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39
Global Minimum for Active Contour Models: A Minimal Path Approach
, 1997
"... A new boundary detection approach for shape modeling is presented. It detects the global minimum of an active contour model’s energy between two end points. Initialization is made easier and the curve is not trapped at a local minimum by spurious edges. We modify the “snake” energy by including the ..."
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Cited by 238 (70 self)
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A new boundary detection approach for shape modeling is presented. It detects the global minimum of an active contour model’s energy between two end points. Initialization is made easier and the curve is not trapped at a local minimum by spurious edges. We modify the “snake” energy by including the internal regularization term in the external potential term. Our method is based on finding a path of minimal length in a Riemannian metric. We then make use of a new efficient numerical method to find this shortest path. It is shown that the proposed energy, though based only on a potential integrated along the curve, imposes a regularization effect like snakes. We explore the relation between the maximum curvature along the resulting contour and the potential generated from the image. The method is capable to close contours, given only one point on the objects’ boundary by using a topologybased saddle search routine. We show examples of our method applied to real aerial and medical images.
A parametric deformable model to fit unstructured 3D data
, 1995
"... Recovery of unstructured 3D data with deformable models has been the subject of many studies over the last ten years. In particular, in medical image understanding, deformable models are useful to get a precise representation of anatomical structures. However, general deformable models involve large ..."
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Cited by 54 (1 self)
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Recovery of unstructured 3D data with deformable models has been the subject of many studies over the last ten years. In particular, in medical image understanding, deformable models are useful to get a precise representation of anatomical structures. However, general deformable models involve large linear systems to solve when dealing with high resolution 3D images. The advantage of parametric deformable models like superquadrics is their small number of parameters to describe a shape combined with a better robustness in the presence of noise or sparse data. Also, at the expense of a reasonable number of additional parameters, free form deformations provide a much closer fit and a volumetric deformation field. This article introduces such a model to fit unstructured 3D points with a parametric deformable surface based on a superquadric fit followed by a free form deformation to describe the cardiac left ventricle. We present the mathematical and algorithmic details of the method, as wel...
Tracking And Motion Analysis Of The Left Ventricle With Deformable Superquadrics
 Medical Image Analysis
, 1996
"... We present a new approach to analyse the deformation of the left ventricle of the heart based on a parametric model that gives a compact representation of a set of points in a 3D image. We present a strategy for tracking surfaces in a sequence of 3D cardiac images. Following tracking, we then i ..."
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Cited by 53 (8 self)
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We present a new approach to analyse the deformation of the left ventricle of the heart based on a parametric model that gives a compact representation of a set of points in a 3D image. We present a strategy for tracking surfaces in a sequence of 3D cardiac images. Following tracking, we then infer quantitative parameters which characterize: left ventricle motion, volume of left ventricle, ejection fraction, amplitude and twist component of cardiac motion. We explain the computation of these parameters using our model. Experimental results are shown in time sequences of two modalities of medical images, nuclear medicine and Xray computed tomography (CT). Video sequences presenting these results are on the CDROM.
Analysis of HalfQuadratic Minimization Methods for Signal and Image Recovery
, 2003
"... Abstract. We address the minimization of regularized convex cost functions which are customarily used for edgepreserving restoration and reconstruction of signals and images. In order to accelerate computation, the multiplicative and the additive halfquadratic reformulation of the original cost ..."
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Cited by 52 (8 self)
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Abstract. We address the minimization of regularized convex cost functions which are customarily used for edgepreserving restoration and reconstruction of signals and images. In order to accelerate computation, the multiplicative and the additive halfquadratic reformulation of the original costfunction have been pioneered in Geman and Reynolds [IEEE Trans. Pattern Anal. Machine Intelligence, 14 (1992), pp. 367–383] and Geman and Yang [IEEE Trans. Image Process., 4 (1995), pp. 932–946]. The alternate minimization of the resultant (augmented) costfunctions has a simple explicit form. The goal of this paper is to provide a systematic analysis of the convergence rate achieved by these methods. For the multiplicative and additive halfquadratic regularizations, we determine their upper bounds for their rootconvergence factors. The bound for the multiplicative form is seen to be always smaller than the bound for the additive form. Experiments show that the number of iterations required for convergence for the multiplicative form is always less than that for the additive form. However, the computational cost of each iteration is much higher for the multiplicative form than for the additive form. The global assessment is that minimization using the additive form of halfquadratic regularization is faster than using the multiplicative form. When the additive form is applicable, it is hence recommended. Extensive experiments demonstrate that in our MATLAB implementation, both methods are substantially faster (in terms of computational times) than the standard MATLAB Optimization Toolbox routines used in our comparison study.
Image Sequence Analysis via Partial Differential Equations
, 1999
"... This article deals with the problem of restoring and motion segmenting noisy image sequences with a static background. Usually, motion segmentation and image restoration are considered separately in image sequence restoration. Moreover, motion segmentation is often noise sensitive. In this article, ..."
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Cited by 50 (3 self)
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This article deals with the problem of restoring and motion segmenting noisy image sequences with a static background. Usually, motion segmentation and image restoration are considered separately in image sequence restoration. Moreover, motion segmentation is often noise sensitive. In this article, the motion segmentation and the image restoration parts are performed in a coupled way, allowing the motion segmentation part to positively influence the restoration part and viceversa. This is the key of our approach that allows to deal simultaneously with the problem of restoration and motion segmentation. To this end, we propose a theoretically justified optimization problem that permits to take into account both requirements. The model is theoretically justified. Existence and unicity are proved in the space of bounded variations. A suitable numerical scheme based on half quadratic minimization is then proposed and its convergence and stability demonstrated. Experimental results obtaine...
Convex halfquadratic criteria and interacting auxiliary variables for image restoration
 IEEE Trans. Image Processing
, 2001
"... © 2001 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other w ..."
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Cited by 48 (13 self)
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© 2001 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE. Abstract—This paper deals with convex halfquadratic criteria and associated minimization algorithms for the purpose of image restoration. It brings a number of original elements within a unified mathematical presentation based on convex duality. Firstly, Geman and Yang’s [1] and Geman and Reynolds’s [2] constructions are revisited, with a view to establish convexity properties of the resulting halfquadratic augmented criteria, when the original nonquadratic criterion is already convex. Secondly, a family of convex Gibbsian energies that incorporate interacting auxiliary variables is revealed as a potentially fruitful extension of Geman and Reynolds’s construction. Index Terms—Convex duality, coordinate descent algorithms, edgepreserving restoration, Gibbs–Markov models, line processes. I.
Orthonormal Vector Sets Regularization with PDE’s and Applications
, 2002
"... 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 ..."
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Cited by 44 (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 φfunctionals. 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.
Unsupervised Contour Representation and Estimation Using BSplines and a Minimum Description Length Criterion
 IEEE Trans. on Image Processing
, 2000
"... This paper describes a new approach to adaptive estimation of parametric deformable contours based on Bspline representations. The problem is formulated in a statistical framework with the likelihood function being derived from a region based image model. The parameters of the image model, the con ..."
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Cited by 38 (3 self)
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This paper describes a new approach to adaptive estimation of parametric deformable contours based on Bspline representations. The problem is formulated in a statistical framework with the likelihood function being derived from a region based image model. The parameters of the image model, the contour parameters, and the Bspline parameterization order (i.e., the number of control points) are all considered unknown. The parameterization order is estimated via a minimum description length (MDL) type criterion. A deterministic iterative algorithm is developed to implement the derived contour estimation criterion. The result is an unsupervised parametric deformable contour: it adapts its degree of smoothness/complexity (number of control points) and it also estimates the observation (image) model parameters. The experiments reported in the paper, performed on synthetic and real (medical) images, confirm the adequacy and good performance of the approach.
Nonlinear Operators in Image Restoration
 In Proceedings of the International Conference on Computer Vision and Pattern Recognition
, 1997
"... We firstly present a variational approach such that during image restoration, edges detected in the original image are being preserved, and then we compare in a second part, the mathematical foundation of this method with respect to some of the well known methods recently proposed in the literature ..."
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Cited by 30 (13 self)
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We firstly present a variational approach such that during image restoration, edges detected in the original image are being preserved, and then we compare in a second part, the mathematical foundation of this method with respect to some of the well known methods recently proposed in the literature within the class of PDE based algorithms (anisotropic diffusion, mean curvature motion, min/max flow technique,...). The performance of our approach is carefully examined and compared to the classical methods. Experimental results on synthetic and real images will illustrate the capabilities of all the studied approaches.
Domain decomposition methods for linear inverse problems with sparsity constraints
, 2007
"... Quantities of interest appearing in concrete applications often possess sparse expansions with respect to a preassigned frame. Recently, there were introduced sparsity measures which are typically constructed on the basis of weighted ℓ1 norms of frame coefficients. One can model the reconstruction o ..."
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Cited by 25 (6 self)
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Quantities of interest appearing in concrete applications often possess sparse expansions with respect to a preassigned frame. Recently, there were introduced sparsity measures which are typically constructed on the basis of weighted ℓ1 norms of frame coefficients. One can model the reconstruction of a sparse vector from noisy linear measurements as the minimization of the functional defined by the sum of the discrepancy with respect to the data and the weighted ℓ1norm of suitable frame coefficients. Thresholded Landweber iterations were proposed for the solution of the variational problem. Despite of its simplicity which makes it very attractive to users, this algorithm converges slowly. In this paper we investigate methods to accelerate significantly the convergence. We introduce and analyze sequential and parallel iterative algorithms based on alternating subspace corrections for the solution of the linear inverse problem with sparsity constraints. We prove their norm convergence to minimizers of the functional. We compare the computational cost and the behavior of these new algorithms with respect to the thresholded Landweber iterations.