Metrical loss functions for image segmentation based on convolutional neural networks

Abstract

Image segmentation remains a fundamental challenge in computer vision, with neural network training heavily dependent on appropriate loss functions. While common losses such as Dice are widely used, they lack rigorous mathematical foundations as proper distance metrics and do not fully capture the geometric structure of partitions. We introduce a weighted metric for comparing segmentation based on partition theory that satisfies all metric axioms, including the triangle inequality. The proposed metric compares partitions through symmetric difference and intersection operations, incorporating both spatial structure and semantic features via a weight function characterizing region properties, such as color, texture and other. We prove that the proposed functional forms a proper metric space on weighted partitions under specified conditions, with particular emphasis on establishing the triangle inequality. Experimental verification on synthetic segmentation tasks demonstrates feasibility, although practical implementation faces challenges, such as the need for differentiated segment extraction, which can be solved using the Straight-Through Estimator approximation. The triangle inequality property opens up opportunities for hierarchical approaches to segmentation and efficient partition search. This work bridges the gap between geometric clustering theory and deep learning-based segmentation, providing a theoretically grounded alternative to heuristic loss functions and also experimentally proves the possibility of using the proposed metric as a loss function when training convolutional neural networks.