Research On Statistical Shape Analysis And Mixture Modelling

With replied mathematics as foundation, pattern recognition is the branch of science dealing with classification and recognition employing computer technologies. Statistics and differential geometry are versatile and effective tools in modeling complicated phenomenon, thus they have been broadly applied in shape analysis and unsupervised learning of mixture modeling.Based on the theories of statistics end differential geometry, this dissertation investigates two aspects: (a) systematic studies of problems associated with shape analysis based on differential manifold (b) theories and methodology of unsupervised learning for mixture models, with particular emphasis on how to choose number of dusters k. The main research works and contributions of this dissertation are outlined as follows:For the part of shape analysis based on differential manifold, we study the theoretical framework of shape analysis on simple dosed planar curves. In traditional algorithms, statistical shape analysis relies on mathematical models built from level sets or landmarks, and then employs principal component analysis to learn parameters of critical points. However, all those algorithms suffer from certain level of human intervention or lack of flexibility of topological deformation. Therefore we need to construct a unified shape space with topological invariance to overcome those shortcomings. In such shape spaces, key parameters of shapes can be learned from trained sets with appropriate probability models, then shape recognition is achieved after one shape can evolve into other unknown shapes.In this dissertation, we build a differential manifold of infinite dimensions with arc length as function for simple dosed planar curves, and all those concepts are borrowed from differential geometry. Variation of shapes is represented as actions of Lee group on those manifolds. Invariance of rotation, translation and scale are achieved by action of low di mension group, and the smooth evolution of shapes is realized with actions on high dimension of differential homeomorphic groups, that is, resolving the geodesic path which prescribes continuous deformation between two shapes. The concept of "geodesic paths" is borrowed from differential geometry, which describes the track of transporting points in least di stance al ong curved surfaces.Widely accepted as a remarkably effective tool for statistical shape analysis, mixture modeling is also extended in this thews with theories and implementations of unsupervised learning. One of the key problems in mixture modeling is how to choose number of dusters (estimate k), while many dassical mixture modeling techniques such as maximum likelihood method and Bayesi an method require k to be known beforehand. However k is not known for most cases, and is estimated from trained sets. Thus estimation of kis the major challenge of optimizing algorithm for finite mixture modeling and must be well resolved before proceeding to estimation of other key parameters. Traditional algorithms attack this problem by adding a criterion function to original parameter estimation algorithm, which tries many different k (from k_min to k_max) and compares the criteria of likelihood for different k. Eventually the "k" is chosen as the optimized duster number when certain criterion is met. This approach has to estimate parameters for many different k, leading to very demanding computation time when the density function of mixture model is T-distribution. We propose a new algorithm to reduce computation time, by Rival Penalized Expectation-Maximization (RPEM) algorithm in the case of T mixture model. The idea is to include penalty term in expectation maximization algorithm (EM), that is, by introducing a special weight in likelihood function. When the initial parameters are updated in EM algorithm, points are updated as the winner points and secondary winner points, where the winner get positive learning rate while the secondary winner get negative ones, thus making part of the initial seeding points converge to the actual centers of duster and other seeds are pushed further away from the centers. This new algorithm can provide very good estimate of kvalue with only once parameter estimating, thus featuring high performance.