The Averaged Hausdorff Distances in Multi-Objective Optimization: A Review

Abstract

A brief but comprehensive review of the averaged Hausdorff distances that have recentlybeen introduced as quality indicators in multi-objective optimization problems (MOPs) is presented.First, we introduce all the necessary preliminaries, definitions, and known properties of thesedistances in order to provide a stat-of-the-art overview of their behavior from a theoretical pointof view. The presentation treats separately the definitions of the (p, q)-distances GDp,q, IGDp,q, and Δp,q for finite sets and their generalization for arbitrary measurable sets that covers as an importantexample the case of continuous sets. Among the presented results, we highlight the rigorousconsideration of metric properties of these definitions, including a proof of the triangle inequalityfor distances between disjoint subsets when p, q ≥ 1, and the study of the behavior of associatedindicators with respect to the notion of compliance to Pareto optimality. Illustration of these resultsin particular situations are also provided. Finally, we discuss a collection of examples and numericalresults obtained for the discrete and continuous incarnations of these distances that allow for anevaluation of their usefulness in concrete situations and for some interesting conclusions at the end,justifying their use and further study.

Mapping a historic neighbourhood through user-generated content: the case of Alfama, Lisbon (Portugal)

“…While IoU provides a measure of the overlap extent between shapes, the Frechet and Hausdorff distances are common curve similarity metrics. Applied to our case, the former is the smallest of the maximum pairwise distances between vertices of different shapes and the latter is the maximum distance found between a vertex of a shape and the nearest point in the other shape (Bogoya et al, 2019;Lyu et al, 2021). Lastly, we used the shape of the two old parishes as the estimated "ground-truth" of Alfama to determine, for each dataset, values of F-scores and distances towards its centroid.…”

Feasibility of Synthetic Computed Tomography Images Generated from Magnetic Resonance Imaging Scans Using Various Deep Learning Methods in the Planning of Radiation Therapy for Prostate Cancer

“…HUs of the sCTs generated from the three networks were compared with those of the original simulation CT images by calculating the mean error (ME) and the MAE. Dice similarity coefficients (DSCs) and Hausdorff distances for body, bony structure, rectal balloon, and soft tissue, segmented in each original CT and sCT image were calculated to compare the structural metrics between the sCTs and the original simulation CT images according to the networks [ 26 ]. The peak signal-to-noise ratio (PSNR) was also computed to measure the reconstruction quality.…”

Non-Epsilon Dominated Evolutionary Algorithm for the Set of Approximate Solutions

“…Thus, it becomes harder for the algorithms to put pressure towards the set of interest. Our proposed algorithm aims to couple with these drawbacks, and we will compare it with the P Q, -NSGA-II and P Q, -MOEA in terms of the ∆ 2 indicator in both decision and objective space [13,52]. The ∆ 2 indicator measures the Averaged Hausdorff distance between a reference set and an approximation.…”