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43
An iterative regularization method for total variationbased image restoration
 MULTISCALE MODEL. SIMUL.
, 2005
"... We introduce a new iterative regularization procedure for inverse problems based on the use of Bregman distances, with particular focus on problems arising in image processing. We are motivated by the problem of restoring noisy and blurry images via variational methods by using total variation regu ..."
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Cited by 195 (29 self)
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We introduce a new iterative regularization procedure for inverse problems based on the use of Bregman distances, with particular focus on problems arising in image processing. We are motivated by the problem of restoring noisy and blurry images via variational methods by using total variation regularization. We obtain rigorous convergence results and effective stopping criteria for the general procedure. The numerical results for denoising appear to give significant improvement over standard models, and preliminary results for deblurring/denoising are very encouraging.
O.: Regularizing flows for constrained matrixvalued images
 J. Math. Imaging Vision
, 2004
"... Abstract. 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 differentialgeometric framework to define PDEs acting on some manifol ..."
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Cited by 47 (8 self)
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Abstract. 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 differentialgeometric framework to define PDEs acting on some 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 structurepreserving integrators that respect intrinsically the constraints geometry. The efficiency and versatility of this approach are illustrated through the anisotropic smoothing of diffusion tensor volumes in medical imaging. Note: This is the draft
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.
Noise removal using smoothed normals and surface fitting
 IEEE T. Image Process
"... Abstract—In this work, we use partial differential equation techniques to remove noise from digital images. The removal is done in two steps. We first use a totalvariation filter to smooth the normal vectors of the level curves of a noise image. After this, we try to find a surface to fit the smoot ..."
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Cited by 39 (11 self)
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Abstract—In this work, we use partial differential equation techniques to remove noise from digital images. The removal is done in two steps. We first use a totalvariation filter to smooth the normal vectors of the level curves of a noise image. After this, we try to find a surface to fit the smoothed normal vectors. For each of these two stages, the problem is reduced to a nonlinear partial differential equation. Finite difference schemes are used to solve these equations. A broad range of numerical examples are given in the paper. Index Terms—Anisotropic diffusion, image denoising, nonlinear partial differential equations (PDEs), normal processing. I.
Channel smoothing: Efficient robust smoothing of lowlevel signal features
 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE
, 2006
"... In this paper, we present a new and efficient method to implement robust smoothing of lowlevel signal features: Bspline channel smoothing. This method consists of three steps: encoding of the signal features into channels, averaging of the channels, and decoding of the channels. We show that line ..."
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Cited by 36 (22 self)
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In this paper, we present a new and efficient method to implement robust smoothing of lowlevel signal features: Bspline channel smoothing. This method consists of three steps: encoding of the signal features into channels, averaging of the channels, and decoding of the channels. We show that linear smoothing of channels is equivalent to robust smoothing of the signal features if we make use of quadratic Bsplines to generate the channels. The linear decoding from Bspline channels allows the derivation of a robust error norm, which is very similar to Tukey’s biweight error norm. We compare channel smoothing with three other robust smoothing techniques: nonlinear diffusion, bilateral filtering, and meanshift filtering, both theoretically and on a 2D orientationdata smoothing task. Channel smoothing is found to be superior in four respects: It has a lower computational complexity, it is easy to implement, it chooses the global minimum error instead of the nearest local minimum, and it can also be used on nonlinear spaces, such as orientation space.
Solving variational problems and partial differential equations mapping into general target manifolds
 J. Comput. Phys
, 2004
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Variational models for image colorization via Chromaticity and Brightness decomposition
 IEEE TRANS. IMAGE PROC
, 2006
"... Colorization refers to an image processing task which recovers color of gray scale images when only small regions with color are given. We propose a couple of variational models using chromaticity color component to colorize black and white images. We first consider Total Variation minimizing (TV) ..."
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Cited by 19 (1 self)
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Colorization refers to an image processing task which recovers color of gray scale images when only small regions with color are given. We propose a couple of variational models using chromaticity color component to colorize black and white images. We first consider Total Variation minimizing (TV) colorization which is an extension from TV inpainting to color using chromaticity model. Secondly, we further modify our model to weighted harmonic maps for colorization. This model adds edge information from the brightness data, while it reconstructs smooth color values for each homogeneous region. We introduce penalized versions of the variational models, we analyze their convergence properties, and we present numerical results including extension to texture colorization.
A convergent and constraintpreserving finite element method for the pharmonic flow into spheres
 SIAM J. NUMER. ANAL
, 2007
"... An explicit fully discrete finite element method, which satisfies the nonconvex side constraint at every node, is developed for approximating the pharmonic flow for p 2 (1;1). Convergence of the method is established under certain conditions on the domain and mesh. Computational examples are presen ..."
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Cited by 13 (8 self)
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An explicit fully discrete finite element method, which satisfies the nonconvex side constraint at every node, is developed for approximating the pharmonic flow for p 2 (1;1). Convergence of the method is established under certain conditions on the domain and mesh. Computational examples are presented to demonstrate finitetime blowups and qualitative geometric changes of weak solutions of the pharmonic flow.
A TVStokes denoising algorithm
"... In this paper, we propose a twostep algorithm for denoising digital images with additive noise. Observing that the isophote directions of an image correspond to an incompressible velocity field, we impose the constraint of zero divergence on the tangential field. Combined with an energy minimizat ..."
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Cited by 8 (2 self)
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In this paper, we propose a twostep algorithm for denoising digital images with additive noise. Observing that the isophote directions of an image correspond to an incompressible velocity field, we impose the constraint of zero divergence on the tangential field. Combined with an energy minimization problem corresponding to the smoothing of tangential vectors, this constraint gives rise to a nonlinear Stokes equation where the nonlinearity is in the viscosity function. Once the isophote directions are found, an image is reconstructed that fits those directions by solving another nonlinear partial differential equation. In both steps, we use finite difference schemes to solve. We present several numerical examples to show the effectiveness of our approach.
Anisotropic smoothing using double orientations
, 2009
"... Abstract. To improve the quality of image restoration methods directional information has recently been involved in the restoration process. In this paper, we propose a two step procedure for denoising images that is particularly suited to recover sharp vertices and X junctions in the presence of h ..."
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Cited by 7 (0 self)
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Abstract. To improve the quality of image restoration methods directional information has recently been involved in the restoration process. In this paper, we propose a two step procedure for denoising images that is particularly suited to recover sharp vertices and X junctions in the presence of heavy noise. In the first step, we estimate the (smoothed) orientations of the image structures, where we find the double orientations at vertices and X junctions using a model of Aach et al. Based on shape preservation considerations this directional information is then applied to establish an energy functional which is minimized in the second step. We discuss the behavior of our new method in comparison with single direction approaches appearing, e.g., when using the classical structure tensor of Förstner and Gülch and demonstrate the very good performance of our method by numerical examples. 1