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The real advantage of using a wavelet transform for image data compression is the power of adapting to local statistics of the pixels. In hyperspectral data, many but not all spectral planes are well correlated. In each spectral plane, the spatial data is composed of patches of relatively smooth areas segmented by edges. The smooth areas can be well compressed by a relatively long wavelet transform with a large number of vanishing moments. However, for the regions around edges, shorter wavelet transforms are preferable. Despite the fact that the local statistics of both the spectral and spatial data change from pixel to pixel, almost all known image data compression algorithms use only one wavelet transform for the entire dataset. For example, the current international still image data compression standard, JPEG2000, has adopted the 5/3 wavelet transform as the default standard for lossless image data compression for all images. There is not a single wavelet filter that performs uniformly better than the others. Thus, it would be beneficial to use many types of wavelet filters based on local activities of the image. The selected wavelet transform can thus be best adapted to the content of the image locally. In this paper, we have derived a fast adaptive lifting scheme that can easily switch wavelet filters from one to the other. The adaptation is performed on a pixel-by-pixel basis, and it does not need any bookkeeping overhead. It is known that the lifting scheme is a fast and powerful tool to implement all wavelet transforms. Especially for integer-to-integer wavelet transforms, the lifting scheme is an indispensable tool for lossless image compression. Taking advantage of our newly developed lossless lifting scheme, the fast adaptive lifting algorithm presented in this paper not only saves two lifting steps but also improves accuracy compared to the conventional lifting scheme for lossless data compression. Moreover, our simulation results for ten two-dimensional images have shown that the fast adaptive lifting scheme outperforms both of the lossless wavelet tranforms used in JPEG2000 and the S+P transform in lossless SPIHT algorithm.
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