Introducing multi‐dimensional ‘hybrid wavelets’ for enhanced evaluation of hyperspectral image cubes and multi‐way data sets

Abstract

Multi-dimensional wavelet transforms (WTs) have been introduced for efficient data compression in order to accelerate chemometric calculations and to reduce requirements for data storage space. For hyphenated measurement techniques or hyperspectral imaging this wavelet compression becomes vital because such sensors acquire unprecedented amounts of information in short periods of time. Conventional, multi-dimensional wavelet compression uses the same wavelet for all dimensions. However, from a mathematical perspective there is no need for this restriction as the transforms in different dimensions are independent from each other. This manuscript presents multidimensional 'hybrid wavelet transforms', which utilize different wavelet types for different dimensions. Thus, hybrid wavelets optimize wavelet compression by adjusting WTs to the different types of data sets.In this manuscript we demonstrate that hybrid wavelets improve acceleration factors compared to conventional multi-dimensional wavelet compression and determine more precise models. Combinations of Haar, Daub4, Daub6, Daub8, Daub10, Daub12, Daub14, Daub16, Daub18 and Daub20 wavelets are considered in this study. Data obtained with two different experimental techniques are used for assessing hybrid wavelet compression: (i) a data cube obtained by means of mid-infrared hyperspectral imaging; (ii) data acquired by excitation-emission matrix (EEM) fluorescence spectroscopy in photo-catalytic studies. For the hyperspectral data cube hybrid wavelets were found, which are superior regarding acceleration and model quality to all 10 conventional WTs (Haar-Daub20). For the EEM example this was achieved in 9 out of 10 cases; thus in 19 out of 20 investigated cases hybrid WTs were found to be superior to conventional wavelet compression.

Accelerating kernel principal component analysis (KPCA) by utilizing two‐dimensional wavelet compression: applications to spectroscopic imaging

“…Thus, it is computationally more efficient to always use the smaller covariance matrix when applying PCA. However, the large covariance matrix (see Equation (23) in the beginning of the appendix) must be used when applying KPCA. The reason for this is that KPCA can extract a number of PCs that exceeds the number of variables (wavelength positions) if the number of samples (spectra) is greater [1,2].…”

“…New approaches are required that enable diagonalization of such large matrices on personal computers within reasonable time. Already available wavelet-based compression methods [22][23][24][25] cannot be used since they load the full dataset and compress it while holding the entire dataset in memory. Here, this must be avoided at all times simply because of datasets sizes.…”

“…In order to ensure a meaningful wavelet compression in the X-dimension [22][23][24][25], the same wavelet coefficients in the X-direction have to be removed from all rows of K. If this is not ensured, the subsequent WTs in the Y-dimension would incorporate wavelet coefficients belonging to different positions of the X-dimension WTs. This would cause the final result to be meaningless.…”

“…In order to handle large amounts of data in a timely fashion, multi-dimensional wavelet transformations (WTs) [13][14][15][16][17][18][19][20][21] have been employed as an effective method of data compression prior to chemometric computations [22][23][24][25] (and references therein). These techniques facilitate reductions in memory requirements and computation times.…”

“….,M of the Gram matrix K. The latter, a 1,. . .,M , will be used to determine the eigenvectors v i¼1;ÁÁÁ;M (Equation (28)) of the covariance matrix C H ; the v i¼1;ÁÁÁ;M will then play the role of PCs in PCA (compare Equation (27) to Equation (23)). Typically, there are only R < M relevant eigenvalues; the remaining eigenvalues are so close to zero that they are negligible from a spectroscopic perspective.…”