Digital computation of fractional Fourier and linear canonical transforms and sparse image representation

Abstract

Fast and accurate digital computation of the fractional Fourier transform (FRT) and linear canonical transforms (LCT) are of utmost importance in order to deploy them in real world applications and systems. The algorithms in O(NlogN) to obtain the samples of the transform from the samples of the input function are presented for several different types of FRTs and LCTs, both in 1D and 2D forms. To apply them in image processing we consider the problem of obtaining sparse transform domains for images. Sparse recovery tries to reconstruct images that are sparse in a linear transform domain, from an underdeter- mined measurement set. The success of sparse recovery relies on the knowledge of domains in which compressible representations of the image can be obtained. In this work, we consider two- and three-dimensional images, and investigate the effects of the fractional Fourier (FRT) and linear canonical transforms (LCT) in obtaining sparser transform domains. For 2D images, we investigate direct transforming versus several patching strategies. For the 3D case, we consider biomedical images, and compare several different strategies such as taking 2D slices and optimizing for each slice and direct 3D transforming.