American Institute of Mathematical Sciences

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February  2020, 14(1): 27-52. doi: 10.3934/ipi.2019062

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

Yun Chen 1,2,†, , Jiasheng Huang 1, , Si Li 3, , Yao Lu 4,†, and  Yuesheng Xu 2,,

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

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

Received  November 2018 Revised  May 2019 Published  November 2019

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.

Keywords: SPECT reconstruction, unstructured grid, integral equation, regularization, fixed-point proximity algorithm.
Mathematics Subject Classification: Primary: 94A08, 65R20, 65F22.
Citation: Yun Chen, Jiasheng Huang, Si Li, Yao Lu, Yuesheng Xu. A content-adaptive unstructured grid based integral equation method with the TV regularization for SPECT reconstruction. Inverse Problems and Imaging, 2020, 14 (1) : 27-52. doi: 10.3934/ipi.2019062
References:
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H. Bauschke and P. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, CMS Books in Mathematics, Springer, New York, 2011. doi: 10.1007/978-1-4419-9467-7.

[2]

R. Boutchko, A. Sitek and G. Gullberg, Practical implementation of tetrahedral mesh reconstruction in emission tomography, Phys. Med. Biol., 58 (2013), 3001-3022.  doi: 10.1088/0031-9155/58/9/3001.

[3]

J. Brankov, Y. Yang and M. Wernick, Tomographic image reconstruction based on a content-adaptive mesh model, IEEE Trans. Medical Imaging, 23 (2004), 202-212.  doi: 10.1109/TMI.2003.822822.

[4]

R. Chan, T. Chan and A. Yip, Numerical methods and applications in total variation image restoration, in Handbook of Mathematical Methods in Imaging, Springer, New York, 2015, 1501–1537. doi: 10.1007/978-3-642-27795-5_24-5.

[5]

Z. Chen, C. A. Micchelli and Y. Xu, Multiscale Methods for Fredholm Integral Equations, Cambridge Monographs on Applied and Computational Mathematics, 28, Cambridge University Press, Cambridge, 2015. doi: 10.1017/CBO9781316216637.

[6]

W. Dahmen, R. Schneider and Y. Xu, Nonlinear functionals of wavelet expansions-adaptive reconstruction and fast evaluation, Numer. Math., 86 (2000), 49-101.  doi: 10.1007/PL00005403.

[7]

A. Dempster, N. Laird and D. Rubin, Maximum likelihood from incomplete data via the EM algorithm, J. Roy. Statist. Soc. Ser. B, 39 (1977), 1-38.  doi: 10.1111/j.2517-6161.1977.tb01600.x.

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H. Edelsbrunner, Geometry and Topology for Mesh Generation, Cambridge Monographs on Applied and Computational Mathematics, 7, Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511530067.

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J. Fessler and A. Hero, Penalized maximum-likelihood image reconstruction using space-alternating generalized EM algorithms, IEEE Trans. Image Processing, 4 (1995), 1417-1429.  doi: 10.1109/SSBI.2002.1233983.

[10]

H. Hudson and R. Larkin, Accelerated image reconstruction using ordered subsets of projection data, IEEE Trans. Medical Imaging, 13 (1994), 601-609.  doi: 10.1109/42.363108.

[11]

Y. Jiang, S. Li and Y. Xu, A higher-order polynomial method for SPECT reconstruction, IEEE Trans. Medical Imaging, 38 (2019), 1271-1283.  doi: 10.1109/TMI.2018.2881919.

[12]

A. Krol, S. Li, L. Shen and Y. Xu, Preconditioned alternating projection algorithms for maximum a posterori ECT reconstruction, Inverse Problems, 28 (2012), 34pp. doi: 10.1088/0266-5611/28/11/115005.

[13]

K. Lange and R. Carson, EM reconstruction algorithms for emission and transmission tomography, J. Comput. Assisted Tomography, 8 (1984), 306-316. 

[14]

Q. Li, L. Shen, Y. Xu and N. Zhang, Multi-step fixed-point proximity algorithms for solving a class of optimization problems arising from image processing, Adv. Comput. Math., 41 (2015), 387-422.  doi: 10.1007/s10444-014-9363-2.

[15]

S. Li, J. Zhang, A. Krol, C. Schmidtlein, D. Feiglin and Y. Xu, Preconditioned alternating projection algorithm for solving the penalized-likelihood SPECT reconstruction problem, Physica Medica: European J. Medical Physics, 38 (2017), 23-35.  doi: 10.1016/j.ejmp.2017.05.001.

[16]

Y. Liu, L. Shen, Y. Xu and H. Yang, A collocation method solving integral equation models for image restoration, J. Integral Equations Appl., 28 (2016), 263-307.  doi: 10.1216/JIE-2016-28-2-263.

[17]

W. Long, Y. Lu, L. Shen and Y. Xu, High-resolution image reconstruction: an env$_{\ell^1}$/TV model and a fixed-point proximity algorithm, Int. J. Numer. Anal. Model., 14 (2017), 255-282. 

[18]

Y. Lu, L. Shen and Y. Xu, Integral equation models for image restoration: High accuracy methods and fast algorithms, Inverse Problems, 26 (2010), 32pp. doi: 10.1088/0266-5611/26/4/045006.

[19]

Y. Lu, H. Ye, Y. Xu, X. Hu, L. Vogelsang, L. Shen, D. Feiglin, E. Lipson and A. Krol, Expectation maximization SPECT reconstruction with a content adaptive singularity-based mesh-domain image model, Proceedings of SPIE: Physics of Medical Imaging, 6913, 2008. doi: 10.1117/12.773082.

[20]

F. Massanes and J. Brankov, Motion compensated reconstruction of 4D SPECT using parallel computation and deformable content adaptive mesh, IEEE Nuclear Science Symposium and Medical Imaging Conference, 2014, 1–4. doi: 10.1109/NSSMIC.2014.7431022.

[21]

F. Massanes and J. Brankov, Calculations of a SPECT projection operator on a graphical processing unit, Proceedings of SPIE: Physics of Medical Imaging, 8313, 2012, 1–6. doi: 10.1117/12.911951.

[22]

C. A. Micchelli, L. Shen and Y. Xu, Proximity algorithms for image models: Denoising, Inverse Problems, 27 (2011), 30pp. doi: 10.1088/0266-5611/27/4/045009.

[23]

V. Panin, G. Zeng and G. Gullberg, Total variation regulated EM algorithm, IEEE Trans. Nuclear Science, 46 (1999), 2202-2210.  doi: 10.1109/23.819305.

[24]

P. Persson, Mesh Generation for Implict Geometries, Ph.D thesis, Massachusetts Institute of Technology, 2005.

[25]

L. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D, 60 (1992), 259-268.  doi: 10.1016/0167-2789(92)90242-F.

[26]

A. Saalehi and A. Borthwick, Quadtree and octree grid generation, Internat. J. Engineering, 9 (1996).

[27]

A. Sawatzky, (Nonlocal) Total Variation in Medical Imaging, Ph.D thesis, Westfälische Wilhelms-Universität Münster, 2011.

[28]

A. Sitek, R. Huesman and G. Gullberg, Tomographic recostruction using an adaptive tetrahedral mesh defined by a point cloud, IEEE Trans. Medical Imaging, 25 (2006), 1172-1179.  doi: 10.1109/TMI.2006.879319.

[29]

G. Strang, Piecewise polynomials and the finite element method, Bull. Amer. Math. Soc., 79 (1973), 1128-1137.  doi: 10.1090/S0002-9904-1973-13351-8.

[30]

M. Wernick and J. Aarsvold, Emission Tomography: the Fundamentals of PET and SPECT, Elsevier Academic Press, San Diego, 2004.

[31]

L. Westover, Footprint evaluation for volume rendering, ACM Siggraph Computer Graphics, 24 (1990), 367-376.  doi: 10.1145/97879.97919.

[32]

Z. Wu, S. Li, X. Zeng, Y. Xu and A. Krol, Reducing staircasing artifacts in SPECT reconstruction by an infimal convolution regularization, J. Comput. Math., 34 (2016), 626-647.  doi: 10.4208/jcm.1607-m2016-0537.

[33]

Y. Xu and Q. Zou, Tree wavelet approximations with applications, Sci. China Ser. A, 48 (2005), 680-702.  doi: 10.1360/04ys0173.

show all references

References:
[1]

H. Bauschke and P. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, CMS Books in Mathematics, Springer, New York, 2011. doi: 10.1007/978-1-4419-9467-7.

[2]

R. Boutchko, A. Sitek and G. Gullberg, Practical implementation of tetrahedral mesh reconstruction in emission tomography, Phys. Med. Biol., 58 (2013), 3001-3022.  doi: 10.1088/0031-9155/58/9/3001.

[3]

J. Brankov, Y. Yang and M. Wernick, Tomographic image reconstruction based on a content-adaptive mesh model, IEEE Trans. Medical Imaging, 23 (2004), 202-212.  doi: 10.1109/TMI.2003.822822.

[4]

R. Chan, T. Chan and A. Yip, Numerical methods and applications in total variation image restoration, in Handbook of Mathematical Methods in Imaging, Springer, New York, 2015, 1501–1537. doi: 10.1007/978-3-642-27795-5_24-5.

[5]

Z. Chen, C. A. Micchelli and Y. Xu, Multiscale Methods for Fredholm Integral Equations, Cambridge Monographs on Applied and Computational Mathematics, 28, Cambridge University Press, Cambridge, 2015. doi: 10.1017/CBO9781316216637.

[6]

W. Dahmen, R. Schneider and Y. Xu, Nonlinear functionals of wavelet expansions-adaptive reconstruction and fast evaluation, Numer. Math., 86 (2000), 49-101.  doi: 10.1007/PL00005403.

[7]

A. Dempster, N. Laird and D. Rubin, Maximum likelihood from incomplete data via the EM algorithm, J. Roy. Statist. Soc. Ser. B, 39 (1977), 1-38.  doi: 10.1111/j.2517-6161.1977.tb01600.x.

[8]

H. Edelsbrunner, Geometry and Topology for Mesh Generation, Cambridge Monographs on Applied and Computational Mathematics, 7, Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511530067.

[9]

J. Fessler and A. Hero, Penalized maximum-likelihood image reconstruction using space-alternating generalized EM algorithms, IEEE Trans. Image Processing, 4 (1995), 1417-1429.  doi: 10.1109/SSBI.2002.1233983.

[10]

H. Hudson and R. Larkin, Accelerated image reconstruction using ordered subsets of projection data, IEEE Trans. Medical Imaging, 13 (1994), 601-609.  doi: 10.1109/42.363108.

[11]

Y. Jiang, S. Li and Y. Xu, A higher-order polynomial method for SPECT reconstruction, IEEE Trans. Medical Imaging, 38 (2019), 1271-1283.  doi: 10.1109/TMI.2018.2881919.

[12]

A. Krol, S. Li, L. Shen and Y. Xu, Preconditioned alternating projection algorithms for maximum a posterori ECT reconstruction, Inverse Problems, 28 (2012), 34pp. doi: 10.1088/0266-5611/28/11/115005.

[13]

K. Lange and R. Carson, EM reconstruction algorithms for emission and transmission tomography, J. Comput. Assisted Tomography, 8 (1984), 306-316. 

[14]

Q. Li, L. Shen, Y. Xu and N. Zhang, Multi-step fixed-point proximity algorithms for solving a class of optimization problems arising from image processing, Adv. Comput. Math., 41 (2015), 387-422.  doi: 10.1007/s10444-014-9363-2.

[15]

S. Li, J. Zhang, A. Krol, C. Schmidtlein, D. Feiglin and Y. Xu, Preconditioned alternating projection algorithm for solving the penalized-likelihood SPECT reconstruction problem, Physica Medica: European J. Medical Physics, 38 (2017), 23-35.  doi: 10.1016/j.ejmp.2017.05.001.

[16]

Y. Liu, L. Shen, Y. Xu and H. Yang, A collocation method solving integral equation models for image restoration, J. Integral Equations Appl., 28 (2016), 263-307.  doi: 10.1216/JIE-2016-28-2-263.

[17]

W. Long, Y. Lu, L. Shen and Y. Xu, High-resolution image reconstruction: an env$_{\ell^1}$/TV model and a fixed-point proximity algorithm, Int. J. Numer. Anal. Model., 14 (2017), 255-282. 

[18]

Y. Lu, L. Shen and Y. Xu, Integral equation models for image restoration: High accuracy methods and fast algorithms, Inverse Problems, 26 (2010), 32pp. doi: 10.1088/0266-5611/26/4/045006.

[19]

Y. Lu, H. Ye, Y. Xu, X. Hu, L. Vogelsang, L. Shen, D. Feiglin, E. Lipson and A. Krol, Expectation maximization SPECT reconstruction with a content adaptive singularity-based mesh-domain image model, Proceedings of SPIE: Physics of Medical Imaging, 6913, 2008. doi: 10.1117/12.773082.

[20]

F. Massanes and J. Brankov, Motion compensated reconstruction of 4D SPECT using parallel computation and deformable content adaptive mesh, IEEE Nuclear Science Symposium and Medical Imaging Conference, 2014, 1–4. doi: 10.1109/NSSMIC.2014.7431022.

[21]

F. Massanes and J. Brankov, Calculations of a SPECT projection operator on a graphical processing unit, Proceedings of SPIE: Physics of Medical Imaging, 8313, 2012, 1–6. doi: 10.1117/12.911951.

[22]

C. A. Micchelli, L. Shen and Y. Xu, Proximity algorithms for image models: Denoising, Inverse Problems, 27 (2011), 30pp. doi: 10.1088/0266-5611/27/4/045009.

[23]

V. Panin, G. Zeng and G. Gullberg, Total variation regulated EM algorithm, IEEE Trans. Nuclear Science, 46 (1999), 2202-2210.  doi: 10.1109/23.819305.

[24]

P. Persson, Mesh Generation for Implict Geometries, Ph.D thesis, Massachusetts Institute of Technology, 2005.

[25]

L. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D, 60 (1992), 259-268.  doi: 10.1016/0167-2789(92)90242-F.

[26]

A. Saalehi and A. Borthwick, Quadtree and octree grid generation, Internat. J. Engineering, 9 (1996).

[27]

A. Sawatzky, (Nonlocal) Total Variation in Medical Imaging, Ph.D thesis, Westfälische Wilhelms-Universität Münster, 2011.

[28]

A. Sitek, R. Huesman and G. Gullberg, Tomographic recostruction using an adaptive tetrahedral mesh defined by a point cloud, IEEE Trans. Medical Imaging, 25 (2006), 1172-1179.  doi: 10.1109/TMI.2006.879319.

[29]

G. Strang, Piecewise polynomials and the finite element method, Bull. Amer. Math. Soc., 79 (1973), 1128-1137.  doi: 10.1090/S0002-9904-1973-13351-8.

[30]

M. Wernick and J. Aarsvold, Emission Tomography: the Fundamentals of PET and SPECT, Elsevier Academic Press, San Diego, 2004.

[31]

L. Westover, Footprint evaluation for volume rendering, ACM Siggraph Computer Graphics, 24 (1990), 367-376.  doi: 10.1145/97879.97919.

[32]

Z. Wu, S. Li, X. Zeng, Y. Xu and A. Krol, Reducing staircasing artifacts in SPECT reconstruction by an infimal convolution regularization, J. Comput. Math., 34 (2016), 626-647.  doi: 10.4208/jcm.1607-m2016-0537.

[33]

Y. Xu and Q. Zou, Tree wavelet approximations with applications, Sci. China Ser. A, 48 (2005), 680-702.  doi: 10.1360/04ys0173.

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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$ 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
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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NMSE CNR FWHM
DTV 2.71 15.79 0.97
RUG 2.35 17.17 0.90
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