During the last few years, many efforts have been done in integrating different informations in a variational framework to segment images. Recent works on curve propagation were able to incorporate stochastic informations [14,10] and prior knowledge on shapes [3,11]. The information inserted in these studies is most of the time extracted offline. Meanwhile, other approaches have proposed to extract region information during the segmentation process itself [2,4,13]. Following these new approaches and extending the work in [10] to vector-valued images, we propose in this paper an entirely variational framework to approach the segmentation problem. Both, the image partition and the statistical parameters for each region are unkown. After a brief reminder on recent segmenting methods, we will present a variational formulation obtained from a bayesian model. After that, we will show two different differentiations driving to the same evolution equations. Detailed studies on gray and color images of the 2-phase case will follow. And we will finish on an application to tracking which shows benefits of our dynamical framework. WMVC'02 Article INRIA Research Report

Nonlinear diffusion equations are now widely used to restore and enhance images. They allow to eliminate noise and artifacts while preserving large global features, such as object contours. In this context, we propose a differential-geometric framework to define regularizing PDEs acting on manifold constrained datasets. We consider the case of images taking value into matrix manifolds defined by orthogonal and spectral constraints. We directly incorporate the geometry and natural metric of the underlying configuration space (viewed as a Lie group or a homogeneous space) in the design of the corresponding flows. Our numerical implementation relies on structure-preserving integrators that respect intrinsically the constraints geometry. This approach was applied to the anisotropic smoothing of diffusion tensor volumes in medical imaging. ECCV'02 Article

We address the problem of regularizing fields of diffusion tensors (i.e symmetric and positive-definite matrices) using PDE's and variational tools. We consider the minimization of a general regularizing functional under orthonormal constraints, introduced with Lagrange multipliers. Accurate numerical schemes are then provided and we compare this approach with a classical reprojection-step method. The application of interest considered here is the regularization of noisy DT-MRI images, in order to construct a coherent fiber network of the whitte matter. CVPR'01 Article

This work proposes a variational based approach to regularize fields of orthonormal vector sets, using constraint-preserving anisotropic diffusion PDE's. Each point of these fields is defined by multiple orthogonal and unitary vectors and can indeed represent a lot of interesting orientation features such 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 Phi-functionals. This leads to a set of coupled vector-valued 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 motion and DT-MRI (Diffusion Tensor MRI) datasets. First, presented at VLSM'01 Workshop, Vancouver/Canada, July 13, 2001 VLSM'01 Article IJCV'02 Article RFIA'02 - In French

Here, we propose a new vector image restoration PDE which removes the noise and enhances blurred vector contours, thanks to a vector generalisation of scalar $\Phi$-function diffusions and shock filters. A local and geometric approach is proposed, which uses pertinent vector informations. Finally, we extend this equation to constrained norm evolutions, in order to restore direction fields and chromaticity noise on color images.- Presented at SCIA'01 Conference, Bergen/Norway, 11-14 Jun 2001 - SCIA'01 Article IEEE Signal Processing Magazine 2002

In this work, we propose a new model for supervised texture segmentation under a region/boundary-based curve propagation model. We pursue two main objectives; the first is to propose a complete framework for texture analysis and modeling that combines existing approaches in this domain and provides a reliable texture description model with a limited set of parameters. This model must be general to describe a wide variety of texture patterns. The second objective is to unify boundary-based and region-based segmentation using a variational approach. International Journal on Computer Vision, Volume 50, Number 3, pages 237-252, Dec. 2002 Article

A novel variational method that unify boundary and region-based segmentation modules under the Geodesic Active Region framework is presented. A statistical analysis over the observed density function using a mixture of Gaussian elements, is first used to indicate the number of the different regions and their intensity properties. Then, the boundary information is determined using a probabilistic edge detector, while the region information is given directly from the observed image using the conditional probability density functions of the mixture model. The defined objective function is minimized using a gradient-descent method implemented through the use of the level-set method. The changes of topology of the curves during their evolution are naturally handled thanks to the level set implementation, while a coupled multi-phase propagation is adopted that increases the robustness and the convergence rate by introducing a coupled system of equations for the different level set functions. ECCV'2000 Article

In this work, we propose a new framework for tracking multiple non-rigid moving objects using a region/boundary-based curve propagation model.The objective of this work is to provide a complete tracking model that can deal successfully with the most difficult tracking cases, i.e non-rigid objects and non-rigid movements.

Coupled PDE's are introduced for vector image restoration and applyed to color images. Diffusion-reaction terms are developed to deal with both the debluring and denoising problems. Slides of the thesis presentation

A In this work, we illustrate the results obtained using a variational approach devised for the purpose of image restoration with the constraint to preserve the edges within the original image.the restoration problem is set as a regularization and minimization of a non quadratic functional

We propose an energy based method to estimate a dense disparity map between 3 images while preserving its discontinuities accordingly to the image boundaries that may be present. We first derive a simplified expression for the disparity that allows us to easily estimate it from a stereo pair of images. Dicsontinuities are preserved using a regularization constraint term based on the Nagel and Enkelmann operator to better estimate and preserve the discontinuities. We assume that the epipolar geometry is known, and we include this information in the energy model. We derive the associated Euler-Lagrange equation and we approach the solution of the underlying partial differential equation using a gradient descent method. In order to reduce the risk to be trapped within some irrelevant local minima, we use a focusing strategy based on a linear Gaussian scale space.

We present a variational approach to dense stereo reconstruction which combines powerful tools such as regularization and multi-scale processing to estimate directly depth from a number of stereo images, while preserving depth discontinuities. The problem is set as a regularization and minimization of a non quadratic functional.

In this work, we provide a totally new approach to deal with the important problem of matching 2D curves from a stereo pair of images We use the framework of energy minimization and express our problem as a geodesic active contour based approach.

Autocalibration in the case of constant and varying intrinsic parameters. This paper deals with a fundamental problem in motion and stereo analysis, namely that of determining the camera intrinsic calibration parameters. Two methods are proposed that follows the autocalibration paradigm, according to which calibration is achieved not with the aid of a calibration pattern but by observing a number of image features in a set of successive images. The two proposed methods relie upon the Singular Value Decomposition of the fundamental matrix, which leads to a particularly simple form of the Kruppa equations. Autocalibration in the case of constant and varying intrinsic parameters are considered. Experimental results from extensive simulations and several image sequences, taking into account the uncertainty of the measurements, demonstrate the effectiveness of the proposed methods in accurately estimating the intrinsic calibration matrices. It is also shown that the computed intrinsic calibration matrices are sufficient for recovering 3D measurement, 3 motion and 3D Recosntruction from uncalibrated images.

Edges, corners and vertices are strong and useful features in computer vision. The work illustreted here deals with the development of an efficient model based approach in order to detect and characterize precisely these important features. The key of our approach is first to propose some efficient models associated to each of these features and second to efficiently extract and characterize these features directly from the image.

This work deals with the development of a parametric model based method to locate and characterize precisely important curved features such as ellipses and B-splines based curves. The method uses all the grey level information of the pixels contained within a window around the feature of interest and produces the complete parametric model that best approximates in a mean-square sense the observed grey level image intensities within the working area.

Feature Extraction Using Parametric Snakes - Energy Based Methods for 2D Curve Tracking and Applications

Stereo Matching, Reconstruction and Refinement of 3-{D} Curves Using Deformable Contours - Energy Based Methods for Ronstruction, and Refinement of 3{D} Curves and Applications

Using Canny's Criteria to Derive a Recursively Implemented Optimal Edge Detector - Fast Algorithms for Low-Level Vision. - An Efficient Method to Build an Early Image Description - - A Computational Approach For Corner And Vertex Detection etc..