B. Merriman, J. Bence, and S. Osher, "Diffusion generated motion by mean curvature," in J. E. Taylor, Editor, Computational Crystal Growers Workshop, pp. 73-83, American Mathematical Society, Providence, Rhode Island, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Such a restriction can produce an inefficient method for mean curvature flows because very small time steps must be used whenever a fine spatial mesh is applied. A method based on the model of diffusion dependent motion of level sets has recently been proposed by Merriman, Bence and Osher [47, 48]. We shall refer to this method as the MBO method (cf. 25] although the name DGCDM algorithm has also been used [48] The specifics of this method are elaborated upon in subsequent chapters, so will not be repeated here. This method naturally handles complicated topological changes with ....

....of the rest of the thesis follows. In Chapter 2, algorithms describing the MBO method for two phase and multiple phase problems are given. This is followed by a discussion on how to select the step size, of the method. For the case of the finite difference discretizations originally proposed [47], the selection of an appropriate time stepping scheme is discussed and several limitations of the method are identified. In Chapter 3, a new, spectral method for the realization of the MBO method is proposed and described in detail. A spatial discretization is given and an efficient quadrature ....

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B. Merriman, J. Bence, and S. Osher. Diffusion generated motion by mean curvature. In J.E. Taylor, editor, Computational Crystal Growers Workshop, pages 73--83. American Mathematical Society, Providence, Rhode Island, 1992.

....we recover the TV norm for a scalar valued function. 3 Other Approaches Other approaches to the vector valued restoration problem, e.g. anisotropic diffusion [2, 3, 5, 18, 24, 30] edge detection and segmentation [2, 5, 12, 16, 27, 32] as well as segmentation methods related to level set methods [4, 15, 17, 22, 23] can be found in the literature. Here we will briefly describe the general anisotropic diffusion approach, and Sapiro Ringach s [22, 23, 24] approach in particular. We view the restoration problem in terms of solving the nonlinear optimization problem min Phi2BV( Omega Gamma k Phik g TV ....

Barry Merriman, James Bence, and Stanley Osher. Diffusion Generated Motion by Mean Curvature. Technical Report CAM 92-18, UCLA Department of Mathematics, April 1992.

....of applications, one wants to follow the motion of a front that moves with some curvature dependent speed. For the special case of pure mean curvature flow, junctions of moving surfaces have been treated by alternately diffusing and sharpening the characteristic functions for each phase region [8, 9]. In this work, we generalize this diffusion generated approach to allow for a normal velocity equal to a positive multiple of the mean curvature, of the interface plus the difference in bulk energies. In two dimensions, the simplest model that we consider involves three curves meeting at a ....

....thus require an extremely fine mesh (at least locally) to resolve this layer. To address these concerns for the case of pure mean curvature flow (i.e. fl ij = 1 and e i = 0) a method (MBO) based on the model of diffusiondependent motion of level sets was proposed by Merriman, Bence and Osher [8, 9]. This method naturally handles complicated topological changes with junctions in several dimensions. Furthermore, this method can be made very efficient by discretizing in space using a Fourier spectral basis and using a quadrature to determine the Fourier coefficients at each step [11, 13] ....

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B. Merriman, J. Bence, and S. Osher. Diffusion generated motion by mean curvature. In J.E. Taylor, editor, Computational Crystal Growers Workshop, pages 73--83. American Mathematical Society, Providence, Rhode Island, 1992. REFERENCES 35

....[CGM96] 2. 4 Other Approaches Other approaches to vector valued image processing, e.g. anisotropic diffusion [ALM92, AM94a, CLM92, PM90, SR96, WG94] edge detection and segmentation [ALM92, CLM92, LC91, Nev77, TP86, Zen86] as well as segmentation methods related to level set methods [CDK95, MBO92, OS88, Sap95, Sap96], can be found in the literature. Here we will briefly describe the general anisotropic diffusion approach, and Sapiro Ringach s approach in particular [Sap95, Sap96, SR96] Notice that solving the nonlinear system of equations (2.6) is a form of anisotropic diffusion. Prior to comparing TV n,m ....

Barry Merriman, James Bence, and Stanley Osher. "Diffusion Generated Motion by Mean Curvature." Technical Report CAM 92-18, UCLA Department of Mathematics, April 1992.

....recently developed to simulate geometrical motions of curves and surfaces. We refer to [4] for a survey of many approaches to defining geometric motion of interfaces. More recently, there has been some work in which a level set formulation is introduced to study the evolution of triple junctions [17]. There have also been several discretized models developed more specifically for the motion of grain boundaries using the Potts model (see [10] and references therein) vertex and boundary dynamics models (see [14] and references therein) meanfields theories (see [6] and references therein) as ....

B. Merriman, J. Bence and S. Osher, Diffusion Generated Motion by Mean Curvature, preprint.

....and the noiseless image. IV. Other Approaches Other approaches to vector valued image processing, e.g. anisotropic diffusion [15] 16] 17] 18] 11] 19] edge detection and segmentation [15] 17] 7] 8] 20] 21] as well as segmentation methods related to level set methods [22] [23], 24] 9] 10] can be found in the literature. Here we will briefly describe the general anisotropic diffusion approach, and Sapiro Ringach s approach in particular [9] 10] 11] Notice that solving the nonlinear system of equations (6) is a form of anisotropic diffusion. Before comparing ....

Barry Merriman, James Bence, and Stanley Osher, "Diffusion Generated Motion by Mean Curvature," Tech. Rep. CAM 92-18, UCLA Department of Mathematics, April 1992.

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B. Merriman, J. Bence, and S. Osher, "Diffusion generated motion by mean curvature," in J. E. Taylor, Editor, Computational Crystal Growers Workshop, pp. 73-83, American Mathematical Society, Providence, Rhode Island, 1992.

....as well. The maximum principle appears to be a natural mathematical translation of causality. Koenderink once again made a major contribution into the PDE s arena when he suggested to add a thresholding operation to the process of Gaussian filtering. As later suggested by Merriman, Bence and Osher [118, 119] and by Ruuth, Merriman and sher [158] and proved by a number of groups [8, 53, 85, 86] this leads to a curvature motion geometric PDE, one of the most famous among geometric PDE s. In [160] Ruuth et al. extended it to diffusion generated motion of curves in Solving a vector heat equation and ....

Merriman, B., Bence, J. and Osher, S., Diffusion generated motion by mean curvature, in J. E. Taylor, Editor, Computational Crystal Growers Workshop, pp. 73-83, American Mathematical Society, Providence, Rhode Island, 1992.

....evolves as the interface (perhaps inducing topological changes) In the general (at least three phase) multiphase case a new methodology is needed. In [12] Merriman, Bence, and Osher first extended the level set method to compute the motion of multiphase junctions. Also in that paper, and in [10, 11] a simple method based on the diffusion of characteristic functions followed by a simple reassignment step, was shown to be appropriate for the motion of multiple junctions corresponding to pure mean curvature flow. More general motion involving rather arbitrary functions of curvature, perhaps ....

B. Merriman, J. Bence, and S. Osher, Diffusion generated motion by mean curvature, in AMS Selected Lectures in Mathematics, The Computational Crystal Grower's Workshop, edited by J. Taylor (AMS, Providence, RI, (

....for texture mapping (and not just texture synthesis) This was done for triangulated surfaces in [3, 20, 28] and we plan to extend this to implicit surfaces via the implicit framework here introduced. We are also interested in investigating threshold dynamics and convolution generated motions [32, 37, 44] for implicit surfaces. Finally, the use of this framework for regularization in inverse problems is of interest as well. These issues will be reported on elsewhere. 14 BERTALM IO, CHENG, OSHER AND SAPIRO ACKNOWLEDGMENT We thank Facundo M emoli for interesting conversations during this work. ....

B. Merriman, J. Bence, and S. Osher, "Diffusion generated motion by mean curvature," in J. E. Taylor, Editor, Computational Crystal Growers Workshop, pp. 73-83, American Mathematical Society, Providence, Rhode Island, 1992.

....for motion by mean curvature of objects of arbitrary codimension, as well as generalizations that allow for a large class of velocity laws. 1 Introduction Diffusion generated motion by mean curvature is a particularly simple and robust algorithm for producing motion by mean curvature of a surface [16, 17]. The major goal underlying this work is to generalize this algorithm from surfaces (dimension d Gamma 1 inside R d ) to objects of arbitrary dimension k inside R d . To guide the generalization and connect it to a class of interesting problems, we concentrate on the special case of ....

....motion. Since the width of the front is O(ffl) the only remedy is to use a mesh spacing which is much less than ffl, which can be impractical numerically [17] To overcome this limitation in the case of surface motion, an algorithm based on an idealization of reaction diffusion was presented in [16, 17]. This algorithm is described in detail in Section 2.2, but it essentially consists of 1 INTRODUCTION 4 moving a set boundary by alternately diffusing the set i.e. applying the linear diffusion evolution equation to the set s characteristic function for a short time and then recovering a ....

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B. Merriman, J. Bence, and S. Osher. Diffusion generated motion by mean curvature. In J.E. Taylor, editor, Computational Crystal Growers Workshop, pages 73--83. American Mathematical Society, Providence, Rhode Island, 1992. Also available as UCLA CAM Report 92-18, April 1992.

....does not apply to the motion of triple point junctions. The level set method was ultimately extended to handle the motion of multiple junctions [15, 30] but the modifications were non trivial. 2 BACKGROUND 4 In the course of investigating the multiple junction problem, Merriman, Bence and Osher [14, 15] developed the surprisingly simple diffusion generated motion by mean curvature algorithm, which gave mean curvature motion without computing curvature. It also automatically captured topological change and had a direct extension to motion of triple point junctions and arbitrary networks of ....

....more traditional surface evolution discretizations. Moreover, the discretization extends unchanged to the motion of multiple junctions, while other approaches tend to require more complicated implementations to accommodate such features. In the original presentation of diffusion generated motion [14], it was pointed out that the diffusive evolution is equivalent to convolution with a Gaussian kernel, and that convolution with any other similarly scaled spherically symmetric kernel would also generate motion by mean curvature. It was also pointed out that this provided a means of ....

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B. Merriman, J. Bence, and S. Osher. Diffusion generated motion by mean curvature. In J.E. Taylor, editor, Computational Crystal Growers Workshop, pages 73--83. American Mathematical Society, Providence, Rhode Island, 1992. Also available as UCLA CAM Report 92-18, April 1992.

....algorithm was our original motivation for considering convolutions as a means of generating interface motion. 3. 1 Diffusion Generated Motion by Mean Curvature A particularly simple convolution based algorithm exists for moving an interface with a normal speed equal to its mean curvature [18, 19]. If the initial region has characteristic function , the updated region at a time Deltat is ( x : K( x) 1 2 ) 2) where K is a Gaussian of width p Deltat, K( x) 1 4 Deltat exp Gamma 1 4 Deltat j xj 2 Diffusion generated refers to the fact that convolution with the ....

....procedure can be described informally as diffusing the set for a short time, and then thresholding at the 1 2 level to obtain a new set. It is intuitively clear that such a diffusion will cause a curvaturedependent blurring of the set boundary, and a formal analysis of the diffusion equation [17, 18, 19] shows this should result in precisely motion by mean curvature. Indeed, a variety of rigorous proofs have been given to show this 3 CONVOLUTION GENERATED MOTION 10 simple algorithm converges to motion by mean curvature in any number of dimensions as the time step size goes to zero [6, 2, 15] ....

[Article contains additional citation context not shown here]

B. Merriman, J. Bence, and S. Osher. Diffusion generated motion by mean curvature. In J.E. Taylor, editor, Computational Crystal Growers Workshop, pages 73--83. American Mathematical Society, Providence, Rhode Island, 1992. Also available as UCLA CAM Report 92-18, April 1992.

....to a singular perturbation involving a small parameter ffl will lead to incorrect answers as in [Ko] without the use of adaptive grids [NPV] This is unnecessary in order for the level set model to function. An interesting variant of the level set method for geometry based motion was introduced in [MBO1] as diffusion generated motion, and has now been generalized to forms known as convolution generated motion or threshold dynamics. This method splits the reaction diffusion model into two highly simplified steps. For an overview of this approach, see [RM] 2 The Level Set Dictionary and ....

B. Merriman, J. Bence and S. Osher, "Diffusion Generated Motion by Mean Curvature", in AMS Selected Lectures in Math., The Comput. Crystal Grower's Workshop, edited by J. Taylor (Am. Math Soc., Providence, RI, 1993), p. 73.

....this stiffness due to a singular perturbation involving a small parameter ffl will lead to incorrect answers as in [25] without the use of adaptive grids [30] This is not an issue in the level set approach. An interesting variant of the level set method for geometry based motion was introduced in [26] as diffusion generated motion, and has now been generalized to forms known as convolution generated motion or threshold dynamics. This method splits the reaction diffusion approach into two highly simplified steps. For an overview of this approach, see [35] 3.2. THE LEVEL SET DICTIONARY Here ....

B. Merriman, J. Bence and S. Osher, "Diffusion Generated Motion by Mean Curvature", AMS Selected Lectures in Math., The Comput. Crystal Grower's Workshop, edited by J. Taylor (Am. Math Soc., Providence, RI, 1993), p. 73.

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J. Bence, B. Merriman, S. Osher. Diffusion generated motion by mean curvature. Preprint UCLA 1992.

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