Image Analysis on Symmetric Positive Definite Manifolds

Abstract

Over the last two decades, the research community has witnessed extensive research growth in the field of analysing and understanding scenes. Automatic scene analysis can support many critical applications, from person reidentification as an advanced security tool, to real-time action classification as an assistive technology for disabled patients. However, building effective systems is still a challenge due to the presence of occlusion, varying illumination, varying pose and other factors encountered in the practical environment. To deal with the real world environment, which is naturally not free from noise, a recent trend in computer vision is to represent a given image through a covariance matrix of a set of extracted features. Covariance matrices are robust to noise and are well known to be compact and informative feature descriptors. Non-singular covariance matrices are naturally symmetric positive definite (SPD) matrices which form connected Riemannian manifolds. As such, their underlying distance and similarity functions might not be accurately defined in Euclidean space, and consequently the Riemannian geometry needs to be considered in order to solve scene analysing tasks. The traditional methods of analysing such manifolds require embedding them in Euclidean spaces, a process which can be interpreted as warping the feature space. However, embedding manifolds is not free from drawbacks and it can lead to limitations, as the manifold structure may not be accurately preserved. In this work we propose three methods for analysing SPD matrices on Riemannian manifolds that unlike traditional methods respect the underlying structure of a given image, while considering the computational complexity of the learning algorithm. While all three methods offer strong solutions for the task of image analysis over SPD manifolds that outperform state-of-the-art methods, each of them tends to tackle one vision application better than the rest. This is owed to the existing differences between each vision application. Although all of these vision tasks can be categorised as a image classification problem, each application offers unique challenges, such as very limited training data, strong pose variation etc. To be more specific, the 1first proposed method outperforms the rest of the proposed methods in face recognition; the extension of the second method performs very well in the task of person re-identification; and the last proposed method outperforms other two in the task of texture recognition. This work addresses the challenge of analysing SPD manifolds using the below proposed methods: 1. Graph-Embedding Discriminant Analysis 2. Relational Divergence Based Classification 3. Random Projections The first method proposes to embed Riemannian manifolds into Reproducing Kernel Hilbert Spaces (RKHS) and then tackle the problem of discriminant analysis on the Hilbert space. To achieve an efficient machinery, we present a graph-based local discriminant analysis that utilises within-class and between-class similarity graphs to characterise intra-class compactness and inter-class separability. Experiments on face recognition, texture classification and person re-identification indicate that the proposed method obtains marked improvement in discrimination accuracy in comparison to several state-of-the-art methods. The second proposed method suggests direct classification on the Manifold by presenting each SPD matrix through its similarity vector with the number of other SPD matrices. In addition, to speed up the process, the proposed method employs the recently introduced Stein divergence. Classification problems on manifolds are then effectively converted into the problem of finding appropriate machinery over the space of similarities. Experiments on face recognition, texture classification and person re-identification show that in comparison to well-known methods, the proposed approach obtains a significant improvement in image classification, while also being several orders of magnitude faster. The third proposed algorithm proposes to project SPD matrices using a set of random projection hyperplanes over an RKHS into a random projection space, which leads to representing each matrix as a vector of projection coefficients. Experiments on face recognition, person re-identification and texture classification show that the proposed approach, in comparison to well-known methods, obtains a significant improvement in image classification, while also being relatively faster. Experiments and comparative evaluations on standard datasets from a variety of image analysis applications suggest that the three proposed algorithms obtain considerably better results (both qualitatively and quantitatively) than other well-known techniques available in the literature. While all three proposed methods have been designed to work for scenes analysis, the experiment result suggest that based on the nature of the given application (i.e., the number of points in the training set), one of the proposed algorithms might be favoured over the rest.