Methods and systems for modeling objects and object image data using medial atoms

[0002] This application claims the benefit of U.S. Provisional Patent Application No. 60/326,623 filed Sep. 28, 2001, the disclosure of which is incorporated herein by reference in its entirety.

[0003] The present invention relates to methods and systems for modeling objects and object image data. More particularly, the present invention relates to methods and systems for modeling objects and object image data using medial atoms.

[0004] In the fields of medicine, computer aided design, and computer aided modeling, it is often necessary to generate computer models of real objects. For example, in the field of medicine, images of anatomical structures are obtained using magnetic resonance imaging (MRI) and computer tomography (CT). Both MRI and CT images are used for diagnostic and treatment purposes. For example, in the field of radiation oncology, CT images are used by physicians to determine the appropriate radiation dosage amount and dosage area for patients undergoing radiation therapy. In radiation therapy, it is desirable that the dosage amount be tailored to the size of the tumor being treated and focused only on the area of the tumor being treated since radiation can harm healthy cells.

[0005] Conventional MRI- and CT-based methods for determining radiation therapy dosage areas and amounts involve examining MRI and CT images, manually drawing a map of the dosage area on the images, and making the corresponding marks on the patient on which to focus the radiation beam. While these manual methods are somewhat effective in determining the appropriate dosage amount, it is desirable to improve the accuracy of these methods and decrease the time required to plan a dosage amount. Thus, radiation therapy is one field that would benefit from improved object and image modeling techniques.

[0006] On example of an object and image modeling system is disclosed in U.S. Pat. No. 5,926,568 to Chaney et al. (hereinafter, the '568 patent), the disclosure of which is incorporated herein by reference in its entirety. According to the '568 patent, automatic image recognition of standard shapes is performed using deformable segments, the changes of which are measurable against a deformed model corresponding to an object in a subsequent image. Statistical correlation techniques optimize the match to further refine the shape of the subsequent image. While the methods and systems disclosed in the '568 patent decrease the time and improve the accuracy of image object matching, the image and object models disclosed therein are two-dimensional. Thus, improved tools for modeling three-dimensional structures and for deforming three-dimensional models into image data are needed.

[0007] Conventional computer-based modeling methods for three-dimensional structures involve constructing meshes that represent the surface of the object being modeled. Each point on the surface of a mesh is represented by a point in a model-independent coordinate system, such as a Cartesian coordinate system, a polar coordinate system, or a spherical coordinate system. Representing each point on the surface of a mesh with a model-independent coordinate system greatly increases the difficulty in comparing models to target images, determining correspondence between models, and deforming the surfaces of models. For example, it may be desirable to perform natural actions, such as increasing or decreasing the girth of a model, bending the model, elongating the model, or twisting the model. The mathematical computations required for performing these actions using a model-independent coordinate system greatly increase processing time required for these operations because each point must be independently moved. Accordingly, there exists a long felt need for improved methods and systems for modeling objects and object image data and matching models to target image data.

[0008] According to one aspect of the invention, a method for modeling objects and object image data using medial atoms is disclosed.

[0009] As used herein, the term object refers to a real-world structure, such as an anatomical structure, desired to be modeled. The term object image data refers to a set of data collected by a sensor from an object and stored on a computer. Examples of object image data include CT scan data, MRI data, x-ray data, digital photographs, or any other type of data collected from the real world that can be represented by a set of pixel intensities and positions.

[0010] As used herein, the term medial atom refers to a collection of vectors having predetermined relationships with respect to each other and with respect to one or more medial axes in a model. The term medial axis, as used herein, refers to a set of points equidistant from tangent points on opposite surfaces of a model and located at the intersections of orthogonal lines from the tangent points within the surfaces.

[0011] Medial atoms may be grouped together to form models. As used herein, the terms model and stock model are each used to refer to a medial atom representation of an object or of object image data. A medial atom model may include a single figure or multiple figures. As used herein, the term figure refers to a component or a sub-component of a model. For example, some models may have only a single figure. An example of an object that can be represented by a single figure is an object with a relatively simple shape, such as a kidney. An example of an object that may require multiple figures for accurate modeling is a human hand. A medial atom based model of a hand may include a main figure consisting of the palm of the hand and subfigures consisting of each finger of the hand.

[0012] According to another aspect of the invention, a stock model may be automatically deformed to match target image data by altering one or more of the medial atoms. Exemplary alterations that may be performed on the medial atoms included resizing the medial atoms to increase or decrease the girth of the stock model, rotating the medial atoms to twist the surface of the stock model, and moving the medial atoms to bend or elongate the stock model. Because the present invention allows natural operations, such as elongating, bending, rotating, twisting, and increasing or decreasing girth on a stock model, the amount of time and processing required to deform the model into target image data is reduced.

[0013] Accordingly, it is an object of the invention to provide methods and systems for generating stock models using medial atoms and for automatically deforming the models to match image data using natural shape-based operations.

[0014] It is another object of the invention to provide methods and systems for generating stock models using a model-based coordinate system.

[0015] Some of the objects of the invention having been stated hereinabove, other objects will become evident as the description proceeds when taken in connection with the accompanying drawings as best described hereinbelow.

[0016] Preferred embodiments of the invention will now be explained with reference to the accompanying drawings of which:

[0017] FIG. 1 is a block diagram of a general purpose computing platform on which the methods and systems of the present invention may be implemented;

[0018] FIGS. 2A and 2B are vector diagrams of medial atoms according to embodiments of the present invention;

[0019] FIG. 3A is a perspective view of a model of a slab-like figure created using medial atoms according to an embodiment of the present invention;

[0020] FIG. 3B is a perspective view of the surface implied by the medial atom model illustrated in FIG. 3A;

[0021] FIG. 4A is a perspective view of a medial atom model of a tube-like figure according to an embodiment of the present invention;

[0022] FIG. 4B is a perspective view of a tube-like surface implied by the medial atom model illustrated in FIG. 4A;

[0023] FIG. 5A is a perspective view of a three-dimensional medial atom model of object or image data according to an embodiment of the present invention;

[0024] FIG. 5B is a perspective view of the medial atom model illustrated in FIG. 5A including a coarse mesh joining the sails of the medial atoms according to an embodiment of the present invention;

[0025] FIG. 5C is a perspective view of an end mesh implied by the medial atom model illustrated in FIG. 5A;

[0026] FIG. 5D is a perspective view of an interpolated surface generated based on the medial atom structure illustrated in FIG. 5A;

[0027] FIGS. 6A and 6B are side views of medial atom models illustrating a model-based coordinate system according to an embodiment of the present invention;

[0028] FIG. 6C is a side view illustrating the representation of one medial atom model in the model-based coordinate system of another medial atom model according to an embodiment of the present invention;

[0029] FIG. 7 is a flow chart illustrating exemplary steps for modeling a 3-D object or image data using medial atoms and for deforming the model into image data according to an embodiment of the present invention;

[0030] FIGS. 8A through 8E illustrate exemplary viewing planes that may be displayed to a user to facilitate modeling of three-dimensional image data using medial atoms according to an embodiment of the present invention; and

[0031] FIG. 9 is a side view of pre- and post-deformation medial atom models according to an embodiment of the present invention.

[0032] The present invention includes methods and systems for modeling objects and object image data using medial atoms. The methods and systems of the present invention can be implemented in hardware, firmware, software, or any combination thereof. In one exemplary embodiment, the methods and systems for image modeling objects and object image data using medial atoms may be implemented as application software adapted to execute on a general purpose computer. FIG. 1 illustrates an exemplary operating environment for the methods and systems for modeling objects and object image data using medial atoms according to an embodiment of the present invention. Referring to FIG. 1, computer 100 includes a microprocessor 102, network interfaces 104, I/O interfaces 105, disk controllers/drives 106, and memory 108 connected via bus 110. Microprocessor 102 may be any type of general-purpose processor suitable for executing application software. An exemplary microprocessor suitable for use with embodiments of the present invention is any of the Pentium family of processors available from Intel Corporation.

[0033] Network interfaces 104 may include one or more network adapter cards that communicate with one or more remote computers 111 via network 112. For example, network interfaces 104 may be Ethernet or ATM cards.

[0034] I/O interfaces 105 may provide serial ports for communicating with external devices, such as display device 113, mouse 114, and keyboard 116. I/O interfaces 105 may also include interfaces for other types of input and output devices, such as microphones and speakers.

[0035] Disk controllers/drives 106 may include hardware components for reading to and writing from storage devices, such as removable disks 118, optical disks 120, and fixed disk 122.

[0036] Memory 108 may include random access memory 124 and read only memory 126 for storing instructions to be executed by microprocessor 102. According to the present invention, memory 108 may store an instance of medial atom modeling software 128 for modeling objects and object image data using medial atoms. Medial atom modeling software 128 may also automatically deform medial atom models into image data. Exemplary modeling and deformation operations performed by media atom modeling software 128 will be described in more detail below.

[0037] As stated above, a medial atom is a collection of vectors having predetermined relationships with respect to each other and with respect to one or more medial axes. FIGS. 2A and 2B illustrate examples of medial atoms according to embodiments of the present invention. Referring to FIG. 2A, medial atom 200 is characterized by a first position X that represents the location of the medial atom along a medial axis in a figure. Medial atom 200 is also characterized by boundary pointing vectors {right arrow over (p)} and {right arrow over (s)}. The vectors {right arrow over (p)} and {right arrow over (s)} are analogous to port and starboard sails on a sailing vessel and are therefore referred to herein as port and starboard sails of a medial atom. The vectors {right arrow over (p)} and {right arrow over (s)} are separated by an angle 2. The vectors {right arrow over (p)} and {right arrow over (s)} terminate at tangent points along a boundary 202 of a model image. The length of each of vectors {right arrow over (p)} and {right arrow over (s)} is equal to r. A vector {right arrow over (b)} perpendicular to the vector {right arrow over (b)} and points out of the page in FIG. 2A. Yet another vector {right arrow over (n)} is perpendicular to the vectors and {right arrow over (b)} and {right arrow over (b)} in the atom plane. The vectors and {right arrow over (b)}, {right arrow over (p)}, {right arrow over (s)}, {right arrow over (n)}, {right arrow over (b)}, and define a frame for medial atom 200.

[0038] In FIG. 2A, medial atom 200 may represent a slab-like section of the interior of an image. {right arrow over (p)} and {right arrow over (s)} provide links between the point X on the medial axis and the intersection with boundary 202. This collection of vectors gives approximation as to the position of the boundary and its relationship with regard to the normal. The implied figure section in FIG. 2A is slab-like and centered on the head of the atom sails. In FIG. 2B, medial atom 200 includes the same vectors as medial atom 200 illustrated in FIG. 2A. However, boundary 202 represents the end of an image. Accordingly, an additional parameter is required to describe the curvature of boundary 202 at the end of the image. In one exemplary embodiment, is equal to 1/cos .

[0039] Media atom modeling software 128 illustrated in FIG. 1 may include a modeling tool that allows the user to select medial atoms, such as those illustrated in FIGS. 2A and 2B, and use the medial atoms to create stock models, which may be compared to image data. Accordingly, medial atoms may have graphical representations similar to those illustrated in FIGS. 2A and 2B, as well as internal mathematical representations indicating the sizes, lengths, and angular separations of the various vectors.

[0040] Discreet quantities of medial atoms can be used to model two- and three-dimensional objects and object image data. Two generic types of three-dimensional models that may be formed using medial atoms include slabs and tubes. FIG. 3A illustrates an example of medial atoms connected to imply a slab-like surface. In FIG. 3A, a plurality of medial atoms 200 are connected to each other via a plurality of medial axes 300. The port and starboard sail vectors emanate from the center point (point X in FIGS. 2A and 2B) to imply a boundary for a three-dimensional model. FIG. 3B illustrates the boundary implied by medial atoms 200 in FIG. 3A. As can be seen from FIG. 3B, medial atoms 200 can be connected together to form three-dimensional slab-like models of objects or object image data.

[0041] FIG. 4A illustrates an example of medial atoms connected together to form a tube-like surface. In FIG. 4A, back-to-back medial atoms 200 are aligned along a common, straight medial axis 300. The port and starboard sail vectors of medial atoms 200 imply a tube-like surface. FIG. 4B illustrates the tube-like surface implied by the medial atoms in FIG. 4A. Thus, FIGS. 4A and 4B illustrate that medial atoms can be combined to form models of three-dimensional tube-like objects and object image data. The structures illustrated in FIGS. 3A-4B can be modified and combined to form models of more complex images and objects, as will be described in more detail below.

[0042] In order to represent points on the surface of a figure that do not directly contact one of the vectors in a medial atom, interpolation is required. Interpolation may be achieved by applying a variant of subdivision surface methods, as described in Catmull E. and J. Clark, Recursively Generated B-Spline Surfaces on Arbitrary Topological Meshes, Computer Aided Design 10: 183-188 (1978), the disclosure of which is incorporated herein by reference in its entirety, to the mesh of implied boundary positions in normals given at the sail ends (including the crest sails). The variant, described in detail in Thall, A., Fast C² Interpolating Subdivision Surfaces Using Iterative Inversion of Stationary Subdivision Rules, University of North Carolina Computer Science Department Technical Report TR02-001, located at web page http://www.cs.unc.edu/Research/MIDAG/pubs/papers/, the disclosure of which is incorporated herein by reference in its entirety, makes the subdivision surface match the position and the normal of the sail ends to within predetermined tolerances. The boundary position is C² smooth everywhere except isolated points corresponding to the atoms at the corner of the mesh. From this surface, Thall's method allows the calculation of interpolated medial atoms.

[0043] FIGS. 5A-5D illustrate the modeling of a single figure using medial atoms and the corresponding interpolation. Referring to FIG. 5A, a discreet set of medial atoms 200 for a single figure model are shown. The dashed lines interconnecting the medial atoms represent medial axes for the model. In FIG. 5B, boundary lines are shown interconnecting the sail points of each of medial atoms 200 to imply a coarse mesh for the object being modeled.

[0044] At the ends of the model, the medial atoms define the smoothness of the boundary surface. FIG. 5C illustrates one of the ends of the surface in FIGS. 5A and 5B. In FIG. 5C, the length of the n vectors of each medial atom define the smoothness of the end of the object being modeled.

[0045] FIG. 5D represents an interpolated boundary mesh generated based on the model in FIG. 5B. The boundary illustrated in FIG. 5D touches the ends of the port and starboard sail vectors illustrated in FIGS. 5A-5C. The points on the surface illustrated in FIG. 5D that do not directly touch one of the medial atom vectors are determined based on interpolation using Thall's method described above. Thus, by using interpolation, medial atom modeling software according to an embodiment of the present invention can produce smooth three-dimensional surface models.

[0046] According to an important aspect of the invention, model can be represented using a model-based coordinate system. As used herein, term model-based coordinate system refers to a coordinate system that is based on the medial atom model, rather than a model-independent coordinate system, such as Cartesian coordinate system. If a medial atom model includes multiple figures, each figure is preferably represented in its own figure-based coordinate system. For example, in a medial atom model of a hand, the palm figure and each finger figure may be represented in its own figure-based coordinate system.

[0047] In one exemplary embodiment of the invention, each point in a medial-atom-based model can be represented by model-based coordinates u, v, and t or u, v, t, and , depending on whether the point is inside or outside of the model. The parameter u represents the position along the medial axis. The parameter v represents the side of the medial axis on which the point appears. The parameter t represents the distance in figural width along the appropriate medial sail in the specified position. Using a model-based coordinate system allows a determination of correspondence between different models. For example, a point on a boundary of a model can be compared to a corresponding point before the model is deformed. In addition, an intensity in a target image at a model-based coordinate relative to a putatively deformed model can be compared to the intensity in a training image at a corresponding model-based coordinate relative to the un-deformed model.

[0048] FIGS. 6A and 6B illustrate modeling objects in model-based coordinates according to an embodiment of the present invention. Referring to FIG. 6A, model 600 is a two-dimensional medial atom model before deformation. Model 602 is a medial atom model after deformation. A point 604 on the surface of medial atom model 600 characterized by parameters (u, v, t) can be tracked and compared to the corresponding point 604 in post-deformation model 602. Similarly, in FIG. 6B, pre-deformation model 600 includes a point 606 outside of its boundaries and another point 608 inside of its boundaries characterized by model-based coordinates (u₁, v₁, t₁, ₁) and (u₂, v₂, t₂, ₂), respectively. These coordinates can be used to locate the corresponding point in post-deformation model 602.

[0049] According to another aspect of the invention, the model-based coordinate system may be used to represent one model in another model's coordinate system and vice versa. For example, it may be desirable to track the relative positions of multiple real world objects with respect to each other. One example in which such modeling may be useful is in tracking organs in the abdomen of a human. In this situation it may be desirable to track the position of one organ relative to one or more other organs. According to the present invention, a model of each organ may be generated using medial atoms as described above. Next, the medial atom model of one organ can be placed in the model-based coordinate system of another organ model and vice versa. Representing medial atom models in each other's coordinate systems facilitates tracking of movement and orientation of objects being modeled relative to each other.

[0050] FIGS. 6C illustrates an example of representing a medial atom model in the coordinate system of another medial atom model. In FIG. 6C, a first medial atom model 600, labeled Model 1, is represented by coordinates (u₂, v₂, t₂, ₂), which are medial atom coordinates of another medial atom model 602, labeled Model 2. Similarly, a point on the surface of Model 2 is represented by medial coordinates (u₁, v₁, t₁, ₁), which are medial coordinates of Model 1. Thus, by allowing medial atom models to be represented in each other's coordinate systems, the present invention facilitates tracking of changes between medial atom models.

[0051] Now that the concepts of medial atoms, modeling image data using medial atoms, and model-based coordinate systems have been introduced, exemplary steps modeling objects and images using medial atoms and for automatically deforming models to match image data will now be described. FIG. 7 is a flow chart illustrating exemplary steps that may be performed in modeling objects and image data using medial atoms according to an embodiment of the present invention to an embodiment of the present invention. Referring to FIG. 7, in step ST1 the user selects medial atoms to be used in a model. The number of medial atoms depends on the complexity and the desired granularity of the model. For example, it would take more medial atoms to model a complex structure, such as a hand, than to model a non-complex structure, such as a kidney. The entity being modeled may be a real world object, a photograph of a real world object, or digitized image data.

[0052] In step ST2, the user selects parameters for each of the medial atoms. Selecting parameters for each of the medial atoms includes choosing the links of the various vectors in the medial atoms and the angles between the vectors. In step ST3, the medial atoms are connected together along a medial access as described above. In step ST4, an implied boundary is generated based on the medial atoms. The implied boundary may be generated using interpolation between sail points, as described above.

[0053] Once the model has been created, the model can be used for a variety of purposes. One purpose is to measure changes in time of image data or of real-world objects. For example, if it is desirable to determine how a person's kidney changes over time, an image, such as CT scan, may be obtained a first instance in time. A medial atom model may be generated based on the first image data using steps ST1-ST4 described above. At a later time, another set of image data of the kidney may be obtained. It may be desirable to determine whether the kidney has changed and how and where the kidney has changed. In order to perform such an operation, the model generated based on the first image data is automatically deformed into the image data generated at a later point and time. (step ST5). A method for automatically deforming a model into image data will be described in more detail below.

[0054] In order to allow a user to model image data using medial atoms, the image data must be displayed to the user in a format that facilitates such modeling. If the target image data is three-dimensional, two-dimensional slices of the target image data may be presented to the user in order to allow the user to select the proper links for medial atom vectors. FIG. 8A illustrates exemplary target image data that medial atom modeling software 128 may display to a user and the corresponding medial atom model. In FIG. 8A, target image plane 800 represents a cross section of image data being modeled. The user selects a medial atom 200 and stretches the sails of medial atom 200 to contact boundaries of the slice of target image data 200 being modeled. This may be accomplished by clicking on medial atom 200 and elongating the sails of medial atom 200 until they touch what the user believes to be the boundary of the image data.

[0055] The result of selecting multiple medial atoms to model the image data is a medial atom model 802. FIG. 8B illustrates model 802 and views of sail half planes 804 and atom plane 806 for a medial atom 200. Displaying the sail half planes to the user may also allow the user to more accurately model the image data using medial atoms. FIG. 8C illustrates a view of an atom plane that may be displayed in two dimensions to the user. FIGS. 8D and 8E represent port sail and starboard sail planes and the corresponding port and starboard vectors of the medial atom that may be displayed to the user to help the user model image data using medial atoms. Thus, by displaying target image data to the user in various planes, an image modeling tool according to the present invention allows the user to accurately model image data using medial atoms.

[0056] Once a user has modeled an image using medial atoms, medial atom modeling software 128 of the present invention automatically deforms the user defined model into the target image data.

[0057] Medial atom modeling software 128 according to the present invention automatically deforms medial atom models into image data using model-directed segmentation of objects in volume data. The deformation begins with a manually chosen initial similarity transform of the model. To meet the efficiency requirements of accurate segmentation, the segmentation process then follows a number of stages of segmentation at successively smaller levels of scale. Table 1 shown below illustrates exemplary steps that may be performed in deforming a medial atom model into the target image data. At each scale level the model is the result of the next larger scale level, and medial atom modeling software 128 optimizes an objective function of the same form: the sum of a geometric typicality metric (described below) and a geometry-to-image match metric (described below). At each scale level there is a type of geometric transformation chosen appropriate to that scale and having no more than about 9 parameters.

[0058] The deformation strategy, from a model to a candidate obtained by geometrically transforming the model, follows two basic geometric principles:

[0059] 1) In both the geometric typicality (a measure of how similar a model is to a typical geometry) and the model-to-image match metrics, all geometry is in figurally related terms. Thus,

[0060] model-relative and candidate-relative positions correspond when they have common figural coordinates, and

[0061] all distances are r-proportional.

[0062] 2) Calculating geometric typicality at any scale level is done in terms of the relations relevant to that scale, i.e., relative to its values predicted by the previous, next larger, scale and by its neighbors at its scale. The neighborhood of a medial atom is made up of its immediately adjacent atoms, and the neighborhood of a boundary tile vertex is made up of the adjacent boundary tile vertices.

[0063] To describe the algorithm in detail, a number of definitions for the various transformations will now be described. The main two transformations are similarity transformations and elongation transformations. A similarity transformation involves scaling, rotating, and twisting a medial atom model to match target image data. An elongation transformation involves elongating the medial atom model to match the target image data. Similarity and elongation transformations are successively applied to a model as a whole and then to individual figures within the model. The application of the various transformations to different levels of a model is referred to herein as segmentation. The steps in performing a segmentation of a medial atom model to match target image data will now be described in detail.

TABLE 1
Geometry by Scale Level
Scale
Level	Geometric	Transformation	Primitive	Transformatio	Neighbors
k	entity	Sk	zik	Parameters ik	N(zik)
1	Model	Similarity	Model compl	7: 3D sim transf	None
	complex		Pose	params
2	Model	Similarity	Model pose	7: 3D sim transf	Adjacent objec
				params
3	Main figure	Similarity plus	Figure pose	8: 3D sim transf	Adjacent figure
		Elongation		params,
				1 elongation pa
3	Subfigure	Similarity in figural	Figural pose	6: 4 2D sim tran	Adjacent figure
		coordinates of its ho	in host's cor	params, 1	possibly attach
		figural boundary, plu	& elongation	hinging param,	to same host
		hinging and elongati		1 elongation pa
4	Through sec	Medial atom change	Medial atom	8 (or 9): medial	2-4 adjacent
	of figure		Value	params ( for	medial atoms
	(medial atom			external atoms)
5	Boundary	Displacement along	Boundary	1: displacement	Adjacent
	vertex	medially implied nor	Vertex positi	param	boundary verti

[0064] The process begins with a model z that is manually translated, rotated, and uniformly scaled into the image data by the user to produce an initial model z⁰. z⁰ is successively transformed through a number of scale levels into deformed models z^k until z⁵ is the final segmentation. The details and descriptions of the primitives, their neighbor relations, and the associated transformations at each scale level are given in Table 1.

[0065] Let z^k be the geometric representation at scale level k. Let z^k_i be the representation of the i^th primitive at scale level k. At all scale levels k4, each z^k_i is represented as a collection of medial atoms, and a geometric transformation on z^k_i is computed by applying that transformation to each medial atom in its representation. Each primitive z^k_i for k>1 has a small set of neighbors N(z^k_i) at scale level k and a parent primitive P(z^k_i) at scale level k¹ but of the same type as z^k_i. That is, for k4 P(z^k_i) is represented as a superset of the set representing z^k_i, and for k5 the parent of a boundary vertex is the corresponding vertex on the medially implied surface with zero displacement. Also associated with scale level k is a type of transformation S^k such that z^k_iS^k P(z^k_i). Let the parameters ^k_i be the parameters of the particular transformation S^k applied to P(z^k_i) at scale level k¹ to produce z^k_i at scale level k.

[0066] The similarity transform S consisting of translation by t, rotation O and uniform scaling applied to a medial atom mx, r, F, produces SmOxt,r,OF,. Figural elongation by v leaves fixed the medial atoms at a specified atom row i (the hinge end for subfigures) and successively produces translations and rotations of the remaining atoms in terms of the atoms in the previously treated, adjacent row i, as follows: $S^3\left(v\right){\underset{\_}{m}}_{\mathrm{ij}}=\left\{x_{i^{-}j}+v\left(x_{\mathrm{ij}}-x_{i^{-}j}\right),r_{\mathrm{ij}},{\left({\underset{\_}{F}}_{\mathrm{ij}}{\underset{\_}{F}}_{i^{-}j}^{-1}\right)}^v{\underset{\_}{F}}_{i^{-}j},{}_{\mathrm{ij}}\right\}$

[0067] The subfigure transformation applies a similarity transform to each of the atoms in the hinge. This transformation, however, is not in Euclidean coordinates but in the figural coordinates of the boundary of the parent. The transformation to Euclidean coordinates is not essential in explaining the present invention, so its details are not described herein. The medial atom transformation S⁴ translation by t, rotation O, r scaling , and object angle change applied to a medial atom mx, r, F, produces S⁴(t,O,,)mxt,r, OF,. The boundary displacement transformation r applied to a boundary vertex with position y, medial radial width r, and medially implied normal {right arrow over (n)} yields the position yr{right arrow over (n)}.

[0068] The algorithm for segmentation successively modifies z^k1 to produce z^k. In doing so it passes through the various primitives z^k_i in z^k and for each i optimizes an objective function H(z^k, z^k1, I)w^kGeomdiff(z^k, z^k1)Match z^k, I). Geomdiff(z^k, z^k1) measures the geometric difference between z^k and z^k1, and thereby Geomdiff(z^k, z^k1) measures the geometric typicality of z^k at scale level k. Match(z^k, I) measures the match between the geometric description z^k and the target image I. Both Geomdiff(z^k, z^k1), and Match(z^k, I) are measured in reference to the object boundaries B^k and B^k1, respectively implied by z^k and z^k1. The weight w^k of the geometric typicality is chosen by the user.

[0069] For any medial representation z, the boundary B is computed as a mesh of quadrilateral tiles as follows, with each boundary tile vertex being known both with regards to its figural coordinates u and its Euclidean coordinates y. For a particular figure, u(u,v,t), as described above. When one figure is an attached subfigure of a host figure, with the attachment along the v coordinate of the subfigure, there is a blend region whose boundary has coordinates u(v,w,t), where v and t are the figural coordinates of the subfigure and w 1,1 moves along the blend from the curve on the subfigure terminating the blend (w1) to the curve on the host figure terminating the blend (w1). This blending procedure is detailed in Liu, G. S., S. M. Pizer, S. Joshi, A. G. Gash, P. T. Fletcher and Q. Han, Representation and Segmentation of Multifigure Objects via M-Reps, University of North Carolina Computer Science Department technical report TR02-037, located at web page http://www.cs.unc.edu/Research/MIDAG/pubs/papers/, the disclosure of which is incorporated by reference in its entirety.

[0070] As mentioned in above, the computation of B is accomplished by a variation of the technique described in the above-referenced Catmull publication of the mesh of quadrilateral tiles (or, in general, tiles formed by any polygon) formed from the two (or three sail ends of the medial atoms in z. Thall's variation produces a limit surface that iteratively approaches a surface interpolating in position to sail ends and with a normal interpolating the respective sail vectors. That surface is a B-spline at all but finitely many points on the surface. Medial atom modeling software 128 gives control of the number of iterations and of a tolerance on the normal and thus of the closeness of the interpolations. A method for extending this approach to the blend region between two subfigures is presently under evaluation.

[0071] Geomdiff(z^k, z^k1) is computed as the sum of two terms, one term measuring the difference between the boundary implied by z^k and the boundary implied by z^k1, and, in situations when N(z^k_i) is not empty, another term measuring the difference between boundary implied by z^k and that implied by z^k with z^k_i replaced by its prediction from its neighbors, with the prediction based on neighbor relations in P(z^k_i). The second term enforces a local shape consistency with the model and depends on the fact that figural geometry allows a geometric primitive to be known in the coordinate system of a neighboring primitive. The weight between the neighbor term and the parent term in the geometrical typicality measure is set by the user. In one experiment used to test medial atom modeling software 128, the neighbor term weight was 0.0 in the medial atom stage and 1.0 in the boundary offset stage.

[0072] The prediction of the value of one geometric primitive z^k_j in a model from another z^k_i at the same scale level using the transformation S^k is defined as follows. Choose the parameters of S^k such that S^k applied to the z^k subset of z^k1 is as close as possible to z^k in the vicinity of z^k_j. Apply that S^k to z^k to give predictions (S^k z^k)_j. Those predictions depend on the prediction of one medial atom by another. Medial atom z⁴_jx_j, r_j, F_j, _j predicts medial atom z⁴_ix_i, r_i, F_i, _i by recording T(x_jx_i)/r_j, (r_jr_j)/r_j, F_jF_i¹, _ji, where F_jF_i¹ is the rotation that takes frame F_i into F_j. T takes z⁴_i into z⁴_j and when applied to a modified z⁴_i produces a predicted z⁴_j.

[0073] The boundary difference Bdiff(z₁, z₂) between two models z₁ and z₂ is given by the following average r-proportional distance between boundary points that correspond according to their figural coordinates, although it could involve points with common figural coordinates other than at the boundary and it will in the future involve probabilistic rather than geometric distance measures. Bdiff $\left({\underset{\_}{z}}_1,{\underset{\_}{z}}_2\right)=\left[-{}_{B_2}\frac{{{\underset{\_}{y}}_1-{\underset{\_}{y}}_2}^2}{2{\left(r\left({\underset{\_}{y}}_2\right)\right)}^2}\underset{\_}{y}\right]/\mathrm{area}\left(B_2\right).$

[0074] The r-value is that given by the model at the present scale level, i.e., the parent of the primitive being transformed. The normalization of distance by medial radius r makes the comparison invariant to uniform scaling of both the model and the deformed model for the local geometric component being adjusted at that scale level.

[0075] Finally, the geometry to image match measure Match(z^k, I) between the geometric description z^k and the target image I is given by ${}_{-{}_{\max }}^{{}_{\max }}{}_{B^k}G\left(\right)I_{\mathrm{template}}\left(y,\right)\hat{I}\left(y^{},\right)\underset{\_}{y}$

[0076] G()I_template(y,)(y, r)d yd where y and y are boundary points in B(z^k) and B(z_template^k) that agree in figural coordinates, G() is a Gaussian in , is the target image/rms-normalized with Gaussian weighting in the boundary-centered collar _max, _maxfor the deformed model candidate (see FIG. 7), and the template image I_template and the associated model z_template are described below.

[0077] In summary, for a full segmentation of a multi-model complex, there is first a similarity transformation of the whole complex, then a similarity transform of each model, then for each of the figures in turn (with parent figures optimized before subfigures) first a similarity-like transform that for protrusion and indentation figures respects their being on the surface of their parent, then modification of all parameters of each medial atom. After all of these transformations are complete, there is finally the optimization of the dense boundary vertices implied by the medial stages. Since only the segmentation of single figure objects is described herein, there are three stages beyond the initialization: the figural stage, the medial atom (figural section) stage, and the boundary offset stage.

[0078] For all of the stages with multiple primitives, such as the medial atom stage and the boundary stage, the strategy of iterative conditional modes is followed, so the algorithm cycles among the atoms in the figure or boundary in random order until the group converges. The geometric transformation of a boundary vertex modifies only its position along its normal 1 parameter; the normal direction changes as a result of the shift, thus affecting the next iteration of the boundary transformation.

[0079] Multiscale segmentation by deformable models requires many applications of optimization of the objective function. The optimization must be done at many scale levels and for increasingly many geometric primitives as the scale becomes smaller. Efficient optimization is thus necessary. Both evolutionary approaches and a conjugate gradient approach to optimization have been tried. The significant speed advantages of the conjugate gradient method are utilizable if one can make the objective function void of nonglobal optima for the range of the parameters being adjusted that is guaranteed by the previous scale level. Accordingly, the objective functions used herein are designed to have as broad optima as possible and chosen the fineness of scale levels and intra-level stages to guarantee that each stage or level produces a result within the bump-free breadth of the main optimum of the next stage or level.

[0080] When the target image is noisy and the object contrast is low, the interstep fineness requirement described above requires multiple substages of image blurring within a scale level. That is, at the first substage the target image must be first blurred before being used in the geometry to image match term. At later substages the blurring that is used decreases.

[0081] The objective function used to evaluate embodiments of the present invention may be a log posterior probability. If a log posterior probability is used for the objective function, both terms in the objective function will be probabilistic, as determined by a set of training images. These terms then would be a log prior for the geometric typicality term and a log likelihood for the geometry to image match term. In this situation there is no issue weighting the geometric typicality and geometry to image match terms. However, at present the geometric typicality term is measured in r-proportional squared distances from model-predicted positions and the geometry to image match term is measured in rams-proportional intensity squared units resulting from the correlation of a template image and the target image, normalized by local variability in these image intensities. While this strategy allows the objective function to change little with image intensity scaling or with geometric scaling, it leaves the necessity of setting the relative weight between the geometric typicality term and the geometry to image match term. In medial atom modeling software 128 this weight may be interactively set.

[0082] The remainder of this section consists of a subsection detailing the geometry-to-image match term of the objective function, followed by a section detailing the boundary offset stage of the optimization.

[0083] It is useful to compute the match between geometry and the image based on a model template. Such a match is enabled by comparing the template image I_template and the target image data I at corresponding positions in figural coordinates, at figural coordinates determined in the model. The template is presently determined from a single training image I_template, in which the model z has been deformed to produce z_template by applying the m-reps deformation method through the medial atom scale level (level 4) on the characteristic image corresponding to a user-approved segmentation. In the present implementation, the template is defined only in a mask region defined by a set of figural coordinates, each with a weight of a Gaussian in its figural distance-to-boundary, , about the model-implied boundary. An exemplary standard deviation that may be used is equal to of the half-width of a collar surrounding the model-implied boundary. The mask is selectable as a collar symmetrically placed about the boundary up to a user-chosen multiple of r from the boundary or as the union of the object interior with the collar, a possibility especially easily allowed by a medial representation.

[0084] FIG. 9 illustrates an image 900 both before and after deformation. In FIG. 7, a boundary collar 902 defines a symmetric border around implied boundary 904. The mask is chosen by subdividing the boundary positions affected by the transformation with a fixed mesh of figural coordinates (u,v) and then choosing spatial positions to be spaced along each medial sail (implied boundary normal) at that (u,v). These along-sail positions are equally spaced in the figural distance coordinate up to a plus or minus a fixed cutoff value _max chosen at modeling time. An exemplary cutoff value that may be used is 0.3, so the standard deviation of the weighting Gaussian in the intensity correlation is 0.15.

[0085] The template-to-image match measure is selectable in medial atom modeling software 128 from among a normalized correlation measure, with weights, and a mutual information measure, with weights, but for all the examples here the correlation measure has been used and the weight in all mask voxels is unity. The correlation measure used is an average, over the boundary sample points, of the along sail intensity profile correlations at these sample points. For the geometry to correspond to the volume integral of these point-to-corresponding-point correlations, each profile must be weighted by the boundary surface area between it and its neighboring sample points, and the profile must be weighted by its r-proportional length. In addition, as indicated above, each product is weighted in the correlation by a Gaussian in from the boundary. Also, to make the intensity profiles insensitive to offsets and linear compression in the intensity scale, the template is offset to a mean of zero and both the template and the target image are rms-normalized. The template's rms value is computed within the mask in the training image, and the target image's rms value is computed for a region corresponding to a blurred version of the mask after the manual placement of the model.

[0086] In medial atom modeling software 128, the template is selectable from among a derivative of Gaussian and the intensity values in the training image in the region. In each case, the template is normalized by being offset by the mean intensity in the mask and normalized in rms value.

[0087] The derivative of Gaussian template for model-to-image match is built in figural coordinates in the space of the model, i.e., the space of the training image. That is, each along-sail template profile, after the Gaussian mask weighting, is a derivative of a Gaussian with a fixed standard deviation in the figural coordinate , or equivalently an r-proportional standard deviation in Euclidean distance. The value 0.1 is chosen as the value of the standard deviation in . Since this template is associated with the target image via common figural coordinates, in effect the template in the target image space is not a derivative of 3D Gaussian but a warped derivative of 3D Gaussian, with the template's standard deviation in spatial terms increases with the figural width.

[0088] The boundary deformation stage is similar to active surfaces, except that the geometric typicality term consists not only of a term measuring the closeness of each boundary offset to that at each of the neighboring boundary positions but also a term measuring the log probability of these offsets in the medially based prior. Since the tolerance of the medially implied boundary is r-proportional, the log Gaussian medially based prior, conditional on the medial estimate, is proportional to the negative square of the r-normalized distance to the medially implied boundary (See Chen, D. T., S. M. Pizer, J. M. Whitted, Using Multiscale Medial Models to Guide Volume Visualization. Tech report TR99-014, Dept. of Comp. Sci., Univ. of North Carolina at Chapel Hill, (1999).

[0089] The method of Joshi, S., S. Pizer, P. T. Fletcher, A. Thall, G. Tracton (2001). Multi-scale 3-D Deformable Model Segmentation Based on Medial Description, Information Processing in Medical Imaging 2001 (IPMI '01), Lecture Notes in Computer Science, Springer 2082: 64-77 (2001), with which the segmentation is completed, uses this combined geometric typicality measure, and its boundary to image match measure is a log probability based on the object and its background each having normal intensity distributions.

[0090] The steps described above for applying similarity and elongation transformations to successive levels in a medial atom model allow natural deformations, such as model thickening, thinning, twisting, rotating, elongating, etc., to be performed. Performing such transformations using medial atoms in figural coordinates greatly reduces the computational effort required to perform such transformations over conventional modeling methods that utilize model-independent coordinates. As a result, the medial atom modeling and deformation methods of the present invention are faster and less labor-intensive than conventional three-dimensional modeling techniques.

[0091] It will be understood that various details of the invention may be changed without departing from the scope of the invention. Furthermore, the foregoing description is for the purpose of illustration only, and not for the purpose of limitationthe invention being defined by the claims.