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Hierarchical decomposition of data using Haar and Legendre scaling functions as well as multiresolution compression and decomposition of data using hyperbolic 3D Haar wavelets, Battle-Lemarie wavelets, and biorthogonal wavelets have been used in the past to visually explore large volumetric data sets. In this work we explore the use of Legendre wavelets for efficient volumetric compression and rendering of data. There are several advantages of using Legendre wavelets. First, by using wavelets rather than scaling functions, we gain the advantages associated with multiresolution decomposition of data. This includes efficient exploration of data at different levels of detail and advantages of incremental rendering of data and progressive transmission. Second, the main advantage of these wavelets over other wavelet models arises from the fact that they do not overlap and therefore require filters of only unit length. In contrast, Battle-Lemarie wavelets require filters of infinite length. Similarly, if B-spline wavelets are used, they will require filters of infinite length as well. Biorthogonal wavelets require filters of finite length; however Legendre wavelets use filters of only unit length. This results in relatively simple and efficient computation. We use coherent projection method and L2 error criterion to compress and render the volumetric data, although the model is flexible to accommodate other volumetric rendering techniques and other error criteria. The Legendre wavelet model for volumetric data compression and rendering has been implemented. The system has been used for visual data exploration of several large volumetric data sets. Detailed statistical measures of compression ratios, rendering time, and associated errors have been derived for different threshold values of many volumetric data sets. Although for lossless compression Legendre wavelet model requires much more time and space, it clearly outperforms Haar wavelet model in compression and image quality for lossy compression with very small L2 errors. This characteristic is very helpful in visual exploration of data.
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