Numerical Evaluation of the Image Space Reconstruction Algorithm
Tomohiro Aoyagi
a
and Kouichi Ohtsubo
Faculty of Information Science and Arts, Toyo University, 2100 Kujirai, Saitama, Japan
Keywords: X-ray CT, PET, Image Reconstruction, ISRA, Steepest Descent.
Abstract: In medical imaging modality, such as X-ray computerized tomography (CT), positron emission tomography
(PET) and single photon emission computed tomography (SPECT), image reconstruction from projection is
to produce an image of a two-dimensional object from estimates of its line integrals along a finite number of
lines of known locations. The method of tomographic image reconstruction from projection can be formulated
with the Fredholm integral equation of the first kind, mathematically. It is necessary to solve the equation.
But it is difficult in general to seek the strict solution. By discretizing the image reconstruction problem, we
applied the image space reconstruction algorithm (ISRA) to the problem and evaluated the image quality. We
computed the normalized mean square error (NMSE) in reconstructed image. We have shown that the error
decreases with increasing the number of detectors, views and iterations. In addition, the effect of the relaxation
parameter, the weighting factor and the noise to the reconstructed image are analysed.
1 INTRODUCTION
In medical imaging modality, such as X-ray
computerized tomography (CT), positron emission
tomography (PET) and single photon emission
computed tomography (SPECT), image
reconstruction from projection is to produce an image
of a two-dimensional object from estimates of its line
integrals along a finite number of lines of known
locations (Herman, 2009; Kak et al., 1998; Imiya,
1985). PET or SPECT is intrinsically a three-
dimensional imaging technique and determines the
distribution of a radiopharmaceutical in the interior of
an object by measuring the radiation outside the
object in a tomographic fashion (Bendriem et al.,
1998). The method of tomographic image
reconstruction from projection can be formulated by
the Fredholm integral equation of the first kind,
mathematically. Since observed data can be
discretized experimentally, it is necessary to
discretize the equation to solve it with digital
processing. Because of the ill-posed nature, it is
difficult to solve strictly this integral equation. Up to
now many image reconstruction methods have been
proposed by the research development regardless of
imaging modality (Stark, 1987; Natterer et al., 2001).
In general inverse problems, the regularization of
a
https://orcid.org/0000-0002-7268-9826
linear ill-posed problems has been derived and
revealed the properties (Daniel 2021; Ronny et al.,
2019; Simon et al., 2022).
It is possible to divide image reconstruction
methods into two methods, transform and iterative.
Transform methods are based on discrete
implementations of analytic solution and give a one-
step solution which is directly calculated from the
observed data. Iterative method can incorporate the
discrete nature of the data sampling and
reconstruction problem and typically some statistical
model of the data acquisition process.
The image space reconstruction algorithm (ISRA)
which is one of the iterative algebraic reconstruction
methods, has been shown to be a non-negative least
squares estimator and was introduced as an
alternative image reconstruction method for PET
(Depierro, 1987; Iniyatharasi et al., 2015). By
modifying the weighted least squares objective
function, a more general form of the ISRA has been
derived and the relation between ISRA and the
maximum likelihood expectation maximization (ML-
EM) has been revealed and shown the convergence
property (Depierro, 1993; Reader, 2011).
However, the effect of discretizing an image
reconstruction model and its parameter have been not
revealed sufficiently. In this paper, by discretizing the