Methods of X-ray CT image reconstruction from few projections

To improve the safety (low dose) and the productivity (fast acquisition) of a X-ray CT system, we want to reconstruct a high quality image from a small number of projections. The classical reconstruction algorithms generally fail since the reconstruction procedure is unstable and suffers from artifacts. A new approach based on the recently developed 'Compressed Sensing' (CS) theory assumes that the unknown image is in some sense 'sparse' or 'compressible', and the reconstruction is formulated through a non linear optimization problem (TV/l1 minimization) by enhancing the sparsity. Using the pixel (or voxel in 3D) as basis, to apply the CS framework in CT one usually needs a 'sparsifying' transform, and combines it with the 'X-ray projector' which applies on the pixel image. In this thesis, we have adapted a 'CT-friendly' radial basis of Gaussian family called 'blob' to the CS-CT framework. The blob has better space-frequency localization properties than the pixel, and many operations, such as the X-ray transform, can be evaluated analytically and are highly parallelizable (on GPU platform). Compared to the classical Kaisser-Bessel blob, the new basis has a multi-scale structure: an image is the sum of dilated and translated radial Mexican hat functions. The typical medical objects are compressible under this basis, so the sparse representation system used in the ordinary CS algorithms is no more needed. 2D simulations show that the existing TV and l1 algorithms are more efficient and the reconstructions have better visual quality than the equivalent approach based on the pixel or wavelet basis. The new approach has also been validated on 2D experimental data, where we have observed that in general the number of projections can be reduced to about 50%, without compromising the image quality. (author)