1. INTRODUCTION AND MOTIVATION

1
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_at_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.
  
  Shannons
  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 results of
• FRI sampling 2 in higher dimensions using
  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, and Quadrate domains (e.g.
  circles, ellipses, cardioids) in 2-D, and
• Radon projections for sampling 2-D polynomials
  with polygonal boundaries and n-dimensional
  bilevel-convex polytopes. The key feature of
  reconstruction algorithms is annihilating filter
  method (Pronys 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).