Analysis for the Weakly Pareto Optimum in Multiobjective-Based Hyperspectral Band Selection

Band selection refers to finding the most representative channels from hyperspectral images. Usually, certain objective functions are designed and combined via regularization terms. A possible drawback of these methods is that they can only generate one solution in a single run with a given band number. To overcome this problem, multiobjective (MO)-based methods, which were able to simultaneously obtain a series of subsets with different band numbers, were investigated for band selection. However, because the range of band selection problem is discrete, recently proposed weighted Tchebycheff (WT)-based MO methods may suffer weakly Pareto optimal problem. In this case, the solutions for each band number will be nonunique and no optimal solution exists. Decision makers have to manually select a unique solution for each band number. In this paper, we provide a theoretical analysis about the weakly Pareto optimal problem in band selection, and quantitatively give the boundary conditions. Moreover, we further summarize the suggestions which will help users avoid the weakly Pareto optimal problem. According to these criteria, we develop a new adaptive-penalty-based boundary intersection (APBI) framework to improve the MO algorithm in hyperspectral band selection. APBI mainly includes two advantages: 1) avoiding weakly Pareto optimum and 2) reducing the sensibility of the penalty factor. The theoretical analysis is further validated by contrast experiments. The results demonstrate that the weakly Pareto optimal solutions really exist in WT methods, while APBI can overcome this problem.