that are used for pose recovery by camera 104 are space points P₂₀ through P₂₇ belonging to television 302. Space points P₂₀ through P₂₄ belong to display screen 304. They correspond to its corners and to a designated pixel. Space points P₂₅ through P₂₇ are high contrast features of television 302 including its markings and a corner. Knowledge of these optical features can be obtained by direct measurement prior to implementing the reduced homography H of the invention or they may be obtained from the specifications supplied by the manufacturer of television 302. Optionally, separate or additional optical features, such as point sources (e.g., LEDs or even IR LEDs) can be provided at suitable locations on television 302 (e.g., around screen 304).

[0245] During operation, the best fit of measured data to unknown pose at time t=t₂ is determined by the optimization method of the previous section, or by another optimization approach. The eigenvector θ* found in the process provides the actual best estimates for the six parameters that are its components. Given those, we will now examine the recovery of camera pose with respect to television 302 and its screen 304.

[0246] First, in unknown pose at t=t₂ we apply the optimization procedure introduced in the prior section. The eigenvector θ* we find, yields the best estimation values for our transposed and reduced homography H^T as expressed by estimation matrix Θ. To recall, Eq. 14 shows that the estimation values correspond to entries of 2×2 C sub-matrix and the components of two-dimensional b vector as follows:

[0247] $\begin{array}{cc}\Theta =\left(\begin{array}{ccc}{\theta }_1& {\theta }_2& {\theta }_3\\ {\theta }_4& {\theta }_5& {\theta }_6\end{array}\right)=\left(\begin{array}{cc}C& \stackrel{\_}{b}\end{array}\right).& \left(\mathrm{Eq}.14\right)\end{array}$

[0248] We can now use this estimation matrix Θ to explicitly recover a number of useful pose parameters, as well as other parameters that are related to the pose of camera 104. Note that it will not always be necessary to extract all pose parameters and the scaling factor κ to obtain the desired information.

[0249] Frequently, the most important pose information of camera 104 relates to a pointer 312 on screen 304. Specifically, it is very convenient in many applications to draw pointer 312 at the location where the optical axis OA of camera 104 intersects screen 304, or, equivalently, the X_w-Y_w plane of world coordinates (X_w,Y_w,Z_w). Of course, optical axis OA remains collinear with Z_c-axis of camera coordinates as defined in the present convention irrespective of pose assumed by camera 104 (see, e.g., FIG. 7). This must therefore be true in the unknown pose at time t=t₂. Meanwhile, in the canonical pose obtaining at time t=t_o in the case shown in FIG. 10A, pointer 312 must be at the origin of world coordinates (X_w,Y_w,Z_w), as indicated by the dashed circle.

[0250] Referring now to FIG. 10B, we see an isometric view of just the relevant aspects of FIG. 10A as they relate to the recovery of the location of pointer 312 on screen 304. To further simplify the explanation, screen coordinates (XS,YS) are chosen such that they coincide with world coordinate axes X_w and Y_w. Screen origin Os is therefore also coincident with the origin of world coordinates (X_w,Y_w,Z_w). Note that in some conventions the screen origin is chosen in a corner, e.g., the upper left corner of screen 304 and in those situations a displacement between the coordinate systems will have to be accounted for by a corresponding coordinate transformation.

[0251] In the canonical pose, as indicated above, camera Z_c-axis is aligned with world Z_w-axis and points at screen origin Os. In this pose, the location of pointer 312 in screen coordinates is just (0,0) (at the origin), as indicated. Viewpoint O is also at the prescribed offset distance d from screen origin Os.

[0252] Unknown rotation and translation, e.g., a hand gesture, executed by the human user places smart phone 102, and more precisely its camera 104 into the unknown pose at time t=t₂, in which viewpoint O is designated with a prime, i.e., O′. The camera coordinates that visualize the orientation of camera 104 in the unknown pose are also denoted with primes, namely (X_c′,Y_c′,Z_c′). (Note that we use the prime notation to stay consistent with the theoretical sections in which ideal parameters in the unknown pose were primed and were thus distinguished from the measured ones that bear a “hat” and the canonical ones that bear no marking.)

[0253] In the unknown pose, optical axis OA extending along rotated camera axis Z_c′ intersects screen 304 at unknown location (x_s,y_s) in screen coordinates, as indicated in FIG. 10B. Location (x_s,y_s) is thus the model or ideal location where pointer 312 should be drawn. Because of the constraint on the motion of camera 104 necessary for practicing our reduced homography we know that viewpoint O′ is still at distance d to the plane (XS-YS) of screen 304. Pointer 312 as seen by the camera from the unknown pose is represented by vector m_s′. However, vector m_s′ which extends along the camera axis from viewpoint O′ in the unknown pose to the unknown location of pointer 312 on screen 304 (i.e., m_s′ extends along optical axis OA) is m′_s=(0,0,d).

[0254] The second Euler rotation angle, namely tilt θ, is visualized explicitly in FIG. 10B. In fact, tilt angle θ is the angle between the p-vector p that is perpendicular to the screen plane (see FIG. 8 and corresponding teachings for the definition of p-vector). By also explicitly drawing offset d between unknown position of viewpoint O′ and screen 304 we see that it is parallel to p-vector p. In fact, tilt θ is also clearly the angle between offset d and optical axis OA of rotated camera Z_c′ axis in the unknown pose.

[0255] According to the present teachings, transposed and reduced homography H^T recovered in the form of estimation matrix Θ contains all the necessary information to recover the position (x_s,y_s) of pointer 312 on screen 304 in the unknown pose of camera 104. In terms of the reduced homography, we know that its application to vector m_s=(x_s,y_s,d) in canonical pose should map it to vector m_s′=(0,0,d) with the corresponding scaling factor κ, as expressed by Eq. 13 (see also Eq. 12). In fact, by substituting the estimation matrix Θ found during the optimization procedure in the place of the transpose of reduced homography H^T, we obtain from Eq. 13:
m_s′=κ(C b) m_s.  (Eq. 13′)

[0256] Written explicitly with vectors we care about, Eq. 13′ becomes:

[0257] ${\stackrel{\_}{m}}_s^{\prime }=\left(\begin{array}{c}0\\ 0\end{array}\right)=\kappa \left(\begin{array}{cc}C& \stackrel{\_}{b}\end{array}\right)\left(\begin{array}{c}{x}_s\\ y_s\\ d\end{array}\right).$

[0258] At this point we see a great advantage of the reduced representation of the invention. Namely, the z-component of vector m_s′ does not matter and is dropped from consideration. The only entries that remain are those we really care about, namely those corresponding to the location of pointer 312 on screen 304.

[0259] Because the map is to ideal vector (0,0,d) we know that this mapping from the point of view of camera 104 is a scale-invariant property. Thus, in the case of recovery of pointer 312 we can drop scale factor κ. Now, solving for pointer 312 on screen 304, we obtain the simple equation:

[0260] $\begin{array}{cc}\left(\begin{array}{c}0\\ 0\end{array}\right)=\left(\begin{array}{cc}C& \stackrel{\_}{b}\end{array}\right)\left(\begin{array}{c}{x}_s\\ y_s\\ d\end{array}\right)=C\left(\begin{array}{c}{x}_s\\ y_s\end{array}\right)+d\stackrel{\_}{b}.& \left(\mathrm{Eq}.28\right)\end{array}$

[0261] To solve this linear equation for our two-dimensional vector (x_s,y_s) we subtract vector d b. Then we multiply by the inverse of matrix C, i.e., by C⁻¹, taking advantage of the property that any matrix times its inverse is the identity. Note that unlike reduced homography H, which is a 2 by 3 matrix and thus has no inverse, matrix C is a non-singular 2 by 2 matrix and thus has an inverse. The position of pointer 312 on screen 304 satisfies the following equation:

[0262] $\begin{array}{cc}\left(\begin{array}{c}{x}_s\\ y_s\end{array}\right)=-{\mathrm{dC}}^{-1}\stackrel{\_}{b}.& \left(\mathrm{Eq}.29A\right)\end{array}$

[0263] To get the actual numerical answer, we need to substitute for the entries of matrix C and vector b the estimation values obtained during the optimization procedure. Just to denote this in the final numerical result, we will denote the estimation values taken from the eigenvector θ* with “hats” (i.e., θ*=({circumflex over (θ)}₁,{circumflex over (θ)}₂,{circumflex over (θ)}₃,{circumflex over (θ)}₄,{circumflex over (θ)}₅,{circumflex over (θ)}₆) and write:

[0264] $\begin{array}{cc}\left(\begin{array}{c}{\hat{x}}_s\\ {\hat{y}}_s\end{array}\right)=-{d\left(\begin{array}{cc}{\hat{\theta }}_1& {\hat{\theta }}_2\\ {\hat{\theta }}_4& {\hat{\theta }}_5\end{array}\right)}^{-1}\left(\begin{array}{c}{\hat{\theta }}_3\\ {\hat{\theta }}_6\end{array}\right).& \left(\mathrm{Eq}.29B\right)\end{array}$

[0265] Persons skilled in the art will recognize that this is a very desirable manner of recovering pointer 312, because it can be implemented without having to perform any extraneous computations such as determining scale factor κ.

[0266] Of course, in many applications the position of pointer 312 on screen 304 is not all the information that is desired. To illustrate how the rotation angles φ,θ,ψ are recovered, we turn to the isometric diagram of FIG. 10C, which again shows just the relevant aspects of FIG. 10A as they relate to the recovery of rotation angles of camera 104 in the unknown pose. Specifically, FIG. 10C shows the geometric meaning of angle θ, which is also the second Euler rotation angle in the convention we have chosen herein.

[0267] Before recovering the rotation angles to which camera 104 was subject by the user in moving from the canonical to the unknown pose, let us first examine sub-matrix C and vector b more closely. Examining them will help us better understand their properties and the pose parameters that we will be recovering.

[0268] We start with 2×2 sub-matrix C. The matrices whose composition led to sub-matrix C and vector b were due to the transpose of the modified or reduced rotation matrix R_r^T involved in the transpose of the reduced homography H^T of the present invention. Specifically, prior to trigonometric substitutions in Eq. 11 we find that in terms of the Euler angles sub-matrix C is just:

[0269] $\begin{array}{cc}C=\left(\begin{array}{cc}{\theta }_1& {\theta }_2\\ {\theta }_4& {\theta }_5\end{array}\right)=\left(\begin{array}{cc}\begin{array}{c}\cos \varphi \cos \psi -\\ \cos \theta \sin \varphi \sin \psi \end{array}& \begin{array}{c}\cos \psi \sin \varphi +\\ \cos \mathrm{\theta cos}\varphi \sin \psi \end{array}\\ \begin{array}{c}-\cos \theta \cos \mathrm{\psi sin}\varphi -\\ \cos \varphi \sin \psi \end{array}& \begin{array}{c}\cos \theta \cos \mathrm{\varphi cos}\psi -\\ \sin \varphi \sin \psi \end{array}\end{array}\right).& \left(\mathrm{Eq}.30A\right)\end{array}$

[0270] Note that these entries are exactly the same as those in the upper left 2×2 block matrix of reduced rotation matrix R_r^T. In fact, sub-matrix C is produced by the composition of upper left 2×2 block matrices of the composition R^T(ψ)R^T(θ)R^T(φ) that makes up our reduced rotation matrix R_r^T (see Eq. 10A). Hence, sub-matrix C can also be rewritten as the composition of these 2×2 block matrices as follows:

[0271] $\begin{array}{cc}C=\left(\begin{array}{cc}\cos \psi & \sin \psi \\ -\sin \psi & \cos \psi \end{array}\right)\left(\begin{array}{cc}1& 0\\ 0& \cos \theta \end{array}\right)\left(\begin{array}{cc}\cos \varphi & \sin \varphi \\ -\sin \varphi & \cos \varphi \end{array}\right).& \left(\mathrm{Eq}.30B\right)\end{array}$

[0272] By applying the rule of linear algebra that the determinant of a composition is equal to the product of determinants of the component matrices we find that the determinant of sub-matrix C is:

[0273] $\begin{array}{cc}\mathrm{Det}\left(C\right)=\mathrm{Det}\left(\begin{array}{cc}\cos \psi & \sin \psi \\ -\sin \psi & \cos \psi \end{array}\right)\mathrm{Det}\left(\begin{array}{cc}1& 0\\ 0& \cos \theta \end{array}\right)\mathrm{Det}\left(\begin{array}{cc}\cos \varphi & \sin \varphi \\ -\sin \varphi & \cos \varphi \end{array}\right)=\cos \theta & \left(\mathrm{Eq}.31\right)\end{array}$

[0274] Clearly, the reduced rotation representation of the present invention resulting in sub-matrix C no longer obeys the rule for rotation matrices that their determinant be equal to one (see Eq. 3). The rule that the transpose be equal to the inverse is also not true for sub-matrix C (see also Eq. 3). However, the useful conclusion from this examination is that the determinant of sub-matrix C is equal to cos θ, which is the cosine of rotation angle θ and in terms of the best estimates from computed estimation matrix Θ this is equal to:
cos θ=θ₁θ₅−θ₂θ₄.  (Eq. 32)

[0275] Because of the ambiguity in sign and in scaling, Eq. 32 is not by itself sufficient to recover angle θ. However, we can use it as one of the equations from which some aspects of pose can be recovered. We should bear in mind as well, however, that our estimation matrix was computed under the constraint that ∥ θ∥=1 (see Eq. 24). Therefore, the property of Eq. 32 is not explicitly satisfied.

[0276] In turning back to FIG. 10C we see the corresponding geometric meaning of rotation angle θ and of its cosine cos θ. Specifically, angle θ is the angle between offset d, which is perpendicular to screen 304, and the optical axis OA in unknown pose. More precisely, optical axis OA in extends from viewpoint O′ in the unknown pose to pointer location ( x_s, y_s) that we have recovered in the previous section.

[0277] Now, rotation angle θ is seen to be the cone angle of a cone 314. Geometrically, cone 314 represents the set of all possible unknown poses in which a vector from viewpoint O′ goes to pointer location (x_s,y_s) on screen 304. Because of the condition imposed by offset distance d, only vectors on cone 314 that start on a section parallel to screen 304 at offset d are possible solutions. That section is represented by circle 316. Thus, viewpoint O′ at any location on circle 316 can produce line OA that goes from viewpoint O′ in unknown pose to pointer 312 on screen 304. The cosine cos θ of rotation angle θ is related to the radius of circle 316. Specifically, the radius of circle 316 is just d|tan θ| as indicated in FIG. 10C. Since Eq. 32 gives us an expression for cos θ, and tan²θ=(1−cos²θ)/cos²θ, we can recover cone 314, circle 316 and angle θ up to sign. This information can be sufficient in some practical applications.

[0278] To recover rotation angles φ,θ,ψ we need to revert back to the mathematics. Specifically, we need to finish our analysis of sub-matrix C we review its form after the trigonometric substitutions using sums and differences of rotation angles φ and ψ (see Eq. 11). In this form we see that sub-matrix C represents an improper rotation and a reflection as follows:

[0279] $\begin{array}{cc}C=\frac{1-\cos \theta }{2}\left(\begin{array}{cc}\cos \left(\varphi -\psi \right)& \sin \left(\varphi -\psi \right)\\ \sin \left(\varphi -\psi \right)& -\cos \left(\varphi -\psi \right)\end{array}\right)+\frac{1+\cos \theta }{2}\left(\begin{array}{cc}\cos \left(\varphi +\psi \right)& \sin \left(\varphi +\psi \right)\\ -\sin \left(\varphi +\psi \right)& \cos \left(\varphi +\psi \right)\end{array}\right).& \left(\mathrm{Eq}.33\right)\end{array}$

[0280] The first term in Eq. 33 represents an improper rotation (reflection along y followed by rotation) and the second term is a proper rotation.

[0281] Turning now to vector b, we note that it can be derived from Eq. 12 and that it contains the two non-zero entries of reduced rotation matrix R_r^T (see Eq. 10C) such that:

[0282] $\begin{array}{cc}\stackrel{\_}{b}=-C\left(\begin{array}{c}\delta x/d\\ \delta y/d\end{array}\right)+\frac{d-\delta z}{d}\left(\begin{array}{c}\sin \psi \\ \cos \psi \end{array}\right)\sin \theta & \left(\mathrm{Eq}.34\right)\end{array}$

[0283] Note that under the condition that the motion of camera 104 be confined to offset distance d from screen 304, δz is zero, and hence Eq. 34 reduces to:

[0284] $\stackrel{\_}{b}=-C\left(\begin{array}{c}\delta x/d\\ \delta y/d\end{array}\right)+\left(\begin{array}{c}\sin \psi \\ \cos \psi \end{array}\right)\sin \theta .$

[0285] Also note, that with no displacement at all, i.e., when δx and δy are zero, vector b further reduces to just the sine and cosine terms. With the insights gained from the analysis of sub-matrix C and vector b we continue to other equations that we can formulate to recover the rotation angles φ,θ,ψ.

[0286] We first note that the determinant Det∥ΘΘ^T∥ we initially invoked in our optimization condition in the theory section can be directly computed. Specifically, we obtain for the product of the estimation matrices:

[0287] $\begin{array}{cc}{\mathrm{\Theta \Theta }}^T=\left(\begin{array}{cc}C& \stackrel{\_}{b}\end{array}\right)\left[\frac{C^{\prime }}{b^{\prime }}\right]={\mathrm{CC}}^{\prime }+\stackrel{\_}{b}\stackrel{\_}{b^{\prime }}& \left(\mathrm{Eq}.35\right)\end{array}$

[0288] From the equation for pointer recovery (Eq. 29A), we can substitute for b b′ in terms of sub-matrix C, whose value we have already found to be cos θ from Eq. 31, and pointer position. We will call the latter just (x_x,y_s) to keep the notation simple, and now we get for b b′:

[0289] $\begin{array}{cc}\stackrel{\_}{b}\stackrel{\_}{b^{\prime }}={\left(1/d\right)}^{2}C\left(\begin{array}{c}{x}_s\\ y_s\end{array}\right)\left(\begin{array}{cc}{x}_s& y_s\end{array}\right)C^{\prime }.& \left(\mathrm{Eq}.36\right)\end{array}$

[0290] Now we write ΘΘ^T just in terms of quantities we know, by substituting b b′ from Eq. 36 into Eq. 35 and combining terms as follows:

[0291] $\begin{array}{cc}{\mathrm{\Theta \Theta }}^T=C\left[\begin{array}{cc}1+{\left(x_s/d\right)}^2& \left(x_s/d\right)\left(y_s/d\right)\\ \left(y_s/d\right)\left(x_s/d\right)& 1+{\left(y_s/d\right)}^2\end{array}\right]C^{\prime }.& \left(\mathrm{Eq}.37\right)\end{array}$

[0292] We now compute the determinant of Eq. 37 (substituting cos θ for the determinant of C) to yield:

[0293] $\begin{array}{cc}\mathrm{Det}{\mathrm{\Theta \Theta }}^T={\cos }^2\theta \left(1+\frac{x_s^2+y_s^2}{d^2}\right).& \left(\mathrm{Eq}.38\right)\end{array}$

[0294] We should bear in mind, however, that our estimation matrix was computed under the constraint that ∥ θ∥=1 (see Eq. 24). Therefore, the property of Eq. 38 is not explicitly satisfied.

[0295] There are several other useful combinations of estimation parameters θ_i that will be helpful in recovering the rotation angles. All of these can be computed directly from equations presented above with the use of trigonometric identities. We will now list them as properties for later use:

[0296] $\begin{array}{cc}{\theta }_1{\theta }_2+{\theta }_4{\theta }_5={\sin }^2\mathrm{\theta sin}\varphi \cos \varphi & \left(\mathrm{Prop}.I\right)\\ {\theta }_1{\theta }_4+{\theta }_2{\theta }_5=-{\sin }^2\theta \sin \mathrm{\psi cos}\psi & \left(\mathrm{Prop}.\mathrm{II}\right)\\ {\theta }_1^2+{\theta }_4^2={\cos }^2\varphi +{\sin }^2\varphi {\cos }^2\theta & \left(\mathrm{Prop}.\mathrm{III}\right)\\ {\theta }_2^2+{\theta }_5^2={\sin }^2\varphi +{\cos }^2\varphi {\cos }^2\theta & \left(\mathrm{Prop}.\mathrm{IV}\right)\\ {\theta }_1^2+{\theta }_2^2+{\theta }_4^2+{\theta }_5^2=1+{\cos }^2\theta & \left(\mathrm{Prop}.V\right)\\ \frac{{\theta }_2-{\theta }_4}{{\theta }_1+{\theta }_5}=\tan \left(\varphi +\psi \right)& \left(\mathrm{Prop}.\mathrm{VI}\right)\end{array}$