TY - JOUR
T1 - An Integral Equation Model for PET Imaging
AU - Tang , Xinhuang
AU - Schmidtlein , Charles Ross
AU - Li , Si
AU - Xu , Yuesheng
JO - International Journal of Numerical Analysis and Modeling
VL - 6
SP - 834
EP - 864
PY - 2021
DA - 2021/11
SN - 18
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/ijnam/19952.html
KW - Positron emission tomography, integral equation model, image reconstruction.
AB - <p style="text-align: justify;">Positron emission tomography (PET) is traditionally modeled as discrete systems.
Such models may be viewed as piecewise constant approximations of the underlying continuous
model for the physical processes and geometry of the PET imaging. Due to the low accuracy of
piecewise constant approximations, discrete models introduce an irreducible modeling error which
fundamentally limits the quality of reconstructed images. To address this bottleneck, we propose
an integral equation model for the PET imaging based on the physical and geometrical considerations, which describes accurately the true coincidences. We show that the proposed integral
equation model is equivalent to the existing idealized model in terms of line integrals which is accurate but not suitable for numerical approximation. The proposed model allows us to discretize
it using higher accuracy approximation methods. In particular, we discretize the integral equation
by using the collocation principle with piecewise linear polynomials. The discretization leads to
new ill-conditioned discrete systems for the PET reconstruction, which are further regularized by a
novel wavelet-based regularizer. The resulting non-smooth optimization problem is then solved by
a preconditioned proximity fixed-point algorithm. Convergence of the algorithm is established for
a range of parameters involved in the algorithm. The proposed integral equation model combined
with the discretization, regularization, and optimization algorithm provides a new PET image
reconstruction method. Numerical results reveal that the proposed model substantially outperforms the conventional discrete model in terms of the consistency to simulated projection data
and reconstructed image quality. This indicates that the proposed integral equation model with
appropriate discretization and regularizer can significantly reduce modeling errors and suppress
noise, which leads to improved image quality and projection data estimation.</p>