On a new reconstruction algorithm in emission tomography

A new SPECT reconstruction algorithm based on the Novikov explicit inversion formula

“…In addition, when necessary, the Fourier spectrum of the signal is also multiplied by Ŵ (ρ) in this step of computations. We utilize this implementation of the Hilbert transform throughout the algorithm to compute values of the functions h c ϕ i (p j ) and h s ϕ i (p j ), and to evaluate the modified projections m ϕ i (p j ), see equation (8).…”

“…The variety of existing SPECT algorithms can be split into a family of analytical methods [3][4][5][6][7][8][9] and a wide class of iterative techniques (see e.g. [13][14][15][16][17][18][19]).…”

“…First analytic methods for the attenuation correction in SPECT were proposed two decades ago by Bellini et al [3] and Tretiak and Metz [4]; both algorithms assumed that the attenuation coefficient was constant inside a patient's body and that the cross section of the body was convex. Two more analytic methods were introduced almost a decade later in [5] and [6] (an algorithm similar to [5] was independently developed in [7,8]). A scheme generalizing all of the above-mentioned approaches has been proposed in [9]; a general analytic approach leading to a variety of reconstruction formulae was given in [10].…”

Generalized transforms of Radon type and their applications

“…However, due to the infinite dimensionality of the co-kernel of the operator, one has a huge freedom in modifying inversion formulas. This freedom (in the exponential transform case) has been used to select the most stable inversion algorithms [61,138]. This still needs to be done for the attenuated transform (see [85] for initial considerations).…”