A Fourier approach to the natural pixel discretization of brain single-photon emission computed tomography

We apply the natural pixel (NP) approach to the single-photon emission computed tomography (SPECT) problem. The NP approach allows us to split the tomographic problem into two sub-problems. The first is a linear inverse problem. The data are the measured projections and the linear operator is described by a Gram matrix, and provides a set of coefficients. The second consists of computing the solution of the tomographic problem as a linear combination of the elements of the NP basis with respect to the coefficients obtained by solving the first problem, and it provides a solution for any given grid of points. The spatially varying geometric response of the system is taken into account by properly choosing the elements of the basis. The rotational invariance shown by the elements of the considered basis induces a block circulant structure in the Gram matrix. This structure can be used to reduce the computational efforts needed for solving the inverse problem. In particular, we diagonalize (blockwise) the Gram matrix by applying the discrete Fourier transform and we solve the inverse problem in the frequency domain associated with the rotation angles. We develop numerical validation with synthetic data in order to test the performance of the NP approach and to assess the reliability of the results. A reconstruction of a two-dimensional image requires 45-94 s, which is an acceptable time for clinical purposes. Finally, we apply the method to acquired clinical data that consist of a three-dimensional brain scan. (C) 2002 John Wiley Sons, Inc.