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156
A geometrical framework for low level vision
 IEEE Trans. on Image Processing
, 1998
"... Abstract—We introduce a new geometrical framework based on which natural flows for image scale space and enhancement are presented. We consider intensity images as surfaces in the space. The image is, thereby, a twodimensional (2D) surface in threedimensional (3D) space for graylevel images, an ..."
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Cited by 223 (35 self)
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Abstract—We introduce a new geometrical framework based on which natural flows for image scale space and enhancement are presented. We consider intensity images as surfaces in the space. The image is, thereby, a twodimensional (2D) surface in threedimensional (3D) space for graylevel images, and 2D surfaces in five dimensions for color images. The new formulation unifies many classical schemes and algorithms via a simple scaling of the intensity contrast, and results in new and efficient schemes. Extensions to multidimensional signals become natural and lead to powerful denoising and scale space algorithms. Index Terms — Color image processing, image enhancement, image smoothing, nonlinear image diffusion, scalespace. I.
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 179 (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.
Analysis versus synthesis in signal priors
, 2005
"... The concept of prior probability for signals plays a key role in the successful solution of many inverse problems. Much of the literature on this topic can be divided between analysisbased and synthesisbased priors. Analysisbased priors assign probability to a signal through various forward measu ..."
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Cited by 147 (17 self)
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The concept of prior probability for signals plays a key role in the successful solution of many inverse problems. Much of the literature on this topic can be divided between analysisbased and synthesisbased priors. Analysisbased priors assign probability to a signal through various forward measurements of it, while synthesisbased priors seek a reconstruction of the signal as a combination of atom signals. In this paper we describe these two prior classes, focusing on the distinction between them. We show that although when reducing to the complete and undercomplete formulations the two become equivalent, in their more interesting overcomplete formulation the two types depart. Focusing on the ℓ1 denoising case, we present several ways of comparing the two types of priors, establishing the existence of an unbridgeable gap between them. 1.
Enhancing Sparsity by Reweighted ℓ1 Minimization
, 2007
"... It is now well understood that (1) it is possible to reconstruct sparse signals exactly from what appear to be highly incomplete sets of linear measurements and (2) that this can be done by constrained ℓ1 minimization. In this paper, we study a novel method for sparse signal recovery that in many si ..."
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Cited by 146 (5 self)
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It is now well understood that (1) it is possible to reconstruct sparse signals exactly from what appear to be highly incomplete sets of linear measurements and (2) that this can be done by constrained ℓ1 minimization. In this paper, we study a novel method for sparse signal recovery that in many situations outperforms ℓ1 minimization in the sense that substantially fewer measurements are needed for exact recovery. The algorithm consists of solving a sequence of weighted ℓ1minimization problems where the weights used for the next iteration are computed from the value of the current solution. We present a series of experiments demonstrating the remarkable performance and broad applicability of this algorithm in the areas of sparse signal recovery, statistical estimation, error correction and image processing. Interestingly, superior gains are also achieved when our method is applied to recover signals with assumed nearsparsity in overcomplete representations—not by reweighting the ℓ1 norm of the coefficient sequence as is common, but by reweighting the ℓ1 norm of the transformed object. An immediate consequence is the possibility of highly efficient data acquisition protocols by improving on a technique known as compressed sensing.
Images as embedding maps and minimal surfaces: Movies, color, texture, and volumetric medical images
 INT. J. COMPUT. VIS
, 2000
"... We extend the geometric framework introduced in Sochen et al. (IEEE Trans. on Image Processing, 7(3):310–318, 1998) for image enhancement. We analyze and propose enhancement techniques that selectively smooth images while preserving either the multichannel edges or the orientationdependent textu ..."
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Cited by 113 (23 self)
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We extend the geometric framework introduced in Sochen et al. (IEEE Trans. on Image Processing, 7(3):310–318, 1998) for image enhancement. We analyze and propose enhancement techniques that selectively smooth images while preserving either the multichannel edges or the orientationdependent texture features in them. Images are treated as manifolds in a featurespace. This geometrical interpretation lead to a general way for grey level, color, movies, volumetric medical data, and colortexture image enhancement. We first review our framework in which the Polyakov action from highenergy physics is used to develop a minimization procedure through a geometric flow for images. Here we show that the geometric flow, based on manifold volume minimization, yields a novel enhancement procedure for color images. We apply the geometric framework and the general Beltrami flow to featurepreserving denoising of images in various spaces. Next, we introduce a new method for color and texture enhancement. Motivated by Gabor’s geometric image sharpening method (Gabor, Laboratory Investigation, 14(6):801–807, 1965), we present a geometric sharpening procedure for color images with texture. It is based on inverse diffusion across the multichannel edge, and diffusion along the edge.
Active Contours without Edges for VectorValued Images
 Journal of Visual Communication and Image Representation
, 2000
"... this paper, we propose an active contour algorithm for object detection in vectorvalued images (such as RGB or multispectral). The model is an extension of the scalar ChanVese algorithm to the vectorvalued case [1]. The model minimizes a MumfordShah functional over the length of the contour, ..."
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Cited by 106 (12 self)
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this paper, we propose an active contour algorithm for object detection in vectorvalued images (such as RGB or multispectral). The model is an extension of the scalar ChanVese algorithm to the vectorvalued case [1]. The model minimizes a MumfordShah functional over the length of the contour, plus the sum of the fitting error over each component of the vectorvalued image. Like the ChanVese model, our vectorvalued model can detect edges both with or without gradient. We show examples where our model detects vectorvalued objects which are undetectable in any scalar representation. For instance, objects with different missing parts in different channels are completely detected (such as occlusion). Also, in color images, objects which are invisible in each channel or in intensity can be detected by our algorithm. Finally, the model is robust with respect to noise, requiring no a priori denoising step. C 2000 Academic Press Key Words: vectorvalued images; active contours; level sets; segmentation; PDEs; object detection
Edgepreserving and Scaledependent Properties of Total Variation Regularization
 Inverse Problems
, 2000
"... We give and prove two new and fundamental properties of total variation minimizing function regularization (TV Regularization): 1) edge locations of function (e.g. image) features tend to be preserved, and under certain conditions, are preserved exactly ; 2) intensity change experienced by individua ..."
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Cited by 100 (5 self)
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We give and prove two new and fundamental properties of total variation minimizing function regularization (TV Regularization): 1) edge locations of function (e.g. image) features tend to be preserved, and under certain conditions, are preserved exactly ; 2) intensity change experienced by individual features is inversely proportional to the scale of each feature. More generally, we describe both qualitatively and quantitatively the exact eects of TV Regularization in R 1 , R 2 and R 3 . We give and prove exact analytic solutions to the nonlinear TV Regularization problem for simple but important cases, which can be used to better understand the eects of TV Regularization for more general cases. The formulae we give describe the eect of TV Regularization when applied to noisecontaminated radially symmetric image features. These formulae also accurately predict the eects of TV Regularization when it is applied to more general functions. Our results help explain how and why TV...
Robust SuperResolution
 in In Proc. of the IEEE Workshop on Applications of Computer Vision
, 2001
"... A robust approach for super resolution is presented, which is especially valuable in the presence of outliers. Such outliers may be due to motion erros, inaccurate blur models, noise, moving objects, motion blur etc. This tobusiness is needed since superresolution methods are very sensitive to such ..."
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Cited by 92 (0 self)
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A robust approach for super resolution is presented, which is especially valuable in the presence of outliers. Such outliers may be due to motion erros, inaccurate blur models, noise, moving objects, motion blur etc. This tobusiness is needed since superresolution methods are very sensitive to such errors.
Efficient schemes for total variation minimization under constraints in image processing
, 2007
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Recent developments in total variation image restoration
 In Mathematical Models of Computer Vision
, 2005
"... ABSTRACT Since their introduction in a classic paper by Rudin, Osher and Fatemi [26], total variation minimizing models have become one of the most popular and successful methodology for image restoration. More recently, there has been a resurgence of interest and exciting new developments, some ext ..."
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Cited by 71 (2 self)
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ABSTRACT Since their introduction in a classic paper by Rudin, Osher and Fatemi [26], total variation minimizing models have become one of the most popular and successful methodology for image restoration. More recently, there has been a resurgence of interest and exciting new developments, some extending the applicabilities to inpainting, blind deconvolution and vectorvalued images, while others offer improvements in better preservation of contrast, geometry and textures, in ameliorating the staircasing effect, and in exploiting the multiscale nature of the models. In addition, new computational methods have been proposed with improved computational speed and robustness. We shall review some of these recent developments. 1