Linear and kernel methods for multi- and hypervariate change detection

Abstract

The iteratively re-weighted multivariate alteration detection (IR-MAD) algorithm may be used both for unsuper- vised change detection in multi- and hyperspectral remote sensing imagery as well as for automatic radiometric normalization of multi- or hypervariate multitemporal image sequences. Principal component analysis (PCA) as well as maximum autocorrelation factor (MAF) and minimum noise fraction (MNF) analyses of IR-MAD images, both linear and kernel-based (which are nonlinear), may further enhance change signals relative to no-change background. The kernel versions are based on a dual formulation, also termed Q-mode analysis, in which the data enter into the analysis via inner products in the Gram matrix only. In the kernel version the inner products of the original data are replaced by inner products between nonlinear mappings into higher dimensional feature space. Via kernel substitution, also known as the kernel trick, these inner products between the mappings are in turn replaced by a kernel function and all quantities needed in the analysis are expressed in terms of the kernel function. This means that we need not know the nonlinear mappings explicitly. Kernel principal component analysis (PCA), kernel MAF and kernel MNF analyses handle nonlinearities by implicitly transforming data into high (even in¯nite) dimensional feature space via the kernel function and then performing a linear analysis in that space. In image analysis the Gram matrix is often prohibitively large (its size is the number of pixels in the image squared). In this case we may sub-sample the image and carry out the kernel eigenvalue analysis on a set of training data samples only. To obtain a transformed version of the entire image we then project all pixels, which we call the test data, mapped nonlinearly onto the primal eigenvectors. IDL (Interactive Data Language) implementations of IR-MAD, automatic radiometric normalization and kernel PCA/MAF/MNF transformations have been written which function as transparent and fully integrated extensions of the ENVI remote sensing image analysis environment. Also, Matlab code exists which allows for fast data exploration and experimentation with smaller datasets. Computationally demanding kernelization of test data with training data and kernel image projections have been programmed to run on massively parallel CUDA-enabled graphics processors, when available, giving a tenfold speed enhancement. The software will be available from the authors' websites in the near future. A data example shows the application to bi-temporal RapidEye data covering the Garzweiler open pit mine in the Ruhr area in Germany.