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Efficient and Reliable Schemes for Nonlinear Diffusion Filtering
 IEEE Transactions on Image Processing
, 1998
"... Nonlinear diffusion filtering is usually performed with explicit schemes. They are only stable for very small time steps, which leads to poor efficiency and limits their practical use. Based on a recent discrete nonlinear diffusion scalespace framework we present semiimplicit schemes which are sta ..."
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Cited by 167 (18 self)
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Nonlinear diffusion filtering is usually performed with explicit schemes. They are only stable for very small time steps, which leads to poor efficiency and limits their practical use. Based on a recent discrete nonlinear diffusion scalespace framework we present semiimplicit schemes which are stable for all time steps. These novel schemes use an additive operator splitting (AOS) which guarantees equal treatment of all coordinate axes. They can be implemented easily in arbitrary dimensions, have good rotational invariance and reveal a computational complexity and memory requirement which is linear in the number of pixels. Examples demonstrate that, under typical accuracy requirements, AOS schemes are at least ten times more efficient than the widelyused explicit schemes.
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 83 (2 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 79 (7 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...
Applications of Nonlinear Diffusion in Image Processing and Computer Vision
, 2001
"... Nonlinear diffusion processes can be found in many recent methods for image processing and computer vision. In this article, four applications are surveyed: nonlinear diffusion filtering, variational image regularization, optic flow estimation, and geodesic active contours. For each of these techniq ..."
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Cited by 24 (2 self)
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Nonlinear diffusion processes can be found in many recent methods for image processing and computer vision. In this article, four applications are surveyed: nonlinear diffusion filtering, variational image regularization, optic flow estimation, and geodesic active contours. For each of these techniques we explain the main ideas, discuss theoretical properties and present an appropriate numerical scheme. The numerical schemes are based on additive operator splittings (AOS). In contrast to
Efficient Image Segmentation Using Partial Differential Equations and Morphology
 Pattern Recognition
, 1998
"... The goal of this paper is to present segmentation algorithms which combine regularization by nonlinear partial differential equations (PDEs) with a watershed transformation with region merging. We develop efficient algorithms for two wellfounded PDE methods. They use an additive operator splitting ( ..."
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Cited by 19 (1 self)
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The goal of this paper is to present segmentation algorithms which combine regularization by nonlinear partial differential equations (PDEs) with a watershed transformation with region merging. We develop efficient algorithms for two wellfounded PDE methods. They use an additive operator splitting (AOS) leading to recursive and separable filters. Further speedup can be obtained by embedding AOS schemes into a pyramid framework. Examples are presented which demonstrate that the preprocessing by these PDE techniques eases and stabilizes the segmentation. The typical CPU time for segmenting a 256 2 image on a workstation is less than 2 seconds. Key Words: Nonlinear diffusion, additive operator splitting, Gaussian pyramid, watershed segmentation, region merging CR Subject Classification: I.4.6, I.4.3, I.4.4. 1 Introduction Segmentation is one of the bottlenecks of many image analysis and computer vision tasks ranging from medical image processing to robot navigation. Ideally it sho...
ScaleSpace Properties of Regularization Methods
 ScaleSpace Theories in Computer Vision. Second International Conference, ScaleSpace ’99, Corfu
"... . In this paper we show that regularization methods form a scalespace where the regularization parameter serves as scale. In analogy to nonlinear diffusion filtering we establish continuity with respect to scale, causality in terms of a maximumminimum principle, simplification properties by means ..."
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Cited by 11 (0 self)
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. In this paper we show that regularization methods form a scalespace where the regularization parameter serves as scale. In analogy to nonlinear diffusion filtering we establish continuity with respect to scale, causality in terms of a maximumminimum principle, simplification properties by means of Lyapunov functionals and convergence to a constant steadystate. We identify nonlinear regularization with a single implicit time step of a diffusion process. This implies that iterated regularization with small regularization parameters is a numerical realization of a diffusion filter. We present numerical experiments in two and three space dimensions illustrating the scalespace behaviour of regularization methods. 1 Introduction There has often been a fruitful interaction between linear scalespace techniques and regularization methods. Torre and Poggio [29] emphasized that differentiation is illposed in the sense of Hadamard, and applying suitable regularization strategies approxim...
Recursive Separable Schemes for Nonlinear Diffusion Filters
, 1997
"... Poor efficiency is a typical problem of nonlinear diffusion filtering, when the simple and popular explicit (Eulerforward) scheme is used: for stability reasons very small time step sizes are necessary. In order to overcome this shortcoming, a novel type of semiimplicit schemes is studied, socall ..."
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Cited by 11 (4 self)
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Poor efficiency is a typical problem of nonlinear diffusion filtering, when the simple and popular explicit (Eulerforward) scheme is used: for stability reasons very small time step sizes are necessary. In order to overcome this shortcoming, a novel type of semiimplicit schemes is studied, socalled additive operator splitting (AOS) methods. They share the advantages of explicit and (semi)implicit schemes by combining simplicity with absolute stability. They are reliable, since they satisfy recently established criteria for discrete nonlinear diffusion scalespaces. Their efficiency is due to the fact that they can be separated into onedimensional processes, for which a fast recursive algorithm with linear complexity is available. AOS schemes reveal good rotational invariance and they are symmetric with respect to all axes. Examples demonstrate that, under typical accuracy requirements, they are at least ten times more efficient than explicit schemes.
NonLinear ScaleSpaces Isomorphic to the Linear Case
 Journal of Mathematical Imaging and Vision
, 1999
"... An innite dimensional class of isomorphisms is considered, relating a particular class of nonlinear scalespaces to the wellestablished linear case. The nonlinearity pertains to an invertible mapping of greyvalues, which can be adapted so as to account for external knowledge. This is particularly ..."
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Cited by 5 (2 self)
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An innite dimensional class of isomorphisms is considered, relating a particular class of nonlinear scalespaces to the wellestablished linear case. The nonlinearity pertains to an invertible mapping of greyvalues, which can be adapted so as to account for external knowledge. This is particularly interesting for applications such as segmentation in medical imaging, whereby one is in possession of a model relating tissue types to image greyvalues. It is moreover of interest in dening a consistent scalespace representation of vectorvalued and multispectral images. Keywords: linear/nonlinear/morphological scalespace theory. 1 Introduction We consider an innite dimensional class of pseudolinear scalespace representations, in which the members are isomorphically related to the linear case by a transformation of greyvalues. Although the nonlinearity can be \transformed away" in theory, it may be prudent not to do so in a practical situation whereby one is in possession of a pri...
Robust multiscale deformable registration of 3D ultrasound images
 International Journal of Image and Graphics
, 2003
"... In this paper, we embed the minimization scheme of an automatic 3D nonrigid registration method in a multiscale framework. The initial model formulation was expressed as a robust multiresolution and multigrid minimization scheme. At the finest level of the multiresolution pyramid, we introduce a fo ..."
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Cited by 5 (0 self)
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In this paper, we embed the minimization scheme of an automatic 3D nonrigid registration method in a multiscale framework. The initial model formulation was expressed as a robust multiresolution and multigrid minimization scheme. At the finest level of the multiresolution pyramid, we introduce a focusing strategy from coarsetofine scales which leads to an improvement in the accuracy of the registration process. A focusing strategy has been tested for a linear and a nonlinear scalespace. Results on real 3D ultrasound images are discussed.