Title: 1. INTRODUCTION AND MOTIVATION
1SHAPES 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).