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Along with a review of some of the mathematical foundations of computed tomography, the article contains new results on derivation of reconstruction formulas in a general setting encompassing all standard formulas; discussion and examples of the role of the point spread function with recipes for producing suitable ones; formulas for, and examples of, the reconstruction of certain functions of the attenuation coefficient, e.g., sharpened versions of it, some of them with the property that reconstruction at a point requires only the attenuation along rays meeting a small neighborhood of the point.

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Given a pair of biorthogonal, compactly supported multiwavelets, we present an algorithm for raising their approximation orders to any desired level, using one lifting step and one dual lifting step. Free parameters in the algorithm are explicitly identified, and can be used to optimize the result with respect to other criteria.

In [1], Beylkin et al. introduced a wavelet-based algorithm that approximates integral or matrix operators of a certain type by highly sparse matrices, as the basis for efficient approximate calculations. The wavelets best suited for achieving the highest possible compression with this algorithm are Daubechies wavelets, while Coiflets lead to a faster decomposition algorithm at slightly lesser compression. We observe that the same algorithm can be based on biorthogonal instead of orthogonal wavelets, and derive two classes of biorthogonal wavelets that achieve high compression and high decomposition speed, respectively. In numerical experiments, these biorthogonal wavelets achieved both higher compression and higher speed than their wavelet counterparts, at comparable accuracy.

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