Similarity is one of the most fundamental notions encountered in problems practically in every branch of science, and is especially crucial in image sciences such as computer vision and pattern recognition. The need to quantify similarity or dissimilarity of some data is central to broad categories of problems involving comparison, search, matching, alignment, or reconstruction. The most common way to model a similarity is using metrics (distances). Such constructions are well-studied in the field of metric geometry, and there exist numerous computational algorithms allowing, for example, to represent one metric using another by means of isometric embeddings.
However, in many applications such a model appears to be too restrictive: many types of similarity are non-metric; it is not always possible to model the similarity precisely or completely e.g. due to missing data; some objects might be mutually incomparable e.g. if they are coming from different modalities. Such deficiencies of the metric similarity model are especially pronounced in large-scale computer vision, pattern recognition, and medical imaging applications.
The ambitious goal of this project is to introduce a paradigm shift in the way we model and compute similarity. We will develop a unifying framework of computational similarity geometry that extends the theoretical metric model, and will allow developing efficient numerical and computational tools for the representation and computation of generic similarity models. The methods will be developed all the way from mathematical concepts to efficiently implemented code and will be applied to todays most important and challenging problems in Internet-scale computer vision and pattern recognition, shape analysis, and medical imaging.