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}@imperial.ac.uk

This research was funded in part by EPSRC (UK). 1

1. IntroductionSampling 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 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 (Prony’s method).

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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).

Perfect reconstruction ?

*

Signal SamplesKernel

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].

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2. Sampling Framework

The generic 2-D sampling setup (can be extended in n-D as well).

)/,/(),( yxxya TyTxyxh

j k

yxxyyx kTyjTxTT )/,/(),(

)/,/(),,(, kTyjTxyxgS yxxykj

Input signal Sampling

),( yxg

SamplesAcquisition device

Sampling kernel kjS ,

),( yxha

),( yx TT

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Sampling kernels and moments from samples

We consider the kernels that satisfy Strang-Fix conditions, and therefore, reproduce polynomials up to certain degree That is,

,),(,

,

yxkyjxC xykjZj Zk

where and are the known coefficients. Scaling functions (from wavelet theory) and B-splines are examples of valid kernels.

This makes it is possible to retrieve the continuous geometric moments of the original signal from its samples:

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0,0,kjC

1,0,kjC

0,1,kjC

),(3 yxxy

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 and 1 using B-spline kernel

Polynomial of degree 0

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3. Tomographic ApproachNow 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:

),( yxg

))sin()cos((),(),( yxtyxgtRg

),( yxg

In fact, the Radon transform projections are obtained from the observed samples .)/,/(),,(, kTyjTxyxgS yxxykj

The reconstruction of the original signal is achieved by back-projection.

),( yxg

),( tRg

),( tRg

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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

j k

kjjkj yxbyxg ,),(

1. Each Radon projection is a 1-D piecewise polynomial of max. degree R and it can be decomposed into a stream of at most N differentiated Diracs using (R+1)-order derivatives.

),( tRg

),()1( tRd gRt

2. Using Radon-moment connection of [3], we compute the moments of the differentiated Diracs from sample difference

),()1( tRd gRt

n

.,)1(

, kjR

kj SDR

j kkjkj

n

nn

gRtn

CRn

Mnnn

dtttRd

,,,

0

,,0

)1(

)(sin)(cos

......2,1,0,ˆ)(sin)(cos),(

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3. Using these moments and the annihilating filter method [1], we retrieve Dirac locations and weights and therefore the projection itself [2].

n

it ria ,),( tRg

4. 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].

it

5. From the knowledge of the convex closure and max(R,N+1) Radon projections , we can determine the coefficients , and thus, the polynomial g(x,y) by solving a system of linear equations.

),( tRg

),( yxg

2/)1(ˆ RRR kjb ,R̂

Note that the sampling kernel must reproduce polynomials at least up to degree in this case.

),( yxxy1)1)(12( RN

The AFBP algorithm can be extended for n-dimensional Diracs and bilevel-convex polytopes as well.

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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 transform 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:].

),( yxg

),( tRg

),( tRg

1

0 0

)(,

)1( )(),(N

i

R

ri

rrig

R ttatRd 10

Simulation: Reconstruction of 2-D polynomial of degree R-1=0.

Original signal Samples Reconst. of corner points

Difference samples .,)2(

, kjkj SDR

0 4/

2/ )2(tan 1

),( yxg kjS ,

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4. References1. 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.

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