The inverse problem of a Gaussian convolution and its application to the finite size of the measurement chambers/detectors in photon and proton dosimetry

A Gaussian convolution kernel K is deduced as a Green's function of a Lie operator series. The deconvolution of a Gaussian kernel is developed by the inverse Green's function K-1. A practical application is the deconvolution of measured profiles D-m(x) of photons and protons with finite detector size to determine the profiles D-p(x) of point-detectors or Monte Carlo Bragg curves of protons. The presented algorithms work if D-m(x) is either an analytical function or only given in a numerical form. Some approximation methods of the deconvolution are compared (differential operator expansion to analytical adaptations of 2 x 2 cm(2) and 4 x 4 cm(2) profiles, Hermite expansions to measured 6 x 6 cm(2) and 20 x 20 cm(2) profiles and Bragg curves of 80/180 MeV protons, FFT to an analytical 4 x 4 cm(2) profile). The inverse problem may imply ill-posed problems, and, in particular, the use of FFT may be susceptible to them.