Nearness to Local Subspace Algorithm for Subspace and Motion Segmentation

There is a growing interest in computer science, engineering, and mathematics for modeling signals in terms of union of subspaces and manifolds. Subspace segmentation and clustering of high dimensional data drawn from a union of subspaces are especially important with many practical applications in computer vision, image and signal processing, communications, and information theory. This paper presents a clustering algorithm for high dimensional data that comes from a union of lower dimensional subspaces of equal and known dimensions. Such cases occur in many data clustering problems, such as motion segmentation and face recognition. The algorithm is reliable in the presence of noise, and applied to the Hopkins 155 Dataset, it generates the best results to date for motion segmentation. The two motion, three motion, and overall segmentation rates for the video sequences are 99.43%, 98.69%, and 99.24%, respectively

Improved image analysis by maximised statistical use of geometry-shape constraints

Identifying the underlying models in a set of data points contaminated by noise and outliers, leads to a highly complex multi-model fitting problem. This problem can be posed as a clustering problem by the construction of higher order affinities between data points into a hypergraph, which can then be partitioned using spectral clustering. Calculating the weights of all hyperedges is computationally expensive. Hence an approximation is required. In this thesis, the aim is to find an efficient and effective approximation that produces an excellent segmentation outcome. Firstly, the effect of hyperedge sizes on the speed and accuracy of the clustering is investigated. Almost all previous work on hypergraph clustering in computer vision, has considered the smallest possible hyperedge size, due to the lack of research into the potential benefits of large hyperedges and effective algorithms to generate them. In this thesis, it is shown that large hyperedges are better from both theoretical and empirical standpoints. The efficiency of this technique on various higher-order grouping problems is investigated. In particular, we show that our approach improves the accuracy and efficiency of motion segmentation from dense, long-term, trajectories. A shortcoming of the above approach is that the probability of a generated sample being impure increases as the size of the sample increases. To address this issue, a novel guided sampling strategy for large hyperedges, based on the concept of minimizing the largest residual, is also included. It is proposed to guide each sample by optimizing over a $k$\textsuperscript{th} order statistics based cost function. Samples are generated using a greedy algorithm coupled with a data sub-sampling strategy. The experimental analysis shows that this proposed step is both accurate and computationally efficient compared to state-of-the-art robust multi-model fitting techniques. However, the optimization method for guiding samples involves hard-to-tune parameters. Thus a sampling method is eventually developed that significantly facilitates solving the segmentation problem using a new form of the Markov-Chain-Monte-Carlo (MCMC) method to efficiently sample from hyperedge distribution. To sample from the above distribution effectively, the proposed Markov Chain includes new types of long and short jumps to perform exploration and exploitation of all structures. Unlike common sampling methods, this method does not require any specific prior knowledge about the distribution of models. The output set of samples leads to a clustering solution by which the final model parameters for each segment are obtained. The overall method competes favorably with the state-of-the-art both in terms of computation power and segmentation accuracy