Graph-based transforms have been shown to be powerful tools in terms of image energy compaction. However, when the support increases to best capture signal dependencies, the computation of the basis functions becomes rapidly untractable. This problem is in particular compelling for high dimensional imaging data such as light fields. The use of local transforms with limited supports is a way to cope with this computational difficulty. Unfortunately, the locality of the support may not allow us to fully exploit long term signal dependencies present in both the spatial and angular dimensions in the case of light fields. This paper describes sampling and prediction schemes with local graph-based transforms enabling to efficiently compact the signal energy and exploit dependencies beyond the local graph support. The proposed approach is investigated and is shown to be very efficient in the context of spatio-angular transforms for quasi-lossless compression of light fields.
While the aforementioned sampling theorem @cite_1 has been proposed for band-limited signals, we extend those equations to our problem in the following section. More precisely, we deal with signals (i.e. Color Signals) that might not be necessarily band-limited on the underlying graph supports (i.e. Super-Rays).
@cite_1
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AbstractIn [1] a sampling theorem for a certain class of signals with ﬁnite rate of innovation (which includesfor example stream of Diracs) has been developed. In essence, such non band-limited signals can besampled at or above the rate of innovation. In the present paper, we consider the case of such signalswhen noise is present. Clearly, the ﬁnite rate of innovation property is lost, but if the signal-to-noise ratio(SNR) is sufﬁcient, several methods are possible to reconstruct the signal while sampling well below theNyquist rate. We thus explore the trade-offs between SNR, sampling rate, computational complexity andreconstruction quality. Applications of such methods can be found in acquisition and processing of signalsin high bandwidth communications, like ultra wide band communication [2]. I. I NTRODUCTION Band-limited functions are just an example of signals that are speciﬁed by a ﬁxed number ofsamples per unit of time, namely, if a signal x(t) is band-limited to [ ! m ;! m ], then the famoussampling theorem by Shannon [3] states that x(nT) with T < ˇ=! The purpose of this investigation is to examine four special nonuniform sampling processes in detail, and to deduce some interesting properties of bandwidth-limited signals. The main results are contained in four generalized sampling theorems. These theorems not only contain the nature of determination (unique-specification, over-specification, and underspecification) of signals but also include explicit reconstruction formulas. From the reconstruction formulas, the complexity and accuracy of the nonuniform sampling processes discussed can be estimated. In addition, these theorems lead to observations regarding the allowable shapes, the "prediction," and the "energy" of bandwidth-limited signals in general. A "minimum-energy" signal is introduced which has certain advantages as compared to the ordinary "time-limited" signals when a finite number of sample values are given. Finally, a statement due to Cauchy on the sampling of bandwidth-limited signals is generalized to include a wider class of nonuniform sample point distributions and modified to give more exact information regarding the nature of determination of signals. In this paper, we consider the problem of sampling signals that are nonband-limited but have finite number of degrees of freedom per unit of time and call this number the rate of innovation. Streams of Diracs and piecewise polynomials are the examples of such signals, and thus are known as signals with finite rate of innovation (FRI). We know that the classical ("band-limited sine") sampling theory does not enable perfect reconstruction of such signals from their samples since they are not band-limited. However, the recent results on FRI sampling suggest that it is possible to sample and perfectly reconstruct such nonband-limited signals using a rich class of kernels. In this paper, we extend those results in higher dimensions using compactly supported kernels that reproduce polynomials (satisfy Strang-Fix conditions). In fact, the polynomial reproduction property of the kernel makes it possible to obtain the continuous moments of the signal from its samples. Using these moments and the annihilating filter method (Prony's method), the innovative part of the signal, and therefore, the signal itself is perfectly reconstructed. In particular, we present local (directional-derivatives-based) and global (complex-moments-based, Radon-transform-based) sampling schemes for classes of FRI signals such as sets of Diracs, bilevel, and planar polygons, quadrature domains (e.g., circles, ellipses, and cardioids), 2D polynomials with polygonal boundaries, and n-dimensional Diracs and convex polytopes. This work has been explored in a promising way in super-resolution algorithms and distributed compression, and might find its applications in photogrammetry, computer graphics, and machine vision. Shannon’s sampling theorem is one of the most powerful results in signal analysis. The aim of this overview is to show that one of its roots is a basic paper of de la Vallee Poussin of 1908. The historical development of sampling theory from 1908 to the present, especially the matter dealing with not necessarily band-limited functions (which includes the duration-limited case actually studied in 1908), is sketched. Emphasis is put on the study of error estimates, as well as on the delicate point-wise behavior of sampling sums at discontinuity points of the signal to be reconstructed.