Image compression can be performed in the original spatial domain or in a transform domain. In the latter case, the image is first transformed, and a subsequent compression operation is applied in the transform domain. An example is the conventional cosine transform method used in the standard JPEG (Joint Photographic Experts Group) algorithm. The use of the wavelet transform for image compression has drawn significant attention since the publication of the works by Daubechies  and Mallat . The primary advantage of the wavelet transform compared with the cosine transform is that the wavelet transform is localized in both the spatial and frequency domains; therefore, the transformation of a given signal will contain both spatial and frequency information of that signal. On the other hand, the cosine transform basis extends infinitely, with the result that the spatial information is spread out over the whole frequency domain. Because of this property, wavelet image compression has shown promising results on medical images. Although compression techniques are mainly applied to two-dimensional (2D) images, the increasing availability of three-dimensional (3D) CT and MR volume data sets raises the need for 3D compression techniques [3,4]. This chapter presents compression with 3D wavelet transforms and discusses the selection of wavelet filters. The compression outcomes of 3D data sets with JPEG and 2D wavelet transform are also compared to the compression obtained with the 3D wavelet transform approach.
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