gms | German Medical Science

48th Meeting of the Particle Therapy Co-Operative Group

Particle Therapy Co-Operative Group (PTCOG)

28.09. - 03.10.2009, Heidelberg

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

Meeting Abstract

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  • W. Ulmer - Medical Systems, Varian International AG, Baden, Switzerland

PTCOG 48. Meeting of the Particle Therapy Co-Operative Group. Heidelberg, 28.09.-03.10.2009. Düsseldorf: German Medical Science GMS Publishing House; 2009. Doc09ptcog212

doi: 10.3205/09ptcog212, urn:nbn:de:0183-09ptcog2121

Published: September 24, 2009

© 2009 Ulmer.
This is an Open Access article distributed under the terms of the Creative Commons Attribution License ( You are free: to Share – to copy, distribute and transmit the work, provided the original author and source are credited.



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. c0 + c1 + c2 = 1 and Ksum = c0K(σ0) + c1K(σ1) + c2K(σ2). Each coefficient satisfies ck > 0. In case 2 (Mexican hut) the property c0 > 1 holds, and c1 = 1 – c0 has to be accounted, i.e. c1 < 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.