Multivariate Gray Model-Based BEMD for Hyperspectral Image Classification

Bidimensional empirical mode decomposition (BEMD) has been one of the core activities in image processing due to its fully data-driven and self-adaptive nature. Unfortunately, this promising technique is sensitive to boundary effect. In this paper, a new method inspired by the multivariate gray model (MGM), namely <formula formulatype="inline"><tex Notation="TeX">${\rm GM}(1, N)$</tex></formula>, is developed for boundary extension of the BEMD. Specifically, our contribution is threefold. First, focusing on evaluating the model coefficients and convolution integral, which are key elements in reducing the prediction error of the <formula formulatype="inline"><tex Notation="TeX">${\rm GM}(1, N)$</tex></formula>, we replace the existing (composite) trapezoidal rule with (composite) Simpson rule and deduce an alternative MGM, termed as <formula formulatype="inline"><tex Notation="TeX">${\rm S\hbox{-}GM}(1,N)$</tex></formula>. Second, the given image is extended by the GM(1, 3) or S-GM(1, 3) <formula formulatype="inline"><tex Notation="TeX">$(N=3)$</tex></formula>, whose characteristic data series and relative data series are, respectively, derived from the pixel values and coordinates of the image. Consequently, the extended image is decomposed into several bidimensional intrinsic mode functions (BIMFs) and a residue whose corresponding parts are extracted as the decomposition results of the original image. Finally, the proposed boundary effect mitigation methods are applied in the hyperspectral image classification. In greater detail, the BIMFs obtained by various BEMD methods are taken as features of the hyperspectral dataset whereas the widely used <formula formulatype="inline"><tex Notation="TeX">$k$</tex></formula>-nearest neighbors (<formula formulatype="inline"><tex Notation="TeX">$k$</tex> </formula>-NN) as well as the support vector machine, whose optimal parameters are selected by the genetic algorithm, are adopted as classifiers. Extensive experiments and comparisons with other generally acknowledged methods confirm that the proposed methods achieve promising improvement in the classification performance.