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Hyperspectral & Data Acquisition Systems
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Multiscale Analysis of Hyperspectral Data
Using Wavelets for Spectral Feature Extraction
4.2 Wavelet Transform Modulus
Figures 4(a), (b) and (c) show the wavelet transform of the spectral curves with respect to the three materials. They were calculated with , where is a Gaussian function. The position parameter u and the log2 of scale s vary respectively along the horizontal and vertical axes. Black, gray and white points represent positive, zero and negative wavelet coefficients respectively. Figures 5(a), (b) and (c) show the results of extracting modulus maxima from . It can be seen that all singularities can be detected easily along the maxima line from coarse to fine scales. The maxima lines detected along the scale space correspond to the main absorption bands of the spectral curve. The 3D views of the wavelet transform are respectively shown in figures 6(a), (b) and (c). The absorption bands create large amplitude coefficients in their cone of influence. In figure 7(a), (b) and (c), eight absorption spectral features are marked with gray rectangles A… E. The left and right borders of the rectangles are determined by the maxima lines. The converged position of a maxima line with negative wavelet coefficients correspond to the left border of a rectangle, then the right border of the rectangle will be determined by the next maxima line with positive wavelet coefficients. Because the spectral curves of these three materials are very similar in shape, the extracted spectral features C, D, E, F, and G almost have the same positions (See table 1.). However, these three materials can be differentiated using the features A, B, and H, which are represent small variations of the spectral curves.
4.3 Lipschitz Exponent
Figure 8 shows the results of Lipschitz exponents calculated at each singularity point for the test data. Because the Lipschitz regular was estimated under the assumption of the compact support wavelet with one vanishing moment, the values of should satisfy for isolated singularity. The negative Lipschitz exponent values indicate the corresponding points possessing high-frequency oscillations in their neighborhood.
The spectral curves for soybean and corn shown in figure 5 look quite similar in shape and their main absorption bands corresponding to the low-frequency components are nearly the same. However, one would easily discover the differences when a scale factor is introduced to overlap these two curves. Figure 9 obviously shows that the high-frequency variations in the corn spectrum are larger than the soybean spectrum in the 21st, 76th and 153rd bands. Therefore, these two different materials can be distinguished easily with the Lipschitz exponents. Table 1. shows that the spectral features C, D, E, F, and G have the same positions in spectral curves, but their Lipschitz exponents are different. The variations will be useful to distinguish a spectral curve. For example, the grass spectral curve at the right boder of E feature is smoother than the other two curves at the same location, so its Lipschitz exponent is larger than the other two.
Table1. Results of spectral absorption positions
5. Discussion
The high spectral resolution of hyperspectral data provides the ability for diagnostic identification of various materials. In order to increase the classification performance, feature extraction is pre-processed to substantially remove the redundant information without sacrificing significant information. In this study, we transferred the hyperspectral data into the scale-space plane using the wavelet transform to extract important spectral features. The wavelet transform can focus on localized signal structures by scaling and dilating a wavelet. The absorption bands of spectral curves are thus detected automatically by the wavelet transform modulus maxima, and the Lipschitz exponents are estimated at each singularity point in the spectral curve from the decay of the wavelet transform amplitude. The local frequency variances provide useful information about the oscillations in the hyperspectral curve for each pixel. Various materials can be distinguished by the differences in the local frequency variations. This new method generates features that are more meaningful and is more stable than other known methods for spectral feature extraction. In particular, the method proposed in this paper will be helpful for spectral analysis that reduces the multidimensional hyperspectral data to a smaller number of essential features that can be both automatically processed and physically interpreted.
Aaknowledgements
The authors would like to thank the National Science Council of Republic of China, for support of this research project: NSC88-2211-E006-084.
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