Inverse problem of linear combinations of Gaussian convolution kernels (deconvolution) and some applications to proton/photon dosimetry and image processing

The deconvolution of a single Gaussian kernel is extended to a sum of Gaussian kernels with positive coefficients (case 1) and to a Mexican hat (case 2). In case 1 the normalization requires the sum of the normalized Gaussian kernels to be always 1, i.e. c0 + c1 + c2 + · · · = 1. Each coefficient satisfies ck > 0. In case 2 (Mexican hat) the properties c0 + c1 = 1 with c0 > 1and c1 < 0 hold; c1 = 1 − c0 has to be accounted for the normalization. We discuss examples of the deconvolution of both cases. Case 1 is considered in an analysis of transverse profiles (protons and photons) and in the deconvolution of CT images to eliminate scatter. Case 2 is applied to a proton Bragg curve measured by an ionization chamber and a diode detector. In the domain of the Bragg peak there is a different physical behavior between both measurement methods. (Some figures in this article are in colour only in the electronic version)