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80
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.
A Review of Nonlinear Diffusion Filtering
, 1997
"... . This paper gives an overview of scalespace and image enhancement techniques which are based on parabolic partial differential equations in divergence form. In the nonlinear setting this filter class allows to integrate apriori knowledge into the evolution. We sketch basic ideas behind the differ ..."
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Cited by 100 (10 self)
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. This paper gives an overview of scalespace and image enhancement techniques which are based on parabolic partial differential equations in divergence form. In the nonlinear setting this filter class allows to integrate apriori knowledge into the evolution. We sketch basic ideas behind the different filter models, discuss their theoretical foundations and scalespace properties, discrete aspects, suitable algorithms, generalizations, and applications. 1 Introduction During the last decade nonlinear diffusion filters have become a powerful and wellfounded tool in multiscale image analysis. These models allow to include apriori knowledge into the scalespace evolution, and they lead to an image simplification which simultaneously preserves or even enhances semantically important information such as edges, lines, or flowlike structures. Many papers have appeared proposing different models, investigating their theoretical foundations, and describing interesting applications. For a n...
A Theoretical Framework for Convex Regularizers in PDEBased Computation of Image Motion
, 2000
"... Many differential methods for the recovery of the optic flow field from an image sequence can be expressed in terms of a variational problem where the optic flow minimizes some energy. Typically, these energy functionals consist of two terms: a data term, which requires e.g. that a brightness consta ..."
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Cited by 99 (25 self)
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Many differential methods for the recovery of the optic flow field from an image sequence can be expressed in terms of a variational problem where the optic flow minimizes some energy. Typically, these energy functionals consist of two terms: a data term, which requires e.g. that a brightness constancy assumption holds, and a regularizer that encourages global or piecewise smoothness of the flow field. In this paper we present a systematic classification of rotation invariant convex regularizers by exploring their connection to diffusion filters for multichannel images. This taxonomy provides a unifying framework for datadriven and flowdriven, isotropic and anisotropic, as well as spatial and spatiotemporal regularizers. While some of these techniques are classic methods from the literature, others are derived here for the first time. We prove that all these methods are wellposed: they posses a unique solution that depends in a continuous way on the initial data. An interesting structural relation between isotropic and anisotropic flowdriven regularizers is identified, and a design criterion is proposed for constructing anisotropic flowdriven regularizers in a simple and direct way from isotropic ones. Its use is illustrated by several examples.
Anisotropic Diffusion of Surfaces and Functions on Surfaces
, 2003
"... We present a unified anisotropic geometric diffusion PDE model for smoothing (fairing) out noise both in triangulated twomanifold surface meshes in R³ and functions defined on these surface meshes, while enhancing curve features on both by careful choice of an anisotropic diffusion tensor. We combin ..."
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Cited by 78 (7 self)
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We present a unified anisotropic geometric diffusion PDE model for smoothing (fairing) out noise both in triangulated twomanifold surface meshes in R³ and functions defined on these surface meshes, while enhancing curve features on both by careful choice of an anisotropic diffusion tensor. We combine the C¹ 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 66 (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.
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 55 (5 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.
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 38 (5 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.
Selection of Optimal Stopping Time for Nonlinear Diffusion Filtering
, 2001
"... We develop a novel timeselection strategy for iterative image restoration techniques: the stopping time is chosen so that the correlation of signal and noise in the filtered image is minimised. The new method is applicable to any images where the noise to be removed is uncorrelated with the signal; ..."
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Cited by 36 (2 self)
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We develop a novel timeselection strategy for iterative image restoration techniques: the stopping time is chosen so that the correlation of signal and noise in the filtered image is minimised. The new method is applicable to any images where the noise to be removed is uncorrelated with the signal; no other knowledge (e.g. the noise variance, training data etc.) is needed. We test the performance of our time estimation procedure experimentally, and demonstrate that it yields nearoptimal results for a wide range of noise levels and for various filtering methods.
B.: Levelset methods for tensorvalued images
 Proc. Second IEEE Workshop on Geometric and Level Set Methods in Computer Vision
, 2003
"... Tensorvalued data are becoming more and more important as input for todays image analysis problems. This has been caused by a number of applications including diffusion tensor (DT) MRI and physical measurements of anisotropic behaviour such as stressstrain relationships, interia and permittivi ..."
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Cited by 32 (5 self)
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Tensorvalued data are becoming more and more important as input for todays image analysis problems. This has been caused by a number of applications including diffusion tensor (DT) MRI and physical measurements of anisotropic behaviour such as stressstrain relationships, interia and permittivity tensors. Consequently, there arises the need to filter and segment such tensor fields. In this paper we extend three important level set methods to tensorvalued data. To this end we first generalise Di Zenzo’s concept of a structure tensor for vectorvalued images to tensorvalued data. This allows us to derive formulations of mean curvature motion and selfsnakes in the case of tensorvalued images. We prove that these processes maintain positive semidefiniteness if the initial matrix data are positive semidefinite. Finally we present a geodesic active contour model for segmenting tensor fields. Since it incorporates information from all channels, it gives a contour representation that is highly robust under noise. 1
Anisotropic Diffusion of Subdivision Surfaces and Functions on Surfaces
 ACM TRANSACTIONS ON GRAPHICS
, 2002
"... We present a unified anisotropic geometric diffusion PDE model for smoothing (fairing) out noise both in triangulated 2manifold surface meshes in R³ 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 29 (7 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 R³ and functions defined on these surface meshes, while enhancing curve features on both by careful choice of an anisotropic diffusion tensor. We combine the C¹ 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