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FIGURE 5 Block diagrams of image compression and decompression using the 3D wavelet transform.

that is, they can be expressed by the product of three ID functions, one for each dimension. The scaling function has the form

^(x, y, z T2(x, y, z T3(x, y, z T4(x, y, z T5(x, y, z T6(x, y, z T7(x, y, z where ^(x), ^(y), and ^(z

eight components: /1 is the low (smooth) resolution portion of the image data, and the remaining blocks are high (detailed) resolution components. As Fig. 6 indicates, can be further decomposed into eight smaller volumes labeled _/2 and _/2's. The detailed images // on level 1 contain higher frequency components than those _/2' of level 2.

With properly chosen wavelet functions, the low-resolution component in the m level is 1/(23)m of the original image size after the transformation, but contains about 90% of the total energy in the m level. It is clear that the high-resolution components are spread into different decomposition levels. For these reasons, the wavelet transform components provide an efficient representation of the original image for compression purposes. Different levels of representation can be encoded differently to achieve a desired compression ratio.

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