Lowpass Filter For Increasing Class Separability

In remote sensing, the number of training samples is often limited. For hyperspectral data, it becomes more difficult to obtain accurate estimates of class statistics because of the small ratio of the training sample size to dimensionality. Generally speaking, classification performance depends on four factors: class separability, the training sample size, dimensionality, and classifier type (or discriminant function). To improve classification performance, attention is often focused on seeking improvements on the factors other than class separability because class separability is usually considered inherent and predetermined. The objective of this paper is to call attention to the fact that class separability can be increased. The lowpass filter is proposed as a means for increasing class separability if a data set consists of multi-pixel objects. In addition, an analysis procedure is proposed in the following order: the lowpass filter, the EM algorithm, feature extraction, and a maximum likelihood classifier. Experiments with hyperspectral data show that increasing class separability compensates for the loss of the classification accuracy caused by the poor statistics estimation due to the small ratio of the training sample size to dimensionality. INTRODUCTION When the number of training samples is relatively small compared to the dimensionality, maximum likelihood estimates of parameters have large variances, leading to a large classification error [1]. Quite often, the small training sample size problem is encountered in hyperspectral data analysis. Although class separability usually increases as dimensionality increases, the growth of classification accuracy due to high class separability often fail to compensate for the loss of the accuracy of parameter estimation. As a result, a peaking phenomenon appears in the relation of classification accuracy versus dimensionality. This is often referred to as the Hughes phenomenon [2]. Several methods have been proposed for mitigating the Hughes phenomenon. Examples include LeaveOne-Out Covariance Estimation [3], and the EM algorithm [4]. Each has provided a certain degree of improvement by reducing dimensionality, selecting classifier types, and increasing the effective number of training samples, respectively. In this paper, a new approach is proposed, from the aspect of class separability. EFFECT OF CLASS SEPARABILITY ON THE HUGHES PHENOMENON The classification performance is usually evaluated by classification accuracy. Consider two equally likely classes that are characterized by normal distributions. The number of training samples for each class is assumed to be finite and fixed. By means of simulation [5], an asymptotic expression for classification errors is given