Transform coding techniques for lossy hyperspectral data compression

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IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING (SUBMITTED DEC. 2005) 15 100 90 80 PSNR (dB) 70 60 50 40 KLT1D−DWT2D DWT1D2D DCT1D−DWT2D 30 50 55 60 65 70 75 80 85 90 95 100 % Transformed coefficients set to zero Fig. 8. Performance comparison of different transforms, with a separable transform in the spectral direction: KLT, DWT and DCT. in Sect. II-F.8. In the following we propose a low-complexity version of the KLT that alleviates this problem with virtually no performance loss with respect to the full-complexity transform. In particular, in Sect. IV-A we define the low-complexity one-dimensional KLT; in Sect. IV-B we define our proposed 3D transform based on the low-complexity KLT, and evaluate its energy compaction capability followed the same procedure used with the other transforms; in Sect. IV-C we provide a breif overview of JPEG 2000, and in Sect. IV-D we describe the integration of the proposed transform within Part 2 of JPEG 2000 [26]. A. One-dimensional transform The KLT applies principal components analysis to the spectral dimension evaluating the average correlation matrix over all spectral vectors. For an AVIRIS scene, this amounts to computing and averaging over 300000 such matrices. To simplify this process, we note that convergence of the estimation process may be achieved using fewer matrices. Using the notation defined in Sect. II-F.8, in the proposed low-complexity transform all the processing is not carried out on the complete set of spectral vectors, but rather on a subset of vectors selected at random. Hence, the sample mean vector is defined as M x ′ =[m ′ x 1,m′ x 2,...,m′ x ], where B m ′ x = 1 ∑ k M ′ N ∑i∈I ′ j∈J xk ij , and I and J are sets containing respectively M ′ and N ′ different indexes picked at random in the intervals [1,M] and [1,N], with M ′ ≤ M and N ′ ≤ N. This process is also depicted in Fig. 9, where the different sets of spectral vectors are highlighted.
IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING (SUBMITTED DEC. 2005) 16 - - - - - - - - - - - - - - - - - - - - - - - - - - - - KLT Transform matrix - - - - Correlated components = - - - - Decorrelated components Spectral vector employed in the evaluation of Covariance matrix in the low complexity KLT method Spectral vector employed in the evaluation of Covariance matrix in the full complexity KLT method Fig. 9. Computation of the covariance matrix for the full-complexity and low-complexity KLT. The covariance matrix is obtained as C X ′ = 1 ∑ M ′ N ∑i∈I ′ j∈J (X ij − M x) ′ T (X ij − M x). ′ Itis used to form the eigenvector set C X ′ u′ i = λ′ i u′ i , where u′ i are the eigenvectors associated with the eigenvalues λ ′ i . Aligning the eigenvectors columnwise we obtain the low-complexity KLT matrix V ′ . The transformed vector is computed as Y ij =(V ′ ) T (X ij − M x). ′ We also denote as ρ = M ′ N ′ the percentage of spectral vectors employed to evaluate the covariance matrix. Obviously, the smaller is ρ, the lower is the complexity of this KLT. The complexity of the first stage of the new transform, i.e. the evaluation of the covariance matrix, becomes O(ρB 2 MN), i.e. it is reduced by a factor ρ, as the number of covariance matrices to be computed decreases linearly with ρ; the other two terms remain unchanged. Therefore, the proposed scheme is able to significantly reduce the complexity of the first stage, while no advantage is achieved in the third stage, because the resulting transform matrix does not exhibit any specific structure that can be exploited to reduce its complexity. Some numerical results on the complexity of the complete JPEG 2000 based algorithm are given in Sect. V-A. Fig. 10 shows an example of covariance matrix of all the spectral vectors. This matrix is clearly symmetric, and is depicted as an image with 256 grey levels. The tone is proportional to the absolute value of the correlation. The elements of the main diagonal have very high values, as also a large number of the main diagonal neighbors do, since a high degree of inter-band correlation is present. It can be seen that some elements of matrix are close to zero, because some pairs of bands are poorly MN
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Transform coding techniques for lossy hyperspectral data compression

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