Wavelets for computationally efficient hyperspectral derivative analysis

Smoothing followed by a derivative operation is often used in the analysis of hyperspectral signatures. The width of the smoothing and/or derivative operator can greatly affect the utility of the method. If one is unsure of the appropriate width or would like to conduct analysis for several widths, scale-space images can be used. This paper shows how the wavelet transform modulus-maxima method can be used to formalize and generalize the smoothing followed by derivative analysis and how the wavelet transform can be used to greatly decrease computational costs of the analysis. The Mallat/Zhong wavelet algorithm is compared to the traditional method, convolution with Gaussian derivative filters, for computing scale-space images. Both methods are compared on two points: 1) computational expense and 2) resulting scalar decompositions. The results show that the wavelet algorithm can greatly reduce the computational expense while practically no differences exist in the subsequent scalar decompositions. The analysis is conducted on a database of hyperspectral signatures, namely, hyperspectral digital image collection experiment (HYDICE) signatures. The reduction in computational expense is by a factor of about 30, and the average Euclidean distance between resulting scale-space images is on the order of 0.02.