Showing papers on "Anisotropic diffusion published in 1998"
Book•
1 Jan 1998
TL;DR: This work states that all scale-spaces fulllling a few fairly natural axioms are governed by parabolic PDEs with the original image as initial condition, which means that, if one image is brighter than another, then this order is preserved during the entire scale-space evolution.
Abstract: Preface Through many centuries physics has been one of the most fruitful sources of inspiration for mathematics. As a consequence, mathematics has become an economic language providing a few basic principles which allow to explain a large variety of physical phenomena. Many of them are described in terms of partial diierential equations (PDEs). In recent years, however, mathematics also has been stimulated by other novel elds such as image processing. Goals like image segmentation, multiscale image representation, or image restoration cause a lot of challenging mathematical questions. Nevertheless, these problems frequently have been tackled with a pool of heuristical recipes. Since the treatment of digital images requires very much computing power, these methods had to be fairly simple. With the tremendous advances in computer technology in the last decade, it has become possible to apply more sophisticated techniques such as PDE-based methods which have been inspired by physical processes. Among these techniques, parabolic PDEs have found a lot of attention for smoothing and restoration purposes, see e.g. 113]. To restore images these equations frequently arise from gradient descent methods applied to variational problems. Image smoothing by parabolic PDEs is closely related to the scale-space concept where one embeds the original image into a family of subsequently simpler , more global representations of it. This idea plays a fundamental role for extracting semantically important information. The pioneering work of Alvarez, Guichard, Lions and Morel 11] has demonstrated that all scale-spaces fulllling a few fairly natural axioms are governed by parabolic PDEs with the original image as initial condition. Within this framework, two classes can be justiied in a rigorous way as scale-spaces: the linear diiusion equation with constant dif-fusivity and nonlinear so-called morphological PDEs. All these methods satisfy a monotony axiom as smoothing requirement which states that, if one image is brighter than another, then this order is preserved during the entire scale-space evolution. An interesting class of parabolic equations which pursue both scale-space and restoration intentions is given by nonlinear diiusion lters. Methods of this type have been proposed for the rst time by Perona and Malik in 1987 190]. In v vi PREFACE order to smooth the image and to simultaneously enhance semantically important features such as edges, they apply a diiusion process whose diiusivity is steered by local image properties. These lters are diicult to analyse mathematically , as they may act locally like a backward diiusion process. …
2,872 citations
TL;DR: It is shown that anisotropic diffusion can be seen as a robust estimation procedure that estimates a piecewise smooth image from a noisy input image and the connection to the error norm and influence function in the robust estimation framework leads to a new "edge-stopping" function based on Tukey's biweight robust estimator that preserves sharper boundaries than previous formulations and improves the automatic stopping of the diffusion.
Abstract: 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 "edge-stopping" 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 "edge-stopping" 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 line processes allows us to develop new anisotropic diffusion equations that result in a qualitative improvement in the continuity of edges.
1,490 citations
TL;DR: A method is proposed to define diffusions of orientation-like quantities and it is shown how such orientation diffusions contain a nonlinearity that is reminiscent of edge-process and anisotropic diffusion.
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 vector-valued 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 orientation-like 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 edge-process and anisotropic diffusion. A number of open questions are proposed.
251 citations
TL;DR: The multigrid implementation provides an efficient hierarchical relaxation method that facilitates the application of anisotropic diffusion to time-critical processes and provides rapid intraregion smoothing and reduction of artifacts due to the elimination of low-frequency error.
Abstract: A multigrid anisotropic diffusion algorithm for image processing is presented. The multigrid implementation provides an efficient hierarchical relaxation method that facilitates the application of anisotropic diffusion to time-critical processes. Through a multigrid V-cycle, the anisotropic diffusion equations are successively transferred to coarser grids and used in a coarse-to-fine error correction scheme. When a coarse grid with a trivial solution is reached, the coarse grid estimates of the residual error can be propagated to the original grid and used to refine the solution. The main benefits of the multigrid approach are rapid intraregion smoothing and reduction of artifacts due to the elimination of low-frequency error. The theory of multigrid anisotropic diffusion is developed. Then, the intergrid transfer functions, relaxation techniques, diffusion coefficients, and boundary conditions are discussed. The analysis includes the examination of the storage requirements, the computational cost, and the solution quality. Finally, experimental results are reported that demonstrate the effectiveness of the multigrid approach.
146 citations
TL;DR: In this paper, a simple diffusion model was proposed to account for the essential features of surface relief grating formation based on the anisotropic diffusion of azobenzene-dyes in polymer matrices further to their photoinduced trans-cis isomerization cycles.
139 citations
TL;DR: A class of time-delay anisotropic diffusion models for image restoration lead to asymptotic states that are selected on the basis of a contrast parameter and bear some analogy with neural networks with Hebbian dynamical learning rules.
Abstract: We present a class of time-delay anisotropic diffusion models for image restoration. These models lead to asymptotic states that are selected on the basis of a contrast parameter and bear some analogy with neural networks with Hebbian dynamical learning rules. Numerical examples show that these models are efficient in removing even high levels of noise, while allowing an accurate tracking of the edges.
72 citations
TL;DR: The solution structure, backbone dynamics and rotational diffusion of the Rhodobacter capsulatus cytochrome c2 have been determined using heteronuclear NMR spectroscopy and the mobility parameters extracted show a quantitative improvement with respect to the model-free analysis assuming isotropic reorientation.
66 citations
TL;DR: A coarse-grid CNN approach is proposed that is capable of calculating an acceptable noise-level estimate and controlling the fine-grid anisotropic diffusion models and a combined geometrical-statistical approach has also been developed for filtering both the impulse and additive Gaussian noise while preserving the image structure.
Abstract: In this paper, we develop a common cellular neural network framework for various adaptive non-linear filters based on robust statistic and geometry-driven diffusion paradigms. The base models of both approaches are defined as difference-controlled non-linear CNN templates, while the self-adjusting property is ensured by simple analogic (analog and logic) CNN algorithms. Two adaptive strategies are shown for the order statistic class. When applied to the images distorted by impulse noise both give more visually pleasing results with lower-frequency weighted mean square error than the median base model. Generalizing a variational approach we derive the constrained anisotropic diffusion, where the output of the geometry-driven diffusion model is forced to stay close to a pre-defined morphological constraint. We propose a coarse-grid CNN approach that is capable of calculating an acceptable noise-level estimate (proportional to the variance of the Gaussian noise) and controlling the fine-grid anisotropic diffusion models. A combined geometrical-statistical approach has also been developed for filtering both the impulse and additive Gaussian noise while preserving the image structure. We briefly discuss how these methods can be embedded into a more complex algorithm performing edge detection and image segmentation. The design strategies are analysed primarily from VLSI implementation point of view; therefore all non-linear cell interactions of the CNN architecture are reduced to two fundamental non-linearities, to a sigmoid type and a radial basis function. The proposed non-linear characteristics can be approximated with simple piecewise-linear functions of the voltage difference of neighbouring cells. The simplification makes it possible to convert all space-invariant non-linear templates of this study to a standard instruction set of the CNN Universal Machine, where each instruction is coded by at most a dozen analog numbers. Examples and simulation results are given throughout the text using various intensity images.
66 citations
4 Jan 1998
TL;DR: A novel local scale controlled piecewise linear diffusion for selective smoothing and edge detection is presented and shows anisotropic, nonlinear diffusion equation using diffusion coefficients/tensors that continuously depend on the gradient is not necessary to achieve sharp, distorted, stable edge detection across many scales.
Abstract: A novel local scale controlled piecewise linear diffusion for selective smoothing and edge detection is presented. The diffusion stops at the place and time determined by the minimum reliable local scale and a spatial variant, anisotropic local noise estimate. It shows anisotropic, nonlinear diffusion equation using diffusion coefficients/tensors that continuously depend on the gradient is not necessary to achieve sharp, distorted, stable edge detection across many scales. The new diffusion is anisotropic and asymmetric only at places it needs to be, i.e., at significant edges. It not only does not diffuse across significant edges, but also enhances edges. It advances geometry-driven diffusion because it is a piecewise linear model rather than a full nonlinear model, thus it is simple to implement and analyze, and avoids the difficulties and problems associated with nonlinear diffusion. It advances local scale control by introducing spatial variant, anisotropic local noise estimation, and local stopping of diffusion. The original local scale control was based on the unrealistic assumption of uniformly distributed noise independent of the image signal. The local noise estimate significantly improves local scale control.
40 citations
TL;DR: A modified version of the anisotropic diffusion algorithm is proposed as a precise edge-preserving smoothing technique modified by using boundary edges and a linking algorithm for boundary edges based on a directional potential function is incorporated.
39 citations
2 Jun 1998
TL;DR: This article proposes a theoretically justified optimization problem that permits to take into account both requirements of restoring and segmenting noisy image sequences with a static background and proposes a suitable numerical scheme based on half quadratic minimization.
Abstract: This article deals with the problem of restoring and segmenting noisy image sequences with a static background. Usually, motion segmentation and image restoration are tackled separately in image sequence restoration. Moreover, segmentation is often noise sensitive. In this article, the motion segmentation and the image restoration parts are performed in a coupled way, allowing the motion segmentation part to positively influence the restoration part and vice-versa. This is the key of our approach that allows to deal simultaneously with the problem of restoration and motion segmentation. To this end, we propose a theoretically justified optimization problem that permits to take into account both requirements. A suitable numerical scheme based on half quadratic minimization is then proposed and its stability demonstrated. Experimental results obtained on noisy synthetic data and real images will illustrate the capabilities of this original and promising approach.
4 Oct 1998
TL;DR: The three dimensional anisotropic diffusion equation utilizes the fact that consecutive frames of high correlation can be obtained in video sequences to remove noise inVideo sequences.
Abstract: In this paper a three dimensional anisotropic diffusion equation is proposed to remove noise in video sequences. The three dimensional anisotropic diffusion equation utilizes the fact that consecutive frames of high correlation can be obtained in video sequences. It will be shown that the three dimensional diffusion equation is more suitable for video sequences than the two dimensional diffusion equation.
4 Oct 1998
TL;DR: A linear transformation is applied to the prior term but, in addition, it is required that the eigenvalues of the transformation have specific properties so that diffusion is allowed only along the direction perpendicular to the local image gradient.
Abstract: Inverse problems encountered in image processing and computer vision are often ill-posed. Whether set in a Bayesian or energy-based context, such problems require prior assumptions expressed through an a priori probability or a regularization term, respectively. In some cases, the prior term exhibits partial dependence on the observations (e.g., images) that is often ignored to simplify modeling and computations. We review methods that take this dependence into account and we propose a new formulation of the prior term that blends some other simple approaches. Similarly to others, we apply a linear transformation to the prior term but, in addition, we require that the eigenvalues of the transformation have specific properties. These properties are chosen so that diffusion is allowed only along the direction perpendicular to the local image gradient. If the gradient magnitude is small, isotropic diffusion is performed. We apply this formulation to stereoscopic disparity estimation and we show several experimental results; improvements over a standard approach are clear.
TL;DR: Mean curvature diffusion is shown to be a position vector diffusion, tending to scalar diffusion as a flat image region is approached, and providing noise removal by steepest descent surface minimization.
1 Apr 1998
TL;DR: A novel locally adaptive conductance for the geometry-driven diffusion (GDD) filtering is proposed, based on a measure of the neighborhood unhomogeneity adopted from the optimal orientation detection of linear symmetry, for nonuniform diffusion filtering of magnetic resonance (MR) tomograms.
Abstract: The paper deals with a nonuniform diffusion filtering of magnetic resonance (MR) tomograms. Alternative digital schemes for discrete implementation of the nonuniform diffusion equations are analyzed and tested. A novel locally adaptive conductance for the geometry-driven diffusion (GDD) filtering is proposed. It is based on a measure of the neighborhood unhomogeneity adopted from the optimal orientation detection of linear symmetry. The algorithm performance is evaluated on the basis of pseudoartificial 2D MR brain phantom and using the signal-to-noise ratio, as well as HC measure, developed for image discrimination characterization. Three filtering methods are applied to MR images acquired by the fast 3D FLASH sequence. The results obtained are quantitatively and visually compared and discussed.
1 Dec 1998
TL;DR: In this paper, a class of fourth order partial differential equations (PDE) is proposed to optimize the trade-off between noise removal and edge preservation, and the time evolution of these PDEs seeks to minimize a functional which is an increasing function of the absolute value of the Laplacian of the image intensity function, hence it generates a family of images of increasing degree of smoothness.
Abstract: A class of fourth order partial differential equations (PDE) are proposed to optimize the trade-off between noise removal and edge preservation. The time evolution of these PDEs seeks to minimize a functional which is an increasing function of the absolute value of the Laplacian of the image intensity function, hence it generates a family of images of increasing degree of smoothness. Since the Laplacian of a plane image is zero, the stationary points of this functional or this class of PDEs are images whose intensity functions are a union of plane images of various boundaries. This kind of images look more natural than step images which are the stationary points of anisotropic diffusion (second order PDEs), so the proposed PDEs are able to achieve comparable degree of noise removal while avoiding the blocky effects widely seen in images processed by anisotropic diffusion. However, the proposed fourth order PDEs tend to develop speckle artifacts which may be characterized as isolated white and/or black dots, but they can be easily alleviated by simple despeckle algorithms such as the one shown in this paper.
TL;DR: In this paper, the positions, sizes, and number densities of monoatomically high, rectangular, reconstructed Pt islands, formed in the submonolayer coverage regime, have been determined for substrate temperatures in the range T = 318-497 K and adatom deposition rates from R = 4 × 10 -5 to 7 × 10 −3 site -1 s -1.
5 Apr 1998
TL;DR: A new anisotropic diffusion algorithm for enhancing and segmenting multispectral image data and its performance in terms of image entropy reduction and impulse elimination as well as visual quality is demonstrated.
Abstract: This paper introduces a new anisotropic diffusion algorithm for enhancing and segmenting multispectral image data. The algorithm is based upon mean curvature motion. Using a modified image gradient computation, the diffusion method is further improved by allowing the control of feature scale, and the sensitivity to heavy-tailed noise is eliminated. For comparison, a vector distance dissimilarity method is introduced and extended for multi-scale processing. The experiments on remotely sensed imagery and color imagery demonstrate the performance of the algorithms in terms of image entropy reduction and impulse elimination as well as visual quality.
16 Aug 1998
TL;DR: It is argued that for high compression an anisotropic diffusion preprocessing results in better quality of the decoded image, because it removes noise and irrelevant details while preserving the edges.
Abstract: Anisotropic diffusion is an image enhancement method It is a nonlinear process which removes noise and irrelevant details while preserving the edges, ie it "extracts" the essential visual information The paper proposes a useful application of anisotropic diffusion in image data compression We argue that for high compression an anisotropic diffusion preprocessing results in better quality of the decoded image
8 Nov 1998
TL;DR: A fuzzy clustering algorithm is introduced that, using information about the intensity distribution, divides the image domain into regions and assigns every pixel in the image a degree of membership to the clusters, i.e. a probability of belonging to each of the regions.
Abstract: Previously, we proposed a second rank tensor conductance function with an explicit dependence on the space coordinates and the data function. This scheme gives the equations an intrinsic anisotropic character not present in previous approaches, and allows the use of a priori knowledge of the system in multi-feature and multi-dimensional images. In this article we extend this scheme by introducing a fuzzy clustering algorithm that, using information about the intensity distribution, divides the image domain into regions and assigns every pixel in the image a degree of membership to the clusters, i.e. a probability of belonging to each of the regions. For this purpose we employ a fuzzy c-means algorithm in which we introduce a priori knowledge about the system by using a planispheric coordinate system that exploits the approximate elliptic-paraboloidal shape and symmetry of the left ventricle. The fuzzy classification of the image domain provides a measure of the probability that neighbouring pixels belong to the same tissue type, and is therefore incorporated into the diffusion process by means of the conductance function. The clustering is updated at regular intervals during the diffusion process, and the initially coarse segmentation of the image is gradually improved until it converges to a meaningful segmentation of the image regions as the smoothing action of the diffusion process clears the image from noise.
TL;DR: The two parallel and topographically organized subsystems of a boundary contour (BCS) and feature contour system (FCS) are demonstrated to generate an isomorphic representation of brightness distributions, achieving a unique solution for the brightness-from-luminance problem.
TL;DR: In this paper, a wavelet expansion of the image is used to detect edges in an image by solving an anisotropic diffusion equation, which has the intrinsic property that low contrast regions are smoothed and high contrast ones are enhanced.
Abstract: We consider the problem of detection of edges in an image by solving an anisotropic diffusion equation, which has the intrinsic property that low-contrast regions are smoothed and high-contrast ones are enhanced. Since wavelets are known to provide better representation of singularities (i.e., edges), a more efficient scheme than those suggested earlier for solving the diffusion equation is formulated in terms of wavelet expansions of the image. These expansions also provide a natural way of estimating the local contrast, and hence of implementing a space-varying parameterization of the diffusion equation for improved performance. Our method can be viewed as a wavelet counterpart of standard spectral methods for solving partial differential equations. © 1998 John Wiley & Sons, Inc. Int J Imaging Syst Technol, 9, 356–368, 1998
4 Oct 1998
TL;DR: An adaptive nonlinear filtering method based on the nonlinear anisotropic diffusion equation, which has the advantage of flexibility and adaptability and a new degradation effect, not signaled before, called the "pinhole effect", is proposed and evaluated.
Abstract: The authors propose an adaptive nonlinear filtering method based on the nonlinear anisotropic diffusion equation. This new method has the advantage of flexibility and adaptability. Furthermore, a new degradation effect, not signaled before, we call the "pinhole effect", which may result in anisotropic diffusion is introduced and analyzed. A robust solution to this effect is proposed and evaluated through experimental data. The performance of the method is demonstrated on real and synthetic images.
TL;DR: The complex log transformation, variants of which are universally used in mammalian retino-cortical systems, allows the nonlinear diffusion equation to be integrated at exponentially enhanced rates due to the nonuniform mesh spacing inherent in the log domain, providing a means of performing rapid image enhancement using anisotropic diffusion.
Abstract: Many computer and robot vision applications require multi-scale image analysis. Classically, this has been accomplished through the use of a linear scale-space, which is constructed by convolution of visual input with Gaussian kernels of varying size (scale). This has been shown to be equivalent to the solution of a linear diffusion equation on an infinite domain, as the Gaussian is the Green‘s function of such a system (Koenderink, 1984). Recently, much work has been focused on the use of a variable conductance function resulting in anisotropic diffusion described by a nonlinear partial differential equation (PDE). The use of anisotropic diffusion with a conductance coefficient which is a decreasing function of the gradient magnitude has been shown to enhance edges, while decreasing some types of noise (Perona and Malik, 1987). Unfortunately, the solution of the anisotropic diffusion equation requires the numerical integration of a nonlinear PDE which is a costly process when carried out on a uniform mesh such as a typical image. In this paper we show that the complex log transformation, variants of which are universally used in mammalian retino-cortical systems, allows the nonlinear diffusion equation to be integrated at exponentially enhanced rates due to the nonuniform mesh spacing inherent in the log domain. The enhanced integration rates, coupled with the intrinsic compression of the complex log transformation, yields a speed increase of between two and three orders of magnitude, providing a means of performing rapid image enhancement using anisotropic diffusion.
TL;DR: In this article, the homoepitaxial island growth on hexagonally reconstructed Au(100) is studied using molecular dynamics based on a well-tested many-atom interatomic potential.
Abstract: The homoepitaxial island growth on hexagonally reconstructed Au(100) is studied using molecular dynamics based on a well-tested many-atom interatomic potential. Our study reveals that the stable islands of rectangular shape are hexagonally reconstructed in conformity with the patterns of the reconstructed Au(100) surface and suggests the ``magic'' stable width for the reconstructed islands in agreement with experimental observations. Furthermore, our results on the adatom diffusion indicate that the experimentally observed strong anisotropic effect is attributed to the long-range exchange diffusion.
TL;DR: It is shown that the number of iterations required for the solution of the linear algebraic system is proportional to the inverse of the smallest grid element diameter, uniformly in the diffusion parameter and the degree of anisotropy.
Abstract: We consider an elementwise data-parallel finite element procedure, recently proposed by Layton and Rabier [Appl. Math. Lett., 5 (1992), pp. 67--70], [J. Numer. Linear Algebra Appl., 2 (1995), pp. 363--394], applied to singularly perturbed convection-diffusion equations with possibly highly anisotropic diffusion. It is shown that the number of iterations required for the solution of the linear algebraic system is proportional to the inverse of the smallest grid element diameter, uniformly in the diffusion parameter and the degree of anisotropy. This is optimal, since the method can in some cases use only element matrices and load vectors and the algorithm requires only local communication on the physical mesh between adjacent elemental subdomains. Our analysis includes both the usual Galerkin formulation and the streamline upwind finite element formulations. The convergence result holds with conforming elements of any order or elements of arbitrary order from a new family of nonconforming elements.
TL;DR: In this paper, a three dimensional anisotropic diffusion equation is proposed to remove noise in video sequences, which utilizes the fact that consecutive frames of high correlation can be obtained in the video sequences.
Abstract: A three dimensional anisotropic diffusion equation is proposed to remove noise in video sequences. The three dimensional anisotropic diffusion equation utilizes the fact that consecutive frames of high correlation can be obtained in video sequences. It is shown that the three dimensional diffusion equation gives better results than the two dimensional diffusion equation which only deals with still images and slowly smoothes the edge boundaries.
1 Jan 1998
TL;DR: This paper shows how the dot diffusion method can be improved by optimization of the so-called class matrix by taking the human visual characteristics into account and shows that such optimization consistently results in images comparable to error diffusion, without sacrificing the parallelism.
Abstract: The dot diffusion method for digital halftoning has the advantage of parallelism unlike the error diffusion method. However, image quality offered by error diffusion is still regarded as superior to other known methods. In this paper we show how the dot diffusion method can be improved by optimization of the so-called class matrix. By taking the human visual characteristics into account we show that such optimization consistently results in images comparable to error diffusion, without sacrificing the parallelism.
Patent•
14 Sep 1998
TL;DR: In this article, a spatial filter in which a coefficient matrix based on the no-neighbor algorithm in a restoration process has been set performs a spatial filtering process on the image data items sequentially outputted from the 4-line buffer to produce a restored image based on a No-Neighbor algorithm, which enables images to be restored in real time, which produces an image whose luminance distribution is approximate to that of the specimen.
Abstract: A 4-line buffer sequentially takes in image data items and temporarily stores a specific size of image data. A spatial filter in which a coefficient matrix based on the no-neighbor algorithm in a restoration process has been set performs a spatial filtering process on the image data items sequentially outputted from the 4-line buffer to produce a restored image based on the no-neighbor algorithm. This enables images to be restored in real time, which produces an image whose luminance distribution is approximate to that of the specimen.