New , e cient Fourier - reconstruction method for approximate imagereconstruction in spiral cone - beam CT at small cone

We present an approximate algorithm for image reconstruction in spiral cone-beam CT at small cone-angles. The rst step of the algorithm is a rebinning from fanbeam to parallel beam projections, that is performed independently for each detector row. The result of this rebinning procedure is a set of parallel views, where all rays are tilted against the z-axis by their cone angle and the rays within each view have diierent z-positions. After that, as with most of the approximate algorithms, the basic idea is to let each oblique ray contribute to the image with a weight that depends on its distance to the image plane. Because the distance of a ray to the image plane changes along the ray, so does its weight. This is the one important diierence compared to standard 2D reconstruction procedures. However, for small cone angles, the variation of the weight along a ray is rather smooth, so that we can synthesize it as a Fourier series using only few Fourier coeecients. In standard 2D image reconstruction, we can build the 2D Fourier spectrum of the image on a polar grid from the Fourier transforms of the views, where each view contributes to a radial line only, because the weights are constant along the rays. In our algorithm, the weights change along the rays. This variation is modeled as a Fourier series with 2N + 1 coeecients. Hence, the views contribute not only to one but to 2N + 1 lines in the 2D Fourier space of the image. For a given pitch (z-table movement per full rotation normalized to the slicewidth of a single detector row), the Fourier coeecients that take into account the variations of the weights are precalculated before the reconstruction and stored in a table. From the generated 2D image spectrum, we calculate the nal image using a gridding technique to convert the sampling grid to cartesian and nally doing a 2D IFFT. We have evaluated our method using simulated test data for various test cases assuming sampling conditions and image quality requirements typical to medical CT. These experiments have shown that the method is applicable for detector arrays with up to at least 32 rows.