Exploring uses of persistent homology for statistical analysis of landmark-based shape data

A method for the use of persistent homology in the statistical analysis of landmark-based shape data is given. Three-dimensional landmark configurations are used as input for separate filtrations, persistent homology is performed, and persistence diagrams are obtained. Groups of configurations are compared using distances between persistence diagrams combined with dimensionality reduction methods. A three-dimensional landmark-based data set is used from a longitudinal orthodontic study, and the persistent homology method is able to distinguish clinically relevant treatment effects. Comparisons are made with the traditional landmark-based statistical shape analysis methods of Dryden and Mardia, and Euclidean Distance Matrix Analysis.     

Brownian Motion and Ornstein–Uhlenbeck Processes in Planar Shape Space

We discuss Brownian motion and Ornstein–Uhlenbeck processes specified directly in planar shape space. In particular, we obtain the drift and diffusion coefficients of Brownian motion in terms of Kendall shape variables and Goodall–Mardia polar shape variables. Stochastic differential equations are given and the stationary distributions are obtained. By adding in extra drift to a reference figure, Ornstein–Uhlenbeck processes can be studied, for example with stationary distribution given by the complex Watson distribution. The triangle case is studied in particular detail, and some simulations given. Connections with existing work are made, in particular with the diffusion of Euclidean shape. We explore statistical inference for the parameters in the model with an application to cell shape modelling.      

Visualization of Multivariate Density Estimates With Shape Trees

This article introduces graphical tools for visualizing multivariate functions, specializing to the case of visualizing multivariate density estimates. We visualize a density estimate by visualizing a series of its level sets. From each connected part of a level set a shape tree is formed. A shape tree is a tree whose nodes are associated with regions of the level set. With the help of a shape tree we define a transformation of a multivariate set to a univariate function. The shape trees are visualized with the shape plots and the location plot. By studying these plots one may identify the regions of the Euclidean space where the probability mass is concentrated. An application of shape trees to visualize the distribution of stock index returns is presented.     

Fast and Accurate Approximation of the Full Conditional for Gamma Shape Parameters

The gamma distribution arises frequently in Bayesian models, but there is not an easy-to-use conjugate prior for the shape parameter of a gamma. This inconvenience is usually dealt with by using either Metropolis–Hastings moves, rejection sampling methods, or numerical integration. However, in models with a large number of shape parameters, these existing methods are slower or more complicated than one would like, making them burdensome in practice. It turns out that the full conditional distribution of the gamma shape parameter is well approximated by a gamma distribution, even for small sample sizes, when the prior on the shape parameter is also a gamma distribution. This article introduces a quick and easy algorithm for finding a gamma distribution that approximates the full conditional distribution of the shape parameter. We empirically demonstrate the speed and accuracy of the approximation across a wide range of conditions. If exactness is required, the approximation can be used as a proposal distribution for Metropolis–Hastings. Supplementary material for this article is available online.     

四次带参Ball曲线的形状分析

基于包络理论与拓扑映射的方法对四次带参Ball曲线进行了形状分析,得出了曲线上含有奇点,拐点和曲线为局部凸或全局凸的充分必要条件,这些条件完全由控制多边形和形状参数所决定;并进一步讨论了形状参数对形状分布图的影响及其对曲线形状的调节能力.研究表明,四次带参Ball曲线的形状调控能力要优于四次带参Bezier曲线.     

Shape as an Information-Thermodynamical Concept and the Pattern Recognition Problem

A hierarchical classification of different concepts of shape of compact connected sets in R ⁿ (topological, Lipshitz, homotopic, Borsuk an homological shapes) is given. The most general among them is the homological shape. There is only a countable number of homological shapes for connected compact sets in R ⁿ . In the case n = 2 even the number of different Borsuk shapes for connected compact sets is countable. Giving a probability distribution of shapes we can define a shape entropy, a mean shape and shape fluctuations. This enables a formulation of information thermodynamics of shape and its applications to different fields (physics – small systems, chemistry, biophysics, pattern recognition). The paper does not develop yet these applications, its aim is to clear the basic notions.     

Shape regularity conditions for polygonal/polyhedral meshes,exemplified in a discontinuous Galerkin discretization

This article concerns shape regularity conditions on arbitrarily shaped polygonal/polyhedral meshes. In (J. Wang and X. Ye, A weak Galerkin mixed finite element method for second‐order elliptic problems, Math Comp 83 (2014), 2101–2126), a set of shape regularity conditions has been proposed, which allows one to prove important inequalities such as the trace inequality, the inverse inequality, and the approximation property of the L² projection on general polygonal/polyhedral meshes. In this article, we propose a simplified set of conditions which provides similar mesh properties. Our set of conditions has two advantages. First, it allows the existence of “small” edges/faces, as long as the shape of the polygon/polyhedron is regular. Second, coupled with an extra condition, we are now able to deal with nonquasiuniform meshes. As an example, we show that the discontinuous Galerkin method for Laplacian equations on arbitrarily shaped polygonal/polyhedral meshes, satisfying the proposed set of shape regularity conditions, achieves optimal rate of convergence. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 308–325, 2015     

Coupling of FEM and BEM in Shape Optimization

In the present paper we consider the numerical solution of shape optimization problems which arise from shape functionals of integral type over a compact region of the unknown shape, especially L ²-tracking type functionals. The underlying state equation is assumed to satisfy a Poisson equation with Dirichlet boundary conditions. We proof that the shape Hessian is not strictly H ^1/2-coercive at the optimal domain which implies ill-posedness of the optimization problem under consideration. Since the adjoint state depends directly on the state, we propose a coupling of finite element methods (FEM) and boundary element methods (BEM) to realize an efficient first order shape optimization algorithm. FEM is applied in the compact region while the rest is treated by BEM. The coupling of FEM and BEM essentially retains all the structural and computational advantages of treating the free boundary by boundary integral equations.This research has been carried out when the second author stayed at the Department of Mathematics, Utrecht University, The Netherlands, supported by the EU-IHP project Nonlinear Approximation and Adaptivity: Breaking Complexity in Numerical Modelling and Data Representation     

Bivariate distributions via a Pareto conditional distribution and a regression function

Uniqueness of specification of a bivariate distribution by a Pareto conditional and a consistent regression function is investigated. New characterizations of the Mardia bivariate Pareto distribution and the bivariate Pareto conditionals distribution are obtained.     

Shape sensitivity analysis of thin-shell structures

This paper presents explicit formulas for shape sensitivity analysis of thin shell structures. The curvature distribution is the design to be determined. The thin-shell theory employed is the general Koiter model in the Cartesian coordinates. For the shape sensitivity formulation, both the direct differentiation method and the material derivative concept have been used. The two formulations are shown to be equivalent. A computer program based on these formulations has been developed and applied to examples. The shape sensitivity results obtained have been compared to those obtained by finite differencing.     

Shape Morphisms to Transitive G-Spaces

The following problem plays an important role in shape theory: find conditions that guarantee that a shape morphism F:X Y of a topological space X to a topological space Y is generated by a continuous mapping f:X Y. In the present paper, we study this problem in equivariant shape theory and give a solution for shape-equivariant morphisms to transitive G-spaces, where G is a compact group with countable base. As a corollary, we prove a sufficient condition for equivariant shapes of a G-space X to be equal to the group G itself. We also prove some statements concerning equivariant bundles that play the key role in the proof of the main results and are of interest on their own.     

Shape preserving representations and optimality of the Bernstein basis

This paper gives an affirmative answer to a conjecture given in [10]: the Bernstein basis has optimal shape preserving properties among all normalized totally positive bases for the space of polynomials of degree less than or equal ton over a compact interval. There is also a simple test to recognize normalized totally positive bases (which have good shape preserving properties), and the corresponding corner cutting algorithm to generate the Bézier polygon is also included. Among other properties, it is also proved that the Wronskian matrix of a totally positive basis on an interval [a, ) is also totally positive.Both authors were partially supported by DGICYT PS90-0121.     

Multivariate skewness and kurtosis measures with an application in ICA

In this paper skewness and kurtosis characteristics of a multivariate p-dimensional distribution are introduced. The skewness measure is defined as a p-vector while the kurtosis is characterized by a p×p-matrix. The introduced notions are extensions of the corresponding measures of Mardia [K.V. Mardia, Measures of multivariate skewness and kurtosis with applications, Biometrika 57 (1970) 519–530] and Móri, Rohatgi & Székely [T.F. Móri, V.K. Rohatgi, G.J. Székely, On multivariate skewness and kurtosis, Theory Probab. Appl. 38 (1993) 547–551]. Basic properties of the characteristics are examined and compared with both the above-mentioned results in the literature. Expressions for the measures of skewness and kurtosis are derived for the multivariate Laplace distribution. The kurtosis matrix is used in Independent Component Analysis (ICA) where the solution of an eigenvalue problem of the kurtosis matrix determines the transformation matrix of interest [A. Hyvärinen, J. Karhunen, E. Oja, Independent Component Analysis, Wiley, New York, 2001].     

Approximations of Shape Metrics and Application to Shape Warping and Empirical Shape Statistics

This paper proposes a framework for dealing with several problems related to the analysis of shapes. Two related such problems are the definition of the relevant set of shapes and that of defining a metric on it. Following a recent research monograph by Delfour and Zolésio [11], we consider the characteristic functions of the subsets of R² and their distance functions. The L² norm of the difference of characteristic functions, the L and the W^1,2 norms of the difference of distance functions define interesting topologies, in particular the well-known Hausdorff distance. Because of practical considerations arising from the fact that we deal with image shapes defined on finite grids of pixels, we restrict our attention to subsets of ² of positive reach in the sense of Federer [16], with smooth boundaries of bounded curvature. For this particular set of shapes we show that the three previous topologies are equivalent. The next problem we consider is that of warping a shape onto another by infinitesimal gradient descent, minimizing the corresponding distance. Because the distance function involves an inf, it is not differentiable with respect to the shape. We propose a family of smooth approximations of the distance function which are continuous with respect to the Hausdorff topology, and hence with respect to the other two topologies. We compute the corresponding Gâteaux derivatives. They define deformation flows that can be used to warp a shape onto another by solving an initial value problem.We show several examples of this warping and prove properties of our approximations that relate to the existence of local minima. We then use this tool to produce computational definitions of the empirical mean and covariance of a set of shape examples. They yield an analog of the notion of principal modes of variation. We illustrate them on a variety of examples.     

Shape functional optimization with restrictions boosted with machine learning techniques

Shape optimization is a widely used technique in the design phase of a product. Current ongoing improvement policies require a product to fulfill a series of conditions from the perspective of mechanical resistance, fatigue, natural frequency, impact resistance, etc. All these conditions are translated into equality or inequality restrictions which must be satisfied during the optimization process that is necessary in order to determine the optimal shape. This article describes a new method for shape optimization that considers any regular shape as a possible shape, thereby improving on traditional methods limited to straight profiles or profiles established a priori. Our focus is based on using functional techniques and this approach is, based on representing the shape of the object by means of functions belonging to a finite-dimension functional space. In order to resolve this problem, the article proposes an optimization method that uses machine learning techniques for functional data in order to represent the perimeter of the set of feasible functions and to speed up the process of evaluating the restrictions in each iteration of the algorithm. The results demonstrate that the functional approach produces better results in the shape optimization process and that speeding up the algorithm using machine learning techniques ensures that this approach does not negatively affect design process response times.     

多元威布尔分布形状参数相等的似然比检验

将多元威布尔分布形状参数相等的检验转化为多元极值分布尺度参数相等的检验,利用Logistic模型的似然比统计量,给出相关参数为0．3,0．5,0．8时,检验统计量的模拟分位数和功效,指出相关参数越小,似然比统计量的功效越大。     

Shape optimization for Stokes flow

This paper is concerned with a shape sensitivity analysis of a viscous incompressible fluid driven by Stokes equations. The structures of continuous shape gradients with respect to the shape of the variable domain for some given cost functionals are established by introducing the Piola transformation and then deriving the state derivative and its associated adjoint state. Finally we give the finite element approximation of the problem and a gradient type algorithm is effectively used for our problem.     

Shape optimization of a body in the Oseen flow

This article is concerned with a numerical simulation of shape optimization of the Oseen flow around a solid body. The shape gradient for shape optimization problem in a viscous incompressible flow is computed by the velocity method. The flow is governed by the Oseen equations with mixed boundary conditions containing the pressure. The structure of continuous shape gradient of the cost functional is derived by using the differentiability of a minimax formulation involving a Lagrange functional with a function space parametrization technique. A gradient type algorithm is applied to the shape optimization problem. Numerical examples show that our theory is useful for practical purpose and the proposed algorithm is feasible. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

Shape sensitivity analysis via a penalization method

Summary The object of this paper is the development of a penalization technique to compute the shape derivative of cost functionals where the state is the solution of a non-linear equation and/or a linear variational inequality. This type of problem is frequently encountered in Shape Sensitivity Analysis.

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Résumé Cet article présente le calcul des dérivées de forme de fonctionnelles définies sur un domaine géométrique par une méthode de pénalisation. On suppose que l'état est la solution d'une équation non-linéaire ou d'une inéquation linéaire. Ce type de problème est fréquemment rencontré en analyse de sensitivité des formes.

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This research was supported in part by the National Sciences and Engineering Council of Canada Operating Grant A-8730 and a FCAR Grant from the « Ministère de l'Education du Québec ».