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79
Snakes: Active contour models
 INTERNATIONAL JOURNAL OF COMPUTER VISION
, 1988
"... A snake is an energyminimizing spline guided by external constraint forces and influenced by image forces that pull it toward features such as lines and edges. Snakes are active contour models: they lock onto nearby edges, localizing them accurately. Scalespace continuation can be used to enlarge ..."
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Cited by 3277 (18 self)
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A snake is an energyminimizing spline guided by external constraint forces and influenced by image forces that pull it toward features such as lines and edges. Snakes are active contour models: they lock onto nearby edges, localizing them accurately. Scalespace continuation can be used to enlarge the capture region surrounding a feature. Snakes provide a unified account of a number of visual problems, including detection of edges, lines, and subjective contours; motion tracking; and stereo matching. We have used snakes successfully for interactive interpretation, in which userimposed constraint forces guide the snake near features of interest.
Manifold regularization: A geometric framework for learning from labeled and unlabeled examples
 JOURNAL OF MACHINE LEARNING RESEARCH
, 2006
"... We propose a family of learning algorithms based on a new form of regularization that allows us to exploit the geometry of the marginal distribution. We focus on a semisupervised framework that incorporates labeled and unlabeled data in a generalpurpose learner. Some transductive graph learning al ..."
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Cited by 356 (13 self)
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We propose a family of learning algorithms based on a new form of regularization that allows us to exploit the geometry of the marginal distribution. We focus on a semisupervised framework that incorporates labeled and unlabeled data in a generalpurpose learner. Some transductive graph learning algorithms and standard methods including Support Vector Machines and Regularized Least Squares can be obtained as special cases. We utilize properties of Reproducing Kernel Hilbert spaces to prove new Representer theorems that provide theoretical basis for the algorithms. As a result (in contrast to purely graphbased approaches) we obtain a natural outofsample extension to novel examples and so are able to handle both transductive and truly semisupervised settings. We present experimental evidence suggesting that our semisupervised algorithms are able to use unlabeled data effectively. Finally we have a brief discussion of unsupervised and fully supervised learning within our general framework.
Semisupervised discriminant analysis
 in Proc. of the IEEE Int’l Conf. on Comp. Vision (ICCV), Rio De Janeiro
, 2007
"... Linear Discriminant Analysis (LDA) has been a popular method for extracting features which preserve class separability. The projection vectors are commonly obtained by maximizing the between class covariance and simultaneously minimizing the within class covariance. In practice, when there is no suf ..."
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Cited by 48 (2 self)
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Linear Discriminant Analysis (LDA) has been a popular method for extracting features which preserve class separability. The projection vectors are commonly obtained by maximizing the between class covariance and simultaneously minimizing the within class covariance. In practice, when there is no sufficient training samples, the covariance matrix of each class may not be accurately estimated. In this paper, we propose a novel method, called Semisupervised Discriminant Analysis (SDA), which makes use of both labeled and unlabeled samples. The labeled data points are used to maximize the separability between different classes and the unlabeled data points are used to estimate the intrinsic geometric structure of the data. Specifically, we aim to learn a discriminant function which is as smooth as possible on the data manifold. Experimental results on single training image face recognition and relevance feedback image retrieval demonstrate the effectiveness of our algorithm. 1.
Orthonormal Vector Sets Regularization with PDE's and Applications
, 2001
"... We are interested in regularizing fields of orthonormal vector sets, using constraintpreserving anisotropic diffusion PDE's. Each point of such a field is defined by multiple orthogonal and unitary vectors and can indeed represent a lot of interesting orientation features such as direction vec ..."
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Cited by 38 (3 self)
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We are interested in regularizing fields of orthonormal vector sets, using constraintpreserving anisotropic diffusion PDE's. Each point of such a field is defined by multiple orthogonal and unitary vectors and can indeed represent a lot of interesting orientation features such as direction vectors or orthogonal matrices (among other examples). We first develop a general variational framework that solves this regularization problem, thanks to a constrained minimization of phifunctionals. This leads to a set of coupled vectorvalued PDE's preserving the orthonormal constraints. Then, we focus on particular applications of this general framework, including the restoration of noisy direction fields, noisy chromaticity color images, estimated camera motions and DTMRI (Diffusion Tensor MRI) datasets.
Diffusion tensor regularization with constraints preservation
 In IEEE Computer Society Conference on Computer Vision and Pattern Recognition, Kauai Marriott
"... This paper deals with the problem of regularizing noisy fields of diffusion tensors, considered as symmetric and semipositive definite n n matrices (as for instance 2D structure tensors or DTMRI medical images). We first propose a simple anisotropic PDEbased scheme that acts directly on the matr ..."
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Cited by 37 (14 self)
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This paper deals with the problem of regularizing noisy fields of diffusion tensors, considered as symmetric and semipositive definite n n matrices (as for instance 2D structure tensors or DTMRI medical images). We first propose a simple anisotropic PDEbased scheme that acts directly on the matrix coefficients and preserve the semipositive constraint thanks to a specific reprojection step. The limitations of this algorithm lead us to introduce a more effective approach based on constrained spectral regularizations acting on the tensor orientations (eigenvectors) and diffusivities (eigenvalues), while explicitely taking the tensor constraints into account. The regularization of the orientation part uses orthogonal matrices diffusion PDE’s and local vector alignment procedures and will be particularly developed. For the interesting 3D case, a special implementation scheme designed to numerically fit the tensor constraints is also proposed. Experimental results on synthetic and real DTMRI data sets finally illustrates the proposed tensor regularization framework. 1.
A trustregion approach to the regularization of largescale discrete forms of illposed problems
 SISC
, 2000
"... We consider largescale least squares problems where the coefficient matrix comes from the discretization of an operator in an illposed problem, and the righthand side contains noise. Special techniques known as regularization methods are needed to treat these problems in order to control the effe ..."
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Cited by 23 (6 self)
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We consider largescale least squares problems where the coefficient matrix comes from the discretization of an operator in an illposed problem, and the righthand side contains noise. Special techniques known as regularization methods are needed to treat these problems in order to control the effect of the noise on the solution. We pose the regularization problem as a quadratically constrained least squares problem. This formulation is equivalent to Tikhonov regularization, and we note that it is also a special case of the trustregion subproblem from optimization. We analyze the trustregion subproblem in the regularization case, and we consider the nontrivial extensions of a recently developed method for general largescale subproblems that will allow us to handle this case. The method relies on matrixvector products only, has low and fixed storage requirements, and can handle the singularities arising in illposed problems. We present numerical results on test problems, on an
Regularization of Orthonormal Vector Sets using Coupled PDE's
 PROCEEDINGS 1ST IEEE WORKSHOP ON VARIATIONAL AND LEVEL SET METHODS IN COMPUTER VISION
, 2001
"... We address the problem of restoring, while preserving possible discontinuities, fields of noisy orthonormal vector sets, taking the orthonormal constraints explicitly into account. We develop a variational solution for the general case where each image feature may correspond to multiple nD orthogon ..."
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Cited by 17 (7 self)
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We address the problem of restoring, while preserving possible discontinuities, fields of noisy orthonormal vector sets, taking the orthonormal constraints explicitly into account. We develop a variational solution for the general case where each image feature may correspond to multiple nD orthogonal vectors of unit norms. We first formulate the problem in a new variational framework, where discontinuities and orthonormal constraints are preserved by means of constrained minimization and functions regularization, leading to a set of coupled anisotropic diffusion PDE's. A geometric interpretation of the resulting equations, coming from the field of solid mechanics, is proposed for the 3D case. Two interesting restrictions of our framework are also tackled : the regularization of 3D rotation matrices and the Direction diffusion (the parallel with previous works is made). Finally, we present a number of denoising results and applications.
Satellite image deconvolution using complex wavelet packets
 In Proc. of ICIP
, 2000
"... The deconvolution of blurred and noisy satellite images is an illposed inverse problem. Donoho has proposed to deconvolve the image without regularization and to denoise the result in a wavelet basis by thresholding the transformed coefficients. We have developed a new filtering method, consisting ..."
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Cited by 14 (7 self)
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The deconvolution of blurred and noisy satellite images is an illposed inverse problem. Donoho has proposed to deconvolve the image without regularization and to denoise the result in a wavelet basis by thresholding the transformed coefficients. We have developed a new filtering method, consisting of using a complex wavelet packet basis. Herein, the thresholding functions associated to the proposed method are automatically estimated. The estimation is performed within a Bayesian framework, by modeling the subbands using Generalized Gaussian distributions, and by applying the Maximum A Posteriori (MAP) estimator on each coefficient. Compared to real waveletpacketbased algorithms, the proposed method is shift invariant, provides good directionality properties and remains of complexity O(N). Y=HX+NwhereHX=h?X 1.
Image Reconstruction Through Regularization by Envelope Guided Conjugate Gradients
, 1994
"... In this paper we propose a new way to iteratively solve the image reconstruction problem from noisy images or noisy data linearly related to the pixel intensities. This is done by exploiting the relation between Tikhonov regularization and multiobjective optimization to obtain iteratively approximat ..."
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Cited by 13 (2 self)
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In this paper we propose a new way to iteratively solve the image reconstruction problem from noisy images or noisy data linearly related to the pixel intensities. This is done by exploiting the relation between Tikhonov regularization and multiobjective optimization to obtain iteratively approximations to the Tikhonov Lcurve and its corner. Monitoring the change of the approximate Lcurves allows us to adjust the regularization parameter adaptively during a preconditioned conjugate gradient iteration, so that the desired image can be reconstructed with a low number of iterations. Nonnegativity constraints are taken into account automatically. We present test results on image reconstruction in positron emission tomography (PET). Keywords: Tikhonov regularization, multiobjective optimization, illposed, Lcurve, envelope, preconditioned conjugate gradients, image reconstruction, positron emission tomography (PET) 1991 MSC Classification: primary 65F10, secondary 65R30, 68U10, 90C29, 9...
Filtered conjugate residualtype algorithms with applications
 SIAM Journal on Matrix Analysis and Applications
, 2005
"... In a number of applications, certain computations to be done with a given matrix are performed by replacing this matrix by its best low rank approximation. This has the effect of yielding the most relevant part of the desired solution while discarding noise and redundancies. One such application is ..."
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Cited by 11 (6 self)
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In a number of applications, certain computations to be done with a given matrix are performed by replacing this matrix by its best low rank approximation. This has the effect of yielding the most relevant part of the desired solution while discarding noise and redundancies. One such application is that of regularization where the righthand side of the original linear system is noisy or inaccurate while the coefficient matrix is very illconditioned. Solving such linear systems accurately is counterproductive as the noise tends to be amplified. A common remedy is to compute the pseudoinverse solution in which the inverses of the smallest singular values are replaced by zeros or small quantities. A similar procedure is also used in methods related to Principal Component Analysis, such as in Latent Semantic Indexing in information retrieval. Here the lowrank approximation to the original matrix is used to analyze similarities with a given query vector. This paper presents a few conjugategradient like methods to provide solutions to these two types of problems by iterative procedures which utilize only matrixvector products.