Mask Boundary Correction in a Cone Beam Imaging System Using Simplified Filtered Backprojection Image Reconstruction

1. Field of the Invention

The invention relates generally to 3D image reconstruction in a cone beam x-ray imaging system, and more specifically to correction of image inaccuracies caused by mask boundaries when using a simplified 3D backprojection image reconstruction technique.

2. Description of the Prior Art

A filtered backprojection (FBP) cone beam image reconstruction technique is described by Kudo, H. and Saito, T., in their article entitled "Derivation and Implementation of a Cone-Beam Reconstruction Algorithm for Nonplanar Orbits", IEEE Trans.Med. Imag., MI-13 (1994) 196-211, incorporated herein by reference.

Briefly, the FBP technique consists of the following steps at each cone beam view (i.e., at each position of the radiation source as it scans about the object, and at which an imaging detector acquires a corresponding set of projection data):

1. Compute a ID projection (i.e., line integral) of the measured cone beam image acquired on a detector plane 1 at each of a plurality of angles .theta.. This step is illustrated by FIG. 1A for a given angle .theta..sub.1 of a plurality of angles .theta., where the projection 2 at coordinates (r,.theta.) comprises the integrated values of the cone beam image 1 on detector plane 1 along plurality of parallel lines L(r,.theta.) that are normal to angle .theta., each line L being at an incremental distance r from an origin O. Generally, if the detector plane 1 comprises an N by N array of pixels, then the number of angles .theta. is typically given by .pi.N/2.

2. Filter each ID projection in accordance with a d/dr filter, resulting in a new set of values at each of the r,.theta. coordinates, such as shown by filtered projection 6 for the angle .theta..sub.1 in FIG. 1A.

3. Normalize the filtered projections with a normalization function M(r,.theta.). Normalization is needed to take into account the number of times the plane of integration Q(r,.theta.) which intersects the source position and the line L(r,.theta.), intersects the scan path, since the data developed at each scan path intersection creates a contribution to the image reconstruction on the plane Q(r,.theta.).

4. Backproject the filtered projection 6 from each angle .theta. into a 2D object space 7 which coincides with the detector plane 1. This step is illustrated by FIG. 1B, wherein lines 8 spread the value from each r,.theta. coordinate into 2D space 7 in a direction normal to each .theta..

5. Perform a 1D d/dt filtering of the backprojection image formed in 2D space 7 by step 4. The 1D filtering is performed in the direction of the scan path, i.e., along line 10, where t points in the direction of the scan path.

6. Perform a weighted 3D backprojection of the resulting data in 2D space 7 (i.e., from each pixel in the detector) onto a plurality of sample points P in a 3D object volume 12. The density assigned to each point P is weighted by the inverse of the square of the distance between the point and the spatial coordinates of the x-ray source (see Equation (59) of the forenoted Kudo et al article).

The above prior art procedure will be referred to hereinafter as the 6-step process. It is assumed in this process that the entire cone beam image of the object is captured on the detector of the imaging system. Consider a plane Q(r,.theta.), which intersects the object, formed by the source and the line L(r,.theta.) on the detector at angle .theta. and at a distance r from the origin. Ignoring the function M(r,.theta.), the operations 1 through 6 compute the contribution to the reconstructed object density on the plane Q(r,.theta.) from the x-ray data illuminating the plane and its immediate vicinity. Since the 6-step process is detector driven, a contribution from the data illuminating the plane is computed every time the plane intersects the scan path and thus is illuminated by the x-ray beam. Consequently, the function M(r,.theta.) is used after the filter function in step 2 to normalize the results. Normalization is particularly undesirable since it requires pre-computing and storing a 2D array M(r,.theta.) for each source position along an imaging scan path. Since there are usually hundreds, if not thousands of source positions, this type of normalization is both computationally intensive and resource (computer memory) expensive. In the Kudo et al article, however, at page 203 it is noted that in the special case where the scan path is a circle, steps 1-5 can be simplified into a single convolution step, which essentially comprises ramp filtering the cone beam image in the direction of the scan path. This ramp filtering is equivalent to the well-known Feldkamp algorithm for a single circular orbit, such algorithm being described in particular detail in the article by L. A. Feldkamp, L. C. Davis, and J. W. Kress, entitled "Practical cone-beam algorithm" published in the J. Opt. Soc. Am. A. Vol. 1, 1984, pages 612-619, incorporated by reference herein (see in particular the convolution function equations 15 and 16 at page 614, which describe the convolution function as: ##EQU1## The key to this simplification is that in the special case of a circular scan path, the normalization function M(r,.theta.) is a constant, equal to 2. Consequently, the filtered projection at each r,.theta. that results after step 2, can merely be divided by 2 to compensate for the data redundancy.

In my prior U.S. patent application Ser. No 09/106,537 entitled SIMPLIFIED CONE BEAM IMAGE RECONSTRUCTION USING 3D BACKPROJECTION, filed Jun. 29, 1998 and incorporated herein by reference, I described how to use this ramp filter simplification for image reconstruction in a cone beam imaging system having source scan paths other than a single circle, such as a spiral scan path, and furthermore, in a cone beam imaging system having a short detector, i.e., a detector that captures only a portion of the cone beam image at each cone beam view (i.e., at each source position). As described in greater detail in my aforenoted prior U.S. Patent Application, my "3-step" simplified technique comprises:

1) applying a mask to each set of cone beam projection data acquired at each source position, thereby forming a masked data set for each source position,

2) ramp filtering the cone beam projection data inside each masked data set, forming ramp filtered data sets, and

3) subjecting the ramp filtered data sets to a weighted 3D backprojection into a 3D space corresponding to a complete field of view of a region of interest (ROI) of the object, thereby reconstructing in the 3D space a 3D image of the ROI in the object.

A key part of my prior invention is the masking process. My prior invention allows the image reconstruction processing to be greatly speeded-up due to a reduced need for extensive computations, elimination of the requirement for normalization step 3 of the 6-step process (thereby obviating the need for significant memory allocation for normalization factors), and furthermore, the imaging system can use a short detector, i.e., one that does not acquire at each source position a complete view of the ROI of the object.

My prior invention can be considered as applying masking to steps 1 through 5 of the Kudo et al 6-step process, followed by step 6. The application of masking to Kudo et al's steps 1 and 2 is conceptually equivalent to the following operations:

m1) compute the integrals on line segments bounded by the mask, and

m2) compute the difference between the line integrals so computed on adjacent parallel line segments.

It is intended that steps m1 and m2 yield a quantity proportional to the Radon derivative for the portion of plane Q(r,.theta.) defined by the current source position and the prior and the subsequent source positions. The angular ranges for various portions of plane Q(r,.theta.) are illustrated in FIG. 3, and will be described in greater detail later herein.

The operations in steps m1 and m2 are illustrated in FIG. 4. As shown therein, L, L.sub.1 ' and L.sub.2 ' are three closely spaced parallel line segments that are bound by a mask 400, where L is midway between L.sub.1 ' and L.sub.2 '. Line segment L is representative of many such line segments formed at various angles in mask 400, and corresponds to the previously described lines L (r,.theta.) of FIG. 1A and lines 8 of FIG. 1B, as well known to those skilled in this technology, which are used for computing Radon derivative data from the cone beam projection data. In the present technique, the arithmetic difference between the integrals computed for a given pair of line segments L.sub.1 ' and L.sub.2 ' in mask 400 is determined, and corresponds to the Radon derivative of the partial plane defined by the line segment L and the current source position, up to a multiplicative constant. In reality, however, this masking method yields only an approximation of the Radon derivative of the partial plane. This is because, as described in my recently issued U.S. Pat. No. 5,748,697, the Radon derivative for the relevant portion of plane Q(r,.theta.) should be computed as shown in FIG. 5 herein. As evident from FIG. 5, line segment L is the same as that shown in FIG. 4, however line segments L.sub.1 and L.sub.2 are obtained by orthogonal translation of L, as they should be. Accordingly, contrary to what is shown in FIG. 4, the ends of line segments L.sub.1 and L.sub.2 shown in FIG. 5 are not bound by mask 500. The ending of the line segments at the boundaries of the mask is referred to herein as "hard masking". As a consequence of the hard masking, the arithmetic difference between the integrals computed on line segments L'.sub.1 and L'.sub.2 does not yield an exact Radon derivative, i.e., one only differing only by a multiplicative constant.

It would be desirable to improve my simplified filtered backprojection image reconstruction processing technique so as to develop a more exact image reconstruction.

A scanning and data acquisition method and apparatus for three dimensional (3D) computerized tomographic (CT) imaging of a region-of-interest (ROI) in an object, wherein image reconstruction processing is applied to a plurality of sets of 2D cone beam projection data, each set being acquired on a 2D detector at a corresponding plurality of scan path source positions. A first image reconstruction processing step comprises applying a mask to each set of the projection data so that data inside the boundaries of each mask form a corresponding plurality of masked 2D data sets. The next step comprises ramp filtering of each masked 2D data set along a plurality of parallel lines formed therein, to generate a corresponding plurality of filtered 2D data sets. Each filtered 2D data set corresponds to a calculation of a first estimate of Radon derivative data determined from a given set of the 2D cone beam projection data. The next step comprises developing 2D correction data for each of the first estimates of Radon derivative data by processing portions of the given set of cone beam projection data that are adjacent boundaries of the mask. The final step comprises combining each filtered 2D data set and the 2D correction data calculated therefore, in a weighted 3D backprojection manner into a 3D space, thereby reconstructing a 3D image of the ROI in the object.

FIGS. 1A and 1B illustrate the Kudo et. al. prior art 3D backprojection approach for cone beam image reconstruction, previously described.

FIG. 2 illustrates a cone beam imaging apparatus useful for performing image reconstruction in accordance with the principles of the invention.

FIG. 3 illustrates a procedure for data combination when performing image reconstruction in accordance with the present invention.

FIGS. 4 and 5 illustrate hard and soft masking, respectively, each technique useful for developing Radon derivative data from the acquired cone beam projection.

FIG. 6 is useful for understanding the generation of the masks shown in FIGS. 4,5 and 7-10.

FIGS. 7 and 8 illustrate the shape of the masks that are used at source positions near the top and bottom of the spiral scan path shown in FIG. 2.

FIG. 9 illustrates ramp filtering of a masked data set.

FIG. 10 is useful for illustrating the arithmetic difference between hard and soft masking, and for understanding how to develop more accurate Radon derivative data, in accordance with the principles of the invention.

FIG. 2 illustrates a cone beam 3D CT imaging apparatus useful for acquiring and processing acquired projection data in accordance with the principles of the present invention. The illustrated imaging apparatus is constructed and operates substantially in accordance with the same principles described in U.S. Pat. No. 5,257,183 entitled METHOD AND APPARATUS FOR CONVERTING CONE BEAM X-RAY PROJECTION DATA TO PLANAR INTEGRAL AND RECONSTRUCTING A THREE-DIMENSIONAL COMPUTERIZED TOMOGRAPHY (CT) IMAGE OF AN OBJECT issued Oct. 26, 1993, and U.S. Pat. No. 5,453,666 entitled HELICAL AND CIRCLE SCAN REGION OF INTEREST COMPUTERIZED TOMOGRAPHY issued Oct. 31, 1995, both incorporated herein by reference, except as to be specifically described later with respect to implementation of image reconstruction processing in accordance with the present invention.

As shown in FIG. 2, in response to control signals from an appropriately programmed computer 206, a computer controlled manipulator 208 causes a source 210 of a cone or pyramid shaped beam of energy (such as x-rays) and a two-dimensional pixelated detector array 212 to cooperate (scan) at a plurality of discreet, sequentially occurring, adjacent source positions along a pre-defined source scanning path. In the illustrated embodiment the scanning path is shown as a spiral scan path 214 centered on a predetermined axis 215 of an object 216. Other types of scan paths that encircle and traverse object 216 can also be used, however, as will become apparent later, a scan path 214 exhibiting a high degree of symmetry in its parallel projection is preferred.

The only height requirement on the detector is that it should be more than the distance between adjacent turns of a projection of the spiral scan path on the detector. If only a region of interest (ROI) in object 216 is to be imaged, in a preferred embodiment, the known technique of providing a top circle scan T at the top level of the ROI and a bottom circle scan B at the bottom level of the ROI, are added.

As a result of the source/detector cooperation under control of computer 206 and manipulator 208, at each of the source positions along path 214, x-ray energy passes through the field of view of the imaging apparatus, is attenuated by object 216, and a set of projection data corresponding to the sensed x-ray energy falling on the elements (pixels) within detector 212 is developed. Each set of projection data is supplied to a data acquisition system (DAS) 217 which, like the previously described portions of FIG. 2, may operate in a fashion well known to those of ordinary skill in this technology for digitizing and storing of the acquired projection data.

In my above-noted U.S. Pat. Nos. 5,257,183 and 5,453,666, image reconstruction processing 218 is provided by Radon space driven conversions, thereby developing an image reconstruction of object 216 on a display 220. The present inventor has devised a way to incorporate the technique of data combination for ROI reconstruction, with the Kudo et. al. image reconstruction processing simplification for a single circle scan imager, thereby providing a cone beam imaging system that can not only have a spiral scan path, but can also use a short detector. Instead of division by the function M(r,.theta.) as done by Kudo et al., in the present invention normalization of the reconstructed object densities is achieved by dividing the x-ray beam coverage of plane Q(r,.theta.) between the various source positions that illuminate the plane without any overlap. Consequently, the image reconstruction processing in accordance with the invention is greatly speeded-up due to a reduced need for extensive computations, the requirement of significant memory allocation for normalization step 3 is obviated, and furthermore, the imaging system can use a shorter detector, i.e., one that does not acquire at each source position a complete view of the ROI of the object.

More specifically, this concept of the division of x-ray beam coverage is illustrated in FIG. 3, which represents a typical integration plane Q(r,.theta.) intersecting cylindrical object 216 and the spiral scan path 214, which is assumed to wrap around an ROI in object 216 on an imaginary cylinder. An edge view of plane Q is illustrated in FIG. 2. Since a non-vertical plane will intersect a cylinder in an ellipse, the plane Q(r,.theta.) intersects object 216 and the cylindrical spiral scan path 214 in two ellipses, one inside the other.

The intersection of the integration plane Q with the object cylinder is indicated by the smaller ellipse E.sub.1, and its intersection with the scan path cylinder is indicated by the larger ellipse E.sub.2. Since spiral path 214 lies on the scan path cylinder, it intersects the plane Q in points that lie on the ellipse E.sub.2. These source positions are shown as S.sub.1, S.sub.2, and S.sub.3 in the FIG. 3. Similarly, it is easy to see that the top scan path circle intersects the plane in two points T.sub.1 and T.sub.2 which lie at the intersection between E.sub.2 and the top edge of the object's region-of-interest (shaded portion of object 216), and that the bottom circle intersects the plane in the two points B.sub.1 and B.sub.2 which lie at the intersection between E.sub.2 and the bottom edge of the object's region-of-interest. Other integration planes may have more or less spiral scan path intersections, depending upon their orientation, and may not intersect either of the top or the bottom circle scan paths.

As is apparent from FIG. 3, the source positions which illuminate that portion of integration plane Q that lies within the ROI (shaded area 300), are T.sub.2, S.sub.1, S.sub.2, S.sub.3, and B.sub.2. Complete X-ray coverage of region-of-interest 300 of this portion of the integration plane can be achieved by suitably combining the data acquired at these 5 source positions, as indicated in FIG. 3. For example, at T.sub.2 we only use the cone beam data within the angle bound by T.sub.1 T.sub.2 and S.sub.1 T.sub.2, and at S.sub.1 we only use the cone beam data within the angle bound by T.sub.2 S.sub.1 and S.sub.2 S.sub.1. And so on. Five partial planes P1 through P5 are therefore defined by the source positions T.sub.2, S.sub.1, S.sub.2, S.sub.3, and B.sub.2, which do not overlap and together completely cover the portion of plane Q that lies within ROI 300. In this way the totality of the cone beam data from each of the contributing source positions illuminates the entire plane Q(r,.theta.) only once without any overlap. Further details of this data combination technique can be found in my earlier cone beam patents such as U.S. Pat. No. 5,463,666.

Because only specific non-overlapping contributions to the Radon data area developed from the projection data, the function M(r,.theta.) can be set to unity for all cone beam views. Thus, as the cone beam projection data acquired by the detector at each of the source positions is processed, as next described, contributions to each plane intersecting the ROI are developed only once.

As described above in conjunction with FIG. 3, the acquired cone beam projection data should be restricted to the appropriate angular range to avoid data redundancy, and thereby avoid the need for Kudo et al's normalization function M(r,.theta.). For image reconstruction in accordance with the present invention, this is accomplished using a masking process. In general, the masking of acquired cone beam projection data is well known, for example see my earlier U.S. Pat. No. 5,504,792 issued Apr. 2, 1996. FIG. 4 illustrates one such mask 400. Mask 400 consists of a top curve 402 and a bottom curve 404, each curve being formed by cone beam projections of the spiral scan path turn above and the spiral scan path turn below the current source position, onto the detector (212 of FIG. 2). For a flat detector located at the rotation axis such that a line connecting the source to the detector origin is normal to the detector plane, the equation for top curve 402 for the spiral scan path projection is given by: ##EQU2## where x and y are the Cartesian coordinate axes of the detector, with the y axis coinciding with the rotation axis, a is the radius of the spiral, and h is the distance between adjacent spiral turns (the pitch). Bottom curve 404 is a reflection of top curve 402 about the origin, i.e., ##EQU3##

As described in conjunction with FIG. 2, for ROI imaging, circular arc scans are needed at the top and bottom levels. The top circle scan T starts at the angle (.pi.+.alpha.) before the start of the spiral scan, and the bottom circle scan B ends at the angle (.pi.+.alpha.) after the end of the spiral scan, where .alpha. is the fan angle of the x-ray beam. The detailed geometry of the mask used at each source position depends on the location of the source in the scan path. Accordingly, one can divide the spiral scan path into 5 distinct regions, as illustrated in FIG. 6. The first region comprises the last (.pi.+.alpha.) turn of the top circle (clockwise from B to A). The second region comprises the first (.pi.+.alpha.) turn of the spiral (clockwise from A to C). The third region comprises the interior portion of the spiral, i.e. after the first (.pi.+.alpha.) turn and before the last (.pi.+.alpha.) turn. The fourth region comprises the last (.pi.+.alpha.) turn of the spiral (similar to the second region). The fifth region comprises the first (.pi.+.alpha.) turn of the bottom circle (similar to the first region, but at the bottom). The masks for these 5 regions are described in greater detailed below, and are illustrated in FIGS. 7-8. These Figures assume the radiation source rotates in a spiral path from top to bottom in a clockwise direction.

(1) For the last (.pi.+.alpha.) turn of the top circle, see mask 700 of FIG. 7, wherein:

The top curve is a horizontal line at the level of the top circular arc; and

The bottom curve is a reflection of Equation (1) about the origin.

(2) For the first (.pi.+.alpha.) turn of the spiral, see mask 800 of FIG. 8, wherein:

The top curve is the intersection of two curves: the standard top spiral mask, Equation (1), and the cone beam projection of the top circle projected from the source given by the equation: ##EQU4## where 2b is the distance between the top and bottom circles; and The bottom curve is a reflection of Equation (1) about the origin.

(3) For the interior portion of the spiral, see mask 400 of FIG. 4, wherein:

The top curve described by Equation (1); and

The bottom curve is a reflection of Equation (1) about the origin.

(4) For the last (.pi.+.alpha.) turn of the spiral, see mask 800 of FIG. 8, but rotated by 180.degree..

(5) For the first (.pi.+.alpha.) turn of the bottom circle, see mask 700 of FIG. 7, but rotated by 180.degree..

A first step of image reconstruction in accordance with the present invention comprises bounding the set of projection data acquired by the detector at each source position with an appropriate mask, such as one of the masks of FIGS. 4 and 7-8. In accordance with the general principles of masking, data in the set that is outside the mask is changed to a zero value, while data that is inside the mask is left unchanged. After a set of projection data is appropriately masked, it is referred to as a masked set of projection data, or more simply as a masked data set. Since the masks are formed by cone beam projection of the spiral turn above and the turn below the current source position, the masked data set corresponds exactly to the angular range bound by the prior and the subsequent source positions, as required by the data combination principles illustrated in FIG. 3. Computer 206 of FIG. 2 can compute the masks "on-the-fly" during image reconstruction, or they could be pre-calculated and stored in system memory.

Next, the Feldkamp ramp filtering technique is applied to the masked sets of projection data, thereby simplifying steps 1 through 5 of the prior art 6-step process into a single ramp filtering step, as illustrated in FIG. 9. More specifically, each set of masked data is ramp filtered along a plurality of parallel lines (t) formed therein, which lines are parallel with and in the direction of a parallel projection into the data set of a line that is tangent to the scan path at the source position that acquired that data set and in the direction of the next source position. Ramp filtering of each masked data set along lines (t) generates a corresponding plurality of filtered data sets, where the data at each point along each line represents a summation of the Radon data at that point, as would have been developed by steps 1 through 5 of the Kudo et al 6-step process. Accordingly, the only step left to reconstruct the image is 3D backprojection, corresponding to Kudo et al's step 6. This 3D backprojection step can also be accomplished by computer 206, with the resultant image reconstruction appearing on display 220.

Although the single step of ramp filtering is much faster than using the prior art steps 1-5, there is a tradeoff, or price to pay, i.e., the developed Radon data is somewhat less accurate than needed to provide an exact image reconstruction. This is due the fact that when calculating the line integral derivative data in the Kudo et al 6-step process, there is no masking of the projection data. Likewise, in the modification of the Kudo et al process as described in my forenoted parent application U.S. Ser. No. 09/052,281, there is no masking of the projection data, only of the extent of the line segments L for which line integral derivatives are calculated. That is, for example, as described in U.S. Ser. No. 09/052,281, line integrals for the adjacent line segments L.sub.1 and L.sub.2 are not masked, and in fact, detector rows above and below the detector rows defining the line segments L are used to exactly determine these line integrals. Masking only takes place on the line segments L, after the line integrals for line segments L.sub.1 and L.sub.2 have been subtracted to calculate the line integral derivatives for the line segments L. However, as a necessary consequence of ramp filtering of the masked projection data in accordance with an aspect of the present invention, it is implied that the extent of the line segments L.sub.1 and L.sub.2 are also limited by their intersection with the mask. This implication leads to a somewhat less than exact reconstruction, but has the tradeoff of improved image reconstruction speed.

The present invention describes a technique for improving the exactness of the image reconstruction without significantly decreasing the speed of the image reconstruction.

More specifically, by comparing FIG. 4 to FIG. 5, it can be seen that the difference between the corresponding line integrals in the two Figures appears only at the upper and lower mask boundaries, the line integrals on the rest of the line segments are identical. Thus the efficient ramp filtering method disclosed in my aforenoted U.S. patent application Ser. No. 09/106,537 can be used to compute a first estimate of the Radon derivative, implicitly using the integrals on line segments inside the mask, as shown in FIG. 4. Next, correction data is developed using, for example, the method shown in FIG. 5 to compute the difference between line integrals at the mask boundaries. The correction data can then be combined with the first estimate, thereby producing a more exact image reconstruction.

The determination of the correction data will now be described. The effect of the hard masking of FIG. 4 on the calculation of the line integral derivatives for lines L, corresponds to: ##EQU5## The effect of the soft masking of FIG. 5 on the calculation of the line integral derivatives for lines L, corresponds to: ##EQU6## Referring jointly to FIGS. 4 and 10 (where FIG. 10 corresponds to FIG. 5, but with additional line segments .DELTA.L shown on mask 1000 that are useful for indicating the difference in the length of the line segments that result during hard as compared with soft masking), equation 1 can be rewritten as: ##EQU7##

By appropriate collection of terms, equation 3 can be rewritten as: ##EQU8##

Substitution of equation 2 in equation 4 yields, ##EQU9##

By re-arranging terms, it is apparent that the exact Radon derivative data which results from soft masking, may be realized by negatively combining the result of hard masking with correction data (CR), as shown below: ##EQU10##

The combination of the 2D correction data with the 2D first estimate data can be made before the 3D weighted backprojection step (i.e., by combining the 2D data in a common single 2D space, such as space 7 shown in FIG. 1B), or the first estimate and the correction data can be individually subjected to 3D weighted backprojection from their own 2D space into a common 3D space, such as space 12 shown in FIG. 1B. In either case, note that the correction data must be backprojected into a 2D space corresponding to the size of a virtual detector, i.e., one that would have been large enough to acquire a complete field of view of the ROI in the object at a single source position.

Since the correction data only needs to be computed at the two mask boundary curves instead of over the entire detector, a ratio reduction in computation of approximately 2/Nz, where Nz is the number of detector rows, is achieved.

Additionally, because the correction data are only computed for cone beam projection data acquired by detector pixels that intersect the mask boundaries, it becomes practical in a preferred embodiment of the invention to use a pixel spread function table to speed up the computation. To generate the pixel spread function table, for each detector pixel (i,j) intersected by the mask boundary, one assign's a value of unity to the pixel and zero everywhere else. Next, one carries out the operations as indicated in FIG. 5. The resulting values are spread over a "virtual detector" 2D space that has a height that is large enough to cover the entire field of view of the object, which 2D information then constitutes the pixel spread function table for the pixel (i,j). Note that the pixel spread function table covers the virtual detector rather than the actual (short) detector, because backprojection from each line segment should be extended to the entire plane Q(r,.theta.), as demanded by the mathematics of Radon inversion.

Since there are approximately Nx pixels on the mask boundary, and the pixel spread function table for each pixel has Nx by Ny entries (where Nx by Ny are the dimensions of the virtual detector), the size of the entire pixel spread function table is Nx by Nx by Ny. Although the table is quite large, the same table can be used for the mask boundaries of the data acquired from all the source positions in the interior of the spiral path. Furthermore, because the bottom mask boundary is the reflection of the top mask boundary about the origin, i.e. (x,y)(-x,-y), a pixel spread function table only needs to be computed for one of them, achieving a reduction factor of 2 in the size of the table.

As previously described, the masks for source positions near the circle scans for ROI imaging are illustrated in FIGS. 7 and 8. For those pixels on portions 802 and 804 of mask 800, the same pixel spread function table can be used as was used for the upper and lower boundaries of mask 500. It is obvious that all the pixels on the horizontal line 702 in FIG. 7 have the same pixel spread function table, and thus only an additional Nx by Ny entries are needed for line 702. For those pixels that are on the lower boundary 704 in FIG. 7 (which is the projection of the immediately adjacent spiral scan path segment from the source positions on the circle scan path), and the upper boundary 806 in FIG. 8 (which is the projection of the circle scan path from the source positions on the immediately adjacent spiral scan path segment), the correction data can be computed directly using the method illustrated in FIG. 5. In a typical spiral scan most of the views involve the standard spiral mask illustrated in FIG. 4, and the number of views involving the boundaries 704 and 806 is small in comparison. Thus the time spent on computing the corrections related to boundaries 704 and 806 is also small.

Thus, there has been shown and described a novel method and apparatus for increasing the exactness of image reconstruction in a cone beam CT imaging apparatus using a simplified 3D backprojection image reconstruction. Many changes, modifications, variations and other uses and applications of the subject invention will, however, become apparent to those skilled in the art after considering this specification and its accompanying drawings, which disclose preferred embodiments thereof. For example, the ramp filtering can be accomplished by processing the masked data sets in Fourier space, as compared with the illustrated real space filtering. Additionally, as previously noted, other scan paths can be used. All such changes, modifications, variations and other uses and applications which do not depart from the teachings herein are deemed to be covered by this patent, which is limited only by the claims which follow as interpreted in light of the foregoing description.