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

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

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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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