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VectorValued Image Regularization with PDEs: A Common Framework for Different Applications
 IEEE Transactions on Pattern Analysis and Machine Intelligence
, 2003
"... We address the problem of vectorvalued 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 reg ..."
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Cited by 181 (8 self)
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We address the problem of vectorvalued 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 vectorvalued images. They are finally applied on a wide variety of image processing problems, including color image restoration, inpainting, magnification and flow visualization.
The Digital TV Filter and Nonlinear Denoising
 IEEE Trans. Image Process
, 2001
"... Motivated by the classical TV (total variation) restoration model, we propose a new nonlinear filterthe digital TV filter for denoising and enhancing digital images, or more generally, data living on graphs. The digital TV filter is a data dependent lowpass filter, capable of denoising data witho ..."
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Cited by 154 (14 self)
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Motivated by the classical TV (total variation) restoration model, we propose a new nonlinear filterthe digital TV filter for denoising and enhancing digital images, or more generally, data living on graphs. The digital TV filter is a data dependent lowpass filter, capable of denoising data without blurring jumps or edges. In iterations, it solves a global total variational optimization problem, which differs from most statistical filters. Applications are given in the denoising of onedimensional (1D) signals, twodimensional (2D) data with irregular structures, gray scale and color images, and nonflat image features such as chromaticity.
CoherenceEnhancing Diffusion Filtering
, 1999
"... The completion of interrupted lines or the enhancement of flowlike structures is a challenging task in computer vision, human vision, and image processing. We address this problem by presenting a multiscale method in which a nonlinear diffusion filter is steered by the socalled interest operato ..."
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Cited by 137 (3 self)
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The completion of interrupted lines or the enhancement of flowlike structures is a challenging task in computer vision, human vision, and image processing. We address this problem by presenting a multiscale method in which a nonlinear diffusion filter is steered by the socalled interest operator (secondmoment matrix, structure tensor). An mdimensional formulation of this method is analysed with respect to its wellposedness and scalespace properties. An efficient scheme is presented which uses a stabilization by a semiimplicit additive operator splitting (AOS), and the scalespace behaviour of this method is illustrated by applying it to both 2D and 3D images.
Demosaicing: image reconstruction from color ccd samples
 IMAGE PROCESSING, IEEE TRANSACTIONS ON
, 1999
"... A simplified color image formation model is used to construct an algorithm for image reconstruction from CCD sensors samples. The proposed method involves two successive steps. The first is motivated by Cok’s template matching technique, while the second step uses steerable inverse diffusion in co ..."
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Cited by 114 (0 self)
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A simplified color image formation model is used to construct an algorithm for image reconstruction from CCD sensors samples. The proposed method involves two successive steps. The first is motivated by Cok’s template matching technique, while the second step uses steerable inverse diffusion in color. Classical linear signal processing techniques tend to oversmooth the image and result in noticeable color artifacts along edges and sharp features. The question is how should the different color channels support each other to form the best possible reconstruction. Our answer is to let the edges support the color information, and the color channels support the edges, and thereby achieve better perceptual results than those that are bounded by the sampling theoretical limit.
Anisotropic Geometric Diffusion in Surface Processing
, 2000
"... INTRODUCTION Geometric evolution problems for curves and surfaces and especially curvature flow problems are an exciting and already classical mathematical research field. They lead to interesting systems of nonlinear partial differential equations and allow the appropriate mathematical modelling o ..."
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Cited by 110 (0 self)
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INTRODUCTION Geometric evolution problems for curves and surfaces and especially curvature flow problems are an exciting and already classical mathematical research field. They lead to interesting systems of nonlinear partial differential equations and allow the appropriate mathematical modelling of physical processes such as material interface propagation, fluid free boundary motion, crystal growth. On the other hand, curves and surfaces are essential objects in computer aided geometric design and computer graphics. Here, issues are fairing, modelling, deformation, and motion. Recently, geometric evolution problems and variational approaches have entered this research field as well and have turned out to be powerful tools. The aim of our work in the field of surface fairing and surface modelling is to modify "classical" curvature motion in a suitable way and apply it in computer graphics. 2 A GENERAL SCHEME Consider an image I :# R. A well known approach to image processing c
Anisotropic Diffusion of Surfaces and Functions on Surfaces
, 2002
"... We present a unified anisotropic geometric diffusion PDE model for smoothing (fairing) out noise both in triangulated 2manifold surface meshes in IR 3 and functions defined on these surface meshes, while enhancing curve features on both by careful choice of an anisotropic diffusion tensor. We combi ..."
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Cited by 75 (8 self)
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We present a unified anisotropic geometric diffusion PDE model for smoothing (fairing) out noise both in triangulated 2manifold surface meshes in IR 3 and functions defined on these surface meshes, while enhancing curve features on both by careful choice of an anisotropic diffusion tensor. We combine the C 1 limit representation of Loop’s subdivision for triangular surface meshes and vector functions on the surface mesh with the established diffusion model to arrive at a discretized version of the diffusion problem in the spatial direction. The time direction discretization then leads to a sparse linear system of equations. Iteratively, solving the sparse linear system, yields a sequence of faired (smoothed) meshes as well as faired functions.
Fast Anisotropic Smoothing of MultiValued Images using CurvaturePreserving PDE’s
 Research Report “Les Cahiers du GREYC”, No 05/01. Equipe IMAGE/GREYC (CNRS UMR 6072), Février
, 2005
"... We are interested in PDE’s (Partial Differential Equations) in order to smooth multivalued 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 liter ..."
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Cited by 65 (3 self)
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We are interested in PDE’s (Partial Differential Equations) in order to smooth multivalued 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 tensordriven 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 subpixel accuracy and preserves then thin image structures better than classical finitedifferences discretizations. Finally, we illustrate the efficiency of our generic curvaturepreserving 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.
LowLevel Image Processing with the Structure Multivector
, 2002
"... The present thesis deals with twodimensional signal processing for computer vision. The main topic is the development of a sophisticated generalization of the onedimensional analytic signal to two dimensions. Motivated by the fundamental property of the latter, the invariance – equivariance const ..."
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Cited by 65 (12 self)
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The present thesis deals with twodimensional signal processing for computer vision. The main topic is the development of a sophisticated generalization of the onedimensional analytic signal to two dimensions. Motivated by the fundamental property of the latter, the invariance – equivariance constraint, and by its relation to complex analysis and potential theory, a twodimensional approach is derived. This method is called the monogenic signal and it is based on the Riesz transform instead of the Hilbert transform. By means of this linear approach it is possible to estimate the local orientation and the local phase of signals which are projections of onedimensional functions to two dimensions. For general twodimensional signals, however, the monogenic signal has to be further extended, yielding the structure multivector. The latter approach combines the ideas of the structure tensor and the quaternionic analytic signal. A rich feature set can be extracted from the structure multivector, which contains measures for local amplitudes, the local anisotropy, the local orientation, and two local phases. Both, the monogenic signal and the struc
Symmetrical Dense Optical Flow Estimation with Occlusion Detection
 In ECCV
, 2002
"... Traditional techniques of dense optical flowestimation don't generally yield symmetrical solutions: the results will di#er if they are applied between images I1 and I2 or between images I2 and I1 . In this work, we present a method to recover a dense optical flow field map from two images, whil ..."
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Cited by 62 (1 self)
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Traditional techniques of dense optical flowestimation don't generally yield symmetrical solutions: the results will di#er if they are applied between images I1 and I2 or between images I2 and I1 . In this work, we present a method to recover a dense optical flow field map from two images, while explicitely taking into account the symmetry across the images as well as possible occlusions and discontinuities in the flow field. The idea is to consider both displacements vectors from I1 to I2 and I2 to I1 and to minimise an energy functional that explicitely encodes all those properties. This variational problem is then solved using the gradient flowdefined by theEulerLw7458) equations associated to the energy. In order to reduce the risk to be trapped within some irrelevant minimum, a focusing strategy based on a multiresolution technique is used to converge toward the solution. Promising experimental results on both synthetic and real images are presented to illustrate the capabilities of this symmetrical variational approach to recover accurate optical flow. 1
Multiresolution signal decomposition schemes. Part 1: Linear and morphological pyramids
 IEEE TRANSACTIONS ON IMAGE PROCESSING
, 2000
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