A Framework for Discrete Integral Transformations I—The Pseudopolar Fourier Transform

Abstract

The Fourier transform of a continuous function, evaluated at frequencies expressed in polar coordinates, is an important conceptual tool for understanding physical continuum phenomena. An analogous tool, suitable for computations on discrete grids, could be very useful; however, no exact analogue exists in the discrete case. In this paper we present the notion of pseudopolar grid (pp grid) and the pseudopolar Fourier transform (ppFT), which evaluates the discrete Fourier transform at points of the pp grid. The pp grid is a type of concentric-squares grid in which the radial density of squares is twice as high as usual. The pp grid consists of equally spaced samples along rays, where different rays are equally spaced in slope rather than angle. We develop a fast algorithm for the ppFT, with the same complexity order as the Cartesian fast Fourier transform; the algorithm is stable, invertible, requires only one-dimensional operations, and uses no approximate interpolations. We prove that the ppFT is invertible and develop two algorithms for its inversion: iterative and direct, both with complexity O(n 2 log n), where n × n is the size of the reconstructed image. The iterative algorithm applies conjugate gradients to the Gram operator of the ppFT. Since the transform is ill-conditioned, we introduce a preconditioner, which significantly accelerates the convergence. The direct inversion algorithm utilizes the special frequency domain structure of the transform in two steps. First, it resamples the pp grid to a Cartesian frequency grid and then recovers the image from the Cartesian frequency grid.

Issue Information

“…As shown in this work, equally sloped sampling does not contribute to refining image quality nor alleviating the missing wedge problem. As confirmed in (Averbuch et al , 2008), equally sloped sampling is the same as the linogram sampling method published before (Edholm and Herman, ’87). …”

“…Clearly, the EST method is a combination of the linogram method and some routine iterative reconstruction techniques (enhancing object support and positivity). As confirmed in (Averbuch et al , 2008), “Edholm and Herman contributed the fundamental insight that there was a continuum transform that could be discretized compatibly, provided one thinks in terms of slopes rather than angles.” Excluding the well-known iterative enhancement of popular constraints, EST is essentially the same as the linogram method published two decades ago (Edholm and Herman, ’87). Mathematical results were proved in (Edholm and Herman, ’87) to appropriately parameterize the image and Fourier spaces for computational benefits.…”

The discrete diffraction transform

“…The inverse DDT algorithm is a modification of the iterative inverse DRT algorithm from [5]. Since y is not necessarily in the range of the DDT, for example, due to noise or measurement errors, we would like to solve min x F D x − z 2 .…”

“…9.1. We use the same preconditioner as in [5] to improve the convergence rate of the conjugate gradient algorithm. Table 3 shows the performance of the iterative inversion algorithm for random images of size (2N + 1) × (2N + 1) for different values of N .…”

“…Following [5], we use the Fractional Fourier transform rather than USFFT in our algorithm, since it does not include interpolation step, unlike USFFT methods, and therefore is theoretically exact.…”