Micromachining of (hhl) silicon structures: experiments and 3D simulation of etched shapes

The micromachining of various (hhl) silicon plates in a 35% KOH-water etchant is studied. Experimental shapes for membranes and mesa etched with initially circular masks are discussed. Theoretical 3D etched shapes for such microstructures are derived from a numerical simulation based on the tensorial model for the anisotropic wet etching. Experimental and theoretical shapes show a fair agreement, indicating a satisfactory adjustment of the dissolution slowness surface related to the etching of silicon in KOH etchant. The interest of the 3D simulation for designing mask patterns is outlined.     

Shape averaging and its applications to industrial design

A computer-assisted technique called shape averaging is presented. Shape averaging produces an abstraction of the typical representation from a set of shapes. Since the averaging is assumed to preserve the characteristics of the original shapes, the result is useful in predicting trends in form or extracting stereotypes from a group of related shapes. The technique can be used to create new forms by blending global features of existing unrelated shapes. The syntactic averaging of shapes consisting of 2-D planar polygons or of 3-D objects represented by planar contours is examined. An algorithm is presented to determine the correspondence between polygons defined by arbitrary numbers of vertices. Algorithms to extract the mean, the median, and the mode from the shapes are also introduced. Potential applications of shape averaging in design are illustrated     

Structure Preserving Manipulation and Interpolation for Multi‐element 2D Shapes

This paper presents a method that generates natural and intuitive deformations via direct manipulation and smooth interpolation for multi‐element 2D shapes. Observing that the structural relationships between different parts of a multi‐element 2D shape are important for capturing its feature semantics, we introduce a simple structure called a feature frame to represent such relationships. A constrained optimization is solved for shape manipulation to find optimal deformed shapes under user‐specified handle constraints. Based on the feature frame, local feature preservation and structural relationship maintenance are directly encoded into the objective function. Beyond deforming a given multi‐element 2D shape into a new one at each key frame, our method can automatically generate a sequence of natural intermediate deformations by interpolating the shapes between the key frames. The method is computationally efficient, allowing real‐time manipulation and interpolation, as well as generating natural and visually plausible results.     

An Optimization Approach to Improving Collections of Shape Maps

Finding an informative, structure‐preserving map between two shapes has been a long‐standing problem in geometry processing, involving a variety of solution approaches and applications. However, in many cases, we are given not only two related shapes, but a collection of them, and considering each pairwise map independently does not take full advantage of all existing information. For example, a notorious problem with computing shape maps is the ambiguity introduced by the symmetry problem — for two similar shapes which have reflectional symmetry there exist two maps which are equally favorable, and no intrinsic mapping algorithm can distinguish between them based on these two shapes alone. Another prominent issue with shape mapping algorithms is their relative sensitivity to how “similar” two shapes are — good maps are much easier to obtain when shapes are very similar. Given the context of additional shape maps connecting our collection, we propose to add the constraint of global map consistency, requiring that any composition of maps between two shapes should be independent of the path chosen in the network. This requirement can help us choose among the equally good symmetric alternatives, or help us replace a “bad” pairwise map with the composition of a few “good” maps between shapes that in some sense interpolate the original ones. We show how, given a collection of pairwise shape maps, to define an optimization problem whose output is a set of alternative maps, compositions of those given, which are consistent, and individually at times much better than the original. Our method is general, and can work on any collection of shapes, as long as a seed set of good pairwise maps is provided. We demonstrate the effectiveness of our method for improving maps generated by state‐of‐the‐art mapping methods on various shape databases.     

Deformation by examples: a density flow approach

In this article, a shape transformation technique is introduced for deforming objects based on a given deformation example. The example consists of two reference shapes representing two different states of an object. The reference shapes are assumed to morph from one state to the other. The evolution between the two reference shapes determines the shape transformation function. Any given objects can then be deformed by the same transformation. A continuous 4D Radial Basis Function is used to construct a density flow field (an extension of the optical flow in computer vision) representing the shape transformation of the example in 3‐space. Objects embedded in the density flow field are deformed by moving vertices of the objects along the density flow vectors. Additional parameters are introduced to control the process of the deformation. This provides explicit control on the shape of the object obtained in the deformation process. Copyright © 2007 John Wiley & Sons, Ltd.     

Polygonal representation of digital planar curves through dominant point detection—a nonparametric algorithm

We describe a new algorithm that detects a set of feature points on the boundary of an 8-connected shape that constitute the vertices of a polygonal approximation of the shape itself. The set of feature points (nodes) is a ranked subset of the original shape points whose connected left and right arm extents cover the entire shape. Nodes are ranked based on their strength (in terms of their importance to other boundary points), length of support region, and distance from the centroid. The polygon obtained by linking the detected nodes approximates the contour in an intuitive way. The proposed algorithm does not require an input parameter and works well for shapes with features of multiple sizes.     

An SL(2) Invariant Shape Median

Median averaging is a powerful averaging concept on sets of vector data in finite dimensions. A generalization of the median for shapes in the plane is introduced. The underlying distance measure for shapes takes into account the area of the symmetric difference of shapes, where shapes are considered to be invariant with respect to different classes of affine transformations. To obtain a well-posed problem the perimeter is introduced as a geometric prior. Based on this model, an existence result can be established in the class of sets of finite perimeter. As alternative invariance classes other classical transformation groups such as pure translation, rotation, scaling, and shear are investigated. The numerical approximation of median shapes uses a level set approach to describe the shape contour. The level set function and the parameter sets of the group action on every given shape are incorporated in a joint variational functional, which is minimized based on step size controlled, regularized gradient descent. Various applications show in detail the qualitative properties of the median.     

Visual spatio-temporal function-based querying

Visual interfaces are very important for human interactions in cyberworlds. Visual spatio-temporal querying should be one of the basic tools for data mining and retrieval in cyberworlds. In this paper, we propose a novel function-based query model for arbitrary shape spatio-temporal querying. The queries are defined as geometric shapes changing over time. In our model, data are interpreted geometrically as multidimensional points with time dimension or as moving points. The queries are formulated with geometric objects and operations over them to form the query solid changing over time. The proposed query model allows us to pose arbitrary shape spatio-temporal range queries. With the uniform geometric model we integrate visual mining and querying of time-dependent data employing 3D visualization tools. It allows for creating an intuitive visual interface using 2D projections of 3D query shapes. Our approach combines visualization of spatio-temporal data with visualization of the range query formulation employing very compact function-based query model. The implemented visual query system and its visual interface are proposed and described. An example of application of the system in analysis of simulation results in molecular dynamics is considered.     

A Continuum Mechanical Approach to Geodesics in Shape Space

In this paper concepts from continuum mechanics are used to define geodesic paths in the space of shapes, where shapes are implicitly described as boundary contours of objects. The proposed shape metric is derived from a continuum mechanical notion of viscous dissipation. A geodesic path is defined as the family of shapes such that the total amount of viscous dissipation caused by an optimal material transport along the path is minimized. The approach can easily be generalized to shapes given as segment contours of multi-labeled images and to geodesic paths between partially occluded objects. The proposed computational framework for finding such a minimizer is based on the time discretization of a geodesic path as a sequence of pairwise matching problems, which is strictly invariant with respect to rigid body motions and ensures a 1–1 correspondence along the induced flow in shape space. When decreasing the time step size, the proposed model leads to the minimization of the actual geodesic length, where the Hessian of the pairwise matching energy reflects the chosen Riemannian metric on the underlying shape space. If the constraint of pairwise shape correspondence is replaced by the volume of the shape mismatch as a penalty functional, one obtains for decreasing time step size an optical flow term controlling the transport of the shape by the underlying motion field. The method is implemented via a level set representation of shapes, and a finite element approximation is employed as spatial discretization both for the pairwise matching deformations and for the level set representations. The numerical relaxation of the energy is performed via an efficient multi-scale procedure in space and time. Various examples for 2D and 3D shapes underline the effectiveness and robustness of the proposed approach.     

2D-Shape Analysis Using Conformal Mapping

The study of 2D shapes and their similarities is a central problem in the field of vision. It arises in particular from the task of classifying and recognizing objects from their observed silhouette. Defining natural distances between 2D shapes creates a metric space of shapes, whose mathematical structure is inherently relevant to the classification task. One intriguing metric space comes from using conformal mappings of 2D shapes into each other, via the theory of Teichmüller spaces. In this space every simple closed curve in the plane (a “shape”) is represented by a ‘fingerprint’ which is a diffeomorphism of the unit circle to itself (a differentiable and invertible, periodic function). More precisely, every shape defines to a unique equivalence class of such diffeomorphisms up to right multiplication by a Möbius map. The fingerprint does not change if the shape is varied by translations and scaling and any such equivalence class comes from some shape. This coset space, equipped with the infinitesimal Weil-Petersson (WP) Riemannian norm is a metric space. In this space, the shortest path between each two shapes is unique, and is given by a geodesic connecting them. Their distance from each other is given by integrating the WP-norm along that geodesic. In this paper we concentrate on solving the “welding” problem of “sewing” together conformally the interior and exterior of the unit circle, glued on the unit circle by a given diffeomorphism, to obtain the unique 2D shape associated with this diffeomorphism. This will allow us to go back and forth between 2D shapes and their representing diffeomorphisms in this “space of shapes”. We then present an efficient method for computing the unique shortest path, the geodesic of shape morphing between each two end-point shapes. The group of diffeomorphisms of S¹ acts as a group of isometries on the space of shapes and we show how this can be used to define shape transformations, like for instance ‘adding a protruding limb’ to any shape.     

Mixed-aspect fractal surfaces

In order to provide accurate tools to model original surfaces in a Computer Aided Geometric Design context, we develop a formalism based on iterated function systems. This model enables us to represent both smooth and fractal free-form curves and surfaces. But, because of the self-similarity property underlying the iterated function systems, curves and surfaces can only have homogeneous roughness. The aim of our work was to elaborate a method to build parametric shapes (curves, surfaces, …) with a non-uniform local aspect: every point is assigned a “geometric texture” that evolves continuously from a smooth to a rough aspect. The principle is to blend shapes with uniform aspects to define a shape with a variable aspect. A blending function controls the influence of each initial shape. An illustrated application is then built, joining surfaces characterized by different kinds of roughness.     

基于样例学习的面部特征自动标定算法

面部特征标定是人脸识别中的一个关键问题.提出了一种基于样例学习的面部特征自动标定(人脸形状自动提取)方法.该方法是基于下面假设提出来的:人脸图像差和形状差之间存在一种近似的线性关系--相似的人脸图像在较大程度上蕴涵着相似的形状.因此,给定标注了特征点的人脸图像学习集,则任意新的输入人脸图像的面部形状可以采用如下方法估计:测量该人脸图像和训练集中图像的相似度,并将同样的相似度用于该人脸图像形状的重建.即:如果输入人脸图像可以表示为训练图像的优化的线性组合,那么同样的线性组合系数就可以直接用于训练集对应形状的线性组合从而得到输入人脸图像的形状.实验表明,该算法相对于其他传统的特征标定算法具有可比的精度和较快的速度.并且,还将此算法扩展到了多姿态情况下,实现了多姿态人脸图像形状的自动提取.     

Arithmetic operations among shapes using shape numbers

In this work, a notation is given called the Discrete Geometry of Shapes, which describes the forms or shapes of flat regions limited by simply connected curves. A procedure is given that deduces from every region a unique number (its shape number) independent of translation and rotation, and optionally, of size and origin.All the integer numbers contain all the universe of discrete shapes (of course with different precision). In this universe there are shapes such as straight lines, circumferences, ellipses, parabolas, trigonometric functions, graphics of time, absorption waves, etc.The Discrete Geometry of Shapes is one-dimensional. It does not use the definition of equation and function to define shapes in a rectangular co-ordinate plane. With this notation it is possible to generate shapes with any characteristics by generating numerical sequences; also it is possible to do arithmetic operations among shapes. For example, the addition of a square and a circle, the average of a triangle and a circle, the square root of a pentagon, the numerical relations between given shapes, etc.Section V of this work describes the third dimension in the Discrete Geometry of Shapes for surfaces and volumes by means of a vector of shape numbers. It is possible to add surfaces, to divide volumes, to obtain the square root of a volume, etc.The main objective of this notation is the simplification of some mathematical and geometrical processes in this analysis of shapes and surfaces.     

Retrieving Similar Shapes Effectively and Efficiently

In this paper, we address the following problem: given a large collection of shapes and a query shape, retrieve all shapes (from the shape database) that are similar to the query shape. A generalized centroid-radii model is used to model all forms of shapes — convex shapes, concave shapes and shapes with holes. Under the model, a shape is represented by a set of vectors, each obtained from the radii emanating from the centroid of a virtual concentric ring.The model can also facilitate multi-resolution and similarity retrievals. Furthermore, using the model, the shape of an object can be transformed into a point in a high dimensional data space. To speed up the retrieval of similar shapes, we also propose a multi-level R-tree index, called the Nested R-trees (NR-trees). Unlike traditional high-dimensional index structures that index a high-dimensional point as it is (with its full dimension), the NR-trees splits the dimensionality of the point into a set of lower dimensions that are indexed by levels of the NR-trees. We also proposed a quick filtering mechanism to further prune the search space.We implemented a shape retrieval system that employs the generalized centroid-radii model and the NR-trees with the filtering mechanism. Our experimental study shows the effectiveness of the proposed shape model, and the efficiency of the NR-trees. The results also show that the filtering mechanism can significantly reduce the retrieval time.