SqN: A Novel Distance Measure for Multiple Images and Applications

In this thesis, a framework for measuring the similarity between several images in the form of a distance measure, namely SqN, is presented and discussed. The design of SqN naturally fits into the standard formulation of variational image registration and generalizes the concept of a distance measure to more than two images.
Quantifying the similarity of images is an important task in modern image pro- cessing. In the literature, a variety of approaches to measure the similarity of two images are discussed. Applications such as dynamic imaging or histological serial sectioning require the registration of a large number of images. For these cases an extension to a method for similarity measurement of multiple images is a logical next step. The literature is still rather modest on this subject.
Especially for the registration of multiple images, a novel distance measure SqN was developed in this work. The basic idea is to evaluate a feature matrix by its singular values to align the images. For this reason, SqN is based on the Schatten- q-Norms, which are computed using the Singular Value Decomposition (SVD). We derive the distance measure SqN by mainly using three di↵erent geometric ideas to achieve image alignment. These ideas include rank minimization of the feature matrix, volume minimization, and correlation maximization. This results in several special cases, including a close relationship to the Normalized Gradient Fields distance measure when normalized intensity gradients are used as features within the feature matrix. Finally, a correlation-based approach generalizes the introduced ideas and provides a general framework for distance measurement with SqN. This framework comprises three main points, which we verify by practical application to medical image registration problems.
First, SqN can register more than two images with comparable quality as is pos- sible with standard two-image based methods such as the Normalized Gradient Fields distance measure or the sum of squared di↵erences. Secondly, registration of comparable quality can be achieved in less time than with the standard methods mentioned above, which are used in a pairwise registration scenario. Furthermore, the order of the images has no influence on the registration results due to the use of the SVD. Finally, the novel distance measure SqN can be applied to real-world medical image registration problems in practice. Our numerical experiments ha- ve verified all key points so that SqN is a promising distance measure for actual application to solve real medical image registration problems.