# American Institute of Mathematical Sciences

February  2020, 14(1): 117-132. doi: 10.3934/ipi.2019066

## Nonlocal TV-Gaussian prior for Bayesian inverse problems with applications to limited CT reconstruction

 1 School of Mathematical Sciences and Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai 200240, China 2 School of Mathematical Sciences, Tongji University, Shanghai 200082, China 3 Department of Mathematical Sciences, University of Liverpool, Liverpool L69 6ZL, United Kingdom 4 School of Mathematical Sciences, MOE-LSC, and Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai 200240, China

* Corresponding author: Xiaoqun Zhang

Received  April 2019 Revised  August 2019 Published  November 2019

Fund Project: The work is supported by NSFC grants (No. 11771288, 11301337, 91630311) and National key research and development program No. 2017YFB0202902

Bayesian inference methods have been widely applied in inverse problems due to the ability of uncertainty characterization of the estimation. The prior distribution of the unknown plays an essential role in the Bayesian inference, and a good prior distribution can significantly improve the inference results. In this paper, we propose a hybrid prior distribution on combining the nonlocal total variation regularization (NLTV) and the Gaussian distribution, namely NLTG prior. The advantage of this hybrid prior is two-fold. The proposed prior models both texture and geometric structures present in images through the NLTV. The Gaussian reference measure also provides a flexibility of incorporating structure information from a reference image. Some theoretical properties are established for the hybrid prior. We apply the proposed prior to limited tomography reconstruction problem that is difficult due to severe data missing. Both maximum a posteriori and conditional mean estimates are computed through two efficient methods and the numerical experiments validate the advantages and feasibility of the proposed NLTG prior.

Citation: Didi Lv, Qingping Zhou, Jae Kyu Choi, Jinglai Li, Xiaoqun Zhang. Nonlocal TV-Gaussian prior for Bayesian inverse problems with applications to limited CT reconstruction. Inverse Problems & Imaging, 2020, 14 (1) : 117-132. doi: 10.3934/ipi.2019066
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##### References:
XCAT images: origin, ground truth and reference with different noise levels
MAP results with sinogram noise level $5$
MAP results with sinogram noise level $20$
CM results with different sinogram data noise levels and references images. Upper: NLTG; Lower: TG
The 95% confidence interval for different sinogram data noise level and references images. The range of the values is from 0 (black) to 100 (whitest). Upper: NLTG; Lower: TG
MAP results: PSNR and SSIM for different sinogram noise levels and reference images
 Noise Ref. FBP TV TGV TG NLTV NLTG 5 $u_\mathrm{ref}^1$ 13.30/0.21 18.98/0.60 21.21/0.71 29.08/0.88 29.69/0.87 30.71/0.91 $u_\mathrm{ref}^2$ 28.22/0.87 28.42/0.84 28.88/0.86 20 $u_\mathrm{ref}^1$ 9.40/0.06 15.66/0.46 18.26/0.48 23.10/0.54 23.92/0.78 24.72/0.79 $u_\mathrm{ref}^2$ 22.51/0.49 23.13/0.75 23.63/0.74
 Noise Ref. FBP TV TGV TG NLTV NLTG 5 $u_\mathrm{ref}^1$ 13.30/0.21 18.98/0.60 21.21/0.71 29.08/0.88 29.69/0.87 30.71/0.91 $u_\mathrm{ref}^2$ 28.22/0.87 28.42/0.84 28.88/0.86 20 $u_\mathrm{ref}^1$ 9.40/0.06 15.66/0.46 18.26/0.48 23.10/0.54 23.92/0.78 24.72/0.79 $u_\mathrm{ref}^2$ 22.51/0.49 23.13/0.75 23.63/0.74
CM results: SSIM and PSNR for different level of Sinogram noise and reference
 PSNR SSIM Noise Ref. NLTG TG NLTG TG 5 $u_\mathrm{ref}^1$ 27.73 21.44 0.80 0.46 $u_\mathrm{ref}^2$ 27.90 20.95 0.66 0.40 20 $u_\mathrm{ref}^1$ 25.97 19.58 0.62 0.37 $u_\mathrm{ref}^2$ 25.71 18.95 0.59 0.31
 PSNR SSIM Noise Ref. NLTG TG NLTG TG 5 $u_\mathrm{ref}^1$ 27.73 21.44 0.80 0.46 $u_\mathrm{ref}^2$ 27.90 20.95 0.66 0.40 20 $u_\mathrm{ref}^1$ 25.97 19.58 0.62 0.37 $u_\mathrm{ref}^2$ 25.71 18.95 0.59 0.31
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