  Scatter processes of photons lead to blurring of images. Multiple scatter can
usually be described by one Gaussian convolution kernel. This can be a crude
approximation and we need a linear combination of 2/3 Gaussian kernels to
account for tails.If image structures are recorded by appropriate measurements,
these structures are always blurred. The ideal image (source function without
any blurring) is subjected to Gaussian convolutions to yield a blurred image,
which is recorded by a detector array. The inverse problem of this procedure is
the determination of the ideal source image from really determined image. If
the scatter parameters are known, we are able to calculate the idealistic
source structure by a deconvolution. We shall extend it to linear combinations
of two/three Gaussian convolution kernels in order to found applications to
aforementioned image processing, where a single Gaussian kernel would be crude.
In this communication, we shall derive a new deconvolution method for a linear
combination of 2/3 Gaussian kernels with different rms values, namely the
formulation of an inhomogeneous Fredholm integral equation of second kind and
the related Liouville - Neumann series (LNS) to calculate solutions in every
desired order. The LNS solution provides the source function rho in terms of
the Fredholm kernel Kf. We can verify some advantages of LNS in image
processing. Applications of the LNS solution are inverse problems (2 or 3
Gaussian kernels) of image processing in CBCT and IMRT detector arrays of
portal imaging. A particular advantage of LNS is given, if the scatter
functions depend on x,y,z. This fact implies that the scatter functions can be
scaled according to the electron density provided by image reconstruction
procedures.
