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57
A general framework for low level vision
 IEEE TRANS. ON IMAGE PROCESSING
, 1998
"... 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 su ..."
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Cited by 216 (37 self)
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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.
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 111 (24 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.
ForwardandBackward Diffusion Processes for Adaptive Image Enhancement and Denoising
 IEEE Transactions on Image Processing
, 2002
"... Signal and image enhancement is considered in the context of a new type of diffusion process that simultaneously enhances, sharpens and denoises images. The nonlinear diffusion coefficient is locally adjusted according to image features such as edges, textures and moments. As such it can switch the ..."
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Cited by 56 (6 self)
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Signal and image enhancement is considered in the context of a new type of diffusion process that simultaneously enhances, sharpens and denoises images. The nonlinear diffusion coefficient is locally adjusted according to image features such as edges, textures and moments. As such it can switch the diffusion process from a forward to a backward (inverse) mode according to a given set of criteria. This results in a forwardandbackward (FAB) adap tive diffusion process that enhances features while locally denoising smoother segments of the signal or image. The proposed method, using the FAB process, is applied in a superresolution scheme.
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.
Diffusion tensor regularization with constraints preservation
 IN IEEE COMPUTER SOCIETY CONFERENCE ON COMPUTER VISION AND PATTERN RECOGNITION, KAUAI MARRIOTT
"... This paper deals with the problem of regularizing noisy fields of diffusion tensors, considered as symmetric and semipositive definite n n matrices (as for instance 2D structure tensors or DTMRI medical images). We first propose a simple anisotropic PDEbased scheme that acts directly on the matr ..."
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Cited by 43 (12 self)
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This paper deals with the problem of regularizing noisy fields of diffusion tensors, considered as symmetric and semipositive definite n n matrices (as for instance 2D structure tensors or DTMRI medical images). We first propose a simple anisotropic PDEbased 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 DTMRI data sets finally illustrates the proposed tensor regularization framework.
Orientation Diffusion or How to comb a Porcupine
 Journal of Visual Communication and Image Representation
, 2001
"... This paper addresses the problem of feature enhancement in noisy images, when the feature is known to be constrained to a manifold. As an example, we approach the orientation denoising problem via the geometric Beltrami framework for image processing. The feature (orientation) field is represented a ..."
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Cited by 37 (6 self)
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This paper addresses the problem of feature enhancement in noisy images, when the feature is known to be constrained to a manifold. As an example, we approach the orientation denoising problem via the geometric Beltrami framework for image processing. The feature (orientation) field is represented accordingly as the embedding of a two dimensional surface in the spatialfeature manifold. The resulted Beltrami flow is a selective smoothing process that respects the feature constraint. Orientation diffusion is treated as a canonical example where the feature (orientation in this case) space is the unit circle S1. Applications to color analysis are discussed and numerical experiments demonstrate again the power of the Beltrami framework for nontrivial geometries in image processing. C ○ 2002 Elsevier Science (USA) 1.
Modified curvature motion for image smoothing and enhancement
 IEEE Trans. Image Processing
, 1998
"... Abstract—In this paper, we formulate a general modified mean curvature based equation for image smoothing and enhancement. The key idea is to consider the image as a graph in some Rn, and apply a mean curvature type motion to the graph. We will consider some special cases relevant to greyscale and ..."
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Cited by 33 (2 self)
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Abstract—In this paper, we formulate a general modified mean curvature based equation for image smoothing and enhancement. The key idea is to consider the image as a graph in some Rn, and apply a mean curvature type motion to the graph. We will consider some special cases relevant to greyscale and color images. Index Terms—Enhancement, smoothing, mean curvature, partial differential equations. I.
Regularization of Orthonormal Vector Sets using Coupled PDE's
 PROCEEDINGS 1ST IEEE WORKSHOP ON VARIATIONAL AND LEVEL SET METHODS IN COMPUTER VISION
, 2001
"... 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 nD orthogon ..."
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Cited by 18 (6 self)
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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 nD 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.
Modular Solvers for Constrained Image Restoration Problems
, 1997
"... Many problems in image restoration cam be formulated as either an unconstrained nonlinear optimization problem: min u R(u) + 2 kKu \Gamma zk 2 which is the Tikhonov [1] approach, where the regularization parameter is to be determined; or as a noise constrained problem: min u R(u); subjec ..."
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Cited by 16 (9 self)
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Many problems in image restoration cam be formulated as either an unconstrained nonlinear optimization problem: min u R(u) + 2 kKu \Gamma zk 2 which is the Tikhonov [1] approach, where the regularization parameter is to be determined; or as a noise constrained problem: min u R(u); subject to 1 2 kKu \Gamma zk 2 = 1 2 j\Omega joe 2 ; where oe is the (estimated) variance of the the noise. In both formulations z is the measured (noisy, and blurred) image, and K a convolution operator. In practice, it is much easier to develop algorithms for the unconstrained problem, and not always obvious how to adapt such methods to solve the corresponding constrained problem. In this paper, we present a new method which can make use of any existing convergent method for the unconstrained problem to solve the constrained one. The new method is based on a Newton iteration applied to an extended system of nonlinear equations, which couples the constraint and the regularized problem. Th...
GeometricVariational Approach for Color Image Enhancement and Segmentation
 of Lecture Notes in Comp. Sci
, 1999
"... . We merge techniques developed in the Beltrami framework to deal with multichannel, i.e. color images, and the MumfordShah functional for segmentation. The result is a color image enhancement and segmentation algorithm. The generalization of the MumfordShah idea includes a higher dimension a ..."
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Cited by 12 (5 self)
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. We merge techniques developed in the Beltrami framework to deal with multichannel, i.e. color images, and the MumfordShah functional for segmentation. The result is a color image enhancement and segmentation algorithm. The generalization of the MumfordShah idea includes a higher dimension and codimension and a novel smoothing measure for the color components and for the segmenting function which is introduced via the \Gamma convergence approach. We use the \Gamma convergence technique to derive, through the gradient descent method, a system of coupled PDEs for the color coordinates and for the segmenting function. 1 Introduction Segmentation is one of the important tasks of image analysis and much efforts have been consecrated to solve it. One can roughly classify the segmentation methods into two classes: 1) Global, i.e. histogram based techniques, and 2) Local, i.e. edge based techniques. In the second class it was shown that a large number of algorithms, including d...