基于对数极坐标及对称循环矩阵的形状特征描述

在形状检索算法中,满足尺度和旋转不变是基本要求.本文将形状的边界用对数极坐标表示,使得形状的放缩和旋转化为简单的平移.由于计算机读取形状边界信息时与起点有关,当形状旋转时会带来边界点列的循环,影响旋转不变性.为消除边界点列循环带来的影响,本文首先证明"奇数阶对称循环矩阵,当生成元循环时,所得循环矩阵的特征值不变",在这个数学理论基础上,把形状边界点数插值到奇数,构造相应的对称循环矩阵,通过这个循环矩阵的特征值来描述形状特征,由此得到一种具有放缩旋转不变的形状检索新算法.实验表明,本文算法对运动目标和非刚性形变的形状检索具有良好的鲁棒性和快捷的运行速度,这在目标跟踪方面将发挥作用.     

Modeling shape distributions and inferences for assessing differences in shapes

The general class of complex elliptical shape distributions on a complex sphere provides a natural framework for modeling shapes in two dimensions. Such class includes many distributions, e.g., complex Normal, Watson, Bingham, angular central Gaussian and several others. We employ this class of distributions to develop methods for asserting differences in populations of shapes in two dimensions. Maximum likelihood and Bayesian methods for estimation of modal difference are developed along with hypothesis testing and credible regions for average shape difference. The methodology is applied in an example from biometry, where we are interested in detecting shape differences between male and female gorilla skulls.     

Numerical experiments in problems of asymptotic representation theory

The article presents the results of numerical computations of statistics related to Young diagrams, including estimates on the maximum and average (with respect to the Plancherel distribution) dimension of irreducible representations of the symmetric group S _n . The computed limit shapes of two-dimensional and three-dimensional diagrams distributed according to the Richardson statistics are also presented. Bibliography: 14 titles.     

Hypersonic shapes of maximum drag for a given heat-transfer rate

The problem of determining the shape of a blunt, axisymmetric body which maximizes the drag for a given heat-transfer rate and diameter is considered. For blunt bodies, pressure drag predominates and is estimated from modified Newtonian flow considerations. With regard to heat transfer, it is assumed that the body is operating in the range of hypersonic speeds where the radiative heating rate can be neglected with respect to the convective heating rate. The latter is estimated from the boundary-layer analysis due to Lees. Optimum power-law shapes as well as variational shapes are determined and are shown to yield almost identical results. For low-to-moderate values of the convective heat-transfer parameter, the optimum shape is very flat and is approximately a one-half power-law shape; in this range, spherical segments are approximately one-half power-law shapes and, hence, are nearly maximum drag shapes. There exists a maximum value of the convective heat-transfer parameter for which maximum drag shapes exist, and the corresponding optimum shape is a cone, or a power-law shape of exponent unity. This limiting shape is shown to be that which maximizes the convective heat-transfer rate for a given diameter.This research was supported in part by NASA-Manned Spacecraft Center under Contract No. NAS-6963. The authors are indebted to Dr. John J. Bertin for helpful discussions and suggestions concerning heattransfer aspects of this paper.     

Evaluation of Spatial Scan Statistics for Irregularly Shaped Clusters

Spatial scan statistics are commonly used for geographic disease cluster detection and evaluation. We propose and implement a modified version of the simulated annealing spatial scan statistic that incorporates the concept of “non-compactness” in order to penalize clusters that are very irregular in shape. We evaluate its power for the simulated annealing scan and compare it with the circular and elliptic spatial scan statistics. We observe that, with the non-compactness penalty, the simulated annealing method is competitive with the circular and elliptic scan statistic, and both have good power performance. The elliptic scan statistic is computationally faster and is well suited for mildly irregular clusters, but the simulated annealing method deals better with highly irregular cluster shapes. The new method is applied to breast cancer mortality data from northeastern United States.     

Archetypal shapes based on landmarks and extension to handle missing data

Archetype and archetypoid analysis are extended to shapes. The objective is to find representative shapes. Archetypal shapes are pure (extreme) shapes. We focus on the case where the shape of an object is represented by a configuration matrix of landmarks. As shape space is not a vectorial space, we work in the tangent space, the linearized space about the mean shape. Then, each observation is approximated by a convex combination of actual observations (archetypoids) or archetypes, which are a convex combination of observations in the data set. These tools can contribute to the understanding of shapes, as in the usual multivariate case, since they lie somewhere between clustering and matrix factorization methods. A new simplex visualization tool is also proposed to provide a picture of the archetypal analysis results. We also propose new algorithms for performing archetypal analysis with missing data and its extension to incomplete shapes. A well-known data set is used to illustrate the methodologies developed. The proposed methodology is applied to an apparel design problem in children.     

Organic skeleton correspondence using part arrangements

Shape correspondence between semantically similar organic shapes with large shape variations is a diffcult problem in shape analysis. Since part geometries are no longer similar,we claim that the challenge is to extract and compare prominent shape substructures, which are recurring part arrangements among semantically related shapes. Our main premise is that the challenge can be solved more effciently on curve skeleton graphs of shapes, which provide a concise abstraction of shape geometry and structure. Instead of directly searching exponentially many skeleton subgraphs, our method extracts the intrinsic reflectional symmetry axis of the skeleton to guide the generation of subgraphs as part arrangements. For any two subgraphs from two skeletons, their orientations are aligned and their pose variations are normalized for matching. Finally, the matchings of all subgraph pairs are evaluated and accumulated to the skeletal feature node correspondences. The comparison results with the state-of-the-art work show that our method significantly improves the effciency and accuracy of the semantic correspondence between a variety of shapes.     

An information-theoretic framework for robustness

This is a paper about the foundation of robust inference. As a specific example, we consider semiparametric location models that involve a shape parameter. We argue that robust methods result via the selection of a representative shape from a set of allowable shapes. To perform this selection, we need a measure of disparity between the true shape and the shape to be used in the inference. Given such a disparity, we propose to solve a certain minimax problem. The paper discusses in detail the use of the Kullback-Leibler divergence for the selection of shapes. The resulting estimators are shown to have redescending influence functions when the set of allowable shapes contains heavy-tailed members. The paper closes with a brief discussion of the next logical step, namely the representation of a set of shapes by a pair of selected shapes.     

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.     

Likelihood ratio tests for equality of shape under varying degrees of orientation invariance

We consider a problem from image cytometry where the objective is to describe possible changes in the shape and orientation of cellular nuclei after treatment with a toxin. The shapes of nuclei are represented by individual ellipses. It is argued that the shape comparison problem can be formulated as a generalization of a hypothesis test for the equality of covariance matrices. For many cell types, the test statistic should be invariant with respect to orientations of the cells. For other cell types, the test statistic should be equivariant with respect to orientations of the cells, but invariant with respect to orientations of the images. Likelihood ratio tests (LRTs) are derived under a Wishart model. The likelihood maximization uses a new result about the minimization of the determinant of a sum of matrices under individual rotations. The applicability and limitations of these LRTs are demonstrated by means of simulation experiments. The reference distributions of the test statistics under the null hypothesis are obtained using unrestricted and restricted randomization procedures. Justification for the Wishart model is provided using a residual diagnostic method. The scientific implications of the results are considered.     

On shape preserving extension and approximation of functions

We establish the concept of shapes of functions by using partial differential inequalites. Our definition about shapes includes some usual shapes such as convex, subharmonic, etc., and gives many new shapes of functions. The main results show that the shape preserving approximation has close relation to the shape preserving extension. One of our main results shows that if f∈C(Ω) has some shape defined by our definition, then f can be uniformly approximated by polynomials P_n ∈ ℙ_n (n∈ℕ) which have the same shape in Ω, and the degree of the approximation is Cω(f,n^−β) with constants C,β>0.     

Shape space from deformation

The construction of shape spaces is studied from a mathematical and a computational viewpoint. A program is outlined reducing the problem to four tasks: the representation of geometry, the canonical deformation of geometry, the measuring of distance in shape space, and the selection of base shapes. The technical part of this paper focuses on the second task: the specification of a deformation mixing two or more shapes in continuously changing proportions.     

A continuous model of the dynamical systems capable to memorise multiple shapes

This paper proposes the novel approach to the mathematical synthesis of continuous self-organising systems capable to memorise and restore own multiple shapes defined by means of functions of single spatial variable or parametric models in two-dimensional space. The model is based on the certain universal form of the integral operator with the kernel representing the system memory. The technique for memorising shapes uses the composition of singular kernels of integral operators. The whole system is described by the potential function, whose minimisation leads to the non-linear dynamics of shape reconstruction by integro-differential non-linear equations with partial derivatives. The corresponding models are proposed and analysed for both parametric and non-parametric shape definitions. Main features of the proposed model are considered, and the results of numerical simulation are shown in case of three shapes memorising and retrieval. The proposed model can be used in theory of smart materials, artificial intelligence and some other branches of non-linear sciences where the effect of multiple shapes memorising and retrieval appears as the core feature.     

Heuristic approaches to large-scale periodic packing of irregular shapes on a rectangular sheet

The nesting problem is a two-dimensional cutting and packing problem where the small pieces to cut have irregular shapes. A particular case of the nesting problem occurs when congruent copies of one single shape have to fill, as much as possible, a limited sheet. Traditional approaches to the nesting problem have difficulty to tackle with high number of pieces to place. Additionally, if the orientation of the given shape is not a constraint, the general nesting approaches are not particularly successful. This problem arises in practice in several industrial contexts such as footwear, metalware and furniture. A possible approach is the periodic placement of the shapes, in a lattice way. In this paper, we propose three heuristic approaches to solve this particular case of nesting problems. Experimental results are compared with published results in literature and additional results obtained from new instances are also provided.     

On the convergence of some Procrustean averaging algorithms

Euclidean “(size-and-)shape spaces” are spaces of configurations of points in R ^N modulo certain equivalences. In many applications one seeks to average a sample of shapes, or sizes-and-shapes, thought of as points in one of these spaces. This averaging is often done using algorithms based on generalized Procrustes analysis (GPA). These algorithms have been observed in practice to converge rapidly to the Procrustean mean (size-and-)shape, but proofs of convergence have been lacking. We use a general Riemannian averaging (RA) algorithm developed in [Groisser, D. (2004) “Newton's method, zeroes of vector fields, and the Riemannian center of mass”, Adv. Appl. Math. 33, pp. 95–135] to prove convergence of the GPA algorithms for a fairly large open set of initial conditions, and estimate the convergence rate. On size-and-shape spaces the Procrustean mean coincides with the Riemannian average, but not on shape spaces; in the latter context we compare the GPA and RA algorithms and bound the distance between the averages to which they converge.