Nonlinear Mean Shift Algorithm Over Riemannian Manifolds

The clustering problem has been addressed in many contexts and byresearchers in many fields, such as the feature clustering process in imagesegmentation, image retrieval, face recognition and object tracking. The originalMean Shift algorithm is widely applied for nonparametric clustering in vectorspaces. In2005, Mean Shift was extended to be used on a particular class ofRiemannian manifolds, matrix Lie groups. A similar, but more general algorithmwas proposed, which could handle points lying on any Riemannian manifolds, notnecessarily Lie groups. After that, the application of Mean Shift in different classesof manifolds has been developed. To derive a general algorithm, the concept ofnonlinear Mean Shift is proposed based on the various applications in2006. Thetheoretical properties of the algorithm are discussed. A proof for convergence of thenonlinear Mean Shift was given.The main work for this dissertation can be summarized as follows:The theory of Mean Shift is discussed firstly. It is proved by data and imageexperiments that the selection of kernel function, kernel radius and threshold ofMean Shift can influence the performance of the clustering and image segmentation.When making clustering analysis and segmenting images with Mean Shift, differentkernel functions will produce different results. For a kernel function, its kernelradius has an ideal range for good clustering and image segmentation performance.Different thresholds will give distinct performance of image segmentation with thesame kernel function and kernel radius.Mean Shift is used in a large class of frequently occurring non-vector spaces invision. Those data distributed on a manifold space, not on a vector space. So theconcepts, properties and computational details are presented for frequentlyoccurring classes of manifolds such as matrix Lie groups, Grassmann manifoldsand essential manifolds.Finally, considering the limit of original Mean Shift clustering points in avector space, the nonlinear Mean Shift over Riemannian manifolds was introduced.This algorithm is capable of clustering distributed points on many classes ofmanifolds. The convergence was proposed and proved complementarily, which guarantee the validity of this improved algorithm.