On Shape of Plane Elastic Curves

We study shapes of planar arcs and closed contours modeled on elastic curves obtained by bending, stretching or compressing line segments non-uniformly along their extensions. Shapes are represented as elements of a quotient space of curves obtained by identifying those that differ by shape-preserving transformations. The elastic properties of the curves are encoded in Riemannian metrics on these spaces. Geodesics in shape spaces are used to quantify shape divergence and to develop morphing techniques. The shape spaces and metrics constructed are novel and offer an environment for the study of shape statistics. Elasticity leads to shape correspondences and deformations that are more natural and intuitive than those obtained in several existing models. Applications of shape geodesics to the definition and calculation of mean shapes and to the development of shape clustering techniques are also investigated.     

FOLD PROFILER: A MATLAB®—based program for fold shape classification

FOLD PROFILER is a MATLAB code for classifying the shapes of profiles of folded surfaces. The classification is based on the comparison of the natural fold profile with curves representing mathematical functions. The user is offered a choice of four methods, each based on a different type of function: cubic Bezier curves, conic sections, power functions and superellipses. The comparison is carried out by the visual matching of the fold profile displayed on-screen from an imported digital image and computed theoretical curves which are superimposed on the image of the fold. To improve the fit with the real fold shape, the parameters of the theoretical curves are changed by simple mouse actions. The parameters of the mathematical function that best fits the real folds are used to classify the fold shape.FOLD PROFILER allows the rapid implementation of four existing methods for fold shape analysis. The attractiveness of this analytical tool lies in the way it gives an instant visual appreciation of the effect of changing the parameters that are used to classify fold geometry.     

Intrinsic Bayesian Active Contours for Extraction of Object Boundaries in Images

We present a framework for incorporating prior information about high-probability shapes in the process of contour extraction and object recognition in images. Here one studies shapes as elements of an infinite-dimensional, non-linear quotient space, and statistics of shapes are defined and computed intrinsically using differential geometry of this shape space. Prior models on shapes are constructed using probability distributions on tangent bundles of shape spaces. Similar to the past work on active contours, where curves are driven by vector fields based on image gradients and roughness penalties, we incorporate the prior shape knowledge in the form of vector fields on curves. Through experimental results, we demonstrate the use of prior shape models in the estimation of object boundaries, and their success in handling partial obscuration and missing data. Furthermore, we describe the use of this framework in shape-based object recognition or classification.     

Analysis of planar shapes using geodesic paths on shape spaces

For analyzing shapes of planar, closed curves, we propose differential geometric representations of curves using their direction functions and curvature functions. Shapes are represented as elements of infinite-dimensional spaces and their pairwise differences are quantified using the lengths of geodesics connecting them on these spaces. We use a Fourier basis to represent tangents to the shape spaces and then use a gradient-based shooting method to solve for the tangent that connects any two shapes via a geodesic. Using the Surrey fish database, we demonstrate some applications of this approach: 1) interpolation and extrapolations of shape changes, 2) clustering of objects according to their shapes, 3) statistics on shape spaces, and 4) Bayesian extraction of shapes in low-quality images.     

Geometric Flows of Curves in Shape Space for Processing Motion of Deformable Objects

We introduce techniques for the processing of motion and animations of non‐rigid shapes. The idea is to regard animations of deformable objects as curves in shape space. Then, we use the geometric structure on shape space to transfer concepts from curve processing in ?ⁿ to the processing of motion of non‐rigid shapes. Following this principle, we introduce a discrete geometric flow for curves in shape space. The flow iteratively replaces every shape with a weighted average shape of a local neighborhood and thereby globally decreases an energy whose minimizers are discrete geodesics in shape space. Based on the flow, we devise a novel smoothing filter for motions and animations of deformable shapes. By shortening the length in shape space of an animation, it systematically regularizes the deformations between consecutive frames of the animation. The scheme can be used for smoothing and noise removal, e.g., for reducing jittering artifacts in motion capture data. We introduce a reduced‐order method for the computation of the flow. In addition to being efficient for the smoothing of curves, it is a novel scheme for computing geodesics in shape space. We use the scheme to construct non‐linear “Bézier curves” by executing de Casteljau's algorithm in shape space.     

带形状参数的Ball曲线性质及应用

构造了带形状参数的2m+1次Ball基及Ball曲线.它具有和Ball基及曲线同样的性质.通过3次带参数的Ball曲线生成圆形和花瓶的实例说明在不变动控制点的情况下,通过调整形状参数λ值可根据需要达到控制曲线形状的目的.     

Computing a compact spline representation of the medial axis transform of a 2D shape

We present a full pipeline for computing the medial axis transform of an arbitrary 2D shape. The instability of the medial axis transform is overcome by a pruning algorithm guided by a user-defined Hausdorff distance threshold. The stable medial axis transform is then approximated by spline curves in 3D to produce a smooth and compact representation. These spline curves are computed by minimizing the approximation error between the input shape and the shape represented by the medial axis transform. Our results on various 2D shapes suggest that our method is practical and effective, and yields faithful and compact representations of medial axis transforms of 2D shapes.     

A Computational Model of Multidimensional Shape

We develop a computational model of shape that extends existing Riemannian models of curves to multidimensional objects of general topological type. We construct shape spaces equipped with geodesic metrics that measure how costly it is to interpolate two shapes through elastic deformations. The model employs a representation of shape based on the discrete exterior derivative of parametrizations over a finite simplicial complex. We develop algorithms to calculate geodesics and geodesic distances, as well as tools to quantify local shape similarities and contrasts, thus obtaining a formulation that accounts for regional differences and integrates them into a global measure of dissimilarity. The Riemannian shape spaces provide a common framework to treat numerous problems such as the statistical modeling of shapes, the comparison of shapes associated with different individuals or groups, and modeling and simulation of shape dynamics. We give multiple examples of geodesic interpolations and illustrations of the use of the models in brain mapping, particularly, the analysis of anatomical variation based on neuroimaging data.     

Three-dimensional face recognition using shapes of facial curves

We study shapes of facial surfaces for the purpose of face recognition. The main idea is to 1) represent surfaces by unions of level curves, called facial curves, of the depth function and 2) compare shapes of surfaces implicitly using shapes of facial curves. The latter is performed using a differential geometric approach that computes geodesic lengths between closed curves on a shape manifold. These ideas are demonstrated using a nearest-neighbor classifier on two 3D face databases: Florida State University and Notre Dame, highlighting a good recognition performance     

2D shape morphing via automatic feature matching and hierarchical interpolation

The paper presents a new method to interpolate a pair of 2D shapes that are represented by piecewise linear curves. The method addresses two key problems in 2D shape morphing process: feature correspondence and path interpolation. First, a robust feature metric is defined to measure the similarity of a pair of 2D shapes in terms of visual appearance, orientation and relative size. Based on the metric, an optimal problem is defined and solved to associate the features on the source shape with the corresponding ones on the target shape. Then, a two-level hierarchical approach is proposed to solve the corresponding features interpolation trajectory problem. The algorithm decomposes the input shapes into a pair of corresponding coarse polygons and several pairs of corresponding features. Then the corresponding coarse polygons are interpolated in an as-rigid-as-possible plausible way; meanwhile the corresponding features are interpolated using the intrinsic method. Thus interior distortions of the intermediate shapes could be avoided and the feature details on the input shapes could be well preserved. Experimental results show that the method can generate smooth, natural and visually pleasing 2D shape morphing effects.     

带形状参数样条曲线的研究

如何通过调整形状参数修改曲线形状是计算机辅助几何设计中一个有意义的研究课题.为了有效地利用形状参数来调整曲线的形状,增强修改曲线的灵活性,研究了5种带形状参数B样条曲线的表示方法及性质,这些曲线模型都可以通过改变形状参数的取值,调整曲线接近控制多边形的程度,从而得到不同位置的连续曲线,分析了每种造型方法的形状参数对曲线形状的影响,给出了形状参数的适用范围,比较了5种造型方法的特点,通过大量的公式推导和实验,提出了利用形状参数不同取值来表示一些自由曲线的新方法,并用实例进行了说明,实验证明,C-B样条曲线、带形状参数的均匀B样条曲线、带形状参数双曲多项式的均匀B样条曲线、带形状参数三角多项式的均匀B样条曲线都可利用形状参数的特定取值表示一些工业领域常用的自由曲线,这比起用控制顶点表示同样的自由曲线更为简单.     

Generation of Smooth Surfaces by Controlling Curvature Variation

To satisfy a designer's intention for constructing aesthetic shapes such as automotive bodies, we propose a surface generation method. In the surface design process, designers determine shapes according to their great concern for the reflected images of vehicle surroundings, shade lines and highlight lines. Since reflection and shading are affected by changes of surface normal, the curvature variation of the surface, which represents the change of the surface normal, should be smooth and distributed as designers want. The proposed method controls curvature distribution directly by determining a surface shape from an evolute, which is a locus of the curvature center of the generatrix and moves along directrices to form the surface. It first generates evolutes of boundary curves to be generatrices as rational Bezier curves, then interpolates their shapes with the Bezier polygons, and locates the interpolated shape to the corresponding position of the directrices. By applying this method, we have confirmed that a smooth shape is generated from four boundary curves.     

Elastic shapes models for improving segmentation of object boundaries in synthetic aperture sonar images

We present a variational framework for naturally incorporating prior shape knowledge in guidance of active contours for boundary extraction in images. This framework is especially suitable for images collected outside the visible spectrum, where boundary estimation is difficult due to low contrast, low resolution, and presence of noise and clutter. Accordingly, we illustrate this approach using the segmentation of various objects in synthetic aperture sonar (SAS) images of underwater terrains. We use elastic shape analysis of planar curves in which the shapes are considered as elements of a quotient space of an infinite dimensional, non-linear Riemannian manifold. Using geodesic paths under the elastic Riemannian metric, one computes sample mean and covariances of training shapes in each classes and derives statistical models for capturing class-specific shape variability. These models are then used as shape priors in a variational setting to solve for Bayesian estimation of desired contours as follows. In traditional active contour models curves are driven towards minimum of an energy composed of image and smoothing terms. We introduce an additional shape term based on shape models of relevant shape classes. The minimization of this total energy, using iterated gradient-based updates of curves, leads to an improved segmentation of object boundaries. This is demonstrated using a number of shape classes in two large SAS image datasets.     

一种三角网格模型的轮廓生成方法

从三维网格模型中提取轮廓信息是一个具有挑战性的过程。现有的方法一般是基于局部形状特征 分析,如曲面的曲率和相邻面法向之间的夹角,但其特性通常对模型中的局部特征变化敏感。为了解决这个问题, 提出一种基于三维形状几何近似的轮廓提取方法。利用完善的变分几何分割算法来得到一套完整的描述性特征曲 线,首先基于变分几何近似方法划分模型为若干分片;其次提取所有分片的内部特征曲线,并过滤较短的特征曲 线;然后将分片的边界曲线平滑化;最后合并分片边界曲线与特征曲线,并延伸曲线得到闭合的轮廓。该方法的 优点是：在几何近似的基础上结合特征提取方法,使轮廓能够体现三维形状的全局结构。通过在各类网格模型上 进行实验和比较表明,该方法在提取模型轮廓的正确性和完整性方面优于现有方法。     

可变形带洞形状的表示与检测

可变形形状(shape)的表示与检测是图像处理领域的重要研究内容。提出了一种关于可变形带洞形状表示与检测的方法,采用带洞形状多边形表示可变形带洞目标形状,有效地解决了带洞形状中不同封闭曲线之间位置的表示关系;通过在带洞形状多边形中添加辅助边,将每条辅助边看成两条完全不相交的边的方法,将带洞形状多边形转化成不带洞的简单多边形,运用受限Delaunay三角剖分法(CDT)剖分多边形,得到关于带洞形状多边形的完全删除序列,运用非序列动态规划实现可变形带洞形状检测。实验结果表明,与其他相关方法相比,本文方法能够较有效地检测带洞形状目标。     

Shape Indexing and Recognition Based on Regional Analysis

Shape indexing and recognition have received great attention in multimedia processing communities due to the wide range of utilities. In achieving shape representation, most influential methods treat shapes as intrinsic curves, which is not in agreement with the way human vision systems achieve the same task. In this paper, a new framework is developed where a shape is treated as a 2-D region. We first perform an eigen analysis to align P in the standard orientation. Three numbers are generated to indicate the global geometrical nature. Next, according to the two eigen vectors, we partition P into four halves and eight quadrants. Five numbers are then produced for each region to signify its geometrical properties and relation with P. The aggregate of these numbers, 63 in total, is the actual index for P. Recognition is effected by weighted LI distances between shapes. This indexing scheme captures the global geometry of shapes and is resilient to rotations and scales, which are of crucial importance in the perceptive process of human vision systems. It can tolerate occlusions present in most standard shape datasets but is not robust against severe occlusions. Empirical studies conducted on standard synthetic and real-world datasets demonstrate encouraging performances.     

A Stochastic Order of Shape Variability with an Application to Cell Nuclei Involved in Mastitis

Variability of shapes of bidimensional closed curves is a key matter in many fields of research. A statistical order of bidimensional shape variability is introduced in this paper. For such a purpose a special class of random elements is considered. The order is defined on such a class and the main properties of the order are analyzed. Such an order involves the curvature of a special parameterization of bidimensional closed curves. The new order can be used as a basis for implementing statistical procedures, such as hypothesis testing on variability of shapes. An example is developed by means of the image analysis of cell nuclei, namely the shapes of cell nuclei in mastitis-affected cow milk and non-affected cow milk are compared.