1. INTRODUCTION AND MOTIVATION

1. Design the annihilating filter such that The roots of filter are the corner points (z=x+iy)of the convex and bilevel polygon. For a polygon with N corner points, we need 2N complex-moments. Polygons Samples Signal Kernel Samples Perfect reconstruction ? B-spline kernel Polygonal lines Circle Complex-moments and annihilating filter Reconstructed corner points (marked with +) 4. RADON PROJECTIONS Annihilating Filter based Back-Projection (AFBP) algorithm Consider a case when is a 2-D polynomial of degree R-1 inside a convex polygonal closure with N corner points. In this case, • Each Radon projection is a 1-D piecewise polynomial and it can be decomposed into a stream of differentiated Diracs. • Using Radon-moment connection of [3], we compute the moments of the differentiated Diracs from sample difference • Using these moments and the annihilating filter method, we retrieve Dirac locations and weights and therefore the projection itself [2]. • By iterating steps 1, 2 and 3 over N+1 distinct projection angles and then back-projecting the Dirac locations, we retrieve the convex polygonal closure [4]. • From the knowledge of convex closure and max( ,N+1) Radon projections , we can retrieve the 2-D polynomial by solving a system of linear equations. Input signal Acquisition device Sampling Set of samples Sampling kernel Sampling kernels and moments from samples We consider the kernels that satisfy Strang-Fix conditions, and therefore, reproduce polynomials up to certain degree where and are the known coefficients. Scaling functions (from wavelet theory) and B-splines are examples of valid kernels. B-spline of degree 3 Polynomial of degree 0 The sampling kernel must reproduce polynomials at least up to degree Polynomial of degree 1 along x Polynomial of degree 1 along y The polynomial reproduction property of the kernel allows us to retrieve the continuous geometric moments of the original signal from its samples: Moreover, it is straightforward to obtain the complex-moments from the geometric moments using: SHAPES FROM SAMPLES USING MOMENTS AND RADON PROJECTIONS Pancham Shukla and Pier Luigi Dragotti Communications and Signal Processing Group, Electrical and Electronic Engineering Department, Imperial College, Exhibition Road, London SW7 2AZ, United Kingdom. E-mail: {p.shukla, p.dragotti}@imperial.ac.uk • 1. INTRODUCTION AND MOTIVATION • Sampling is a fundamental step in obtaining sparse representation of signals (e.g. images, video) for applications such as coding, communication, and storage. Shannon’s classical sampling theory considers sampling of bandlimited signals using sinc kernel. However, most real-world signals are nonbandlimited, and therefore, it is important to understand the sampling of nonbandlimited signals. Fortunately, recent research on • Sampling signals with finite rate of innovation(FRI) [1] suggests the ways of sampling and perfect reconstruction of many nonbandlimited signals using a rich class of kernels [2]. • In this research, we extend the resultsof • FRI sampling[2] in higher dimensionsusing compactly supported kernels (e.g. B-splines, scaling functions) that reproduce polynomials (satisfy Strang-Fix conditions). We show that it possible to perfectly reconstruct many multidimensional nonbandlimited signals (or shapes) from their samples. In particular, we exploit: • Complex-moments for sampling bilevel-convex polygons, Diracs,andQuadrate domains (e.g. circles, ellipses, cardioids) in 2-D, and • Radon projections for sampling 2-D polynomialswith polygonal boundariesandn-dimensionalbilevel-convexpolytopes. The key feature of reconstruction algorithms is annihilating filter method (Prony’s method). 3. COMPLEX-MOMENTS Note that it is possible to uniquely reconstruct the convex-bilevel polygons and quadrature domains (e.g. circles, ellipses, cardioids) from a finite number of complex-moments [3]. However, in sampling, we retrieve the complex-moments from the samples of 2-D FRI shapes (e.g. polygons, quadrature domains, Diracs, and polygonal lines). 2. SAMPLING FRAMEWORK The generic 2-D sampling setup (can be extended in n-D as well). AFBP reconstruction of 2-D polynomial of degree 0 inside a convex pentagon. AFBP algorithm can be extended for 2-D Diracs, bilevel-convex polygons with polygonal voids, and for n-dimensional bilevel-convex polytopes and Diracs. 5. CONCLUSION This work provides new understanding of perfect reconstruction of nonbandlimited shapes from their samples and finds its application in super-resolution image registration for low cost camera network (see [5] for detail). • 6. REFERENCES • M Vetterli, P Marziliano, and T Blu, ‘Sampling signals with finite rate of innovation,’ IEEE Trans. Sig. Proc., 50(6): 1417-1428, June 2002. • P L Dragotti, M Vetterli, and T Blu, ‘Sampling moments and reconstructing signals of finite rate of innovation: Shannon meets Strang-Fix,’ IEEE Trans. Sig. Proc., Feb 2006, (submitted). • 3. P Milanfar, M Putinar, J Varah, B Gustafsson, and G Golub, ‘Shape reconstruction from moments: theory, algorithms, and applications,’ Proc. SPIE, 4116: 406-416, Nov 2000. • 4. I Maravich and M Vetterli, ‘A sampling theorem for the Radon transform of finite complexity objects,’ Proc. IEEE ICASSP, 1197-1200, May 2002. • 5. L Baboulaz and P L Dragotti, ‘Distributed acquisition and image super-resolution based on continuous moments from samples,’ Proc. IEEE ICIP, Atlanta, 2006 (to appear).