HOW TO TAKE ADVANTAGE OF IN BANDLIMITED

Patrick Vandewalle1 ∗, Luciano Sbaiz1, Joos Vandewalle2 † and Martin Vetterli1,3

1LCAV - School of and Communication Sciences Ecole Polytechnique F´ed´erale de Lausanne (EPFL) - CH-1015 Lausanne, Switzerland 2SCD - Department of (ESAT) Katholieke Universiteit Leuven - Kasteelpark Arenberg 10, B-3001 Heverlee, Belgium 3Department of Electrical Engineering and UC Berkeley - Berkeley CA94720

ABSTRACT pled is aliased, and perfect reconstruction is not pos- sible. In systems, aliasing is normally treated Vetterli et al. [1] showed that perfect reconstruction is as a disturbing signal. That motivates the need for effec- also possible for signals with finite rate of innovation. Cer- tive analog, optical and digital anti-aliasing filters. How- tain types of non-bandlimited signals (like streams of Diracs, ever, aliasing conveys also valuable information on the sig- piecewise , etc.) can be reconstructed from a fi- nal above the . Hence, an effective pro- nite set of samples. Vaidyanathan [2] considers some other cessing of the samples, based on a model of the input signal, sampling schemes for non-bandlimited signals, like the use would allow to virtually increase the sampling frequency of different sampling kernels. An overview of the current using slower and cheaper converters. In this paper, we pre- state in sampling is given by Unser [3]. sent such an algorithm for bandlimited signals that are sam- In this paper, we derive some results for bandlimited sig- pled below twice the maximum signal frequency. Using a nals that are sampled below twice the maximum frequency subspace method in the , we show that of the signal. We will use multiple sets of regular samples these signals can be reconstructed from multiple sets of sam- to reconstruct the original signal exactly. A similar problem ples. The offset between the sets is unknown and can have for discrete-time signals is solved by Marziliano et al. [4] arbitrary values. This approach can be applied to the cre- using combinatorial optimization methods. ation of super-resolution images from sets of low resolution images. In this application, registration parameters have to 2. PROBLEM STATEMENT be computed from aliased images. We show that parameters and high resolution images can be computed precisely, even We define x(t) as a periodic, bandlimited and continuous- when high levels of aliasing are present on the low resolu- time signal with period 1 and maximum frequency L (since tion images. we consider a L is integer). Its continuous time Fourier series is called X[l] (with −L ≤ l ≤ L). x(t) K K 1. INTRODUCTION is regularly sampled at frequency , with integer- valued. This results in a discrete-time signal y0[n]=x[n/k]. Y [k] In general, we say that a signal can be perfectly reconstructed Its discrete-time 0 can be represented X[k] from its samples if it is bandlimited and the sampling fre- as a function of the continuous-time Fourier coefficients quency satisfies the Nyquist criterion, i.e. it is larger than as twice the maximum signal frequency. If the signal is not ∞ K K − 1 bandlimited or the sampling frequency is too low, the sam- Y [k]= X[k + iK] − ≤ k ≤ . 0 with 2 2 i=−∞ ∗The work presented in this paper was supported by the National Com- petence Center in Research on Mobil