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17
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.
Orientation Diffusions
 IEEE Trans. Image Processing
, 1998
"... Abstract—Diffusions are useful for image processing and computer vision because they provide a convenient way of smoothing noisy data, analyzing images at multiple scales, and enhancing discontinuities. A number of diffusions of image brightness have been defined and studied so far; they may be appl ..."
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Cited by 131 (0 self)
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Abstract—Diffusions are useful for image processing and computer vision because they provide a convenient way of smoothing noisy data, analyzing images at multiple scales, and enhancing discontinuities. A number of diffusions of image brightness have been defined and studied so far; they may be applied to scalar and vectorvalued quantities that are naturally associated with intervals of either the real line, or other flat manifolds. Some quantities of interest in computer vision, and other areas of engineering that deal with images, are defined on curved manifolds; typical examples are orientation and hue that are defined on the circle. Generalizing brightness diffusions to orientation is not straightforward, especially in the case where a discrete implementation is sought. An example of what may go wrong is presented. A method is proposed to define diffusions of orientationlike quantities. First a definition in the continuum is discussed, then a discrete orientation diffusion is proposed. The behavior of such diffusions is explored both analytically and experimentally. It is shown how such orientation diffusions contain a nonlinearity that is reminiscent of edgeprocess and anisotropic diffusion. A number of open questions are proposed at the end. Index Terms—Orientation analysis, texture analysis, diffusions, scalespace.
Fast Multiscale Image Segmentation
"... We introduce a fast, multiscale algorithm for image segmentation. Our algorithm uses modern numeric techniques to nd an approximate solution to normalized cut measures in time that is linear in the size of the image with only a few dozen operations per pixel. In just one pass the algorithm provides ..."
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Cited by 121 (13 self)
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We introduce a fast, multiscale algorithm for image segmentation. Our algorithm uses modern numeric techniques to nd an approximate solution to normalized cut measures in time that is linear in the size of the image with only a few dozen operations per pixel. In just one pass the algorithm provides a complete hierarchical decomposition of the image into segments. The algorithm detects the segments by applying a process of recursive coarsening in which the same minimization problem is represented with fewer and fewer variables producing an irregular pyramid. During this coarsening process we may compute additional internal statistics of the emerging segments and use these statistics to facilitate the segmentation process. Once the pyramid is completed it is scanned from the top down to associate pixels close to the boundaries of segments with the appropriate segment. The algorithm is inspired by algebraic multigrid (AMG) solvers of minimization problems of heat or electric networks. We demonstrate the algorithm by applying it to real images.
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...
Fingerprint Enhancement by Shape Adaptation of ScaleSpace Operators with Automatic Scale Selection
"... This work presents two mechanisms for processing fingerprint images; shapeadapted smoothing based on second moment descriptors and automatic scale selection based on normalized derivatives. The shape adaptation procedure adapts the smoothing operation to the local ridge structures, which allows int ..."
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Cited by 38 (9 self)
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This work presents two mechanisms for processing fingerprint images; shapeadapted smoothing based on second moment descriptors and automatic scale selection based on normalized derivatives. The shape adaptation procedure adapts the smoothing operation to the local ridge structures, which allows interrupted ridges to be joined without destroying essential singularities such as branching points and enforces continuity of their directional fields. The scale selection procedure estimates local ridge width and adapts the amount of smoothing to the local amount of noise. In addition, a ridgeness measure is defined, which reflects how well the local image structure agrees with a qualitative ridge model, and is used for spreading the results of shape adaptation into noisy areas. The combined approach makes it possible to resolve fine scale structures in clear areas while reducing the risk of enhancing noise in blurred or fragmented areas. The result is a reliable and adaptively detailed estimate of the ridge orientation field and ridge width, as well as a smoothed greylevel version of the input image. We propose that these general techniques should be of interest to developers of automatic fingerprint identification
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 31 (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.
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...
A unified geometric model for 3D confocal image analysis in cytology
 Lawrence Berkeley National Laboratory, Univ. California, Berkeley
, 1998
"... Abstract—In this paper, we use partialdifferentialequationbased filtering as a preprocessing and post processing strategy for computeraided cytology. We wish to accurately extract and classify the shapes of nuclei from confocal microscopy images, which is a prerequisite to an accurate quantitativ ..."
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Cited by 13 (0 self)
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Abstract—In this paper, we use partialdifferentialequationbased filtering as a preprocessing and post processing strategy for computeraided cytology. We wish to accurately extract and classify the shapes of nuclei from confocal microscopy images, which is a prerequisite to an accurate quantitative intranuclear (genotypic and phenotypic) and internuclear (tissue structure) analysis of tissue and cultured specimens. First, we study the use of a geometrydriven edgepreserving image smoothing mechanism before nuclear segmentation. We show how this filter outperforms other widelyused filters in that it provides higher edge fidelity. Then we apply the same filter, with a different initial condition, to smooth nuclear surfaces and obtain subpixel accuracy. Finally we use another instance of the geometrical filter to correct for misinterpretations of the nuclear surface by the segmentation algorithm. Our prefiltering and post filtering nicely complements our initial segmentation strategy, in that it provides substantial and measurable improvement in the definition of the nuclear surfaces. Index Terms—Cytology, differential geometry, dynamic surfaces, image processing, level sets, Riemannian geometry, segmentation, surface evolution. I.
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.