Sampling reconstruction of N-dimensional bandlimited images after multilinear filtering in fractional Fourier domain

This paper addresses the problem of multidimensional signal reconstruction from generalized samples in fractional Fourier domain including the deterministic case and the stochastic case. The generalized sampling expansion is investigated for the case where the fractional bandlimited input depends on N real variable, i.e., f ( t ) = f ( t 1 , ⋯ , t N ) and is used as a common input to a parallel bank of m independent N dimensional linear fractional Fourier filters Hα,k(u), k = 1 , ⋯ , m . For the deterministic input, the input is assumed to have its N dimensional fractional Fourier transform bandlimited to the frequency rang |ui|≤Ωi, for i = 1 , ⋯ , N . If m, the number of fractional Fourier filters, is written as a product of positive integers in the form m = m 1 m 2 ⋯ m N , and if the fractional bandlimited input f(t) is processed by fractional Fourier filter Hα,k(u)resulting m outputs gk(t), then f(t) can be reconstructed in terms of the samples gk(nT), each output being sampled at the identical rates of Ω 1 csc α / m 1 π , Ω 2 csc α / m 2 π , ⋯ , Ω N csc α / m N π samples/second in t 1 , ⋯ , t N respectively. This contrasts with the rates of Ω 1 csc α / π , Ω 2 csc α / π , ⋯ , Ω N csc α / π in t 1 , ⋯ , t N needed for reconstruction of the unfiltered input f(t). Input sampling expansions in terms of samples of the output filters are given for both deterministic and stochastic inputs, the generalized sampling expansion for random input having the same form as for the deterministic case but interpreted in the mean-square sense. Our formulation and results are general and include derivative sampling and periodic nonuniform sampling in the fractional Fourier domain for multidimensional signals as special case. Finally, the potensional application of the multidimensional generalized sampling is presented to show the advantage of the theory. Especially, the application of multidimensional generalized sampling in the context of the image scaling about image super-resolution is also discussed. The simulation results of image scaling are also presented.