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Scatter processes of photons lead to blurring of images. Multiple scatter canusually be described by one Gaussian convolution kernel. This can be a crudeapproximation and we need a linear combination of 2/3 Gaussian kernels toaccount for tails.If image structures are recorded by appropriate measurements,these structures are always blurred. The ideal image (source function withoutany blurring) is subjected to Gaussian convolutions to yield a blurred image,which is recorded by a detector array. The inverse problem of this procedure isthe determination of the ideal source image from really determined image. Ifthe scatter parameters are known, we are able to calculate the idealisticsource structure by a deconvolution. We shall extend it to linear combinationsof two/three Gaussian convolution kernels in order to found applications toaforementioned image processing, where a single Gaussian kernel would be crude.In this communication, we shall derive a new deconvolution method for a linearcombination of 2/3 Gaussian kernels with different rms values, namely theformulation of an inhomogeneous Fredholm integral equation of second kind andthe related Liouville - Neumann series (LNS) to calculate solutions in everydesired order. The LNS solution provides the source function rho in terms ofthe Fredholm kernel Kf. We can verify some advantages of LNS in imageprocessing. Applications of the LNS solution are inverse problems (2 or 3Gaussian kernels) of image processing in CBCT and IMRT detector arrays ofportal imaging. A particular advantage of LNS is given, if the scatterfunctions depend on x,y,z. This fact implies that the scatter functions can bescaled according to the electron density provided by image reconstructionprocedures.

Deconvolution of a linear combination of Gaussian kernels by an  inhomogeneous Fredholm integral equation of second kind and applications to  image processing