Using Natural Wavelet Bases and Multiscale Stochastic Models for Tomographic Reconstruction

Abstract

Abstract : We use a multiscale natural pixel type representation of an object, originally developed for incomplete data problems, to construct nearly orthonormal basis functions. The coefficients of expansion of an object in these basis functions are obtained as the 1-D wavelet transform of the (strip integral) projections of the object. This enables us to formulate a multiscale tomographic reconstruction technique wherein the object is reconstructed at multiple scales or resolutions. A complete reconstruction is obtained by combining the reconstructions at different scales. The nearly orthonormal behavior of the basis functions results in a system matrix, relating the input (the object coefficients) and the output (the projection data), which is extremely sparse. The system matrix, in addition to being sparse, is well-conditioned and has a symmetric block-Toeplitz structure if the angular projections are uniformly spaced between 0 degrees and 180 degrees. Fast inversion algorithms exist for these matrices. The multiscale reconstruction technique can find applications in object feature recognition directly from projection data, tackling ill-posed imaging problems where the projection data are incomplete and/or noisy, and construction of multiscale stochastic models for which fast estimation algorithms exist. In this paper, we include examples illustrating the above applications of our multiscale reconstruction technique.