Transform coding techniques for lossy hyperspectral data compression

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IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING (SUBMITTED DEC. 2005) 7 A rectangular 2D transform is such that first the complete 1D wavelet transform (i.e., all decomposition levels) is computed in one dimension, and then the complete transform is applied to the second dimension. In 3D, a square transform is obtained by first computing one decomposition level in all dimensions, and then iterating on the LLL cube. Conversely, the rectangular transform is obtained by first applying the complete transform along the first dimension, then along the second one, and finally along the third one. In 3D, hybrid transforms can also be obtained as in [16] by first applying the complete transform in one dimension, and then taking a 2D square transform in the other two dimensions. The obtained transform is referred to as 3D hybrid rectangular/square DWT. F. 3D transforms selected for evaluation The previously described one-dimensional transforms have been combined in various ways to obtain 3D transforms for hyperspectral data. The most relevant combinations are reported in the following. As for filter selection in the DWT and DWPT, the (9,7) biorthogonal wavelet filter pair has been used throughout this work; this filter is known to provide excellent compression performance, and has been selected for inclusion in the JPEG 2000 standard. 1) 3D square DWT: This method is based on the wavelet transform applied in all three dimensions simultaneously. In particular, one level of wavelet decomposition is applied along each of the three dimensions. This procedure is repeated on the obtained LLL cube, as opposed to the rectangular transform described in Sect. II-F.3. As an example, Fig. 2 shows in a pictorial way the subbands obtained by performing three levels of 3D-decomposition on a data cube as described above. The obtained decomposition has cubic subbands in 3D. Fig. 2. Subbands obtained by performing three levels of 3D square DWT on a data cube.
IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING (SUBMITTED DEC. 2005) 8 2) 3D square DWPT: In the 3D square DWPT, one level of wavelet packet decomposition is applied along the three dimensions; then, all the sub-cubes obtained by the wavelet packet decomposition may be further split so as to minimize an appropriate cost function. This procedure is repeated iteratively on each obtained cube for a given number of decomposition levels. 3) Hybrid rectangular/square 3D transform: As hyperspectral data carry a lot of information in the spectral dimension, it is interesting to think of transforms that operate differently in the spectral and spatial directions, in order to match the different nature of those correlations. A way to obtain such multidimensional transform is the 3D hybrid rectangular/square wavelet transform. Fig. 3 depicts a hybrid 3D wavelet transform obtained with three decomposition levels as described. As can be seen, the subbands generated by this transform are rectangles in 2D and parallelepipeds in 3D. The number of subbands is higher than that of the classical square transform; in particular the frequency partitioning is such that high horizontal and low vertical frequencies lie in subbands where the basis functions are short and long in the horizontal and vertical dimensions respectively. The obtained frequency tessellation is finer, and has more radial symmetry than the square transform. Fig. 3. Hybrid rectangular/square 3D DWT subband decomposition of a data cube. 4) Hybrid spectral wavelet packet and spatial square 2D wavelet transform: In this method a DWPT is first applied in the spectral dimension, while a 2D square DWT is applied in the spatial dimension. In other terms, this is equivalent to the hybrid rectangular/square 3D wavelet, in which the spectral DWT is replaced by a spectral DWPT. The cost function for DWPT is minimized in the third dimension considering the whole cube obtained by the 1D packet decomposition, and not the single 1D vectors. In fact, the separated optimization for each single vector could yield different bases. In this case, the transformed data cube would present discontinuities, which would penalize the performance of the next spatial 2D DWT stage. An other advantage is the reduced overhead, since a
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Transform coding techniques for lossy hyperspectral data compression

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