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## Deconvolution of combinations of Gaussian kernels and applications to proton dosimetry and image processing

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Published: | September 24, 2009 |
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Many problems of scatter, energy/range straggling or influence of the finite size of a ionization chamber lead to the convolution with a Gaussian kernel K and to a deconvolution with regard to the inverse problem K^{-1}. If the convolution problem is extended to a sum of Gaussian kernels with different rms-values as considered in lateral scatter of protons, the deconvolution problem has to be extended appropriately. We consider here a sum of Gaussian kernels with positive coefficients (case 1) and to a Mexican hut (case 2). In case 1 the normalization requires that the sum of the normalized Gaussian kernels is always 1, i.e. c_{0} + c_{1} + c_{2} = 1 and K_{sum} = c_{0}K(σ_{0}) + c_{1}K(σ_{1}) + c_{2}K(σ_{2}). Each coefficient satisfies c_{k} > 0. In case 2 (Mexican hut) the property c_{0} > 1 holds, and c_{1} = 1 – c_{0} has to be accounted, i.e. c_{1} < 0.

We discuss examples of the deconvolution of both cases. Case 1 is considered in an analysis of transverse profiles of protons (multiple scatter of Molière) and in the deconvolution of CT images for the elimination of scatter. The second case is applied to a proton Bragg curve measured by an ionization chamber and a diode detector, i.e. to the conversion from the data obtained by the ionization chamber to the diode detector. In the domain of the Bragg peak there is a significantly different physical behavior between both measurement methods. The application to image processing shows the wide scope of the methods.