Fourier space tomographic image reconstruction method
A generalized projection-slice theorem for divergent beam projections is disclosed. The theorem results in a method for processing the Fourier transform of the divergent beam projections at each view acquired by a CT system to the Fourier transform of the object function. Using this method, an inverse Fourier transform may be used to reconstruct tomographic images from the acquired divergent beam projections.
1. A computed tomography imaging system, comprising:
an x-ray source producing divergent beams;
a plurality of detector elements for receiving impinging x-ray beams emanating from the x-ray source for a series of projection views in which the source revolves around an object to be imaged;
an acquisition system for acquiring from the detector elements x-ray attenuation data corresponding to each beam at each of the projection views; and
an image reconstructor that receives the x-ray attenuation data from the acquisition system and performs the following steps to reconstruct an image: (a) backprojecting the x-ray attenuation data for each beam to form an array of data points therealong, (b) weighting each backprojected data point by a weighting factor ω(r), where r is the distance between the backprojected data point and a source location of the divergent beams to form weighted backprojected data points, (c) Fourier transforming and processing an array of data which includes the weighted backprojected data points to form an acquired k-space data set; (d) aligning the acquired k-space data set with a reference k-space, and (e) reconstructing an image from the referenced k-space data by performing an inverse Fourier transformation thereon.
2. The system as recited in claim 1 wherein the image reconstructor repeats steps (a) through (e) a plurality of times with the divergent beam moved to a different source position for each repetition.
3. The system as recited in claim 1 wherein the image reconstructor repeats steps (a) through (d) a plurality of times with the divergent beam moved to a different source position for each repetition.
4. The system as recited in claim 1 wherein each divergent beam is a two-dimensional fan beam of x-rays and the Fourier transformations performed in steps (c) and (e) are two-dimensional Fourier transformations.
5. The system as recited in claim 1 wherein each divergent beam is a three-dimensional cone beam of x-rays and the Fourier transformations performed in steps (c) and (e) are three-dimensional Fourier transformations.
6. The system as recited in claim 1 wherein the aligning step (d) includes:
i) phase shifting the acquired k-space data set; and
ii) weighting the phase shifted k-space data set.
7. The system as recited in claim 1 wherein the weighting factor ω(r) is 1/r.
8. The system as recited in claim 2 wherein the different source positions are located along a source trajectory which extends around the object being imaged.
9. The system as recited in claim 8 wherein the source trajectory is substantially circular.
10. The system as recited in claim 8 wherein the source trajectory is substantially helical.
11. The system as recited in claim 3 wherein the different source positions are located along a source trajectory which extends around an object being imaged.
12. The system as recited in claim 11 wherein the source trajectory is substantially circular.
13. The system as recited in claim 11 wherein the source trajectory is substantially helical.
14. A computed tomography imaging system, comprising:
an x-ray source producing divergent beams;
a plurality of detector elements for receiving impinging x-ray beams emanating from the x-ray source for a series of projection views in which the source revolves around an object to be imaged;
an acquisition system for acquiring from the detector elements x-ray attenuation data corresponding to each beam at each of the projection views; and
an image reconstructor that receives the x-ray attenuation data from the acquisition system and performs the following steps: (a) backprojecting each acquired projection to form a set of backprojected data; (b) weighting each set of backprojected data; (c) Fourier transforming the weighted backprojected data to form acquired k-space data; (d) aligning the acquired k-space data for each acquired projection with a reference k-space; and (e) reconstructing an image from the aligned k-space data.
15. The system as recited in claim 14 wherein step (e) includes performing an inverse Fourier transformation.
16. The system as recited in claim 14 wherein step (d) includes phase shifting the acquired k-space data by an amount which is a function of the location of a divergent beam source at the time the corresponding projection was acquired.
17. The system as recited in claim 16 wherein step (b) employs a weighting factor which is a function of the distance of the backprojected data from the divergent beam source.