Greedy and K-Greedy algoritmhs for multidimensional data association

[EN] The multidimensional assignment (MDA) problem is a combinatorial optimization problem arising in many applications, for instance multitarget tracking (MTT). The objective of an MDA problem of dimension $d\in\Bbb{N}$ is to match groups of $d$ objects in such a way that each measurement is associated with at most one track and each track is associated with at most one measurement from each list, optimizing a certain objective function. It is well known that the MDA problem is NP-hard for $d\geq3$. In this paper five new polynomial time heuristics to solve the MDA problem arising in MTT are presented. They are all based on the semi-greedy approach introduced in earlier research. Experimental results on the accuracy and speed of the proposed algorithms in MTT problems are provided. © 2006 IEEE.This research was supported by a Marie Curie fellowship of the European Community program "Improving Human Research Potential and the Socio-economic Knowledge Base" under Contract HPMI-CT-2002-00221.Perea Rojas Marcos, F.; De Waard, HW. (2011). Greedy and K-Greedy algoritmhs for multidimensional data association. IEEE Transactions on Aerospace and Electronic Systems. 47(3):1915-1925. doi:10.1109/TAES.2011.59372731915192547

Solution Approaches to the Three-index Assignment Problem

This thesis explores the axial Three-Index Assignment Problem (3IAP), also called the Multidimensional Assignment Problem. The problem consists in allocating n jobs to n machines in n factories, such that exactly one job is executed by one machine in one factory at a minimum total cost. The 3IAP is an extension of the classical two-dimensional assignment problem. This combinatorial optimisation problem has been the subject of numerous research endeavours, and proven NP-hard due to its inextricable nature. The study adopts an algorithmic approach to develop swift and e ective methods for solving the problem, focusing on balancing computational e ciency and solution accuracy. The Greedy-Style Procedure (GSP) is a novel heuristic algorithm for solving the 3IAP, guaranteeing feasible solutions in polynomial time. Speci c arrangements of cost matrices can lead to the generation of higher-quality feasible solutions. In addressing the 3IAP, analysing the tie-cases and the matrix ordering led to new variants. Further exploration of cost matrix characteristics has allowed two new heuristic classes to be devised for solving 3IAP. The approach focuses on selecting the best solution within each class, resulting in an optimal or a high-quality approximate solution. Numerical experiments con rm the e ciency of these heuristics, consistently delivering quality feasible solutions in competitive computational times. Moreover, by employing diverse optimisation solvers, we propose and implement two e ective methods to achieve optimal solutions for 3IAP in good CPU times. The study introduces two local search methods based on evolutionary algorithms to solve 3IAP. These approaches explore the solution space through random permutations and the Hungarian method. Building on this, a hybrid genetic algorithm that integrates these local search strategies has been proposed for solving the 3IAP. Implementing the Hybrid Genetic Algorithm (HGA) produces high-quality solutions with reduced computational time, surpassing traditional deterministic approaches. The e ciency of the HGA is demonstrated through experimental results and comparative analyses. On medium to large 3IAP instances, our method delivers comparable or better solutions within a competitive computational time frame. Two potential future developments and expected applications are proposed at the end of this project. The rst extension will examine the correlation between cost matrices and the optimal total cost of the assignment and will investigate the dependence structure of matrices and its inuence on optimal solutions. Copula theory and Sklar's theorem can help with this analysis. The focus will be on understanding the stochastic dependence of cost matrices and their multivariate properties. Furthermore, the impact of variations in cost distributions, is often modelled based on economic sectors. The second extension involves integrating variable costs de ned by speci c probability distributions, enhancing the comprehensive analysis of economic scenarios and their impact on the assignment problem. The study considers various well-de ned probability distributions and highlights more practical applications of the assignment problem in real-world economics. The project's original contribution lies in its algorithmic approach to investigating the 3IAP, which has led to the development of new, fast, and e cient heuristic methods that strategically balance computational speed and the accuracy of the solutions achieved

Occlusion reasoning for multiple object visual tracking

Thesis (Ph.D.)--Boston UniversityOcclusion reasoning for visual object tracking in uncontrolled environments is a challenging problem. It becomes significantly more difficult when dense groups of indistinguishable objects are present in the scene that cause frequent inter-object interactions and occlusions. We present several practical solutions that tackle the inter-object occlusions for video surveillance applications. In particular, this thesis proposes three methods. First, we propose "reconstruction-tracking," an online multi-camera spatial-temporal data association method for tracking large groups of objects imaged with low resolution. As a variant of the well-known Multiple-Hypothesis-Tracker, our approach localizes the positions of objects in 3D space with possibly occluded observations from multiple camera views and performs temporal data association in 3D. Second, we develop "track linking," a class of offline batch processing algorithms for long-term occlusions, where the decision has to be made based on the observations from the entire tracking sequence. We construct a graph representation to characterize occlusion events and propose an efficient graph-based/combinatorial algorithm to resolve occlusions. Third, we propose a novel Bayesian framework where detection and data association are combined into a single module and solved jointly. Almost all traditional tracking systems address the detection and data association tasks separately in sequential order. Such a design implies that the output of the detector has to be reliable in order to make the data association work. Our framework takes advantage of the often complementary nature of the two subproblems, which not only avoids the error propagation issue from which traditional "detection-tracking approaches" suffer but also eschews common heuristics such as "nonmaximum suppression" of hypotheses by modeling the likelihood of the entire image. The thesis describes a substantial number of experiments, involving challenging, notably distinct simulated and real data, including infrared and visible-light data sets recorded ourselves or taken from data sets publicly available. In these videos, the number of objects ranges from a dozen to a hundred per frame in both monocular and multiple views. The experiments demonstrate that our approaches achieve results comparable to those of state-of-the-art approaches

Random multi-index matching problems

The multi-index matching problem (MIMP) generalizes the well known matching problem by going from pairs to d-uplets. We use the cavity method from statistical physics to analyze its properties when the costs of the d-uplets are random. At low temperatures we find for d>2 a frozen glassy phase with vanishing entropy. We also investigate some properties of small samples by enumerating the lowest cost matchings to compare with our theoretical predictions.Comment: 22 pages, 16 figure