Research Of Discrete Shape Manifold Method And Applications

As an old research topic,shape analysis has attracted the researchers’ attention to the geometry,statistics,pattern recognition,computer vision,graphics,biomedicine,and so on.A huge number of shape analysis methods have been proposed,and many applicationoriented techniques have been carried out.Generally speaking,The main application of shape analysis includes shape retrieval,shape visualization,shape data summary,shape classification and so on.The early years of the new millennium saw a renewed focus and energy in the shape analysis areas under the scholars’ efforts of geometry and statistics.The researchers began to seek a framework suitable for different application scenarios,in order to give a unified theoretical explanation for different application scenarios and stimulate new applications based on theoretical exploration.After more than ten years of development,Anuj Srivastava,Eric P.Klassen,and other research groups have built and promoted a framework for theoretical research in shape analysis.The framework is designed as an infinite-dimensional shape manifold so that different applications correspond to different mathematical concepts on shape manifold,such as geodesic,geodesic length,probability model,and so on.An infinite-dimensional shape manifold is an theoretical effective framework.However,there are still some issues should be solved in the engineering application:1.Whether infinite dimensional manifold is also the best choice in engineering application;2.What is the relationship between shape constraints and the basis of tangent space on shape manifold;3.How to improve the retrieval score of shape manifold methods in general shape database.In this thesis,we mainly concentrate on shape expression,shape manifold dimension,shape manifold tangent space base,statistical shape analysis and so on based on the shape manifold theory.The main work and contribution of this thesis includes:(1)As the infinite-dimensional shape manifold is not suitable to engineering application,this thesis propose a finite dimensional shape manifold framework based on discrete curve model.This framework theoretically avoids to approximate the infinite dimensional shape manifold as a finite dimensional manifold.Furthermore,this thesis uses the frame bundle theory to separate coordinate system and coordinate for tangent space.Then,the nonlinear structure group on shape manifold is transformed from the local coordinate to the local coordinate system so that the relationship between local coordinates is linearized.The experiments of geodesic visualization and shape retrieval prove the effectiveness.(2)To study the relationship between shape constructions and the basis of tangent space on shape manifold,this thesis focus on the fundamental issues of shape manifold,such as the definition and the dimension of tangent space of shape manifold,the Riemannian metric definition,the algorithm of geodesics,and so on.Furthermore,the algorithms for statistic shape analysis are proposed to demonstrate the proposed framework’s effectiveness.Finally,a variety of experiments,such as geodesic visualization,shape retrieval,and shape arithmetics,demonstrate the effectiveness and expandable potential of the framework.(3)Generally,manifold-based methods obtain a higher score for shape retrieval in a biological database,while the retrieval score in a general shape database is low.To solve this problem,we propose an integral curve prior.It is based on an important observation that the shapes from biological database generally have low shape complexity,while the complexity of shape from the general shape database is high.A total variation function is used to measure the complexity of shape,and to show integral curve prior will largely reduce the shape complexity.The experiment results on general shape database demonstrate the power of the proposed prior.(4)A novel mechanism is proposed to automatically generate manipulator trajectories,where the manipulator trajectory is formulated as a closed curve on a trajectory manifold.Then,the issue of trajectory generation is explained as the issue of curve generation,i.e.,the data generation on the shape manifold.On the one hand,this study demonstrates that the shapes of curves in higher-dimensional spaces(trajectory manifold)also have an important role to play.On the other hand,the proposed mechanism provides a new solution for manipulator trajectory generation.