Non-Euclidean visual traveling salesman problem

Abstract

Traveling Salesman Problem (TSP) is defined as the task of finding the shortest tour of N cities given intercity costs. Usually the intercity costs are 2D Euclidean distances. In the presence of obstacles or in the case of 3D surfaces, the intercity distances are in general not Euclidean. The TSP with obstacles and on 3D surfaces approximates our everyday visual navigation. There are three questions related to the mechanisms involved in solving TSP: (i) how do subjects find the intercity distances, (ii) how do they determine clusters of cities, and (ii) how do they produce the TSP tour. In our model, the non-Euclidean distances (geodesics) are found by using a non-linear Eikonal equation, i.e. the evolution of interfaces (Sethian, 1999). The geodesic distances are then used as intercity costs in an MST graph pyramid (Haxhimusa et. al., 2007). The original TSP problem is represented by a sequence of problems involving clusters of cities. The hierarchical clustering is performed by using a Boruvka's minimum spanning tree. Close to the top of the pyramid, the original TSP problem is represented at a very coarse level and involves very small number of “cities”. This coarse representation is solved optimally. Expanding this coarse tour in a top-down manner leads to a solution of the original TSP. The new model has an adaptive spatial structure and it simulates visual acuity and visual attention. The model solves the TSP problem sequentially, by moving its attention from city to city. The model's performance will be compared to the performance of human subjects.