TOMOGRAPHIC RECONSTRUCTION AND ESTIMATION BASED ON MULTISCALE NATURAL-PIXEL BASES

We use a natural pixel-type representation of an object, originally de veloped for incomplete data tomography problems, to construct nearly o rthonormal multiscale basis functions. The nearly orthonormal behavior of the multiscale basis functions results in a system matrix, relatin g the input (the object coefficients) and the output (the projection d ata), which is extremely sparse. In addition, the coarsest scale eleme nts of this matrix capture any ill conditioning in the system matrix a rising from the geometry of the imaging system. We exploit this featur e to partition the system matrix by scales and obtain a reconstruction procedure that requires inversion of only a well-conditioned and spar se matrix. This enables us to formulate a tomographic reconstruction t echnique from incomplete data wherein the object is reconstructed at m ultiple scales or resolutions, In case of noisy projection data we ext end our multiscale reconstruction technique to explicitly account for noise by calculating maximum a posteriori probability (MAP) multiscale reconstruction estimates based on a certain self-similar prior on the multiscale object coefficients. The framework for multiscale reconstr uction presented here can find application in regularization of imagin g problems where the projection data are incomplete, irregular, and no isy, and in object feature recognition directly from projection data.