Tomographic Approach for Sampling Multidimensional Signals with Finite Rate of Innovation

1
Tomographic Approach for Sampling
Multidimensional Signals with Finite Rate of
Innovation
Pancham Shukla and Pier Luigi Dragotti
Communications and Signal Processing Group
Electrical and Electronic Engineering Department
Imperial College, London
SW7 2AZ, UK E-mail p.shukla, p.dragotti_at_imperia
l.ac.uk
This research was funded in part by EPSRC (UK).
1
2
1. Introduction
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
acquisition devices are non-ideal. Fortunately,
recent research on Sampling signals with finite
rate of innovation (FRI) 1,2 suggests the ways
of sampling and perfect reconstruction of many
1-D nonbandlimited signals (e.g. Diracs,
Piecewise polynomials) using a rich class of
kernels (e.g. sinc, Gaussian, kernels that
reproduce polynomials and exponentials, kernels
with rational Fourier transform). The
reconstruction is based on annihilating filter
method (Pronys method).
2
3
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 nonbandlimited
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 and moments 3.
3
4
2. Sampling Framework
The generic 2-D sampling setup (can be extended
in n-D as well).
4
5
5
6
Reproduction of 2-D polynomials of degree 0 and 1
using B-spline kernel
Polynomial of degree 0
B-spline of degree 3
Polynomial of degree 1 along y
Polynomial of degree 1 along x
6
7
3. Tomographic Approach
Now we consider sampling of FRI signals
such as 2-D polynomials with convex polygonal
boundaries, and n-D Diracs and bilevel-convex
polytopes using Radon transform and annihilating
filter method.
Radon transform projection of a 2-D
function with compact support is
given by
In fact, the Radon transform projections
are obtained from the observed samples
7
8
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
9
Note that the sampling kernel must
reproduce polynomials at least up to degree
in this case.
The AFBP algorithm can be extended for
n-dimensional Diracs and bilevel-convex polytopes
as well.
9
10
AFBP reconstruction of the 2-D polynomial with
convex polygonal boundary.
(a) The 2-D polynomial of degree R-10
inside convex polygon with N5 corner points. (b)
Radon transform projection Rg(t, theis a 1-D
piecewise polynomial signal of degree R1. (d)
Second order derivative of the projection
is a stream of N differentiated Diracs.
10
11
Simulation Reconstruction of 2-D polynomial of
degree R-10.
Original signal
Samples
Reconst. of corner points
Difference samples
11
12
4. References

1. M Vetterli, P Marziliano, and T Blu, Sampling
   signals with finite rate of innovation, IEEE
   Trans. Sig. Proc., 50(6) 1417-1428, Jun 2002.
2. 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., Jun 2006, accepted.
3. P Milanfar, G Verghese, W Karl, and A Willsky,
   Reconstructing polygons from moments with
   connections to array processing, IEEE Trans.
   Sig. Proc., 43(2) 432-443, Feb 1995.
4. I Maravic and M Vetterli, A sampling theorem for
   the Radon transform of finite complexity
   objects, Proc. IEEE ICASSP, 1197-1200, Orlando,
   Florida, USA, May 2002.
5. P Shukla and P L Dragotti, Sampling schemes for
   2-D signals with finite rate of innovation using
   kernels that reproduce polynomials, Proc. IEEE
   ICIP, Genova, Italy, Sep 2005.

12