Super-resolution through Error Energy Reduction
TL;DR: A computational procedure is devised which must reduce a defined ‘error energy’ which is implicit in the truncated spectrum and it is demonstrated that by so doing, resolution well beyond the diffraction limit is attained.
Abstract: A new view of the problem of continuing a given segment of the spectrum of a finite object is presented. Based on this, the problem is restated in terms of reducing a defined ‘error energy’ which is implicit in the truncated spectrum. A computational procedure, which is readily implemented on general purpose computers, is devised which must reduce this error. It is demonstrated that by so doing, resolution well beyond the diffraction limit is attained. The procedure is shown to be very effective against noisy data.
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Book•
03 Oct 1988TL;DR: This chapter discusses two Dimensional Systems and Mathematical Preliminaries and their applications in Image Analysis and Computer Vision, as well as image reconstruction from Projections and image enhancement.
Abstract: Introduction. 1. Two Dimensional Systems and Mathematical Preliminaries. 2. Image Perception. 3. Image Sampling and Quantization. 4. Image Transforms. 5. Image Representation by Stochastic Models. 6. Image Enhancement. 7. Image Filtering and Restoration. 8. Image Analysis and Computer Vision. 9. Image Reconstruction From Projections. 10. Image Data Compression.
8,504 citations
01 Sep 1982
TL;DR: In this article, a local eigenexpansion is proposed to estimate the spectrum of a stationary time series from a finite sample of the process, which is equivalent to using the weishted average of a series of direct-spectrum estimates based on orthogonal data windows to treat both bias and smoothing problems.
Abstract: In the choice of an estimator for the spectrum of a stationary time series from a finite sample of the process, the problems of bias control and consistency, or "smoothing," are dominant. In this paper we present a new method based on a "local" eigenexpansion to estimate the spectrum in terms of the solution of an integral equation. Computationally this method is equivalent to using the weishted average of a series of direct-spectrum estimates based on orthogonal data windows (discrete prolate spheroidal sequences) to treat both the bias and smoothing problems. Some of the attractive features of this estimate are: there are no arbitrary windows; it is a small sample theory; it is consistent; it provides an analysis-of-variance test for line components; and it has high resolution. We also show relations of this estimate to maximum-likelihood estimates, show that the estimation capacity of the estimate is high, and show applications to coherence and polyspectrum estimates.
3,921 citations
TL;DR: It is shown that various inverse problems in signal recovery can be formulated as the generic problem of minimizing the sum of two convex functions with certain regularity properties, which makes it possible to derive existence, uniqueness, characterization, and stability results in a unified and standardized fashion for a large class of apparently disparate problems.
Abstract: We show that various inverse problems in signal recovery can be formulated as the generic problem of minimizing the sum of two convex functions with certain regularity properties. This formulation makes it possible to derive existence, uniqueness, characterization, and stability results in a unified and standardized fashion for a large class of apparently disparate problems. Recent results on monotone operator splitting methods are applied to establish the convergence of a forward-backward algorithm to solve the generic problem. In turn, we recover, extend, and provide a simplified analysis for a variety of existing iterative methods. Applications to geometry/texture image decomposition schemes are also discussed. A novelty of our framework is to use extensively the notion of a proximity operator, which was introduced by Moreau in the 1960s.
2,645 citations
Cites result from "Super-resolution through Error Ener..."
...In that paper, the problem was to find a signal x in a closed vector subspace C, knowing its projection p onto a closed vector subspace V (hence L = PV and Q = {p}); it was also observed that the standard signal extrapolation schemes of Gerchberg [37] and Papoulis [59] fitted this framework....
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...[37] R. W. Gerchberg, Super-resolution through error energy reduction, Optica Acta, 21 (1974), pp. 709–720....
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TL;DR: A view of the algorithm as a novel optimization method which combines desirable characteristics of both classical optimization and learning-based algorithms is provided and Mathematical results on conditions for uniqueness of sparse solutions are also given.
Abstract: We present a nonparametric algorithm for finding localized energy solutions from limited data. The problem we address is underdetermined, and no prior knowledge of the shape of the region on which the solution is nonzero is assumed. Termed the FOcal Underdetermined System Solver (FOCUSS), the algorithm has two integral parts: a low-resolution initial estimate of the real signal and the iteration process that refines the initial estimate to the final localized energy solution. The iterations are based on weighted norm minimization of the dependent variable with the weights being a function of the preceding iterative solutions. The algorithm is presented as a general estimation tool usable across different applications. A detailed analysis laying the theoretical foundation for the algorithm is given and includes proofs of global and local convergence and a derivation of the rate of convergence. A view of the algorithm as a novel optimization method which combines desirable characteristics of both classical optimization and learning-based algorithms is provided. Mathematical results on conditions for uniqueness of sparse solutions are also given. Applications of the algorithm are illustrated on problems in direction-of-arrival (DOA) estimation and neuromagnetic imaging.
1,864 citations
Cites background from "Super-resolution through Error Ener..."
...Papoulis in [10] and Gerchberg in [11] proposed what...
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TL;DR: In this article, a projection operator onto a closed convex set in Hilbert space is proposed for image restoration from partial data which permits any number of nonlinear constraints of a certain type to be subsumed automatically.
Abstract: A projection operator onto a closed convex set in Hilbert space is one of the few examples of a nonlinear map that can be defined in simple abstract terms. Moreover, it minimizes distance and is nonexpansive, and therefore shares two of the more important properties of ordinary linear orthogonal projections onto closed linear manifolds. In this paper, we exploit the properties of these operators to develop several iterative algorithms for image restoration from partial data which permit any number of nonlinear constraints of a certain type to be subsumed automatically. Their common conceptual basis is as follows. Every known property of an original image f is envisaged as restricting it to lie in a well-defined closed convex set. Thus, m such properties place f in the intersection E0 = Ei of the corresponding closed convex sets E1,E2,···Em. Given only the projection operators Pi onto the individual Ei's, i = 1 → m, we restore f by recursive means. Clearly, in this approach, the realization of the Pi's in a Hilbert space setting is one of the major synthesis problems. Section I describes the geometrical significance of the three main theorems in considerable detail, and most of the underlying ideas are illustrated with the aid of simple diagrams. Section II presents rules for the numerical implementation of 11 specific projection operators which are found to occur frequently in many signal-processing applications, and the Appendix contains proofs of all the major results.
1,116 citations
References
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TL;DR: Good generalized these methods and gave elegant algorithms for which one class of applications is the calculation of Fourier series, applicable to certain problems in which one must multiply an N-vector by an N X N matrix which can be factored into m sparse matrices.
Abstract: An efficient method for the calculation of the interactions of a 2' factorial ex- periment was introduced by Yates and is widely known by his name. The generaliza- tion to 3' was given by Box et al. (1). Good (2) generalized these methods and gave elegant algorithms for which one class of applications is the calculation of Fourier series. In their full generality, Good's methods are applicable to certain problems in which one must multiply an N-vector by an N X N matrix which can be factored into m sparse matrices, where m is proportional to log N. This results inma procedure requiring a number of operations proportional to N log N rather than N2. These methods are applied here to the calculation of complex Fourier series. They are useful in situations where the number of data points is, or can be chosen to be, a highly composite number. The algorithm is here derived and presented in a rather different form. Attention is given to the choice of N. It is also shown how special advantage can be obtained in the use of a binary computer with N = 2' and how the entire calculation can be performed within the array of N data storage locations used for the given Fourier coefficients. Consider the problem of calculating the complex Fourier series N-1 (1) X(j) = EA(k)-Wjk, j = 0 1, * ,N- 1, k=0
11,795 citations
Journal Article•
TL;DR: In this article, an algorithm is presented for the rapid solution of the phase of the complete wave function whose intensity in the diffraction and imaging planes of an imaging system are known.
5,197 citations
TL;DR: In this paper, the authors apply the theory developed in the preceding paper to a number of questions about timelimited and bandlimited signals, and find the signals which do the best job of simultaneous time and frequency concentration.
Abstract: The theory developed in the preceding paper1 is applied to a number of questions about timelimited and bandlimited signals. In particular, if a finite-energy signal is given, the possible proportions of its energy in a finite time interval and a finite frequency band are found, as well as the signals which do the best job of simultaneous time and frequency concentration.
2,498 citations
TL;DR: A communication-theory model for the process of image formation is used and it is found that the most likely object has a maximum entropy and is represented by a restoring formula that is positive and not band limited.
Abstract: Given M sampled image values of an incoherent object, what can be deduced as the most likely object? Using a communication-theory model for the process of image formation, we find that the most likely object has a maximum entropy and is represented by a restoring formula that is positive and not band limited. The derivation is an adaptation to optics of a formulation by Jaynes for unbiased estimates of positive probability functions. The restoring formula is tested, via computer simulation, upon noisy images of objects consisting of random impulses. These are found to be well restored, with resolution often exceeding the Rayleigh limit and with a complete absence of spurious detail. The proviso is that the noise in each image input must not exceed about 40% of the signal image. The restoring method is applied to experimental data consisting of line spectra. Results are consistent with those of the computer simulations.
673 citations
TL;DR: It is shown that two distinctly different objects of finite size cannot have identical images, so that no ambiguous image exists for such objects and diffraction limits resolving power in the sense of only the lack of precision of image measurement imposed by the system noise.
Abstract: The existence of an ultimate absolute limit for resolving power is investigated utilizing the ambiguous image concept, viz., different objects cannot be distinguished if they have identical images. Any absolute limit to the resolving power of an optical system must be based upon the existence of ambiguous images rather than on an arbitrary specification of the precision of image measurement, since precision can always be improved, even at the photon-counting limit. It is shown that for all objects of finite angular size, the image spectrum within the passband of the optical system contains the information necessary to determine the object spectrum throughout the entire frequency domain. Knowledge of the object spectrum implies knowledge of the object. It is shown that two distinctly different objects of finite size cannot have identical images, so that no ambiguous image exists for such objects. Therefore, diffraction limits resolving power in the sense of only the lack of precision of image measurement imposed by the system noise. Equations are derived which describe processing procedures by means of which object detail can be extracted from diffraction images. An illustrative example shows the successful processing of the image of two monochromatic point sources separated by 0.2 of the Rayleigh criterion distance.
349 citations