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 Dm(x) of photons and protons with finite detector size to determine the profiles Dp(x) of point-detectors or Monte Carlo Bragg curves of protons. The presented algorithms work if Dm(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 cm2 and 4 x 4 cm2 profiles, Hermite expansions to measured 6 x 6 cm2 and 20 x 20 cm2 profiles and Bragg curves of 80/180 MeV protons, FFT to an analytical 4 x 4 cm2 profile). The inverse problem may imply ill-posed problems, and, in particular, the use of FFT may be susceptible to them.