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Following the Mallat algorithm, two filters [the lowpass filter (L) and its corresponding highpass filter (H)] are applied to the signal, followed by dyadic decimation removing every other elements of the signal. Thereby halving it is overall length. This is done recursively by reapplying the same procedure to the result of the filter subbands to be an increasingly smoother version of the original vector as shown in Fig.1. In this paper, such 1-D discrete Wavelet transform will be used for reducing hyperspectral data in the spectral domain for each pixel individually. This transform will decompose the hyperspectral of each pixel into a set of composite bands that are linear, weighted combination of the original spectral bands. In order to control the smoothness one of the simplest and must localized Daubechies filter, called DAVB4 has been used. This filter has only four coefficients.

An example of the actual signature of one pixel for 195 bands of the California 94 AVIRIS dataset and different level of lowpass component of wavelet decomposition of this spectral signature is shown in Fig.2. As seen from the Fig.2 as the number of wavelet decomposition levels increases, the structure of the spectral signature become smoother than the structure of original signature.

In the algorithm of wavelet reduction we need to reconstruct the spectral signature to automatically select the number of levels of wavelet decomposition.

While wavelet decomposition involves filtering and downsampling the wavelet reconstruction involve upsampling and filtering. The upsampling process lengthens decomposed spectral data by inserting zeros as highpass component between each element. The inverse wavelet transform is given by:



Fig.1. A dyadic filter tree implementation for a level-3 DWT

Fig.2. Example of a pixel spectral signature and different levels of wavelet decomposition for the lowpass component
3. Wavelet-Based Dimension Reduction

A. General Description of Algorithm
Wavelet-Based reduction can be effectively applied to hyperspectral imagery. Performance of wavelet reduction can be better for larger dimensions. This property is due to very nature wavelet compression, where significant feature of the signal might be lost when the signal is under sampled.

The general description of the wavelet reduction algorithm follows;

1. For each pixel in a hyperspectral scene, the 1-D signal corresponding to its spectral signature is decomposed using Daubechies wavelet.
2. For each hyperspectral pixel, approximation the original spectral is reconstructed using IDWT. The needed level of decomposition for a given pixel is the one that corresponds to producing an acceptable correlation whit the original signature.
3. Combining results from all pixels, the number of the level of decomposition (L) is automatically computed as the lowest level needed after discarding outliers.
4. Using the number of L computed in (3) the reduced output data are composed of all pixels decomposed to level L. Therefore, if the original number of bands was N the output number of bands is N/2 L

B. Automatic Decomposed Level Selection
The correlation between the original spectral signature and the reconstructed spectral approximation is an indicator, which measures the similarity between two spectral signatures and used for selecting how many levels of decomposition can be applied while steel yielding good classification accuracy. The correlation function between the original spectral signature (x) and its reconstructed approximation (y) is shown in


Where N is the original dimension of the signal.

Table.1 shows the similarity between the original spectral signature and its reconstructed approximation of one class for the scene in our image. As seen from the table, as the number of levels of decomposition increases and the signal become more different from the original data, a proportionate decrease in correlation is observed. For each pixel in the hyperspectral scene and for each level of decomposition the correlation between original and reconstructed signal is computed.

All correlation higher than the user-specified threshold contributes to the histogram for that level of decomposition. When all pixels are processed, the lowest level of wavelet decomposition needed to produce such correlation is used for the reminder of the algorithm.

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