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

TL;DR

This paper introduces a new deconvolution method using an inhomogeneous Fredholm integral equation of the second kind and Liouville-Neumann series to recover ideal images blurred by a linear combination of Gaussian kernels, with applications in advanced image processing.

Contribution

It extends deconvolution techniques to linear combinations of multiple Gaussian kernels using Fredholm equations and Liouville-Neumann series, improving image reconstruction accuracy.

Findings

LNS provides an effective solution for deconvolving multiple Gaussian kernels.

The method adapts to scatter functions dependent on spatial coordinates.

Applications demonstrated in CBCT and IMRT detector image processing.

Abstract

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.