Noniterative MAP Reconstruction Using Sparse

Wepresentamethodfornoniterativemaximumapos- teriori (MAP) tomographic reconstruction which is based on the use of sparse matrix representations. Our approach is to precom- pute and store the inverse matrix required for MAP reconstruc- tion. This approach has generally not been used in the past because the inverse matrix is typically large and fully populated (i.e., not sparse). In order to overcome this problem, we introduce two new ideas. The first idea is a novel theory for the lossy source coding of matrix transformations which we refer to as matrix source coding. This theory is based on a distortion metric that reflects the distor- tions produced in the final matrix-vector product, rather than the distortions in the coded matrix itself. The resulting algorithms are shown to require orthonormal transformations of both the mea- surement data and the matrix rows and columns before quantiza- tion and coding. The second idea is a method for efficiently storing and computing the required orthonormal transformations, which we call a sparse-matrix transform (SMT). The SMT is a general- ization of the classical FFT in that it uses butterflies to compute an orthonormal transform; but unlike an FFT, the SMT uses the but- terflies in an irregular pattern, and is numerically designed to best approximate the desired transforms. We demonstrate the potential ofthenoniterativeMAPreconstructionwithexamplesfromoptical tomography. The method requires offline computation to encode the inverse transform. However, once these offline computations are completed, the noniterative MAP algorithm is shown to reduce both storage and computation by well over two orders of magni- tude, as compared to a linear iterative reconstruction methods.