Jorge L. C. Sanz

Image reconstruction from frequency-offset Fourier data

Motivated by the ability of synthetic-aperture radar and related imaging systems to produce images of surprisingly high quality, we consider the problem of reconstructing the magnitude of a complex signal f from samples of the Fourier transform of f located in a small region offset from the origin. It is shown that high-quality speckle reconstructions are possible so long as the phase of f is highly random. In this case, the quality of the reconstruction is insensitive to the location of the known Fourier data, and edges at all orientations are reproduced equally well. A large number of computer examples are presented demonstrating these attributes. Methods for improving image quality are also briefly discussed.

Unified Hilbert space approach to iterative least-squares linear signal restoration

We deal with iterative least-squares solutions of the linear signal-restoration problem g = Af. First, several existing techniques for solving this problem with different underlying models are unified. Specifically, the following are shown to be special cases of a general iterative procedure [ BialyH., Arch. Ration. Mech. Anal.4, 166 ( 1959)] for solving linear operator equations in Hilbert spaces: (1) a Van Cittert-type algorithm for deconvolution of discrete and continuous signals; (2) an iterative procedure for regularization when g is contaminated with noise; (3) a Papoulis–Gerchberg algorithm for extrapolation of continuous signals [ PapoulisA., IEEE Trans. Circuits Syst.CAS-22, 735 ( 1975); GerchbergR. W., Opt. Acta21, 709 ( 1974)]; (4) an iterative algorithm for discrete extrapolation of band-limited infinite-extent discrete signals {and the minimum-norm property of the extrapolation obtained by the iteration [ JainA.RanganathS., IEEE Trans. Acoust. Speech Signal Process. ASSP-29, ( 1981)]}; and (5) a certain iterative procedure for extrapolation of band-limited periodic discrete signals [ TomV., IEEE Trans. Acoust. Speech Signal Process.ASSP-29, 1052 ( 1981)]. The Bialy algorithm also generalizes the Papoulis–Gerchberg iteration to cases in which the ideal low-pass operator is replaced by some other operators. In addition a suitable modification of this general iteration is shown. This technique leads us to new iterative algorithms for band-limited signal extrapolation. In numerical simulations some of these algorithms provide a fast reconstruction of the sought signal.

Radon and Projection Transform-Based Computer Vision

This book is a description of the applicability of projection transforms to computer vision and image processing. It deals with architecture implementations for real-time processing and, in particular, presents novel algorithms suitable for VLSI implementations. The architecture ideas unleash the power of the Radon transform for technological applications of the machine vision.

Some aspects of band-limited signal extrapolation: Models, discrete approximations, and noise

We present some theoretical results on the band-limited signal extrapolation problem. In Section I we describe four basic models for the extrapolation problem. These models are useful in understanding the relationship between the continuous extrapolation problem and some discrete algorithms given in [1] and [2]. One of these models was shown to approximate the continuous band-limited extrapolation problem [3]. Another model is obtained when the discrete Fourier transform (DFT) is used to implement the well-known iterative algorithm given in [4] and [5] which was designed for solving the continuous extrapolation problem; in Section II this model is related to the continuous model by means of an interesting approximation theorem. Also, an important conjecture is presented. Section III shows some approximation results. Specifically, we prove that some discrete-discrete and discrete-continuous extrapolations of noisy signals converge to solutions of a certain continuous-continuous noisy extrapolation problem when the noise η is bounded by a known number, max \eta(x)| leq \epsilon . This convergence is obtained by using normal families of entire functions in ¢nand some other complex analysis tools. We also show that the extrapolation problem is very sensitive to noise even in cases where only small amounts of extrapolation are desired. This result indicates that in the presence of noise, extrapolation techniques should be used judiciously in order to obtain reasonable results.

Discrete and continuous band-limited signal extrapolation

This paper has two main purposes. First, some new algorithms for the extrapolation of multidimensional band-limited sequences are presented. These algorithms extend those given in [1] in two ways: 1) We do not impose any restrictions on either the shape of the region containing the set of samples of the two-dimensional signal to be extrapolated or on the shape of its passband zone. 2) We obtain a class of sequences which are band-limited extrapolations of the given data (including the minimum norm solution of [1] as a special case). The second objective of out paper is to relate the discrete extrapolation problem to the continuous signal extrapolation problem [2]. Specifically, we prove that the solution obtained by using our approach for the discrete extrapolation problem tends to the solution of the continuous problem when the sampling rate used in the known part of the continuous signal to be extrapolated approaches infinity.

Stability of unique Fourier-transform phase reconstruction

The problem of Fourier-transform phase reconstruction from the Fourier-transform magnitude of multidimensional discrete signals is considered. It is well known that, if a discrete finite-extent n-dimensional signal (n ≥ 2) has an irreducible z transform, then the signal is uniquely determined from the magnitude of its Fourier transform. It is also known that this irreducibility condition holds for all multidimensional signals except for a set of signals that has measure zero. We show that this uniqueness condition is stable in the sense that it is not sensitive to noise. Specifically, it is proved that the set of signals whose z transform is reducible is contained in the zero set of a certain multidimensional polynomial. Several important conclusions can be drawn from this characterization, and, in particular, the zero-measure property is obtained as a simple byproduct.

Unique reconstruction of a band-limited multidimensional signal from its phase or magnitude

The mathematical problem of unique recovery of a band-limited multidimensional signal from its phase or its magnitude is considered. Specifically, we show that any irreducible band-limited function f(s1, …, sn), si∈C, i = 1, …, n is uniquely determined, except for trivial associates, from (1) the phase of f(x1, …, xn), xi∈ ℛ, i = 1, …, n, if not all the zeros of f(s1, …, sn) occur in conjugate pairs; or (2) the magnitude of f(x1, …, xn), xi∈ ℛ, i = …, n.

Mathematical Considerations for the Problem of Fourier Transform Phase Retrieval from Magnitude

In this paper, we deal with the problem of retrieving a finite-extent function from the magnitude of its Fourier transform. This so-called phase retrieval problem will first be posed under its different underlying models. We will present a brief review of the main results known in this area for both discrete and continuous phase retrieval models giving special emphasis to the algebraic problem of the uniqueness of the solution. Several important issues which are yet unresolved will be pointed out and discussed. We will then consider the discrete phase retrieval problem as a special case of a more general problem which consists of recovering a real-valued sequence x from the magnitude of the output of a linear distortion:

A unified approach to noniterative linear signal restoration

| Hx | ( j ),\, j = 1, \cdots ,n

The SIMD model of parallel computation

. A number of important results will be obtained for this general setting by means of algebraic-geometric techniques. In particular, the problems of the existence of a solution for phase retrieval, number of feasible solutions, stability of the (essential...