Transform coding techniques for lossy hyperspectral data compression

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IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING (SUBMITTED DEC. 2005) 5 Input H l (z) H h (z) 2 2 H l (z) H h (z) 2 2 Input H l (z) 2 H l (z) H h (z) 2 2 H h (z) 2 H l (z) H h (z) 2 2 Fig. 1. Implementation of wavelet transforms by means of a filter bank scheme with lowpass and highpass filters denoted as H l (z) and H h (z) respectively. (a) DWT and (b) DWPT. The circles denote subsampling by a factor of two. highpass outputs are allowed to be further split into approximation and detail. If both the lowpass and highpass sequences are always split, the system is said to be a complete wavelet packet transform. However, it is not necessary for the transform to be complete; for any given input signal, there exists an optimal choice of highpass and lowpass iterations that captures most of the input signal correlation, which is known as best basis wavelet packet transform. Given an appropriate cost function, a search algorithm adaptively selects the best basis for a given signal. Different cost functions can be employed, e.g. entropy, minimum distortion, minimum number of coefficients above a certain threshold [21], or rate-distortion optimization [22]. Our purpose is to select the best 3D transforms in terms of energy compaction; hence, the cost function in [22] is not suitable because it explicitly takes quantization into account, while our analysis aims at being independent of the specific quantization scheme employed. We have found that the coding gain, which is a performance measure of transform efficiency [20], is a very good and theoretically sound objective function for seeking the best decomposition tree. Assuming a DWPT with l decomposition
IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING (SUBMITTED DEC. 2005) 6 levels, the coding gain G (l−1)→l is defined as G (l−1)→l = ( ∏ N i=1 (σi l )2 ) 1 N where σ l−1 is the standard deviation of the transformed coefficients in a subband at level l − 1, and σ i l are the standard deviations of the i =1,...,N subbands stemming from a further decomposition; note that the denominator is the geometric mean of the subband variances. For example, N is equal to two for a one-dimensional transform, and to four and eight for a 2D and 3D transform respectively. The energy distribution of the transformed subbands is supposed to be highly unbalanced, i.e. a small percentage of the subbands concentrates a high percentage of the total energy. The more unbalanced is the distribution, the lower is the geometric mean. If G (l−1)→l > 1 the decomposition at level l is σ 2 l−1 retained, otherwise the current subband at level l − 1 is kept. This procedure has a very intuitive interpretation in terms of average distortion after reconstruction. Given a total coding rate, high-resolution quantization theory can be applied to the bit-allocation problem among the eight subbands at level l. This yields [23] that the optimal average distortion for a Gaussian source is D = γ( ∏ N i=1 σ i) 1 N 2 −2R , with γ a given constant. Moreover, the distortion if the decomposition at level l − 1 is retained is proportional to σl−1 2 . Therefore, the coding gain is simply the ratio between the distortions respectively obtained by keeping the current representation and performing an additional decomposition step. As a consequence, under the high-rate and Gaussian assumptions, maximizing the coding gain amounts to picking the branch of the decomposition tree that minimizes the average distortion of the reconstructed data. It is worth noting that, since this rate-distortion model embeds the quantizer effect in the constant γ, using the coding gain allows one to obtain the optimal decomposition tree taking quantization into account, but without making any explicit assumption on the employed quantizer. This is very important, as we are interested in comparing only the transforms and not the quantizers, although in a practical compression algorithm the transform will have to be followed by a quantizer. E. Multidimensional extensions of wavelet transforms When the DWT and DWPT have to be applied to 3D data set, multidimensional extensions of either transform are required. In the following we will consider three possible extensions, which are referred to as square, rectangular and hybrid transforms. Our description refers to the DWT; the generalization to DWPT is straightforward. A square 2D transform is such that first one decomposition level is computed in all dimensions. Then, the (multidimensional) approximation subband is considered, and a new iteration is applied to it.
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Transform coding techniques for lossy hyperspectral data compression

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