Analytic reconstruction algorithms in emission tomography with variable attenuation

We present, two analytic reconstruction algorithms applicable in emission tomography with variable attenuation. One of this algorithms is based on the recently discovered explicit inversion formula for the attenuated Radon transform with non-uniform attenuation; it is intended for applications in single-photon emission computed tomography. The second of the methods we present is applicable for approximate inversion of the generalized Radon transform with more general weights for which inversion formulae are not known; inversion of such transforms are required, for instance, in the emission tomography of gases. The latter algorithm is based on an approximate (up to a smoothing term) inversion of the underlying integral operator; in spite of its approximate nature it yields quite accurate reconstructions and exhibits very low sensitivity to noise in data. In fact, when applied to data containing significant level of noise the latter algorithm yields better reconstructions than the first of our mehods (based on theoretically exact inversion formula). We support conclusions of the paper by a number of convincing numerical results.