Results 1  10
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15
Robust Anisotropic Diffusion
, 1998
"... Relations between anisotropic diffusion and robust statistics are described in this paper. Specifically, we show that anisotropic diffusion can be seen as a robust estimation procedure that estimates a piecewise smooth image from a noisy input image. The "edgestopping" function in the anisotropic d ..."
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Cited by 281 (16 self)
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Relations between anisotropic diffusion and robust statistics are described in this paper. Specifically, we show that anisotropic diffusion can be seen as a robust estimation procedure that estimates a piecewise smooth image from a noisy input image. The "edgestopping" function in the anisotropic diffusion equation is closely related to the error norm and influence function in the robust estimation framework. This connection leads to a new "edgestopping" function based on Tukey's biweight robust estimator, that preserves sharper boundaries than previous formulations and improves the automatic stopping of the diffusion. The robust statistical interpretation also provides a means for detecting the boundaries (edges) between the piecewise smooth regions in an image that has been smoothed with anisotropic diffusion. Additionally, we derive a relationship between anisotropic diffusion and regularization with line processes. Adding constraints on the spatial organization of the ...
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 170 (19 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.
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 81 (8 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...
Image Segmentation and Edge Enhancement with Stabilized Inverse Diffusion Equations.
 IEEE Transactions on Image Processing
, 1999
"... We introduce a family of firstorder multidimensional ordinary differential equations (ODEs) with discontinuous righthand sides and demonstrate their applicability in image processing. An equation belonging to this family is an inverse diffusion everywhere except at local extrema, where some stabil ..."
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Cited by 32 (9 self)
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We introduce a family of firstorder multidimensional ordinary differential equations (ODEs) with discontinuous righthand sides and demonstrate their applicability in image processing. An equation belonging to this family is an inverse diffusion everywhere except at local extrema, where some stabilization is introduced. For this reason, we call these equations "stabilized inverse diffusion equations" ("SIDEs"). Existence and uniqueness of solutions, as well as stability, are proven for SIDEs. A SIDE in one spatial dimension may be interpreted as a limiting case of a semidiscretized PeronaMalik equation [14], [15]. In an experimental section, SIDEs are shown to suppress noise while sharpening edges present in the input signal. Their application to image segmentation is also demonstrated.
A semidiscrete nonlinear scalespace theory and its relation to the PeronaMalik paradox
 F. Solina (Ed.), Advances in computer vision
, 1997
"... We discuss a semidiscrete framework for nonlinear diffusion scalespaces, where the image is sampled on a finite grid and the scale parameter is continuous. This leads to a system of nonlinear ordinary differential equations. We investigate conditions under which one can guarantee wellposedness pro ..."
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Cited by 27 (4 self)
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We discuss a semidiscrete framework for nonlinear diffusion scalespaces, where the image is sampled on a finite grid and the scale parameter is continuous. This leads to a system of nonlinear ordinary differential equations. We investigate conditions under which one can guarantee wellposedness properties, an extremum principle, average grey level invariance, smoothing Lyapunov functionals, and convergence to a constant steadystate. These properties are in analogy to previously established results for the continuous setting. Interestingly, this semidiscrete framework helps to explain the socalled PeronaMalik paradox: The PeronaMalik equation is a forwardbackward diffusion equation which is widelyused in image processing since it combines intraregional smoothing with edge enhancement. Although its continuous formulation is regarded to be illposed, it turns out that a spatial discretization is sufficient to create a wellposed semidiscrete diffusion scalespace. We also pro...
Parallel Implementations of AOS Schemes: A Fast Way of Nonlinear Diffusion Filtering
 In Proc. 1997 IEEE International Conference on Image Processing
, 1997
"... In most cases nonlinear diffusion filtering is implemented by means of explicit finite difference schemes. These algorithms are not very efficient, since they are only stable for small time steps. We address this problem by presenting unconditionally stable semiimplicit schemes which are based on a ..."
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Cited by 16 (4 self)
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In most cases nonlinear diffusion filtering is implemented by means of explicit finite difference schemes. These algorithms are not very efficient, since they are only stable for small time steps. We address this problem by presenting unconditionally stable semiimplicit schemes which are based on an additive operator splitting (AOS). They are very efficient since they can be implemented by recursive filtering, and their separability allows a straightforward implementation in any dimension. We analyse their behaviour on a parallel computer and demonstrate that parallel AOS schemes on a modern sharedmemory multiprocessor system with 8 processors allow a speedup of two orders of magnitude in comparison to the widelyused explicit scheme on a single processor. c fl1997 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to server...
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.
Scale Space Analysis by Stabilized Inverse Diffusion Equations
 Lecture Notes In Computer Science: 1252, First International Conference on ScaleSpace Theory in Computer Vision
, 1997
"... . We introduce a family of firstorder multidimensional ordinary differential equations (ODEs) with discontinuous righthand sides and demonstrate their applicability in image processing. An equation belonging to this family is an inverse diffusion everywhere except at local extrema, where some sta ..."
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Cited by 6 (1 self)
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. We introduce a family of firstorder multidimensional ordinary differential equations (ODEs) with discontinuous righthand sides and demonstrate their applicability in image processing. An equation belonging to this family is an inverse diffusion everywhere except at local extrema, where some stabilization is introduced. For this reason, we call these equations "stabilized inverse diffusion equations" ("SIDEs"). A SIDE in one spatial dimension may be interpreted as a limiting case of a semidiscretized PeronaMalik equation [3, 4]. In an experimental section, SIDEs are shown to suppress noise while sharpening edges present in the input signal. Their application to image segmentation is demonstrated. 1 Introduction In this paper we introduce, analyze, and apply a new class of nonlinear image processing algorithms. These algorithms are motivated by the great recent interest in using evolutions specified by partial differential equations (PDE's) as image processing procedures for tas...
Why the PeronaMalik Filter Works
, 1997
"... Although the widelyused PeronaMalik filter is regarded as illposed, straightforward implementations are often surprisingly stable. We give an explanation for this effect by applying a discrete nonlinear scalespace framework: a spatial discretization on a fixed pixel grid gives a wellposed scal ..."
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Cited by 1 (0 self)
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Although the widelyused PeronaMalik filter is regarded as illposed, straightforward implementations are often surprisingly stable. We give an explanation for this effect by applying a discrete nonlinear scalespace framework: a spatial discretization on a fixed pixel grid gives a wellposed scalespace with many imagesimplifying properties, and an explicit time discretization leads to a scheme which does not introduce additional oscillations. This explains why staircasing is essentially the only practically appearing instability. Key words: Nonlinear diffusion, wellposedness, discretization, scalespace. 1 Introduction The nonlinear diffusion filter by Perona and Malik [1] was the starting point of a new area in image processing, where images are filtered by studying their evolutions under nonlinear partial differential equations (PDEs). Unfortunately, this starting point is usually regarded as illposed, because the filter is designed to behave like a backward diffusion acro...
Multiscale Segmentation With VectorValued Nonlinear Diffusions on Arbitrary Graphs
"... Abstract—We propose a novel family of nonlinear diffusion equations and apply it to the problem of segmentation of multivalued images. We show that this family can be viewed as an extension of stabilized inverse diffusion equations (SIDEs) which were proposed for restoration, enhancement, and segmen ..."
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Cited by 1 (0 self)
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Abstract—We propose a novel family of nonlinear diffusion equations and apply it to the problem of segmentation of multivalued images. We show that this family can be viewed as an extension of stabilized inverse diffusion equations (SIDEs) which were proposed for restoration, enhancement, and segmentation of scalarvalued signals and images in [39]. Our new diffusion equations can process vectorvalued images defined on arbitrary graphs which makes them well suited for segmentation. In addition, we introduce novel ways of utilizing the shape information during the diffusion process. We demonstrate the effectiveness of our methods on a large number of segmentation tasks. Index Terms—Nonlinear diffusions, scalespace, segmentation, stabilized inverse diffusion equations (SIDEs), texture.