Complete Algebraic Reconstruction of Piecewise-Smooth Functions from Fourier Data

In this paper we provide a reconstruction algorithm for piecewise-smooth functions with a-priori known smoothness and number of discontinuities, from their Fourier coefficients, posessing the maximal possible asymptotic rate of convergence -- including the positions of the discontinuities and the pointwise values of the function. This algorithm is a modification of our earlier method, which is in turn based on the algebraic method of K.Eckhoff proposed in the 1990s. The key ingredient of the new algorithm is to use a different set of Eckhoff's equations for reconstructing the location of each discontinuity. Instead of consecutive Fourier samples, we propose to use a "decimated" set which is evenly spread throughout the spectrum

Local and global geometry of Prony systems and Fourier reconstruction of piecewise-smooth functions

Many reconstruction problems in signal processing require solution of a certain kind of nonlinear systems of algebraic equations, which we call Prony systems. We study these systems from a general perspective, addressing questions of global solvability and stable inversion. Of special interest are the so-called "near-singular" situations, such as a collision of two closely spaced nodes. We also discuss the problem of reconstructing piecewise-smooth functions from their Fourier coefficients, which is easily reduced by a well-known method of K.Eckhoff to solving a particular Prony system. As we show in the paper, it turns out that a modification of this highly nonlinear method can reconstruct the jump locations and magnitudes of such functions, as well as the pointwise values between the jumps, with the maximal possible accuracy.Comment: arXiv admin note: text overlap with arXiv:1211.068

Decoupling of Fourier Reconstruction System for Shifts of Several Signals

We consider the problem of ``algebraic reconstruction'' of linear combinations of shifts of several signals $f_1,\ldots,f_k$ from the Fourier samples. For each $r=1,\ldots,k$ we choose sampling set $S_r$ to be a subset of the common set of zeroes of the Fourier transforms {\cal F}(f_\l), \ \l \ne r, on which ${\cal F}(f_r)\ne 0$. We show that in this way the reconstruction system is reduced to $k$ separate systems, each including only one of the signals $f_r$. Each of the resulting systems is of a ``generalized Prony'' form. We discuss the problem of unique solvability of such systems, and provide some examples

Decimated generalized Prony systems

We continue studying robustness of solving algebraic systems of Prony type (also known as the exponential fitting systems), which appear prominently in many areas of mathematics, in particular modern "sub-Nyquist" sampling theories. We show that by considering these systems at arithmetic progressions (or "decimating" them), one can achieve better performance in the presence of noise. We also show that the corresponding lower bounds are closely related to well-known estimates, obtained for similar problems but in different contexts

Moment inversion problem for piecewise D-finite functions

We consider the problem of exact reconstruction of univariate functions with jump discontinuities at unknown positions from their moments. These functions are assumed to satisfy an a priori unknown linear homogeneous differential equation with polynomial coefficients on each continuity interval. Therefore, they may be specified by a finite amount of information. This reconstruction problem has practical importance in Signal Processing and other applications. It is somewhat of a ``folklore'' that the sequence of the moments of such ``piecewise D-finite''functions satisfies a linear recurrence relation of bounded order and degree. We derive this recurrence relation explicitly. It turns out that the coefficients of the differential operator which annihilates every piece of the function, as well as the locations of the discontinuities, appear in this recurrence in a precisely controlled manner. This leads to the formulation of a generic algorithm for reconstructing a piecewise D-finite function from its moments. We investigate the conditions for solvability of the resulting linear systems in the general case, as well as analyze a few particular examples. We provide results of numerical simulations for several types of signals, which test the sensitivity of the proposed algorithm to noise

Accuracy of Algebraic Fourier Reconstruction for Shifts of Several Signals

We consider the problem of "algebraic reconstruction" of linear combinations of shifts of several known signals $f_1,\ldots,f_k$ from the Fourier samples. Following \cite{Bat.Sar.Yom2}, for each $j=1,\ldots,k$ we choose sampling set $S_j$ to be a subset of the common set of zeroes of the Fourier transforms ${\cal F}(f_\ell), \ \ell \ne j$, on which ${\cal F}(f_j)\ne 0$. It was shown in \cite{Bat.Sar.Yom2} that in this way the reconstruction system is "decoupled" into $k$ separate systems, each including only one of the signals $f_j$. The resulting systems are of a "generalized Prony" form. However, the sampling sets as above may be non-uniform/not "dense enough" to allow for a unique reconstruction of the shifts and amplitudes. In the present paper we study uniqueness and robustness of non-uniform Fourier sampling of signals as above, investigating sampling of exponential polynomials with purely imaginary exponents. As the main tool we apply a well-known result in Harmonic Analysis: the Tur\'an-Nazarov inequality (\cite{Naz}), and its generalization to discrete sets, obtained in \cite{Fri.Yom}. We illustrate our general approach with examples, and provide some simulation results

A Multiscale Pyramid Transform for Graph Signals

Multiscale transforms designed to process analog and discrete-time signals and images cannot be directly applied to analyze high-dimensional data residing on the vertices of a weighted graph, as they do not capture the intrinsic geometric structure of the underlying graph data domain. In this paper, we adapt the Laplacian pyramid transform for signals on Euclidean domains so that it can be used to analyze high-dimensional data residing on the vertices of a weighted graph. Our approach is to study existing methods and develop new methods for the four fundamental operations of graph downsampling, graph reduction, and filtering and interpolation of signals on graphs. Equipped with appropriate notions of these operations, we leverage the basic multiscale constructs and intuitions from classical signal processing to generate a transform that yields both a multiresolution of graphs and an associated multiresolution of a graph signal on the underlying sequence of graphs.Comment: 16 pages, 13 figure