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!Analytical Reconstruction Techniques
As we have seen ((Integrálgeometria és Integráltranszformációk|#Integrálgeometria és Integráltranszformációk|earlier)) tomographic imaging produces acquired data that is the ((A 2D Radon-transzformáció|#A 2D Radon-transzformáció|Radon transform)) of some physical distribution (e.g. isotope concentration, attenuation coefficient). This chapter discusses how to find the inverse of the Radon transform and with that how raw data can be processed to regain the original distribution we are interested in, i.e. the isotope concentration or the x-ray attenuation coefficient. This transformation is called image reconstruction.
In this chapter we show the inverse of the Radon transform as a mathematical operator, and relating to that the most used numerically stable interpretation that allows for forming algorithms. The most important is the filtered backprojection, that is the most wide spread amongst reconstruction algorithms.
!! Inverse Radon transform: the snag
Johann Radon in his 1917 article gave the inverse of the Radon transform:
{EQUATION(size="80")}$f\left(\mathbf{x} \right)=\frac{1}{{4\pi}^{2}}\int_{{S}^{1}}\int_{{\mathbb{R}}^{1}}\frac{\frac{d}{dt}Rf\left(t,\vartheta \right)}{\mathbf{x}\boldsymbol{\omega}-t}dtd\vartheta {EQUATION}
The integral as we have seen at the ((A Hilbert-transzformált|#A Hilbert-transzformált|Hilbert transfrom)) is meant to be taken as a Cauchy Principal Value (P.V.) as the denominator induces a singularity rendering the expression non-integrable in a Riemann sense. A more serious problem is that, as a consequence, the numerical evaluation of the integral is impossible. We will also see that this inversion formula only holds for two ( or rather even) dimensions, for higher (for odd) dimensions another formula and another numerical algorithm is needed. Similarly to treating the Hilbert transform the same manipulations can help us avoiding the numerical difficulties, this will result in the Filtered Backprojection algorithm. In our discussion we give a different way of deriving the filtered backprojection yielding ways for designing filters.
It can be stated that for an ''f'' function of compact support, that if any nonzero infinite set of projection (angular distribution of the Radon transform) determines uniquely the original distribution but none of the finite sets. It is of course not an unusual statement in numerical mathematics, but it must be investigated how the sampling influences the accuracy of the reconstruction.
The ((A Központi Szelet Tétel|#A Központi Szelet Tétel|next section)) states the central slice theorem giving the basis for filtered backprojection algorithms.