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Level Set Methods in Segmentation of SDOCT Retinal Images
Published in Ayman El-Baz, Jasjit S. Suri, Level Set Method in Medical Imaging Segmentation, 2019
N Padmasini, R Umamaheswari, Yacin Sikkandar Mohamed, Manavi D Sindal
Level set is an implicit deformable model, also called implicit active contours, that can model arbitrarily complex shapes and topological changes. This energy functional usually depends on the image data as well as the characteristic features used to identify the objects to be segmented. One of the primary and classical variational level set models was developed by Chan and Vese (CV model). Their method seeks the desired segmentation as the best piecewise constant approximation to a given image. In their proposed model for active contours to detect objects in a given image, they made use of curve evolution, Mumford–Shah functional for segmentation, and level sets. Their model could detect objects whose boundaries are not necessarily defined by gradient. They minimized the energy which is seen as a particular case of the minimal partition problem. In the level set formulation, the problem becomes a “mean-curvature flow” like evolving the active contour, which will stop on the desired boundary. However, the stopping term does not depend on the gradient of the image, as in the classical active contour models, but is instead related to a particular segmentation of the image. The initial curve can be anywhere in the image, and interior contours are automatically detected [54]. This type of variational level set model also known as the region based model, provides optimal segmentation by minimizing an energy functional. This energy functional usually depends on the image data as well as the characteristic features used to identify the objects to be segmented.
Grain boundary motion in particulate material
Published in Y. Kishino, Powders and Grains 2001, 2020
J.L. Turner, M. Nakagawa, M.T. Lusk
In continuum systems characterized by smooth interfaces with constant surface energy, the area of internal regions shrinks at a constant rate—i.e. the process is governed by mean curvature flow (Upmanyu 1999). Grain coarsening in the solid state exhibits this behavior, and experiments were performed to see if internal grains also shrink at a constant rate in granular assemblies.
Some qualitative properties for the Kirchhoff total variation flow
Published in Applicable Analysis, 2022
The objective of this paper is to discuss the existence and asymptotic behaviour of solutions near the extinction time to the following class of Kirchhoff type problem involving the 1-Laplace operator where is a bounded domain with Lipschitz boundary, , and denotes the 1-Laplace operator. In the mathematical literature, equations in (P)–(1) are also known as very singular diffusion equations, see, for example, Giga et al. [3] and their references. These kinds of problems attracted large attention in the literature due to their applications in image processing, faceted crystal growth, continuum mechanics, for details, see Refs. [3, 4]. In geometry, as it is well known the unit normal of the level set is given formally by , then the mean curvature of this surface is formally given by which turns out that the solution of a parabolic 1-Laplacian problem can be also seen as a solution to the evolution mean curvature flow for the level sets , see Ecker [5] for more details.
Automatic 3D human body landmarks extraction and measurement based on mean curvature skeleton for tailoring
Published in The Journal of The Textile Institute, 2022
Haoyang Xie, Yueqi Zhong, Zhicai Yu, Azmat Hussain, Guanmin Chen
The proposed approach decomposes the 3D human body into 13 parts to enable the subsequent tailoring measurements. The magic number of 13 is coming from extensive experimental observations. In this paper, the segmentation method is based on the skeleton rather than the original mesh. MCS is a kind of contraction-based curve skeleton extraction algorithm, which formulate the skeletonization via the Mean Curvature Flow (MCF). Given a watertight 2D-manifold triangle mesh where denotes the vertices and denotes the edges. MCS drives the MCF on towards the extreme to collapse the geometry and capture a skeletal structure. During the collapsing process, the MCS records the original mesh topology, which enables us to map the skeleton segmentation back to the original mesh efficiently. Compared with other curve skeletons (Au et al., 2008; Jalba et al., 2013), another fabulous advantage is that MCS formally defines a well-centered curve skeleton by explicitly inducing a centered term in the energy function. The well-centered property provides the facility for detecting some landmarks such as elbow, crotch, etc. MCS energy is defined as:
Eigenvalue estimates for the drifting Laplacian and the p-Laplacian on submanifolds of warped products
Published in Applicable Analysis, 2021
Wei Lu, Jing Mao, Chuan-Xi Wu, Ling-Zhong Zeng
Especially, if the prescribed warped product was chosen to be the Euclidean space with the same dimension, Theorem 3.1 would give lower bounds for (i.e. the weighted fundamental tone) of the weighted Laplacian on minimal submanifolds in the Euclidean space. The most interesting connection between minimal submanifolds (of Euclidean spaces) and the weighted Laplacian was initiated by Colding and Minicozzi [11], which is a byproduct of an important and deep study of general mean curvature flow (MCF for short). We will give an interesting and brief introduction to eigenvalue estimates for the weighted Laplacian on minimal submanifolds of Euclidean spaces – see details at the end of Section 3.