August  2022, 16(4): 967-995. doi: 10.3934/ipi.2022007

A variational method for Abel inversion tomography with mixed Poisson-Laplace-Gaussian noise

Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing 100094, China

* Corresponding author: Linghai Kong

Received  April 2021 Revised  January 2022 Published  August 2022 Early access  March 2022

Abel inversion tomography plays an important role in dynamic experiments, while most known studies are started with a single Gaussian assumption. This paper proposes a mixed Poisson-Laplace-Gaussian distribution to characterize the noise in charge-coupled-device (CCD) sensed radiographic data, and develops a multi-convex optimization model to address the reconstruction problem. The proposed model is derived by incorporating varying amplitude Gaussian approximation and expectation maximization algorithm into an infimal convolution process. To solve it numerically, variable splitting and augmented Lagrangian method are integrated into a block coordinate descent framework, in which anisotropic diffusion and additive operator splitting are employed to gain edge preserving and computation efficiency. Supplementarily, a space of functions of adaptive bounded Hessian is introduced to prove the existence and uniqueness of solution to a higher-order regularized, quadratic subproblem. Moreover, a simplified algorithm with higher order regularizer is derived for Poisson noise removal. To illustrate the performance of the proposed algorithms, numerical tests on synthesized and real digital data are performed.

Citation: Linghai Kong, Suhua Wei. A variational method for Abel inversion tomography with mixed Poisson-Laplace-Gaussian noise. Inverse Problems and Imaging, 2022, 16 (4) : 967-995. doi: 10.3934/ipi.2022007
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Figure 1.  Synthesized clean images for our simulations
Figure 2.  Numerical experiments of Poisson noise removal. (a), (d): Noisy version of Fig. 1(a) and Fig. 1(c), respectively. (b), (e): Restored images obtained by the Le-Chartrand-Asaki method. (c), (f): Recovered images obtained by our proposed Algorithm 2 with $ \beta = 0 $
Figure 3.  Poisson denoising result. (a) Noisy version of Fig. 1(d). (b) A restoration result obtained by the Le-Chartrand-Asaki method. (c) A restored image derived by our proposed Algorithm 2
Figure 4.  Numerical experiment of mixed PLG noise removal. (a) A inverse color image of Fig. 1(a). (b) A noisy version of (a) obtained by adding PLG noise. (c) A restored image derived by our proposed Algorithm 1 with $ \mathcal{A} = I $
Figure 5.  Numerical result of PLG noise removal. (a) A noisy image, obtained by adding PLG noise to Fig. 1(c). (b) A restored image obtained by our proposed Algorithm 1 with $ \mathcal{A} = I $. (c) The image of $ w $
Figure 6.  Numerical result of mixed noise removal. (a) A noisy version of Fig. 1(d) by adding mixed Laplace-Gaussian noise. (b) A recovered image obtained by our proposed algorithm 1 with $ \mathcal{A} $ being the identity operator
Figure 7.  (a), (e): Noisy medical images. (b), (f), (c), (g): Restored versions derived by our proposed algorithm 1 with $ \mathcal{A} = I $, $ K_d = 0.01 $ ((b), (f)), $ K_d = 0.018 $ ((c), (g)). (d), (h): Restored versions derived by the TV-rw$ \ell^2 $ algorithm, respectively
Figure 8.  (a) A single projection image obtained by an industrial Computed tomographic system. (b) A recovered version obtained by our proposed algorithm 1 with $ \mathcal{A} = I $. (c) A recovered version derived by the TV-rw$ \ell^2 $ algorithm
Figure 9.  Experiment of tomographic image reconstruction. (a) A noisy version of Fig. 1(c) (SSIM = 0.5118), obtained by adding PLG noise. (b) The middle line slice of (a). (c) The initial guess $ \mathcal{A}^{-1}f $. (d) A reconstructed image derived by our proposed Algorithm 1
Figure 10.  Experiment of tomographic image reconstruction. (a) A noisy version of Fig. 1(c) (SSIM = 0.2690), obtained by adding PLG noise. (b) The axis slice of (a). (c) An initial guess of $ u_0 $. (d) A reconstructed image by our proposed Algorithm 1
Table 1.  Notations of finite difference
$D_r^c u_{i, j}$, $D_z^c u_{i, j}$ $\frac{1}{\triangle r} [u_{i+\frac{1}{2}, j}-u_{i-\frac{1}{2}, j}]$, $\frac{1}{\triangle z}[u_{i, j+\frac{1}{2}}-u_{i, j-\frac{1}{2}}] $
$D_{r}^{\pm}u_{i, j}$, $D_{z}^{\pm}u_{i, j}$ $\pm \frac{1}{\triangle r}[u_{i\pm 1, j}-u_{i, j}]$, $\pm \frac{1}{\triangle z}[u_{i, j\pm 1}-u_{i, j}]$
$D_{rr}u_{i, j}$ $[u_{i-1, j}-2u_{i, j}+u_{i+1, j}]/(\triangle r)^2$
$D_{zz}u_{i, j}$ $[u_{i, j-1}-2u_{i, j}+u_{i, j+1}]/(\triangle z)^2$
$D_{rz}u_{i, j}$ $[u_{i+1, j+1}-u_{i, j+1}-u_{i+1, j}+u_{i, j}]/(\triangle r\triangle z)$
$D_{zr}u_{i, j}$ $[u_{i, j}-u_{i-1, j}-u_{i, j-1}+u_{i-1, j-1}]/(\triangle z\triangle r)$
$D_r^c u_{i, j}$, $D_z^c u_{i, j}$ $\frac{1}{\triangle r} [u_{i+\frac{1}{2}, j}-u_{i-\frac{1}{2}, j}]$, $\frac{1}{\triangle z}[u_{i, j+\frac{1}{2}}-u_{i, j-\frac{1}{2}}] $
$D_{r}^{\pm}u_{i, j}$, $D_{z}^{\pm}u_{i, j}$ $\pm \frac{1}{\triangle r}[u_{i\pm 1, j}-u_{i, j}]$, $\pm \frac{1}{\triangle z}[u_{i, j\pm 1}-u_{i, j}]$
$D_{rr}u_{i, j}$ $[u_{i-1, j}-2u_{i, j}+u_{i+1, j}]/(\triangle r)^2$
$D_{zz}u_{i, j}$ $[u_{i, j-1}-2u_{i, j}+u_{i, j+1}]/(\triangle z)^2$
$D_{rz}u_{i, j}$ $[u_{i+1, j+1}-u_{i, j+1}-u_{i+1, j}+u_{i, j}]/(\triangle r\triangle z)$
$D_{zr}u_{i, j}$ $[u_{i, j}-u_{i-1, j}-u_{i, j-1}+u_{i-1, j-1}]/(\triangle z\triangle r)$
Table 2.  Parameter values for three algorithms: PLG-EM-TVBH, Poisson-TVBH and TV-rw$ \ell^2 $. $ b,\lambda,\Delta t $ are used in all of the algorithms. $ \tau,\beta $, $ \sigma_0^2 $, $ K_d,K_h $ are used in PLG-EM-TVBH and Poisson-TVBH, where the initial guesses, $ (\gamma_i)^0 $, $ (\sigma_i^2)^0 $, $ i = 1,2 $, are specific to PLG-EM-TVBH. $ \varrho $ is specific to TV-rw$ \ell^2 $
Alogrithm parameters $b$ $\lambda$ $\tau$ $\beta$ $\Delta t$ $(\gamma_i)^0$
Considered values $10^{-2}$ 1 0.02 0.8 0.025 0.5
Algorithm parameters $(\sigma_1^2)^0$ $(\sigma_2^2)^0$ $K_d$ $K_h$ $\sigma_0^2$ $\varrho$
Considered values 0.9 0.1 $ 10^{-2}$ $10^{-4}$ 1.5 0.8
Alogrithm parameters $b$ $\lambda$ $\tau$ $\beta$ $\Delta t$ $(\gamma_i)^0$
Considered values $10^{-2}$ 1 0.02 0.8 0.025 0.5
Algorithm parameters $(\sigma_1^2)^0$ $(\sigma_2^2)^0$ $K_d$ $K_h$ $\sigma_0^2$ $\varrho$
Considered values 0.9 0.1 $ 10^{-2}$ $10^{-4}$ 1.5 0.8
Table 3.  Denoising performance
Image Noise SSIM-n SSIM-c SSIM-p
Fig. 2a Poisson 0.8364 0.9883 0.9981
Fig. 2d Poisson 0.8208 0.9761 0.9865
Fig. 3 Poisson 0.6546 0.8973 0.9742
Fig. 4 PLG 0.6090 - 0.9725
Fig. 5 PLG 0.4512 - 0.9418
Fig. 6 LG 0.3451 - 0.9710
Image Noise SSIM-n SSIM-c SSIM-p
Fig. 2a Poisson 0.8364 0.9883 0.9981
Fig. 2d Poisson 0.8208 0.9761 0.9865
Fig. 3 Poisson 0.6546 0.8973 0.9742
Fig. 4 PLG 0.6090 - 0.9725
Fig. 5 PLG 0.4512 - 0.9418
Fig. 6 LG 0.3451 - 0.9710
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