To improve the safety (lower dose) and the productivity (faster acquisition) of an 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 the reconstruction suffers from artifacts. The "Compressed Sensing" (CS) approach supposes that the unknown image is in some sense "sparse" or "compressible", and reoncstructs it through a non linear optimization problem (TV/ℓ¹ minimization) by enhancing the sparsity. Using the pixel/voxel as basis, to apply CS framework in CT one usually needs a "sparsifying" transform, and combine it with the "X-ray projector" applying 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. It have 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 multiscale 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. Simulations (2D) show that the existing TV/L1 algorithms are more efficient and the reconstructions have better visual quality than the equivalent approach based on the pixel/wavelet basis. The new approach has also been validated on experimental data (2D), where we have observed that the number of projections in general can be reduced to about 50%, without compromising the image quality.
