Highly accurate and efficient forward and back projection methods for computed tomography

1. A method for reconstructing an image from projection data by an image reconstruction computing entity, the method comprising:

receiving, by the image reconstruction computing entity, projection data captured by an imaging device;

reconstructing, by the image reconstruction computing entity, an image based at least in part on the projection data using a forward and back projection technique employing a convolutional spline process, wherein the convolution spline process comprises:

computing projected coordinates of 2-D pixels and 3-D voxels on a detector plane,

computing 1-D footprints on the detector plane of the pixels as a first convolutional spline,

computing 2-D footprints on the detector plane of voxels as a second convolutional spline,

approximating 1-D detector blur of pixels by convolving the 1-D footprints with a blur function modeled as a third convolutional spline,

approximating 2-D detector blur of voxels by convolving the 2-D footprints with a blur function modeled using a fourth convolutional spline, and

determining the contributions of at least one of the pixels or the voxels to a detector bin using weighted sums of evaluation of the first, second, third, and fourth convolutional splines according to the projected coordinates; and

providing, by the image reconstruction computing entity, the image such that a user computing entity receives the image, the user computing entity configured to display the image via a user interface thereof.

2. The method of claim 1 , wherein the forward and back projection technique employing the convolutional spline process comprises using a box spline basis for performing at least one of a footprint or detector blur computation.

3. The method of claim 1 , wherein the forward and back projection technique employing the convolution spline process uses an analytical closed-form calculation of the Radon transform of a box spline in a fan-beam, cone-beam and parallel-beam geometric setting with both flat and arc detectors.

4. The method of claim 3 , wherein the analytical closed-form projection of pixel basis in fan-beam geometric setting is a non-separable function: P u,ζ {M Ξ }(s)=M Ξ′(s) (s′(s))=M [ξ′ 1 (s),ξ′ 2 (s)] (s′(s)).

5. The method of claim 3 , wherein the analytical closed-form projection of voxel basis in cone-beam geometric setting is a non-separable function: P u,τ {M Ξ }(s,t)=M B R T Ξ(2,3) (B R T p(2,3))=M [B R T ξ 1 (2,3)B R T ξ 2 ,B R T ξ 3 ] (u,v).

6. The method of claim 3 , wherein analytical closed-form projection of pixel basis in parallel-beam geometric setting measured by a detector of size is a non-separable function: P u,ζ {M Ξ }(s)=M Ξ′(s,p) (s′(s))=M [ξ′ 1 (s),ξ′ 2 (s),ξ] (s).

7. The method of claim 3 , wherein analytical closed-form projection of voxel basis in parallel-beam geometric setting measured by a detector of size τ=[τ s , τ t ] is a non-separable function: R u,τ {M Ξ }(s,t)=M Z(s,t) (s,t)=M [B P T Ξ(2,3),τ s ,τ t ] (s′(s)).

8. The method of claim 3 , wherein analytical closed-form calculation of projection of pixel basis in fan-beam geometric setting measured by a detector of size is a non-separable function: P u,ζ {M Ξ }(s)=M Ξ′(s,p) (s′(s))=M [ξ′ 1 (s),ξ′ 2 (s),ξ′(s,p)] (s′(s)).

9. The method of claim 3 , wherein analytical closed-form calculation of projection of voxel basis in cone-beam geometric setting measured by a detector of size τ=[τ s ,τ t ] is a none-separable function: P u,ζ {M Ξ }(s,t)=M [B R T ξ 1 (2,3) B R T ξ 2 (2,3), B R T ξ 3 (2,3), τ′ s ,τ′ t ](u,v).

10. The method of claim 3 , wherein parameters of the step of computing projected coordinates of pixels and voxels comprises determining a connecting path from a source to the pixels and voxels and computing projected coordinates of the pixels and voxels on the projected plane based at least in part on boundaries of the pixels and the voxels and detector geometry.

11. The method of claim 3 , further comprising determining an effective blur width of detector cells by computing the perspective projection of positions of the detector cells.

12. The method of claim 1 , wherein the imaging device captured the projection data using a low-dose, few view, or limited view data capture technique.

13. An apparatus comprising at least one processor, a communications interface configured for communicating via at least one network, and at least one memory storing computer program code, the at least one memory and the computer program code configured to, with the processor, cause the apparatus to at least:

receive projection data captured by an imaging device;

reconstruct an image based at least in part on the projection data using a forward and back projection technique employing a convolutional spline process, wherein the convolution spline process comprises:

computing projected coordinates of 2-D pixels and 3-D voxels on a detector plane,

computing 1-D footprints on the detector plane of the pixels as a first convolutional spline,

computing 2-D footprints on the detector plane of voxels as a second convolutional spline,

approximating 1-D detector blur of pixels by convolving the 1-D footprints with a blur function modeled as a third convolutional spline,

approximating 2-D detector blur of voxels by convolving the 2-D footprints with a blur function modeled using a fourth convolutional spline, and

determining the contributions of at least one of the pixels or the voxels to a detector bin using weighted sums of evaluation of the first, second, third, and fourth convolutional splines according to the projected coordinates; and

provide the image such that a user computing entity receives the image, the user computing entity configured to display the image via a user interface thereof.

14. The apparatus of claim 13 , wherein the forward and back projection technique employing the convolutional spline process comprises using a box spline basis for performing at least one of a footprint or detector blur computation.

15. The apparatus of claim 13 , wherein the forward and back projection technique employing the convolution spline process uses an analytical closed-form calculation of the Radon transform of a box spline in a fan-beam, cone-beam and parallel-beam geometric setting with both flat and arc detectors.

16. The apparatus of claim 13 , wherein the imaging device captured the projection data using a low-dose, few view, or limited view data capture technique.

17. A computer program product comprising at least one non-transitory computer-readable storage medium having computer-readable program code portions stored therein, the computer-readable program code portions comprising executable portions configured, when executed by a processor of an apparatus, to cause the apparatus to:

receive projection data captured by an imaging device;

reconstruct an image based at least in part on the projection data using a forward and back projection technique employing a convolutional spline process, wherein the convolution spline process comprises:

computing projected coordinates of 2-D pixels and 3-D voxels on a detector plane,

computing 1-D footprints on the detector plane of the pixels as a first convolutional spline,

computing 2-D footprints on the detector plane of voxels as a second convolutional spline,

approximating 1-D detector blur of pixels by convolving the 1-D footprints with a blur function modeled as a third convolutional spline,

approximating 2-D detector blur of voxels by convolving the 2-D footprints with a blur function modeled using a fourth convolutional spline, and

determining the contributions of at least one of the pixels or the voxels to a detector bin using weighted sums of evaluation of the first, second, third, and fourth convolutional splines according to the projected coordinates; and

provide the image such that a user computing entity receives the image, the user computing entity configured to display the image via a user interface thereof.

18. The method of claim 1 , wherein the computed 1-D footprints are exact 1-D footprints on the detector plane and the computed 2-D footprints are exact 2-D footprints on the detector plane.

19. The apparatus of claim 13 , wherein the computed 1-D footprints are exact 1-D footprints on the detector plane and the computed 2-D footprints are exact 2-D footprints on the detector plane.

20. The computer program product of claim 17 , wherein the computed 1-D footprints are exact 1-D footprints on the detector plane and the computed 2-D footprints are exact 2-D footprints on the detector plane.