Threshold dynamics for high order geometric motions

Abstract. We derive a biomembrane model consisting of a fluid enclosed by a lipid membrane. The membrane is characterized by its Canham-Helfrich energy (Willmore energy with area constraint) and acts as a boundary force on the Navier-Stokes system modeling an incompressible fluid. We give a concise description of the model and of the associated numerical scheme. We provide numerical simulations with emphasis on the comparisons between different types of flow: the geometric model which does not take into account the bulk fluid and the biomembrane model for two different regimes of parameters.

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“…It is worth mentioning the related works [22,16], which focus on the minimization of the bending energy without constraints.…”

Section: Introductionmentioning

confidence: 99%

Dynamics of Biomembranes: Effect of the Bulk Fluid

Bonito

Nochetto

Pauletti

2011

Math. Model. Nat. Phenom.

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“…A similar thresholding procedure can be found derived in [28]. A class of algorithms for the high order geometric motion of planar curves following a similar thresholding procedure can be found in [27].…”

Section: Unsupervised Algorithmmentioning

confidence: 99%

Hyperspectral Image Classification Using Graph Clustering Methods

Zhang¹,

Merkurjev²,

Koniges³

et al. 2017

Image Processing on Line

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Hyperspectral imagery is a challenging modality due to the dimension of the pixels which can range from hundreds to over a thousand frequencies depending on the sensor. Most methods in the literature reduce the dimension of the data using a method such as principal component analysis, however this procedure can lose information. More recently methods have been developed to address classification of large datasets in high dimensions. This paper presents two classes of graph-based classification methods for hyperspectral imagery. Using the full dimensionality of the data, we consider a similarity graph based on pairwise comparisons of pixels. The graph is segmented using a pseudospectral algorithm for graph clustering that requires information about the eigenfunctions of the graph Laplacian but does not require computation of the full graph. We develop a parallel version of the Nyström extension method to randomly sample the graph to construct a low rank approximation of the graph Laplacian. With at most a few hundred eigenfunctions, we can implement the clustering method designed to solve a variational problem for a graph-cut-based semi-supervised or unsupervised classification problem. We implement OpenMP directive-based parallelism in our algorithms and show performance improvement and strong, almost ideal, scaling behavior. The method can handle very large datasets including a video sequence with over a million pixels, and the problem of segmenting a data set into a pre-determined number of classes. Source CodeThe reviewed source code and documentation for this algorithm are available from the web page of this article 1 . Compilation and usage instructions are included in the README.txt file of the archive.

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“…After they find the gradient descent equations with respect to u 1 and u 2 , they construct the thresholding numerical scheme to solve the obtained system of parabolic equations. We will adapt ideas by Esedoglu and Tsai to solve our MPLE problem and illustrate the usefulness of this simple method.Some extension of the MBO algorithms appeared in [11,12,21] An efficient algorithm for motion by mean curvature using adaptive grids was proposed in [26].…”

Section: Background On Variational Methods In Image Segmentation and mentioning

confidence: 99%

“…The first variation of the model (11) yields the following gradient descent equation: (12) and the adaptation of the MBO scheme was used to solve it. Similarly to the MBO scheme where the propagation step based on the heat equation is combined with thresholding, Esedoglu and Tsai proposed the following scheme:…”

Section: Background On Variational Methods In Image Segmentation and mentioning

confidence: 99%

Statistical Density Estimation Using Threshold Dynamics for Geometric Motion

Kostic¹,

Bertozzi²

2012

J Sci Comput

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Our goal is to estimate a probability density based on discrete point data via segmentation techniques. Since point data may represent certain activities, such as crime, our method can be successfully used for detecting regions of high activity. In this work we design a binary segmentation version of the well-known Maximum Penalized Likelihood Estimation (MPLE) model, as well as a minimization algorithm based on thresholding dynamics originally proposed by Merriman et al. (The Computational Crystal Growers, pp. 73-83, 1992). We also present some computational examples, including one with actual residential burglary data from the San Fernando Valley.

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Threshold dynamics for high order geometric motions

Cited by 54 publications

References 28 publications

Dynamics of Biomembranes: Effect of the Bulk Fluid

Dynamics of Biomembranes: Effect of the Bulk Fluid

Hyperspectral Image Classification Using Graph Clustering Methods

Statistical Density Estimation Using Threshold Dynamics for Geometric Motion

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