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35
Illposed problems in early vision
 Proceedings of the IEEE
, 1988
"... The first processing stage in computational vision, also called early vision, consists of decoding twodimensional images in terms of properties of 30 surfaces. Early vision includes problems such as the recovery of motion and optical flow, shape from shading, surface interpolation, and edge detect ..."
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Cited by 182 (13 self)
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The first processing stage in computational vision, also called early vision, consists of decoding twodimensional images in terms of properties of 30 surfaces. Early vision includes problems such as the recovery of motion and optical flow, shape from shading, surface interpolation, and edge detection. These are inverse problems, which are often illposed or illconditioned. We review here the relevant mathematical results on illposed and illconditioned problems and introduce the formal aspects of regularization theory in the linear and nonlinear case. Specific topics in early vision and their regularization are then analyzed rigorously, characterizing existence, uniqueness, and stability of solutions.
Robust Solutions To LeastSquares Problems With Uncertain Data
, 1997
"... . We consider leastsquares problems where the coefficient matrices A; b are unknownbutbounded. We minimize the worstcase residual error using (convex) secondorder cone programming, yielding an algorithm with complexity similar to one singular value decomposition of A. The method can be interpret ..."
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Cited by 145 (12 self)
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. We consider leastsquares problems where the coefficient matrices A; b are unknownbutbounded. We minimize the worstcase residual error using (convex) secondorder cone programming, yielding an algorithm with complexity similar to one singular value decomposition of A. The method can be interpreted as a Tikhonov regularization procedure, with the advantage that it provides an exact bound on the robustness of solution, and a rigorous way to compute the regularization parameter. When the perturbation has a known (e.g., Toeplitz) structure, the same problem can be solved in polynomialtime using semidefinite programming (SDP). We also consider the case when A; b are rational functions of an unknownbutbounded perturbation vector. We show how to minimize (via SDP) upper bounds on the optimal worstcase residual. We provide numerical examples, including one from robust identification and one from robust interpolation. Key Words. Leastsquares, uncertainty, robustness, secondorder cone...
Solving illconditioned and singular linear systems: A tutorial on regularization
 SIAM Rev
, 1998
"... Abstract. It is shown that the basic regularization procedures for finding meaningful approximate solutions of illconditioned or singular linear systems can be phrased and analyzed in terms of classical linear algebra that can be taught in any numerical analysis course. Apart from rewriting many kn ..."
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Cited by 83 (2 self)
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Abstract. It is shown that the basic regularization procedures for finding meaningful approximate solutions of illconditioned or singular linear systems can be phrased and analyzed in terms of classical linear algebra that can be taught in any numerical analysis course. Apart from rewriting many known results in a more elegant form, we also derive a new twoparameter family of merit functions for the determination of the regularization parameter. The traditional merit functions from generalized cross validation (GCV) and generalized maximum likelihood (GML) are recovered as special cases.
The LCurve and its Use in the Numerical Treatment of Inverse Problems
 in Computational Inverse Problems in Electrocardiology, ed. P. Johnston, Advances in Computational Bioengineering
, 2000
"... The Lcurve is a loglog plot of the norm of a regularized solution versus the norm of the corresponding residual norm. It is a convenient graphical tool for displaying the tradeoff between the size of a regularized solution and its fit to the given data, as the regularization parameter varies. The ..."
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Cited by 32 (2 self)
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The Lcurve is a loglog plot of the norm of a regularized solution versus the norm of the corresponding residual norm. It is a convenient graphical tool for displaying the tradeoff between the size of a regularized solution and its fit to the given data, as the regularization parameter varies. The Lcurve thus gives insight into the regularizing properties of the underlying regularization method, and it is an aid in choosing an appropriate regularization parameter for the given data. In this chapter we summarize the main properties of the Lcurve, and demonstrate by examples its usefulness and its limitations both as an analysis tool and as a method for choosing the regularization parameter. 1 Introduction Practically all regularization methods for computing stable solutions to inverse problems involve a tradeoff between the "size" of the regularized solution and the quality of the fit that it provides to the given data. What distinguishes the various regularization methods is how...
A Multiple Input Image Restoration Approach
 Journal of Visual Communication and Image Representation
, 1990
"... this paper image restoration applications, where multiple distorted versions of the same original image are available, are considered. A general adaptive restoration algorithm is derived on the basis of a set theoretic regularization technique. The adaptivity of the algorithm is introduced in two wa ..."
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Cited by 17 (5 self)
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this paper image restoration applications, where multiple distorted versions of the same original image are available, are considered. A general adaptive restoration algorithm is derived on the basis of a set theoretic regularization technique. The adaptivity of the algorithm is introduced in two ways: (a) by a constraint ,operator which incorporates properties of the response of the hu man visual system into the restoration process and (b) by a weight matrix which assigns greater importance for the deconvolution process to areas of high spatial activity than to areas of low spatial activity. Different degrees of trust are assigned to the various distorted images depending on the amounts of noise. The proposed algorithm is general and can be used for any type of linear distortion and constraint operators. It can also be used to restore signals other than images. Experimental results obtained by an iterative implementation of the proposed algorithms are pre sented. c 1990 Academic Press, Inc
Analysis And Design Of MinimaxOptimal Interpolators
 IEEE Trans. Signal Proc
, 1998
"... We consider a class of interpolation algorithms, including the leastsquares optimal Yen interpolator, and we derive a closedform expression for the interpolation error for interpolators of this type. The error depends on the eigenvalue distribution of a matrix which is specified for each set of sa ..."
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Cited by 13 (3 self)
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We consider a class of interpolation algorithms, including the leastsquares optimal Yen interpolator, and we derive a closedform expression for the interpolation error for interpolators of this type. The error depends on the eigenvalue distribution of a matrix which is specified for each set of sampling points. The error expression can be used to prove that the Yen interpolator is optimal. The implementation of the Yen algorithm suffers from numerical illconditioning, forcing the use of a regularized, approximate solution. We suggest a new, approximate solution, consisting of a sinckernel interpolator with specially chosen weighting coefficients. The newly designed sinckernel interpolator is compared with the usual sinc interpolator using Jacobian (area) weighting, through numerical simulations. We show that the sinc interpolator with Jacobian weighting works well only when the sampling is nearly uniform. The newly designed sinckernel interpolator is shown to perform better than ...
Stochastic Modeling and Estimation of Multispectral Image Data
 IEEE Trans. Image Processing
, 1995
"... Multispectral images consist of multiple channels, each containing data acquired from a different band within the frequency spectrum. Since most objects emit or reflect energy over a large spectral bandwidth, there usually exists a significant correlation between channels. Due to often harsh imaging ..."
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Cited by 11 (1 self)
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Multispectral images consist of multiple channels, each containing data acquired from a different band within the frequency spectrum. Since most objects emit or reflect energy over a large spectral bandwidth, there usually exists a significant correlation between channels. Due to often harsh imaging environments, the acquired data may be degraded by both blur and noise. Simply applying a monochromatic restoration algorithm to each frequency band ignores the crosschannel correlation present within a multispectral image. A Gibbs prior is proposed for multispectral data modeled as a Markov random field, containing both spatial and spectral cliques. Spatial components of the model use a nonlinear operator to preserve discontinuities within each frequency band, while spectral components incorporate nonstationary crosschannel correlations. The multispectral model is used in a Bayesian algorithm for the restoration of color images, in which the resulting nonlinear estimates are shown to be ...
DirectFourier Reconstruction In Tomography And Synthetic Aperture Radar
 Intl. J. Imaging Sys. and Tech
, 1998
"... We investigate the use of directFourier (DF) image reconstruction in computerized tomography and synthetic aperture radar (SAR). One of our aims is to determine why the convolutionbackprojection (CBP) method is favored over DF methods in tomography, while DF methods are virtually always used in SAR ..."
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Cited by 9 (0 self)
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We investigate the use of directFourier (DF) image reconstruction in computerized tomography and synthetic aperture radar (SAR). One of our aims is to determine why the convolutionbackprojection (CBP) method is favored over DF methods in tomography, while DF methods are virtually always used in SAR. We show that the CBP algorithm is equivalent to DF reconstruction using a Jacobianweighted 2D periodic sinckernel interpolator. This interpolation is not optimal in any sense, which suggests that DF algorithms utilizing optimal interpolators may surpass CBP in image quality. We consider use of two types of DF interpolation: a windowed sinc kernel, and the leastsquares optimal Yen interpolator. Simulations show that reconstructions using the Yen interpolator do not possess the expected visual quality, because of regularization needed to preserve numerical stability. Next, we show that with a concentricsquares sampling scheme, DF interpolation can be performed accurately and efficiently...
Piecewise and Local Image Models for Regularized Image Restoration Using CrossValidation
 IEEE Transactions On Image Processing
, 1999
"... We describe two broad classes of useful and physically meaningful image models that can be used to construct novel smoothing constraints for use in the regularized image restoration problem. The two classes, termed piecewise image models (PIM's) and local image models (LIM's), respectively, capture ..."
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Cited by 8 (3 self)
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We describe two broad classes of useful and physically meaningful image models that can be used to construct novel smoothing constraints for use in the regularized image restoration problem. The two classes, termed piecewise image models (PIM's) and local image models (LIM's), respectively, capture unique image properties that can be adapted to the image and that reflect structurally significant surface characteristics. Members of the PIM and LIM classes are easily formed into regularization operators that replace differentialtype constraints. We also develop an adaptive strategy for selecting the best PIM or LIM for a given problem (from among the defined class), and we explain the construction of the corresponding regularization operators. Considerable attention is also given to determining the regularization parameter via a crossvalidation technique, and also to the selection of an optimization strategy for solving the problem. Several results are provided that illustrate the processes of model selection, parameter selection, and image restoration. The overall approach provides a new viewpoint on the restoration problem through the use of new image models that capture salient image features that are not well represented through traditional approaches.
Image restoration for confocal microscopy: Improving the limits of deconvolution, with application to the visualization of the mammalian hearing organ
 Biophys J
, 2001
"... ABSTRACT Deconvolution algorithms have proven very effective in conventional (widefield) fluorescence microscopy. Their application to confocal microscopy is hampered, in biological experiments, by the presence of important levels of noise in the images and by the lack of a precise knowledge of the ..."
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Cited by 8 (0 self)
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ABSTRACT Deconvolution algorithms have proven very effective in conventional (widefield) fluorescence microscopy. Their application to confocal microscopy is hampered, in biological experiments, by the presence of important levels of noise in the images and by the lack of a precise knowledge of the point spread function (PSF) of the system. We investigate the application of waveletbased processing tools to deal with these problems, in particular wavelet denoising methods, which turn out to be very effective in application to threedimensional confocal images. When used in combination with more classical deconvolution algorithms, these methods provide a robust and efficient restoration scheme allowing one to deal with difficult imaging conditions. To make our approach applicable in practical situations, we measured the PSF of a BioradMRC1024 confocal microscope under a large set of imaging conditions, including in situ acquisitions. As a specific biological application, we present several examples of restorations of threedimensional confocal images acquired inside an intact preparation of the hearing organ. We also provide a quantitative assessment of the gain in quality achieved by waveletaided restorations over classical deconvolution schemes, based on a set of numerical experiments that we performed with test images.