Nonuniform sampling, image recovery from sparse data and the discrete sampling theorem

Abstract

In many applications, sampled data are collected in irregular fashion or are partly lost or unavailable. In these cases, it is necessary to convert irregularly sampled signals to regularly sampled ones or to restore missing data. We address this problem in the framework of a discrete sampling theorem for band-limited discrete signals that have a limited number of nonzero transform coefficients in a certain transform domain. Conditions for the image unique recovery, from sparse samples, are formulated and then analyzed for various transforms. Applications are demonstrated on examples of image superresolution and image reconstruction from sparse projections.

Spectral methods for imputation of missing air quality data

“…Therefore, we conclude that the spectral methods in general, K-SVD and DCT in particular, do present viable tool for data imputation and should be used as the tool of choice in general as it presents the overall best performance. Both the KSVD (Aharon et al 2006) and DCT (Yaroslavsky et al 2009) methods assume band-limited signal, i.e., only a small portion of signal's spectral representation coefficients are non-zero. In most implementations the portion of non-zero coefficients is predetermined.…”

“…The theorem constitutes the new data imputation scheme presented here. Within its context two spectral signal representations are considered: The DCT (Rao et al 1990;Yaroslavsky et al 2009) and the sparse coding K-Cluster Single Variable Decomposition (K-SVD) (Aharon et al 2006). The application of the suggested methods show that they are comparable to the state-of-the-art when imputing short missing sequences and do hold the upper hand when larger chunks of subsequent data are missing.…”

“…The goal then, is to generate out of this incomplete set of K samples, a complete set of N signal samples that secures the most accurate, in a certain metrics-typically L 2 , approximation. The discrete sampling theorem (Yaroslavsky et al 2009) states the terms and conditions a signal must fulfills in the transform domain so its A N samples can be recovered from the available A K samples: Theorem 1 implies that selecting a transform that features the best energy compaction with the smallest number of transform coefficients secures the best approximation of A N for a given subset A K of its samples. The recovery process is based on the following simple iterative procedure (Yaroslavsky et al 2009 ): Algorithm 1.…”

Compression, restoration, resampling, ‘compressive sensing’: fast transforms in digital imaging

“…Therefore one can regard available K samples as being sparsely placed at nodes of a denser sampling lattice with the total amount of nodes K N  . The general framework for recovery of discrete signals from a given set of their arbitrarily taken samples can be formulated as an approximation task in the assumption that continuous signals are represented in computers by their N K  irregularly taken samples and it is believed that if all N samples in a certain regular sampling lattice were known, they would be sufficient for representing those continuous signals ( [ 20], [ 126]). The goal of the processing is generating, out of an incomplete set of K samples, a complete set of N signal samples in such a way as to secure the most accurate, in terms of the reconstruction mean square error (MSE), approximation of the discrete signal, which would be obtained if the continuous signal it is intended to represent were densely sampled at all N positions.…”

Nuclear threat detection with mobile distributed sensor networks

“…We seek a function that coincides with the value reported at each data point (sample's position). Methods for generating this type of function are reviewed in Yaroslavsky et al (2009). Of these methods we select the iterative reconstruction procedure of the Papoulis (1975) type shown in the flow diagram of Fig.…”