Joint reconstruction and low-rank decomposition for dynamic inverse problems

Simon Arridge; Pascal Fernsel; Andreas Hauptmann

doi:10.3934/ipi.2021059

Article Contents

2022, Volume 16, Issue 3: 483-523. Doi: 10.3934/ipi.2021059 open access

This issue Previous Article Next Article Generative imaging and image processing via generative encoder

Joint reconstruction and low-rank decomposition for dynamic inverse problems

1.
Department of Computer Science, University College London, London WC1E 6BT, United Kingdom
2.
Center for Industrial Mathematics, University of Bremen, 28359 Bremen, Germany
3.
Department of Mathematical Sciences, University of Oulu, 90014 Oulu, Finland

^*Corresponding author: Pascal Fernsel
^*Corresponding author: Pascal Fernsel

Received: June 2020

Revised: June 2021

Published: June 2022

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Abstract

A primary interest in dynamic inverse problems is to identify the underlying temporal behaviour of the system from outside measurements. In this work, we consider the case, where the target can be represented by a decomposition of spatial and temporal basis functions and hence can be efficiently represented by a low-rank decomposition. We then propose a joint reconstruction and low-rank decomposition method based on the Nonnegative Matrix Factorisation to obtain the unknown from highly undersampled dynamic measurement data. The proposed framework allows for flexible incorporation of separate regularisers for spatial and temporal features. For the special case of a stationary operator, we can effectively use the decomposition to reduce the computational complexity and obtain a substantial speed-up. The proposed methods are evaluated for three simulated phantoms and we compare the obtained results to a separate low-rank reconstruction and subsequent decomposition approach based on the widely used principal component analysis.

Keywords:

Mathematics Subject Classification: Primary: 15A23, 65K10, 65F22; Secondary: 15A69.

Citation:

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Figure 1. Illustration of a phantom that can be represented by the decomposition in (2). The phantom consists of $ K = 3 $ components: the background and two dynamic components with periodically changing intensity (left and right plot). As such, this phantom can be efficiently represented by a low-rank decomposition considered in this study

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Figure 2. Structure of the NMF in the context of the dynamic Shepp-Logan phantom as shown in Figure 1. Here, the nonnegative spatial and temporal basis functions can be naturally represented by the matrices $ B $ and $ C $

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Figure 3. Results for the dynamic Shepp-Logan phantom considered in Section 3.1.1 with $ \vert \mathcal{I}_t \vert = 6 $ angles per time step and 1% Gaussian noise. Shown are the leading extracted features for the BC model and for BC-X

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Figure 4. Results for the dynamic Shepp-Logan phantom considered in Section 3.1.1 with $ \vert \mathcal{I}_t \vert = 6 $ angles per time step and 1% Gaussian noise. Shown are the leading extracted features for the $\texttt{gradTV_PCA}$ model and for $\texttt{gradTV_NMF}$

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Figure 5. Results for the dynamic Shepp-Logan phantom considered in Section 3.1.1 with $ \vert \mathcal{I}_t \vert = 6 $ angles per time step and 3% Gaussian noise. Shown are the leading extracted features for the BC model and for $\texttt{gradTV_PCA}$

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Figure 6. Results for the dynamic Shepp-Logan phantom considered in Section 3.1.1 with $ \vert \mathcal{I}_t \vert = 3 $ angles per time step and 1% Gaussian noise. Shown are the leading extracted features for the BC model and for BC-X

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Figure 7. Mean PSNR and SSIM values of the reconstructions of the dynamic Shepp-Logan phantom considered in Section 3.1.1 with 1% Gaussian noise for different numbers of projection angles

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Figure 8. Mean PSNR and SSIM values of the reconstructions of the dynamic Shepp-Logan phantom considered in Section 3.1.1 with 3% Gaussian noise for different numbers of projection angles

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Figure 9. Needed time in seconds for the reconstruction and feature extraction of the dynamic Shepp-Logan phantom considered in Section 3.1.1 with 1% Gaussian noise

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Figure 10. Results for the dynamic Shepp-Logan phantom considered in Section 3.1.1 with a stationary operator and 1% Gaussian noise. Shown are the leading extracted features with the sBC model for $ \vert \mathcal{I}_t \vert = 6 $ angles per time step and $ \vert \mathcal{I}_t \vert = 30. $

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Figure 11. Needed time in seconds, mean PSNR and mean SSIM values of the reconstructions of the dynamic Shepp-Logan phantom with 1% Gaussian noise for the stationary case sBC and different numbers of projection angles

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Figure 12. Illustration of the vessel phantom dataset consisting of $ T = 100 $ phantoms of dimension $ 264\times264, $ where the intensity of the blue highlighted area changes according to blue curve on the left

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Figure 13. Results for the vessel phantom with $ \vert \mathcal{I}_t \vert = 12 $ angles per time step and 1% Gaussian noise. Shown are the leading extracted features for the BC model and for BC-X

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Figure 14. Results for the vessel phantom with $ \vert \mathcal{I}_t \vert = 12 $ angles per time step and 1% Gaussian noise. Shown are the leading extracted features for the $\texttt{gradTV_PCA}$ model and for $\texttt{gradTV_NMF}$

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Figure 15. Results for the vessel phantom with $ \vert \mathcal{I}_t \vert = 12 $ angles per time step and 3% Gaussian noise. Shown are the leading extracted features for the BC model and for $\texttt{gradTV_PCA}$

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Figure 16. Selected time steps of the ground truth based on the dynamic Shepp-Logan phantom used in Section 3.1.3 for the numerical experiments. The dynamic parts of the dataset consist of the left ellipse, which periodically change its size, and the right ellipse, which periodically change its intensity

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Figure 17. Results for the dynamic Shepp-Logan phantom considered in Section 3.1.3 with $ \vert \mathcal{I}_t \vert = 12 $ angles per time step and 1% Gaussian noise. Shown are the extracted features for BC and BC-X with $ K = 4 $ and $ K = 3 $ respectively

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Figure 18. Results for the dynamic Shepp-Logan phantom considered in Section 3.1.3 with $ \vert \mathcal{I}_t \vert = 12 $ angles per time step and 1% Gaussian noise. Shown are the 5 leading extracted features for $\texttt{gradTV_PCA}$ and all features for $\texttt{gradTV_NMF}$

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Figure 19. Results for the dynamic Shepp-Logan phantom considered in Section 3.1.3 with $ \vert \mathcal{I}_t \vert = 12 $ angles per time step and 1% Gaussian noise. Shown are the extracted features for BC with $ K = 3 $ and $\texttt{gradTV_NMF}$ with $ K = 5. $

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Figure 20. Mean PSNR and SSIM values of the reconstructions of the dynamic Shepp-Logan phantom considered in Section 3.1.3 with 1% Gaussian noise for different numbers of projection angles

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Figure 21. Illustration of two iteration steps of the MM principle for a cost function $ \mathcal{F} $ with bounded curvature and a surrogate function $ \mathcal{Q}_\mathcal{F}, $ which is strictly convex in the first argument

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Figure 22. Plots of $ \Vert B_{\bullet, k} C_{k, \bullet} \Vert_{\infty} $ with $ K = 10 $ in descending order for the dynamic Shepp-Logan phantom (Figure 22a) and the vessel phantom (Figure 22b) considered in Section 3.1.1 and 3.1.2 with 1% Gaussian noise and the parameters given in Table 3 and 4. In the case of $\texttt{gradTV_PCA}$, the ten leading features with respect to the singular values are considered

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Figure 23. Plots of $ \Vert B_{\bullet, k} C_{k, \bullet} \Vert_{\infty} $ with $ K = 10 $ in descending order for the dynamic Shepp-Logan phantom considered in Section 3.1.3 with 1% Gaussian noise and the parameters given in Table 5. In the case of $\texttt{gradTV_PCA}$, the ten leading features with respect to the singular values are considered

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Algorithm 1 $\texttt{gradTV}$

1: Initialise: $ X $
2: Input: $ \rho_{\text{grad}}, \rho_{\text{thr}}, \rho_{\text{TV}} >0 $
3: repeat
4:   $ X_{\bullet, t} \gets X_{\bullet, t} - \rho_{\text{grad}} (A_t^\intercal A_tX_{\bullet, t} - A_t^\intercal Y_{\bullet, t}) \ \ \ \text{for all}\ t $
5:    $ (U,\Sigma, V) \gets {\text{SVD}}(X) $
6:   $ \Sigma \gets {\text{SOFTTHRESH}}_{\rho_{\text{thr}}}(\Sigma) $
7:   $ X \gets U\Sigma V^\intercal $
8:   $ X\gets \max(X, 0) $
9: until STOPPINGCRITERION satisfied
10: $ X \gets {\text{TVDENOISER}}_{\rho_{\text{TV}}}(X) $
11: return $ X $

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Table 1. Summary and short explanation of the considered algorithms in the experimental section

Algorithm	Description
BC	Joint reconstruction and feature extraction with the NMF model BC without constructing $X$, see algorithm in Theorem 2.3
BC-X	Joint reconstruction and feature extraction with NMF model BC-X and explicit construction of $X$, see algorithm in Theorem 2.2
sBC	Joint reconstruction and feature extraction method with NMF model sBC for stationary operator, see algorithm in Corollary 1
$\texttt{gradTV}$	Low-rank based reconstruction method for $X$, see Algorithm 1
$\texttt{gradTV_PCA}$	Separated reconstruction and feature extraction with Algorithm 1 and subsequent PCA computation
$\texttt{gradTV_NMF}$	Separated reconstruction and feature extraction with Algorithm 1 and subsequent NMF computation based on the model in (6)

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DownLoad: CSV

Table 2. Mean PSNR and SSIM values of the reconstruction results for the vessel phantom for different noise levels and numbers of projection angles. Values in brackets indicate that the dynamic part of the dataset in the corresponding experiment could not be reconstructed sufficiently well

		BC		BC-X		$\texttt{gradTV}$
Noise	$\vert \mathcal{I}_t \vert$	PSNR	SSIM	PSNR	SSIM	PSNR	SSIM
1%	7	(34.130)	(0.9016)	32.969	0.8382	31.463	0.8414
1%	12	35.050	0.9068	33.919	0.8496	34.309	0.8839
3%	12	30.148	0.7484	28.119	0.5708	29.375	0.6698

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DownLoad: CSV

Table 3. Parameter choice of the experiments in Section 3.1.1 for the dynamic Shepp-Logan phantom for 1% and 3% Gaussian noise

	BC		BC-X		$\texttt{gradTV}$
Parameter	1% noise	3% noise	1% noise	3% noise	1% noise	3% noise
$\alpha$	-	-	70	70	-	-
$\mu_C$	0.1	0.1	0.1	0.1	-	-
$\tau$	10	50	6	20	-	-
$\rho_{\text{grad}}$	-	-	-	-	$1\cdot 10^{-3}$	$8\cdot 10^{-4}$
$\rho_{\text{thr}}$	-	-	-	-	$7\cdot 10^{-4}$	$1\cdot 10^{-3}$
$\rho_{\text{TV}}$	-	-	-	-	$1\cdot 10^{-2}$	$2.5\cdot 10^{-2}$
$\tilde{\mu}_{C}$	-	-	-	-	$0.1$	$0.1$

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Table 4. Parameter choice of the experiments in Section 3.1.2 for the vessel phantom for 1% and 3% Gaussian noise

	BC		BC-X		$\texttt{gradTV}$
Parameter	1% noise	3% noise	1% noise	3% noise	1% noise	3% noise
$\alpha$	-	-	$3\cdot 10^{2}$	$3\cdot 10^{2}$	-	-
$\mu_C$	1	1	1	1	-	-
$\tau$	$1.3\cdot 10^{2}$	$4.3\cdot 10^{2}$	$90$	$3\cdot 10^{2}$	-	-
$\rho_{\text{grad}}$	-	-	-	-	$2\cdot 10^{-4}$	$8\cdot 10^{-5}$
$\rho_{\text{thr}}$	-	-	-	-	$2\cdot 10^{-4}$	$2.5\cdot 10^{-4}$
$\rho_{\text{TV}}$	-	-	-	-	$2\cdot 10^{-2}$	$4\cdot 10^{-2}$
$\tilde{\mu}_{C}$	-	-	-	-	$0.1$	$0.1$

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Table 5. Parameter choice of the experiments in Section 3.1.3 for the dynamic Shepp-Logan phantom for 1% Gaussian noise

Parameter	BC	BC-X	gradTV
$\alpha$	-	$70$	-
$\mu_C$	$0.1$	$0.1$	-
$\tau$	$10$	$10$	-
$\rho_{\text{grad}}$	-	-	$1\cdot 10^{-3}$
$\rho_{\text{thr}}$	-	-	$2\cdot 10^{-4}$
$\rho_{\text{TV}}$	-	-	$1\cdot 10^{-2}$
$\tilde{\mu}_{C}$	-	-	$0.1$

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