Wavelet Transform and Tomography: Continuous and Discrete Approaches
In tomography, an image is reconstructed from its projections from different directions. In this chapter, the relationships between the wavelet decomposition of an image and those of its projections are developed. The problem is first analyzed using a continuous formulation. It may be shown that the images reconstructed from the wavelet transform of the projections provide a 2-D wavelet decomposition of the image. Conversely, an appropriate 1-D filtering may be done on the sinogram prior to reconstruction, in order to recover any 2-D continuous wavelet transform. Then, the application of a multiresolution scheme using a pair of discrete filters is studied. We respectively examine the construction of 2-D filters (resp. 1-D filters) from a pair of 1-D (resp. 2-D) exact reconstruction filters. Both the case of dyadic and quincunx decompositions are considered. For illustration, results on an X-ray CT scan medical image are presented.