Transform coding techniques for lossy hyperspectral data compression

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IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING (SUBMITTED DEC. 2005) 9 single spectral decomposition tree has to be transmitted, instead of a separate tree for each spectral vector. Best basis selection using the coding gain yields the decomposition represented in Fig. 4. Consistently with the notion that spectral vectors have a significant information content, the obtained decomposition is finer than the classical dyadic wavelet tree in the high-frequency portion of the spectrum, and almost resembles a Fourier transform. L H L H L H L H L H L H Fig. 4. Best basis for the spectral DWPT. 5) Hybrid spectral wavelet packet transform and spatial wavelet packet transform: In this method a wavelet packet decomposition is applied separately in the spectral and spatial dimensions. The cost function is minimized in the third dimension considering the cube obtained by the 1D DWPT; then, a 2D DWPT is evaluated on each single band. 6) Hybrid spectral discrete wavelet transform and spatial wavelet packet transform: In this method a DWT is applied in the spectral dimension, while a 2D DWPT follows in the spatial dimension. 7) Spectral discrete cosine transform and spatial discrete wavelet transform: This method applies a one-dimensional DCT transform to each spectral vector, and a 2D square DWT on each single band of the obtained transformed cube. 8) Spectral KLT and spatial DWT: This method applies the KLT in the spectral dimension followed by the 2D square DWT along the spatial dimensions. In order to evaluate the transform matrix which optimally decorrelates the spectral dimension, we estimate the covariance matrix of the hyperspectral data cube assuming that each spectral vector, containing the radiance of a pixel at a given spatial location in all the bands, is a realization of the random process that has to be decorrelated.
IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING (SUBMITTED DEC. 2005) 10 In particular, given the hyperspectral data cube, i.e. B bands containing M lines and N samples, we form the M × N column vectors X ij =[x 1 ij ,x2 ij ,...,xB ij ]T , for i =1, ...M and j =1, ...N, where x k ij is the pixel with spatial coordinates (i, j) in band k. We employ the sample mean vector M x =[m x 1,m x 2,...,m x B], with m x k = 1 ∑ M ∑ N MN i=1 j=1 xk ij , as estimate of the ensemble averages of each band. For each spectral vector, we estimate its covariance matrix using one single realization, i.e. C X,i,j = (X ij − M x ) T (X ij − M x ). Finally, we compute the average covariance matrix C X = 1 ∑ M ∑ N MN i=1 j=1 C X,i,j. We solve the eigenvector problem for the symmetric matrix C X , obtaining the eigenvalues λ i and eigenvectors u i that satisfy C X u i = λ i u i . The KLT kernel is a unitary matrix V , whose columns are the eigenvectors u i arranged in descending order of eigenvalue magnitude. This matrix is used to transform each spectral vector, after subtracting its mean value, as Y ij = V T (X ij − M x ). The complexity of the decorrelation transform is the sum of three contributions. The first one is the evaluation of the covariance matrix (O(B 2 MN)); the second one is the solution of the eigenvector problem (O(B 3 ), [24]); the third one is the computation of transform coefficients (O(B 2 MN)). It can be observed that M, N and B are of the same order of magnitude, hence the second term is negligible with the respect to the first and third. The overall computational complexity is very high, and has so far limited the use of the KLT in practical applications. III. RESULTS OF TRANSFORM EVALUATION A. Evaluation procedure All the transforms described in Sect. II-F have been compared in terms of their energy compaction capability, which is a measure of the fraction of signal energy contained in a given number of transform coefficients. It is a very important property in image compression, since it provides an estimation of the effect of quantization in a practical coding scheme. The energy compaction property is evaluated for each transform by performing the following steps: • computing the 3D transform on a few hyperspectral scenes; • zeroing out a given percentage of transform coefficients, taken as those with the smallest magnitude; • computing the inverse 3D transform; • computing the peak signal-to-noise ratio (PSNR) with respect to the original image. Of particular importance is the fact that the coefficients to be zeroed out are taken in arbitrary order within the complete 3D set of transform coefficients, and not on a band by band basis. This is
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Transform coding techniques for lossy hyperspectral data compression

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