Reduced dimension quadratic detection algorithms

Abstract

Hyperspectral images may be collected in tens to hundreds of spectral bands having band widths on the order of 1-10 nanometers. Principal component (PC), maximum-noise-fraction (MNF), and vector quantization (VQ) transforms are used for dimension reduction and subspace selection. The impact of the PC, MNF, and VQ transforms on image quality are measured in terms of mean-squared error, image-plus-noise variance to noise variance, and maximal-angle error, respectively. These transforms are not optimal for detection problems. The signal-to-noise ratio (SNR) is a fundamental parameter for detection and classification. In particular, for additive signals in a normally distributed background, the performance of the matched filter depends on SNR, and the performance of the quadratic anomaly detector depends on SNR and the number of degrees-of-freedom. In this paper we demonstrate the loss in SNR that can occur from the application of the PC, MNF, and VQ transforms. We define a whitened-vector-quantization (WVQ) transform that can be used to reduce the dimension of the data such that the loss in SNR is bounded, and we construct a transform (SSP) that preserves SNR for signals contained in a given subspace such that the dimension of the image of the transform is the dimension of the subspace.