A Model of Human Categorization and Similarity Based Upon Category Theory

Categorization and the judgement of similarity are fundamental in cognition. We propose that these and other activities are based upon an underlying structure of knowledge, or concept representation, in the brain. Further, we propose that this structure can be represented mathematically in a declarative form via category theory, the mathematical theory of structure. We test the resulting mathematical model in an experiment in which human subjects provide judgements of similarity for pairs of line drawings using a numerical scale to represent degrees of similarity. The resulting numerical similarities are compared with those derived from the category-theoretic model by comparing diagrams. The diagrams represent distributed concept structures underlying the line drawings. To compare with a more conventional analysis technique, we also compare the human judgements with those provided by a two-dimensional feature space model equipped with a distance metric for the line drawings. The results are equally favorable for both models. Because of this and the putative explanatory power of the category-theoretic model, we propose that this model is worthy of further exploration as a mathematical model for cognitive science

Hierarchical representation of 2D polygons based on approximate skeletons

No abstract availabl

Trends in Mathematical Imaging and Surface Processing

Motivated both by industrial applications and the challenge of new problems, one observes an increasing interest in the field of image and surface processing over the last years. It has become clear that even though the applications areas differ significantly the methodological overlap is enormous. Even if contributions to the field come from almost any discipline in mathematics, a major role is played by partial differential equations and in particular by geometric and variational modeling and by their numerical counterparts. The aim of the workshop was to gather a group of leading experts coming from mathematics, engineering and computer graphics to cover the main developments

matching, interpolation, and approximation ; a survey

In this survey we consider geometric techniques which have been used to measure the similarity or distance between shapes, as well as to approximate shapes, or interpolate between shapes. Shape is a modality which plays a key role in many disciplines, ranging from computer vision to molecular biology. We focus on algorithmic techniques based on computational geometry that have been developed for shape matching, simplification, and morphing

How to Walk Your Dog in the Mountains with No Magic Leash

We describe a $O(\log n )$-approximation algorithm for computing the homotopic \Frechet distance between two polygonal curves that lie on the boundary of a triangulated topological disk. Prior to this work, algorithms were known only for curves on the Euclidean plane with polygonal obstacles. A key technical ingredient in our analysis is a $O(\log n)$-approximation algorithm for computing the minimum height of a homotopy between two curves. No algorithms were previously known for approximating this parameter. Surprisingly, it is not even known if computing either the homotopic \Frechet distance, or the minimum height of a homotopy, is in NP

Planar Shape Interpolation Based on Local Injective Mapping

在只给出用简单多边形表示的两输入形状的情况下,实现一种简单易用、自然高效的形状插值方法.首先利用基于形状感知的特征匹配算法生成源形状和目标形状之间的匹配;之后在源形状上构造三角剖分,并通过求解映射到目标形状上的尽量刚体的局部单射得到同构三角剖分;最后利用扭曲有界的插值方法得到中间序列.实验结果表明,该方法构造的形变结果能较好地体现源形状和目标形状的特征对应信息,形变过程自然,扭曲较小.This paper presents an efficient and easy-to-use planar shape interpolation method, given two input shapes represented by simple polygons. We firstly used a perception-based feature matching algorithm to match the feature points in the source shape with the target shape, then built compatible triangulations by constructing a locally injective mapping between the source and target shapes. Finally, an interpolation method with bounded distortion was adopted to get intermediate frames. Experimental results show that the interpolation results by our method can well reflect the feature correspondences between the source and the target shapes, and the resultant deformation is visually pleasing with less distortion.国家自然科学基金(61472332);; 中央高校基本科研业务费专项基金(20720140520

Theory and applications of bijective barycentric mappings

Barycentric coordinates provide a convenient way to represent a point inside a triangle as a convex combination of the triangle's vertices, and to linearly interpolate data given at these vertices. Due to their favourable properties, they are commonly applied in geometric modelling, finite element methods, computer graphics, and many other fields. In some of these applications it is desirable to extend the concept of barycentric coordinates from triangles to polygons. Several variants of such generalized barycentric coordinates have been proposed in recent years. An important application of barycentric coordinates consists of barycentric mappings, which allow to naturally warp a source polygon to a corresponding target polygon, or more generally, to create mappings between closed curves or polyhedra. The principal practical application is image warping, which takes as input a control polygon drawn around an image and smoothly warps the image by moving the polygon vertices. A required property of image warping is to avoid fold-overs in the resulting image. The problem of fold-overs is a manifestation of a larger problem related to the lack of bijectivity of the barycentric mapping. Unfortunately, bijectivity of such barycentric mappings can only be guaranteed for the special case of warping between convex polygons or by triangulating the domain and hence renouncing smoothness. In fact, for any barycentric coordinates, it is always possible to construct a pair of polygons such that the barycentric mapping is not bijective. In the first part of this thesis we illustrate three methods to achieve bijective mappings. The first method is based on the intuition that, if two polygons are sufficiently close, then the mapping is close to the identity and hence bijective. This suggests to ``split'' the mapping into several intermediate mappings and to create a composite barycentric mapping which is guaranteed to be bijective between arbitrary polygons, polyhedra, or closed planar curves. We provide theoretical bounds on the bijectivity of the composite mapping related to the norm of the gradient of the coordinates. The fact that the bound depends on the gradient implies that these bounds exist only if the gradient of the coordinates is bounded. We focus on mean value coordinates and analyse the behaviour of their directional derivatives and gradient at the vertices of a polygon. The composition of barycentric mappings for closed planar curves leads to the problem of blending between two planar curves. We suggest to solve it by linearly interpolating the signed curvature and then reconstructing the intermediate curve from the interpolated curvature values. However, when both input curves are closed, this strategy can lead to open intermediate curves. We present a new algorithm for solving this problem, which finds the closed curve whose curvature is closest to the interpolated values. Our method relies on the definition of a suitable metric for measuring the distance between two planar curves and an appropriate discretization of the signed curvature functions. The second method to construct smooth bijective mappings with prescribed behaviour along the domain boundary exploits the properties of harmonic maps. These maps can be approximated in different ways, and we discuss their respective advantages and disadvantages. We further present a simple procedure for reducing their distortion and demonstrate the effectiveness of our approach by providing examples. The last method relies on a reformulation of complex barycentric mappings, which allows us to modify the ``speed'' along the edges to create complex bijective mappings. We provide some initial results and an optimization procedure which creates complex bijective maps. In the second part we provide two main applications of bijective mapping. The first one is in the context of finite elements simulations, where the discretization of the computational domain plays a central role. In the standard discretization, the domain is triangulated with a mesh and its boundary is approximated by a polygon. We present an approach which combines parametric finite elements with smooth bijective mappings, leaving the choice of approximation spaces free. This approach allows to represent arbitrarily complex geometries on coarse meshes with curved edges, regardless of the domain boundary complexity. The main idea is to use a bijective mapping for automatically warping the volume of a simple parametrization domain to the complex computational domain, thus creating a curved mesh of the latter. The second application addresses the meshing problem and the possibility to solve finite element simulations on polygonal meshes. In this context we present several methods to discretize the bijective mapping to create polygonal and piece-wise polynomial meshes

Photorealistic retrieval of occluded facial information using a performance-driven face model

Facial occlusions can cause both human observers and computer algorithms to fail in a variety of important tasks such as facial action analysis and expression classification. This is because the missing information is not reconstructed accurately enough for the purpose of the task in hand. Most current computer methods that are used to tackle this problem implement complex three-dimensional polygonal face models that are generally timeconsuming to produce and unsuitable for photorealistic reconstruction of missing facial features and behaviour. In this thesis, an image-based approach is adopted to solve the occlusion problem. A dynamic computer model of the face is used to retrieve the occluded facial information from the driver faces. The model consists of a set of orthogonal basis actions obtained by application of principal component analysis (PCA) on image changes and motion fields extracted from a sequence of natural facial motion (Cowe 2003). Examples of occlusion affected facial behaviour can then be projected onto the model to compute coefficients of the basis actions and thus produce photorealistic performance-driven animations. Visual inspection shows that the PCA face model recovers aspects of expressions in those areas occluded in the driver sequence, but the expression is generally muted. To further investigate this finding, a database of test sequences affected by a considerable set of artificial and natural occlusions is created. A number of suitable metrics is developed to measure the accuracy of the reconstructions. Regions of the face that are most important for performance-driven mimicry and that seem to carry the best information about global facial configurations are revealed using Bubbles, thus in effect identifying facial areas that are most sensitive to occlusions. Recovery of occluded facial information is enhanced by applying an appropriate scaling factor to the respective coefficients of the basis actions obtained by PCA. This method improves the reconstruction of the facial actions emanating from the occluded areas of the face. However, due to the fact that PCA produces bases that encode composite, correlated actions, such an enhancement also tends to affect actions in non-occluded areas of the face. To avoid this, more localised controls for facial actions are produced using independent component analysis (ICA). Simple projection of the data onto an ICA model is not viable due to the non-orthogonality of the extracted bases. Thus occlusion-affected mimicry is first generated using the PCA model and then enhanced by accordingly manipulating the independent components that are subsequently extracted from the mimicry. This combination of methods yields significant improvements and results in photorealistic reconstructions of occluded facial actions