Sampling Functions and Sparse Reconstruction Methods

SUMMARY In this paper we investigate the effects of different sampling operators on the performance of sparse reconstruction methods. The common paradigm in seismic data processing is to favor regular sampling. We will show, however, that regular sampling often hampers our data recovery efforts. Random sampling, on the other hand, can lead to algorithms where the reconstruction is almost perfect when the underlying spectrum of the signal can be assumed sparse. Also, simple 1D, 2D and 3D synthetic examples are provided to test the sparse reconstruction of signals sampled by various sampling functions. Introduction The sampling theorem is an interesting topic of chief importance in the physical sciences and engineering. Sampling methods can be classified into different groups. The most important distinction is between uniform and nonuniform sampling methods. In the uniform sampling method, a signal is sampled periodically at constant intervals. This leads to recovery conditions based on the well-known Nyquist sampling theorem. An overview of uniform sampling and its properties is given by Unser (2000). Meanwhile, in the nonuniform sampling case samples are picked in an irregular fashion. Belmont (1993) gives a nice explanation of nonuniform sampling operators. Regardless of the adopted sampling method, the truly important step is to reconstruct the original signal from sampled data. Reconstruction algorithms utilize specific assumption about the original signal. The most commonly used assumption is the band-limitation. In other words, band-limited signals can completely be recovered from a wide range of sampling methods. Duijndam et al. (1999) utilized band-limiting assumptions in order to reconstruct the irregularly sampled seismic records in the spatial directions. Another important assumption is the sparseness of a signal in the Fourier domain. Liu and Sacchi (2004) used iteratively updated weighting functions of the data spectrum as a sparsity constraint. Zwartjes and Gisolf (2006) compared the performance of several sparsity norms on seismic data and simply addressed the methods as the Fourier reconstruction. Naghizadeh and Sacchi (2007) used band-limiting operators to reconstruct the low frequency portion of the aliased seismic data. Subsequently, they used Multi-Step Auto-Regressive (MSAR) operator to calculate prediction filters to reconstruct all the frequencies. In this paper we will investigate the effects of various sampling schemes on the performance of sparse reconstruction algorithms. First, by developing a mathematical framework for general sampling operators, we try to understand the differences between uniform and nonuniform sampling operators and their spectra. Later, using simple 1D examples we will …