A Similarity Measure for Image and Volumetric Data Based on Hermann Weyl's Discrepancy

Abstract

The paper focuses on similarity measures for translationally misaligned image and volumetric patterns. For measures based on standard concepts such as cross-correlation, L(p)-norm, and mutual information, monotonicity with respect to the extent of misalignment cannot be guaranteed. In this paper, we introduce a novel distance measure based on Hermann Weyl's discrepancy concept that relies on the evaluation of partial sums. In contrast to standard concepts, in this case, monotonicity, positive-definiteness, and a homogenously linear upper bound with respect to the extent of misalignment can be proven. We show that this monotonicity property is not influenced by the image's frequencies or other characteristics, which makes this new similarity measure useful for similarity-based registration, tracking, and segmentation.