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Robust Anisotropic Diffusion
, 1998
"... Relations between anisotropic diffusion and robust statistics are described in this paper. Specifically, we show that anisotropic diffusion can be seen as a robust estimation procedure that estimates a piecewise smooth image from a noisy input image. The "edgestopping" function in the anisotropic d ..."
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Cited by 278 (16 self)
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Relations between anisotropic diffusion and robust statistics are described in this paper. Specifically, we show that anisotropic diffusion can be seen as a robust estimation procedure that estimates a piecewise smooth image from a noisy input image. The "edgestopping" function in the anisotropic diffusion equation is closely related to the error norm and influence function in the robust estimation framework. This connection leads to a new "edgestopping" function based on Tukey's biweight robust estimator, that preserves sharper boundaries than previous formulations and improves the automatic stopping of the diffusion. The robust statistical interpretation also provides a means for detecting the boundaries (edges) between the piecewise smooth regions in an image that has been smoothed with anisotropic diffusion. Additionally, we derive a relationship between anisotropic diffusion and regularization with line processes. Adding constraints on the spatial organization of the ...
A generalized Gaussian image model for edgepreserving MAP estimation
 IEEE Trans. on Image Processing
, 1993
"... Absfrucf We present a Markov random field model which allows realistic edge modeling while providing stable maximum a posteriori MAP solutions. The proposed model, which we refer to as a generalized Gaussian Markov random field (GGMRF), is named for its similarity to the generalized Gaussian distri ..."
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Cited by 238 (34 self)
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Absfrucf We present a Markov random field model which allows realistic edge modeling while providing stable maximum a posteriori MAP solutions. The proposed model, which we refer to as a generalized Gaussian Markov random field (GGMRF), is named for its similarity to the generalized Gaussian distribution used in robust detection and estimation. The model satisifies several desirable analytical and computational properties for MAP estimation, including continuous dependence of the estimate on the data, invariance of the character of solutions to scaling of data, and a solution which lies at the unique global minimum of the U posteriori loglikeihood function. The GGMRF is demonstrated to be useful for image reconstruction in lowdosage transmission tomography. I.
Geodesic Active Regions and Level Set Methods for Supervised Texture Segmentation
 INTERNATIONAL JOURNAL OF COMPUTER VISION
, 2002
"... This paper presents a novel variational framework to deal with frame partition problems in Computer Vision. This framework exploits boundary and regionbased segmentation modules under a curvebased optimization objective function. The task of supervised texture segmentation is considered to demonst ..."
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Cited by 234 (8 self)
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This paper presents a novel variational framework to deal with frame partition problems in Computer Vision. This framework exploits boundary and regionbased segmentation modules under a curvebased optimization objective function. The task of supervised texture segmentation is considered to demonstrate the potentials of the proposed framework. The textured feature space is generated by filtering the given textured images using isotropic and anisotropic filters, and analyzing their responses as multicomponent conditional probability density functions. The texture segmentation is obtained by unifying region and boundarybased information as an improved Geodesic Active Contour Model. The defined objective function is minimized using a gradientdescent method where a level set approach is used to implement the obtained PDE. According to this PDE, the curve propagation towards the final solution is guided by boundary and regionbased segmentation forces, and is constrained by a regularity force. The level set implementation is performed using a fast front propagation algorithm where topological changes are naturally handled. The performance of our method is demonstrated on a variety of synthetic and real textured frames.
An EM Algorithm for WaveletBased Image Restoration
, 2002
"... This paper introduces an expectationmaximization (EM) algorithm for image restoration (deconvolution) based on a penalized likelihood formulated in the wavelet domain. Regularization is achieved by promoting a reconstruction with lowcomplexity, expressed in terms of the wavelet coecients, taking a ..."
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Cited by 233 (21 self)
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This paper introduces an expectationmaximization (EM) algorithm for image restoration (deconvolution) based on a penalized likelihood formulated in the wavelet domain. Regularization is achieved by promoting a reconstruction with lowcomplexity, expressed in terms of the wavelet coecients, taking advantage of the well known sparsity of wavelet representations. Previous works have investigated waveletbased restoration but, except for certain special cases, the resulting criteria are solved approximately or require very demanding optimization methods. The EM algorithm herein proposed combines the efficient image representation oered by the discrete wavelet transform (DWT) with the diagonalization of the convolution operator obtained in the Fourier domain. The algorithm alternates between an Estep based on the fast Fourier transform (FFT) and a DWTbased Mstep, resulting in an ecient iterative process requiring O(N log N) operations per iteration. Thus, it is the rst image restoration algorithm that optimizes a waveletbased penalized likelihood criterion and has computational complexity comparable to that of standard wavelet denoising or frequency domain deconvolution methods. The convergence behavior of the algorithm is investigated, and it is shown that under mild conditions the algorithm converges to a globally optimal restoration. Moreover, our new approach outperforms several of the best existing methods in benchmark tests, and in some cases is also much less computationally demanding.
The Jackknife and the Bootstrap for General Stationary Observations
, 1989
"... this paper we will always consider statistics TN of the form TN (X 1 ; :::; XN ) = T (ae ..."
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Cited by 225 (2 self)
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this paper we will always consider statistics TN of the form TN (X 1 ; :::; XN ) = T (ae
The development and comparison of robust methods for estimating the fundamental matrix
 International Journal of Computer Vision
, 1997
"... Abstract. This paper has two goals. The first is to develop a variety of robust methods for the computation of the Fundamental Matrix, the calibrationfree representation of camera motion. The methods are drawn from the principal categories of robust estimators, viz. case deletion diagnostics, Mest ..."
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Cited by 220 (9 self)
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Abstract. This paper has two goals. The first is to develop a variety of robust methods for the computation of the Fundamental Matrix, the calibrationfree representation of camera motion. The methods are drawn from the principal categories of robust estimators, viz. case deletion diagnostics, Mestimators and random sampling, and the paper develops the theory required to apply them to nonlinear orthogonal regression problems. Although a considerable amount of interest has focussed on the application of robust estimation in computer vision, the relative merits of the many individual methods are unknown, leaving the potential practitioner to guess at their value. The second goal is therefore to compare and judge the methods. Comparative tests are carried out using correspondences generated both synthetically in a statistically controlled fashion and from feature matching in real imagery. In contrast with previously reported methods the goodness of fit to the synthetic observations is judged not in terms of the fit to the observations per se but in terms of fit to the ground truth. A variety of error measures are examined. The experiments allow a statistically satisfying and quasioptimal method to be synthesized, which is shown to be stable with up to 50 percent outlier contamination, and may still be used if there are more than 50 percent outliers. Performance bounds are established for the method, and a variety of robust methods to estimate the standard deviation of the error and covariance matrix of the parameters are examined. The results of the comparison have broad applicability to vision algorithms where the input data are corrupted not only by noise but also by gross outliers.
An equivalence between sparse approximation and Support Vector Machines
 A.I. Memo 1606, MIT Arti cial Intelligence Laboratory
, 1997
"... This publication can be retrieved by anonymous ftp to publications.ai.mit.edu. The pathname for this publication is: aipublications/15001999/AIM1606.ps.Z This paper shows a relationship between two di erent approximation techniques: the Support Vector Machines (SVM), proposed by V.Vapnik (1995), ..."
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Cited by 205 (7 self)
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This publication can be retrieved by anonymous ftp to publications.ai.mit.edu. The pathname for this publication is: aipublications/15001999/AIM1606.ps.Z This paper shows a relationship between two di erent approximation techniques: the Support Vector Machines (SVM), proposed by V.Vapnik (1995), and a sparse approximation scheme that resembles the Basis Pursuit DeNoising algorithm (Chen, 1995 � Chen, Donoho and Saunders, 1995). SVM is a technique which can be derived from the Structural Risk Minimization Principle (Vapnik, 1982) and can be used to estimate the parameters of several di erent approximation schemes, including Radial Basis Functions, algebraic/trigonometric polynomials, Bsplines, and some forms of Multilayer Perceptrons. Basis Pursuit DeNoising is a sparse approximation technique, in which a function is reconstructed by using a small number of basis functions chosen from a large set (the dictionary). We show that, if the data are noiseless, the modi ed version of Basis Pursuit DeNoising proposed in this paper is equivalent to SVM in the following sense: if applied to the same data set the two techniques give the same solution, which is obtained by solving the same quadratic programming problem. In the appendix we also present a derivation of the SVM technique in the framework of regularization theory, rather than statistical learning theory, establishing a connection between SVM, sparse approximation and regularization theory.
Parameter Estimation Techniques: A Tutorial with Application to Conic Fitting
 Image and Vision Computing
, 1997
"... : Almost all problems in computer vision are related in one form or another to the problem of estimating parameters from noisy data. In this tutorial, we present what is probably the most commonly used techniques for parameter estimation. These include linear leastsquares (pseudoinverse and eigen ..."
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Cited by 196 (6 self)
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: Almost all problems in computer vision are related in one form or another to the problem of estimating parameters from noisy data. In this tutorial, we present what is probably the most commonly used techniques for parameter estimation. These include linear leastsquares (pseudoinverse and eigen analysis); orthogonal leastsquares; gradientweighted leastsquares; biascorrected renormalization; Kalman øltering; and robust techniques (clustering, regression diagnostics, Mestimators, least median of squares). Particular attention has been devoted to discussions about the choice of appropriate minimization criteria and the robustness of the dioeerent techniques. Their application to conic øtting is described. Keywords: Parameter estimation, Leastsquares, Bias correction, Kalman øltering, Robust regression (R#sum# : tsvp) Unite de recherche INRIA SophiaAntipolis 2004 route des Lucioles, BP 93, 06902 SOPHIAANTIPOLIS Cedex (France) Telephone : (33) 93 65 77 77  Telecopie : (33) 9...
Recovering 3D Shape and Motion from Image Streams using NonLinear Least Squares
, 1993
"... The simultaneous recovery of 3D shape and motion from image sequences is one of the more difficult problems in computer vision. Classical approaches to the problem rely on using algebraic techniques to solve for these unknowns given two or more images. More recently, a batch analysis of image stream ..."
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Cited by 192 (35 self)
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The simultaneous recovery of 3D shape and motion from image sequences is one of the more difficult problems in computer vision. Classical approaches to the problem rely on using algebraic techniques to solve for these unknowns given two or more images. More recently, a batch analysis of image streams (the temporal tracks of distinguishable image features) under orthography has resulted in highly accurate reconstructions. We generalize this approach to perspective projection and partial or uncertain tracks by using a nonlinear least squares technique. While our approach requires iteration, it quickly converges to the desired solution, even in the absence of a priori knowledge about the shape or motion. Important features of the algorithm include its ability to handle partial point tracks, to use line segment matches and point matches simultaneously, and to use an objectcentered representation for faster and more accurate structure and motion recovery. We also show how a projective (a...
On the Unification Line Processes, Outlier Rejection, and Robust Statistics with Applications in Early Vision
, 1996
"... The modeling of spatial discontinuities for problems such as surface recovery, segmentation, image reconstruction, and optical flow has been intensely studied in computer vision. While "lineprocess" models of discontinuities have received a great deal of attention, there has been recent interest i ..."
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Cited by 190 (8 self)
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The modeling of spatial discontinuities for problems such as surface recovery, segmentation, image reconstruction, and optical flow has been intensely studied in computer vision. While "lineprocess" models of discontinuities have received a great deal of attention, there has been recent interest in the use of robust statistical techniques to account for discontinuities. This paper unifies the two approaches. To achieve this we generalize the notion of a "line process" to that of an analog "outlier process" and show how a problem formulated in terms of outlier processes can be viewed in terms of robust statistics. We also characterize a class of robust statistical problems for which an equivalent outlierprocess formulation exists and give a straightforward method for converting a robust estimation problem into an outlierprocess formulation. We show how prior assumptions about the spatial structure of outliers can be expressed as constraints on the recovered analog outlier processes and how traditional continuation methods can be extended to the explicit outlierprocess formulation. These results indicate that the outlierprocess approach provides a general framework which subsumes the traditional lineprocess approaches as well as a wide class of robust estimation problems. Examples in surface reconstruction, image segmentation, and optical flow are presented to illustrate the use of outlier processes and to show how the relationship between outlier processes and robust statistics can be exploited. An appendix provides a catalog of common robust error norms and their equivalent outlierprocess formulations.