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Methods of computed tomography are well developed and widely used in medicine and industry. If tomographic data are complete, it is possible to reconstruct the images with submillimeter resolution. If the data are incomplete, tomograms may blur, i.e. their resolution degrades, noise increases and artifacts form. The situation is worst if measurement data are so poor that the system of equations which describe the discrete reconstruction problem appears to be strongly underdetermined. In this situation, images of acceptable quality can be obtained with algorithms that regularize the solution and use a priori information about the object, and do post-processing of reconstructed tomograms also with the use of a priori information, as a rule. This chapter provides two examples demonstrating the reconstruction of the internal structure of an object from strongly incomplete measurement data: few-view computed tomography (FVCT) and diffuse optical tomography (DOT) of strongly scattering media. The problem of reconstruction from a small number of views (<10) arises, for example, in experimental plasma research (Pickalov & Melnikova, 1995) or nondestructive testing (Subbarao et al., 1997). DOT is now deemed to hold much promise for cancer detection (Arridge, 1999; Hawrysz & Sevick-Muraca, 2000; Yodh & Chance, 1995). Here the strong incompleteness of data is caused by the fact that the number of source-receiver relations that define the number of measurements is strictly limited. Despite that these types of tomography use different wavelength bands (X-ray and near infrared) and different mathematical models (linear and non-linear), we think it is not only possible, but also interesting to consider them together because in both cases we successfully use similar reconstruction algorithms and similar post-processing methods. The unique possibility to do that comes from the fact that in case of DOT, we use a simplified reconstruction method (Konovalov et al., 2003; 2006b; 2007; Lyubimov et al., 2002; 2003) reducing the inverse problem to a solution of the integral equation with integration along a conditional photon average trajectory (PAT) – an analog of the Radon transform in projection tomography. In case of FVCT, we use actual data from measurements in a simple experimental radiography setup (Konovalov et al., 2006 ). The FVCT procedure is simulated by rotation of the object from exposure to exposure about the centre of the reconstruction region. For

Algebraic Reconstruction and Post-Processing in Incomplete Data Computed Tomography: from X-rays to Laser Beams