We address the problem of vector-valued image regularization with variational methods and PDE's. From the study of existing formalisms, we propose a unifying framework based on a very local interpretation of the regularization processes. The resulting equations are then specialized into new regularization PDE's and corresponding numerical schemes that respect the local geometry of vector-valued images. They are finally applied on a wide variety of image processing problems, including color image restoration, inpainting, magnification and flow visualization.

In this paper, we present an efficient way to denoise bivariate data like height fields, color pictures or vector fields, while preserving edges and other features. Mixing surface area minimization, graph flow, and nonlinear edge-preservation metrics, our method generalizes previous anisotropic diffusion approaches in image processing, and is applicable to data of arbitrary dimension. Another notable difference is the use of a more robust discrete differential operator, which captures the fundamental surface properties. We demonstrate the method on range images and height fields, as well as greyscale or color images.

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 vice-versa. 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...

Defined as the apparent motion in a sequence of images, the optical flow is very important in the Computer Vision community where its accurate estimation is strongly needed for many applications. It is one of the most studied problem in Computer Vision. In spite of this, not much theoretical analysis has been done. In this article, we first present a review of existing variational methods. Then, we will propose an extended model that will be rigorously justified on the space of functions of bounded variations. Finally, we present an algorithm whose convergence will be carefully demonstrated. Some results showing the capabilities of this method will end that work.

We are interested in regularizing fields of orthonormal vector sets, using constraint-preserving 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 -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 motions and DT-MRI (Di usion Tensor MRI) datasets.

This paper deals with the problem of regularizing noisy fields of diffusion tensors, considered as symmetric and semi-positive definite n n matrices (as for instance 2D structure tensors or DT-MRI medical images). We first propose a simple anisotropic PDE-based 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 DT-MRI data sets finally illustrates the proposed tensor regularization framework.

Abstract. Nonlinear partial differential equations (PDE) are now widely used to regularize images. They allow to eliminate noise and artifacts while preserving large global features, such as object contours. In this context, we propose a geometric framework to design PDE flows actingon constrained datasets. We focus our interest on flows of matrixvalued functions undergoing orthogonal and spectral constraints. The correspondingevolution PDE’s are found by minimization of cost functionals, and depend on the natural metrics of the underlyingconstrained manifolds (viewed as Lie groups or homogeneous spaces). Suitable numerical schemes that fit the constraints are also presented. We illustrate this theoretical framework through a recent and challenging problem in medical imaging: the regularization of diffusion tensor volumes (DT-MRI).

This paper addresses the issue of motion estimation on image sequences. The standard motion equation used to compute the apparent motion of image irradiance patterns is an invariance brightness based hypothesis called the optical flow constraint. Other equations can be used, in particular the extended optical flow constraint, which is a variant of the optical flow constraint, inspired by the fluid mechanic mass conservation principle. In this paper, we propose a physical interpretation of this extended optical flow equation and a new model unifying the optical flow and the extended optical flow constraints. We present results obtained for synthetic and meteorological images. 1.

We are interested in PDE’s (Partial Differential Equations) in order to smooth multi-valued images in an anisotropic manner. Starting from a review of existing anisotropic regularization schemes based on diffusion PDE’s, we point out the pros and cons of the different equations proposed in the literature. Then, we introduce a new tensor-driven PDE, regularizing images while taking the curvatures of specific integral curves into account. We show that this constraint is particularly well suited for the preservation of thin structures in an image restoration process. A direct link is made between our proposed equation and a continuous formulation of the LIC’s (Line Integral Convolutions by Cabral and Leedom [11]). It leads to the design of a very fast and stable algorithm that implements our regularization method, by successive integrations of pixel values along curved integral lines. Besides, the scheme numerically performs with a sub-pixel accuracy and preserves then thin image structures better than classical finite-differences discretizations. Finally, we illustrate the efficiency of our generic curvature-preserving approach- in terms of speed and visual quality- with different comparisons and various applications requiring image smoothing: color images denoising, inpainting and image resizing by nonlinear interpolation.

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 n-D 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.