A mathematical framework for image segmentation evaluation - Ribeiro, Pedro Felipe De Oliveira [UNIFESP] (2020)
The analysis of similarity has interested the scientific community from the early 1900’s, with the pioneering work of Jaccard, to our very day. Many methods and algorithms for improving the correct quantification of similarity are presented by the community every year, in a clear attempt to better understand its behavior. Interestingly, quite fewer papers attempt to develop a rigorous mathematical analysis of similarity. This work goes in the opposite direction, as it aims to provide the community with a very strong mathematical framework to the analysis of similarity. With developments that range from functional analysis and linear spaces to applied statistics, this study presents new results to further the understanding of similarity with a very strong quantitative grasp. We present a complete similarity coefficient generator, that generalizes many very well known similarity coefficients (such as the Dice and the Cosine similarity), alongside with a complete metric space of similarity, with a p-norm defined, and an induced metric. From this distance function we construct a coefficient capable of quantifying the performance of different similarity coefficients when applied to the same similarity problem. Therefore, the presented coefficient is also capable of ranking distinct similarity coefficients according to their adequacy to a particular similarity problem. We also present a function form of the statistical measures of TP, FP, and FP, and prove that exists a relation between similarity and the results of Fourier processes. Therefore, we show that a bridge exists between the fields of similarity on image segmentation and classical analysis.