A content-adaptive unstructured grid based integral equation method with the TV regularization for SPECT reconstruction

Yun Chen; Jiasheng Huang; Si Li; Yao Lu; Yuesheng Xu

doi:10.3934/ipi.2019062

Article Contents

2020, Volume 14, Issue 1: 27-52. Doi: 10.3934/ipi.2019062

This issue Previous Article Artifacts in the inversion of the broken ray transform in the plane Next Article A partial data inverse problem for the convection-diffusion equation

A content-adaptive unstructured grid based integral equation method with the TV regularization for SPECT reconstruction

1.
School of Mathematics, Sun Yat-sen University, Guangdong Province Key Laboratory of Computational Science, Sun Yat-sen University, Guangzhou 510275, China
2.
Department of Mathematics and Statistics, Old Dominion University, Norfolk, VA 23529, USA
3.
School of Computer Science and Technology, Guangdong University of Technology, Guangzhou 510006, China
4.
School of Data and Computer Science, Sun Yat-sen University, Guangdong Province Key Laboratory of Computational Science, Sun Yat-sen University, Guangzhou 510275, China

^* Corresponding author: Yuesheng Xu
^* Corresponding author: Yuesheng Xu

^† Y. Chen and Y. Lu contributed equally to this work

Received: November 2018

Revised: May 2019

Published: November 2019

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Abstract

Existing reconstruction methods for single photon emission computed tomography (SPECT) are most based on discrete models, leading to low accuracy in reconstruction. Reconstruction methods based on integral equation models (IEMs) with a higher order piecewise polynomial discretization on the pixel grid for SEPCT imaging were recently proposed to overcome the accuracy deficiency of the discrete models. Discretization of IEMs based on the pixel grid leads to a system of a large dimension, which may require higher computational costs to solve. We develop a SPECT reconstruction method which employs an IEM of the SPECT data acquisition process and discretizes it on a content-adaptive unstructured grid (CAUG) with the total variation (TV) regularization aiming at reducing computational costs of the integral equation method. Specifically, we design a CAUG of the image domain for the discretization of the IEM, and propose a TV regularization defined on the CAUG for the resulting ill-posed problem. We then apply a preconditioned fixed-point proximity algorithm to solve the resulting non-smooth optimization problem, and provide convergence analysis of the algorithm. Numerical experiments are presented to demonstrate the superiority of the proposed method over the competing methods in terms of suppressing noise, preserving edges and reducing computational costs.

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Mathematics Subject Classification: Primary: 94A08, 65R20, 65F22.

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Figure 1. The content-adaptive unstructured grid (CAUG): (a) The initial estimate; (b) The given attenuation distribution; (c) The quadtree grid $ \mathcal{G}^0 $ generated by the quadtree scheme from $ f_0 $ and $ \mu $; (d) The CAUG from $ \mathcal{G}^0 $ through the force equilibrium method

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Figure 2. The chest phantom and the simulated projection

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Figure 3. CAUGs, where the upper, middle and lower rows are noise-free, low and high noise cases, respectively

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Figure 4. Results via different reconstruction methods on the CAUGs. (a) Projection data with noise-free, low and high noise in the upper, middle and lower rows, respectively; (b) Reconstructions by the ML-EM (ML); (c) Reconstructions by the MAP-EM (MAP); (d) Reconstructions by the PFP$ ^{2} $A with the proposed regularization defined on the CAUG (RUG)

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Figure 5. Results via the PFP$ ^{2} $A on the pixel grid and on the CAUG. (a) Projection data with noise-free, low and high noise in the upper, middle and lower rows, respectively; (b) Reconstructions by the PFP$ ^{2} $A with the discrete TV on the pixel grid (DTV); (c) Reconstructions by the RUG

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Figure 6. (a) The relative error of the iterative sequence varies with computing time (starting with 0.6 seconds) for reconstruction on the CAUG and on the pixel grid from projection data with high noise; (b) Total time of obtaining the reconstructed images, including 6.8 seconds for yielding the initial estimate and 3.6 seconds for generating the CAUG

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Figure 7. Reference image and image quality assessment for the reconstructed results from projection data with high noise. (a) The selected ROI1, ROI2 and line profile are labeled in red, blue and white, respectively; (b) NMSE; (c) Standard deviation for two ROIs; (d) SNR for two ROIs; (e) CNR; (f) FWHM for the line profile, where REF. is the reference image

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Figure 8. Results through reconstruction from projections with size $ 120\times 128 $ by the DTV and the RUG. (a) Projection data with noise-free, low and high noise in the upper, middle and lower rows, respectively; (b) Reconstructions by the DTV; (c) Reconstructions by the RUG

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Table 1. Dimensions of the solution spaces based on different basis functions for images with different sizes

	$ 128\times 128 $	$ 256\times 256 $	$ 512\times 512 $
Pixel-based piecewise constant	16384	65536	262144
Pixel-based piecewise linear	65536	262144	1048576
CAUG-based piecewise linear	2181	8207	18316

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Table 2. Evaluation for the reconstructed images from projection data with high noise in figure 8

	NMSE	CNR	FWHM
DTV	2.71	15.79	0.97
RUG	2.35	17.17	0.90

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