Provided by Cognitive Sciences ePrint Archive
Learning and Nonlinear Models - Revista da Sociedade Brasileira de Redes Neurais, Vol. 1, No. 2, pp. 122-130, 2003
© Sociedade Brasileira de Redes Neurais
TOMOGRAPHIC IMAGE RECONSTRUCTION OF FAN-BEAM PROJECTIONS
WITH EQUIDISTANT DETECTORS USING PARTIALLY CONNECTED NEURAL
NETWORKS
Luciano Frontino de Medeiros
FACINTER/PR
Hamilton Pereira da Silva
COPPE/UFRJ
Eduardo Parente Ribeiro
UFPR
Abstract: We present a neural network approach for tomographic imaging problem using interpolation methods and
fan-beam projections. This approach uses a partially connected neural network especially assembled for solving tomographic
reconstruction with no need of training. We extended the calculations to perform reconstruction with interpolation and to allow
tomography of fan-beam geometry. The main goal is to aggregate speed while maintaining or improving the quality of the
tomographic reconstruction process.
Index Terms: Tomography, reconstruction, neural network, fan-beam, interpolation.
1. Introduction
Neural networks have already been studied for tomographic image reconstruction using multilayer perceptron with
backpropagation training methods [1-3]. Self-organizing neural networks have also been proposed to reconstruct original
image when a limited number of projections is available [4]. In this article, we present an alternative approach to obtain
computerized tomography (CT) images using partially connected neural networks whose synaptic connections are calculated
according to the geometry of problem without the need of training. We have extended the formulation of early works [5,6] to
include interpolation and fan-beam geometry. This configuration is more closely related to the one used on most equipments
where a single x-ray source emits a thin divergent beam and several detectors receive the radiation after attenuation by the
object [7]. If the distance between source and detectors is large enough, the rays can be considered parallel and tomographic
reconstruction assuming parallel rays will produce negligible errors. On the other hand, if the distance is small, the angles
between source and detectors will be representative resulting in a distorted reconstructed image. In figure 1, a simulation with a
synthetic Sheep-Logan phantom was performed to illustrate such effect. If the x-ray measurements were made assuming
parallel geometry the reconstruction with the same model results in a good image (a) but if the detectors were positioned in a
fan-beam fashion the same reconstruction with parallel geometry would provide an image with undesirable artifacts.
Reconstructed image quality can also be improved by using linear interpolation instead of simple truncation which is
what is usually done due to the discrete processing of data. We show how interpolation can be naturally incorporated to
network weights to produce the desired reconstructed image. The proposed partially connected neural network has its weights
previously calculated according to the geometry of the problem including interpolation. This weighed filtered backprojection
implementation can be calculated faster than convention reconstruction with the expense of more memory usage to store the
weights [8]. Hardware implementation of such network could better explore the inherent parallelism of the problem and
achieve a very significant speedup [ 9].
2. Methods
A tomographic projection of an object is obtained when an electromagnetic radiation beam (like x-ray or gamma ray) pass
through the object and projects a shadow in a fence or image intensifier at the opposite side (figure 2). Object essentials
properties like mass rest or specific density raise a resistance, reducing the intensity of the beam, which decays exponentially
according to the total radiopacity along the path of the ray [10,11]:
I=I0exp{-∫L f(x,y)du}
(1)
where f(x,y) represents the absorption coefficient at a point (x,y), L is the path of the ray and u is the distance along L.
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