Tomographic Approach as long as Sampling Multidimensional Signals with Finite Rate of Innovation

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Tomographic Approach as long as Sampling Multidimensional Signals with Finite Rate of Innovation

Spence-Pothitt, Carma, Freelance Writer/Editor has reference to this Academic Journal, PHwiki organized this Journal Tomographic Approach as long as Sampling Multidimensional Signals with Finite Rate of Innovation Pancham Shukla in addition to Pier Luigi Dragotti Communications in addition to Signal Processing Group Electrical in addition to Electronic Engineering Department Imperial College, London SW7 2AZ, UK E-mail: {p.shukla, p.dragotti}@imperial.ac.uk This research was funded in part by EPSRC (UK). 1 1. Introduction Sampling is a fundamental step in obtaining sparse representation of signals (e.g. images, video) as long as applications such as coding, communication, in addition to storage. Shannon’s classical sampling theory considers sampling of b in addition to limited signals using ‘sinc’ kernel. However, most real-world signals are nonb in addition to limited, in addition to acquisition devices are non-ideal. Fortunately, recent research on Sampling signals with finite rate of innovation (FRI) [1,2] suggests the ways of sampling in addition to perfect reconstruction of many 1-D nonb in addition to limited signals (e.g. Diracs, Piecewise polynomials) using a rich class of kernels (e.g. sinc, Gaussian, kernels that reproduce polynomials in addition to exponentials, kernels with rational Fourier trans as long as m). The reconstruction is based on annihilating filter method (Prony’s method). 2 Contribution: We extend the results of FRI sampling [2] in higher dimensions using compactly supported kernels (e.g. B-splines) that reproduce polynomials (i.e., satisfy Strang-Fix conditions). Earlier, we have shown that it possible to perfectly reconstruct many 2-D nonb in addition to limited signals (or shapes) from their samples [5]. In sequel to [5], here we show the sampling of more general FRI signals using the connection between Radon projections in addition to moments [3]. 3

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2. Sampling Framework The generic 2-D sampling setup (can be extended in n-D as well). 4 5 Polynomial of degree 1 along x Polynomial of degree 1 along y B-spline of degree 3 Reproduction of 2-D polynomials of degree 0 in addition to 1 using B-spline kernel Polynomial of degree 0 6

3. Tomographic Approach Now we consider sampling of FRI signals such as 2-D polynomials with convex polygonal boundaries, in addition to n-D Diracs in addition to bilevel-convex polytopes using Radon trans as long as m in addition to annihilating filter method. Radon trans as long as m projection of a 2-D function with compact support is given by: In fact, the Radon trans as long as m projections are obtained from the observed samples 7 Annihilating Filter based Back-Projection (AFBP) algorithm Consider a case when is a 2-D polynomial of max. degree R-1 inside a convex polygonal closure with N corner points. In this case, we observe that 2. Using Radon-moment connection of [3], we compute the moments of the differentiated Diracs from sample difference 8 Note that the sampling kernel must reproduce polynomials at least up to degree in this case. The AFBP algorithm can be extended as long as n-dimensional Diracs in addition to bilevel-convex polytopes as well. 9

AFBP reconstruction of the 2-D polynomial with convex polygonal boundary. (a) The 2-D polynomial of degree R-1=0 inside convex polygon with N=5 corner points. (b) Radon trans as long as m projection Rg(t, theis a 1-D piecewise polynomial signal of degree R=1. (d) Second order derivative of the projection is a stream of N differentiated Diracs:]. 10 Simulation: Reconstruction of 2-D polynomial of degree R-1=0. Original signal Samples Reconst. of corner points Difference samples 11 4. References M Vetterli, P Marziliano, in addition to T Blu, ‘Sampling signals with finite rate of innovation,’ IEEE Trans. Sig. Proc., 50(6): 1417-1428, Jun 2002. P L Dragotti, M Vetterli, in addition to T Blu, ‘Sampling moments in addition to reconstructing signals of finite rate of innovation: Shannon meets Strang-Fix,’ IEEE Trans. Sig. Proc., Jun 2006, accepted. P Milanfar, G Verghese, W Karl, in addition to A Willsky, ‘Reconstructing polygons from moments with connections to array processing,’ IEEE Trans. Sig. Proc., 43(2): 432-443, Feb 1995. I Maravic in addition to M Vetterli, ‘A sampling theorem as long as the Radon trans as long as m of finite complexity objects,’ Proc. IEEE ICASSP, 1197-1200, Orl in addition to o, Florida, USA, May 2002. P Shukla in addition to P L Dragotti, ‘Sampling schemes as long as 2-D signals with finite rate of innovation using kernels that reproduce polynomials,’ Proc. IEEE ICIP, Genova, Italy, Sep 2005. 12

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