Distance Functions: Theory, Algorithms and Applications

Distance functions are at the core of numerous computer vision and machine learning tasks. For example, a multiple view geometry application (such as homography estimation), often includes a descriptor matching step. First, small image patches are selected. Then, descriptors (vectors) are computed to describe these patches. To decide if two descriptors (from two different images) are from the same real-world location the descriptors are compared using a distance function. The choice of the distance function determines both the accuracy and the speed of the method. Distance functions are also used for image retrieval and determine both the percent of similar images returned and the time it takes to return them. In machine learning, knearest neighbor classification performance is effected by the chosen distance function. This thesis introduces several new distance functions and investigates their properties. Special emphasis is on practical applicability with theoretical insights. The thesis presents efficient algorithms to compute several distances. The distances are designed to be robust to common image noise such as occlusion, geometrical transforms, light changes and non-rigid deformations. On the other hand, they are designed to be as distinctive as possible. Our proposed methods have been successfully used both by computer vision researchers and by researchers in other fields. The success of the methods in other fields is probably because the noise characteristics in those fields are similar to image noise characteristics. The first and second parts of the thesis are devoted to the Earth Mover’s Distance (EMD) [1]. The EMD is a generalization of the transportation distance (also known as Monge-Kantorovich, Mallows, 1 Wasserstein and match distance) for non-normalized histograms. We introduce a new generalization of the transportation distance for non-normalized histograms called, ÊMD. Unlike the classic EMD, ÊMD is a metric even when the histograms are non-normalized. Additionally, in practice, ÊMD is often better than EMD. We also show that ÊMD is a generalization of the L1 distance. Histograms of oriented gradient descriptors are ubiquitous tools in numerous computer vision tasks. Unfortunately, the distances that are used to match such descriptors do not take into account the special structure of these histograms. One problem with these distances is that they only compare corresponding bins of histograms. Although there are cross-bin distances that take into account the amount of similarity between all bins, they are not adequate to the histograms of oriented gradients that are computed from images. Additionally, most cross-bin distances are too slow. We present a