Computable invariants for curves and surfaces

During the last decade the trend in image analysis has been to shift from axiomatically derived measures into ones that are extracted empirically from data samples. The problem is that accurate geometric data are often unavailable and thus, for proper data augmentation, the research community has resorted yet again to axiomatic construction of invariant measures for curves and surfaces. For some geometric problems, such as shape classification and matching, it is appealing to adopt learning approaches due to their potential accuracy and computational efficiency. Nevertheless, even within the deep learning arena, geometric invariants offer a natural criterion for the learning process. Using geometric invariants one can overcome the need for annotated data and replace it by a purely geometric measure, leading to an unsupervised or a semisupervised learning schemes. Here, we review such constructions that are useful for the geometric analysis of visual information. The measures we explore include the construction of a scale or similarity invariant arc-length for curves and surfaces, an affine invariant one, resulting spectral geometries, and potential signatures that reflect the result of the discrepancies between the corresponding metric spaces. As an example we study novel signatures for surfaces known as the self functional map, which allow us to translate the problem of shape matching into that of computing distances between matrices.