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158
Diffusion of General Data on NonFlat Manifolds via Harmonic Maps Theory: The Direction Diffusion Case
 Journal Computer Vision
, 2000
"... Abstract. In a number of disciplines, directional data provides a fundamental source of information. A novel framework for isotropic and anisotropic diffusion of directions is presented in this paper. The framework can be applied both to denoise directional data and to obtain multiscale representati ..."
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Cited by 65 (6 self)
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Abstract. In a number of disciplines, directional data provides a fundamental source of information. A novel framework for isotropic and anisotropic diffusion of directions is presented in this paper. The framework can be applied both to denoise directional data and to obtain multiscale representations of it. The basic idea is to apply and extend results from the theory of harmonic maps, and in particular, harmonic maps in liquid crystals. This theory deals with the regularization of vectorial data, while satisfying the intrinsic unit norm constraint of directional data. We show the corresponding variational and partial differential equations formulations for isotropic diffusion, obtained from an L2 norm, and edge preserving diffusion, obtained from an L p norm in general and an L1 norm in particular. In contrast with previous approaches, the framework is valid for directions in any dimensions, supports nonsmooth data, and gives both isotropic and anisotropic formulations. In addition, the framework of harmonic maps here described can be used to diffuse and analyze general image data defined on general nonflat manifolds, that is, functions between two general manifolds. We present a number of theoretical results, open questions, and examples for gradient vectors, optical flow, and color images.
Compressed sensing with quantized measurements
 IEEE Signal Proc. Lett
"... Abstract We consider the problem of estimating a sparse signal from a set of quantized, Gaussian noise corrupted measurements, where each measurement corresponds to an interval of values. We give two methods for (approximately) solving this problem, each based on minimizing a differentiable convex ..."
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Cited by 61 (0 self)
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Abstract We consider the problem of estimating a sparse signal from a set of quantized, Gaussian noise corrupted measurements, where each measurement corresponds to an interval of values. We give two methods for (approximately) solving this problem, each based on minimizing a differentiable convex function plus an ℓ 1 regularization term. Using a first order method developed by Yin et al, we demonstrate the performance of the methods through numerical simulation. We find that, using these methods, compressed sensing can be carried out even when the quantization is very coarse, e.g., 1 or 2 bits per measurement.
Color Image Enhancement via Chromaticity Diffusion
 IEEE Transactions on Image Processing
, 2002
"... A novel approach for color image denoising is proposed in this paper. The algorithm is based on separating the color data into chromaticity and brightness, and then processing each one of these components with partial differential equations or diffusion flows. In the proposed algorithm, each color p ..."
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Cited by 58 (3 self)
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A novel approach for color image denoising is proposed in this paper. The algorithm is based on separating the color data into chromaticity and brightness, and then processing each one of these components with partial differential equations or diffusion flows. In the proposed algorithm, each color pixel is considered as andimensional vector. The vectors' direction, a unit vector, gives the chromaticity, while the magnitude represents the pixel brightness. The chromaticity is processed with a system of coupled diffusion equations adapted from the theory of harmonic maps in liquid crystals. This theory deals with the regularization of vectorial data, while satisfying the intrinsic unit norm constraint of directional data such as chromaticity. Both isotropic and anisotropic diffusion flows are presented for thisdimensional chromaticity diffusion flow. The brightness is processed by a scalar median filter or any of the popular and well established anisotropic diffusion flows for scalar image enhancement. We present the underlying theory, a number of examples, and briefly compare with the current literature. Index TermsBrightness, chromaticity, color image denoising, directions, harmonic maps, isotropic and anisotropic diffusion, liquid crystals, partial differential equations. I.
From high energy physics to low level vision
 Presented in ONR workshop, UCLA
, 1996
"... Abstract. A geometric framework for image scale space, enhancement, and segmentation is presented. We consider intensity images as surfaces in the (x � I) space. The image is thereby a 2D surface in 3D space for gray level images, and a 2D surface in 5D for color images. The new formulation uni es m ..."
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Cited by 57 (23 self)
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Abstract. A geometric framework for image scale space, enhancement, and segmentation is presented. We consider intensity images as surfaces in the (x � I) space. The image is thereby a 2D surface in 3D space for gray level images, and a 2D surface in 5D for color images. The new formulation uni es many classical schemes and algorithms via a simple scaling of the intensity contrast, and results in new and e cient schemes. Extensions to multi dimensional signals become natural and lead to powerful denoising and scale space algorithms. Here, we demonstrate the proposed framework by applying it to denoise and improve graylevel and color images. 1 Introduction: A philosophical point of view In this paper we adopt an action potential that was recently introduced in physics and use it to produce a natural scale space for images as surfaces. It will lead us to the construction of image enhancement procedures for gray and color images. This model also integrates many existing segmentation and scale space procedures
The Perceptual Organization of Texture Flows: A Contextual Inference Approach
 IEEE Trans. Pattern Analysis and Machine Intelligence
, 2003
"... Abstract—Locally parallel dense patterns—sometimes called texture flows—define a perceptually coherent structure of particular significance to perceptual organization. We argue that with applications ranging from image segmentation and edge classification to shading analysis and shape interpretation ..."
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Cited by 54 (23 self)
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Abstract—Locally parallel dense patterns—sometimes called texture flows—define a perceptually coherent structure of particular significance to perceptual organization. We argue that with applications ranging from image segmentation and edge classification to shading analysis and shape interpretation, texture flows deserve attention equal to edge segment grouping and curve completion. This paper develops the notion of texture flow from a geometrical point of view to argue that local measurements of such structures must incorporate two curvatures. We show how basic theoretical considerations lead to a unique model for the local behavior of the flow and to a notion of texture flow “good continuation. ” This, in turn, translates to a specification of consistency constraints between nearby flow measurements which we use for the computation of globally (piecewise) coherent structure through the contextual framework of relaxation labeling. We demonstrate the results on synthetic and natural images. Index Terms—Texture flow, perceptual organization, social conformity of a line, good continuation, texture segmentation, line discontinuities, point singularities, shading flow, local parallelism, orientation diffusion, tangential curvature, normal curvature, relaxation labeling. æ
Diffusion and Regularization of Vector and MatrixValued Images
, 2002
"... The goal of this paper is to present a unified description of diffusion and regularization techniques for vectorvalued as well as matrixvalued data fields. In the vectorvalued setting, we first review a number of existing methods and classify them into linear and nonlinear as well as isotropic an ..."
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Cited by 53 (16 self)
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The goal of this paper is to present a unified description of diffusion and regularization techniques for vectorvalued as well as matrixvalued data fields. In the vectorvalued setting, we first review a number of existing methods and classify them into linear and nonlinear as well as isotropic and anisotropic methods. For these approaches we present corresponding regularization methods. This taxonomy is applied to the design of regularization methods for variational motion analysis in image sequences. Our vectorvalued framework is then extended to the smoothing of positive semidefinite matrix fields. In this context a novel class of anisotropic di usion and regularization methods is derived and it is shown that suitable algorithmic realizations preserve the positive semidefiniteness of the matrix field without any additional constraints. As an application, we present an anisotropic nonlinear structure tensor and illustrate its advantages over the linear structure tensor.
FAST DUAL MINIMIZATION OF THE VECTORIAL TOTAL VARIATION NORM AND APPLICATIONS TO COLOR IMAGE PROCESSING
, 2008
"... Abstract. We propose a regularization algorithm for color/vectorial images which is fast, easy to code and mathematically wellposed. More precisely, the regularization model is based on the dual formulation of the vectorial Total Variation (VTV) norm and it may be regarded as the vectorial extensio ..."
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Cited by 52 (2 self)
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Abstract. We propose a regularization algorithm for color/vectorial images which is fast, easy to code and mathematically wellposed. More precisely, the regularization model is based on the dual formulation of the vectorial Total Variation (VTV) norm and it may be regarded as the vectorial extension of the dual approach defined by Chambolle in [13] for grayscale/scalar images. The proposed model offers several advantages. First, it minimizes the exact VTV norm whereas standard approaches use a regularized norm. Then, the numerical scheme of minimization is straightforward to implement and finally, the number of iterations to reach the solution is low, which gives a fast regularization algorithm. Finally, and maybe more importantly, the proposed VTV minimization scheme can be easily extended to many standard applications. We apply this L 1 vectorial regularization algorithm to the following problems: color inverse scale space, color denoising with the chromaticitybrightness color representation, color image inpainting, color wavelet shrinkage, color image decomposition, color image deblurring, and color denoising on manifolds. Generally speaking, this VTV minimization scheme can be used in problems that required vector field (color, other feature vector) regularization while preserving discontinuities. Keywords: Vectorvalued TV norm, dual formulation, BV space, image denoising, ROF model, inverse scale space, chromaticitybrightness color representation, image decomposition, image inpainting, image deblurring, wavelet shrinkage, denoising on manifold. 1.
Augmented Lagrangian method, dual methods, and split Bregman iteration for ROF, vectorial TV, and high order models
 SIAM Journal on Imaging Sciences
"... E. Fatemi, Physica D, 60(1992), pp. 259–268] based on total variation (TV) minimization has proven to be very useful. A lot of efforts have been devoted to obtain fast numerical schemes and overcome the nondifferentiability of the model. Methods considered to be particularly efficient for the ROF m ..."
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Cited by 51 (10 self)
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E. Fatemi, Physica D, 60(1992), pp. 259–268] based on total variation (TV) minimization has proven to be very useful. A lot of efforts have been devoted to obtain fast numerical schemes and overcome the nondifferentiability of the model. Methods considered to be particularly efficient for the ROF model include the dual methods of ChanGolubMulet (CGM) [T.F. Chan, G.H. Golub, and P. Mulet, SIAM J. Sci. Comput., 20(1999), pp. 1964–1977] and Chambolle [A. Chambolle, J. Math. Imaging
An efficient TVL1 algorithm for deblurring multichannel images corrupted by impulsive noise
 SIAM J. SCI. COMPUT
, 2009
"... We extend the alternating minimization algorithm recently proposed in [38, 39] to the case of recovering blurry multichannel (color) images corrupted by impulsive rather than Gaussian noise. The algorithm minimizes the sum of a multichannel extension of total variation (TV), either isotropic or anis ..."
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Cited by 50 (8 self)
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We extend the alternating minimization algorithm recently proposed in [38, 39] to the case of recovering blurry multichannel (color) images corrupted by impulsive rather than Gaussian noise. The algorithm minimizes the sum of a multichannel extension of total variation (TV), either isotropic or anisotropic, and a data fidelity term measured in the L1norm. We derive the algorithm by applying the wellknown quadratic penalty function technique and prove attractive convergence properties including finite convergence for some variables and global qlinear convergence. Under periodic boundary conditions, the main computational requirements of the algorithm are fast Fourier transforms and a lowcomplexity Gaussian elimination procedure. Numerical results on images with different blurs and impulsive noise are presented to demonstrate the efficiency of the algorithm. In addition, it is numerically compared to an algorithm recently proposed in [20] that uses a linear program and an interior point method for recovering grayscale images.
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...