Dynamic MRI reconstruction via weighted nuclear norm and total variation regularization

Bao-Li Shi; Li-Wen Fu; Meng Yuan; Hao-Hui Zhu; Zhi-Feng Pang

doi:10.3934/ipi.2024044

Article Contents

2025, Volume 19, Issue 3: 539-559. Doi: 10.3934/ipi.2024044

This issue Previous Article A fundamental sequences method for an inverse boundary value problem for the heat equation in double-connected domains Next Article Subspace-based coupled tensor decomposition for hyperspectral blind fusion

Dynamic MRI reconstruction via weighted nuclear norm and total variation regularization

1.
School of Mathematics and Statistics, Henan University, Kaifeng, 475004, China
2.
Department of Ultrasound, Henan Provincial People's Hospital, Zhengzhou, 450003, China
3.
Center for Applied Mathematics of Henan Province, Henan University, Zhengzhou, 450046, China

* Corresponding author: Zhi-Feng Pang
* Corresponding author: Zhi-Feng Pang

Received: November 2023

Revised: July 2024

Early access: November 14, 2024

Published online: November 14, 2024

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Abstract

Dynamic Magnetic Resonance Imaging (dMRI) is a valuable tool in the diagnosis, management and monitoring of numerous diseases. The dimensions of the image determine the number of radio frequency pulses that must be applied, which makes the dMRI scan a time-consuming process. This paper puts forth a new reconstruction model based on the low-rank plus sparse (LS) decomposition technique with the objective of accelerating the imaging speed. In the proposed model, the weighted nuclear norm and the total variation are employed to, respectively, describe the property of low rank and the sparsity. As the model can be transformed into a saddle-point problem, the primal-dual method can be accelerated by utilizing a fast iterative shrinkage thresholding algorithm (FISTA). The results of the numerical experiments demonstrate that the proposed approach is more effective than several other low-rank or sparsity-based dMRI reconstruction methods in terms of visual quality and quantitative metrics, such as the signal-to-error ratio ($ \mathrm{SER} $) and the structural similarity index measure ($ \mathrm{SSIM} $). The experimental results demonstrate that the proposed method contributes to the advancement of research in the field of image reconstruction, including the development of models and numerical algorithms.

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Mathematics Subject Classification: 65K15, 49M37.

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Figure 1. RE curves against the iteration number for different dataset. (a). Pincat dataset. (b). Cardiac perfusion dataset. (c). Breast dataset

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Figure 2. Comparison of reconstructed results based on the different method on Pincat dataset. (a). Original image; (b). Zero-filling; (c). kt-RPCA; (d). ICTGV; (e). LS; (f). SR-LS; (g). kt-SLR; (h). NNTGV; (i). TV; (j). WNNTV. (a1)-(j1). Surface images respond to the red-boxed area in (a)-(j). (a2)-(j2): Pseudo-colour maps of difference images between the original image and the reconstructed images respond to the red-boxed area in (a)-(j)

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Figure 3. Comparison of reconstructed results based on the different method on Cardiac dataset. (a). Original image; (b). Zero-filling; (c). kt-RPCA; (d). ICTGV; (e). LS; (f). SR-LS; (g). kt-SLR; (h). NNTGV; (i). TV; (j). WNNTV. The pseudo-colour maps (a1)-(j1) are zoomed-in plots within the red-boxed region in (a)-(j). (a2)-(j2) zoomed different images in the red box of (a)-(j)

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Figure 4. Comparison of reconstructed results based on the different method on Breast dataset. (a). Original image; (b). Zero-filling; (c). kt-RPCA; (d). ICTGV; (e). LS; (f). SR-LS; (g). kt-SLR; (h). NNTGV; (i). TV; (j). WNNTV. The pseudo-colour maps (a1)-(j1) are zoomed-in plots within the red-boxed region in (a)-(j). (a2)-(j2) are that comparisons on Breast by zooming different images in the red box of first row (a)-(j), respectively

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Table 1. Pseudo-radial sampling optimal parameter values of the proposed model on Pincat, Cardiac and Breast datasets

	Pincat
Sampling	$\alpha$	$\beta$	$\sigma$	$\tau$	$\mu$	iter
Pseudo-radial	3.5$\times10^{-4}$	90	0.020	3.0	1.0	200

	Cardiac
Sampling	$\alpha$	$\beta$	$\sigma$	$\tau$	$\mu$	iter
Pseudo-radial	4.0$\times10^{-4}$	1.5	0.001	50	0.9	200

	Breast
Sampling	$\alpha$	$\beta$	$\sigma$	$\tau$	$\mu$	iter
Pseudo-radial	4$\times10^{-4}$	5	0.001	105	1.0	200

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Table 2. CPU time, $\mathrm{SER} (\uparrow) $ and $\mathrm{SSIM} (\uparrow) $ results for four algorithms based on three datasets

	Pincat			Cardiac			Breast
	Time	SER	SSIM	Time	SER	SSIM	Time	SER	SSIM
PDM	66.50s	38.58	0.9988	90.93s	20.71	0.9650	274.22s	25.25	0.9602
Alg2	58.35s	38.58	0.9988	84.07s	20.71	0.9650	263.03s	25.22	0.9600
Alg3	61.53s	38.58	0.9988	84.66s	20.71	0.9650	274.33s	25.22	0.9578
FPDM	62.08s	38.59	0.9988	85.79s	20.72	0.9650	293.34s	25.26	0.9614

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Table 3. Comparison of $\mathrm{SER} (\uparrow) $ and $\mathrm{SSIM} (\uparrow) $ of different models on three datasets with different sampling spokes

		Pincat
Models	spokes	12	22	32	42	52
Zerofilled	$\mathrm{SER}$	13.51	17.37	20.53	23.12	25.46
	$\mathrm{SSIM}$	0.6715	0.7685	0.8321	0.8732	0.9035
kt-RPCA	$\mathrm{SER}$	22.71	27.98	31.20	33.67	35.57
	$\mathrm{SSIM}$	0.8934	0.9516	0.9738	0.9846	0.9902
ICTGV	$\mathrm{SER}$	20.21	27.25	33.19	36.93	39.51
	$\mathrm{SSIM}$	0.8488	0.9594	0.9895	0.9956	0.9973
LS	$\mathrm{SER}$	16.05	24.20	32.40	36.90	39.71
	$\mathrm{SSIM}$	0.7134	0.8983	0.9769	0.9914	0.9954
SR-LS	$\mathrm{SER}$	{21.62	29.02	34.22	37.15	39.06
	$\mathrm{SSIM}$	0.8810	0.9666	0.9892	0.9943	0.9961
kt-SLR	$\mathrm{SER}$	19.11	30.64	36.83	40.77	43.64
	$\mathrm{SSIM}$	0.8490	0.9884	0.9975	0.9990	0.9995
NNTGV	$\mathrm{SER}$	20.17	27.15	33.31	37.20	39.62
	$\mathrm{SSIM}$	0.8549	0.9618	0.9926	0.9971	0.9983
TV	$\mathrm{SER}$	19.97	28.58	35.91	40.38	43.19
	$\mathrm{SSIM}$	0.8511	0.9733	0.9968	0.9991	0.9996
$\mathbf{WNNTV}$	$\mathrm{SER}$	20.81	30.85	38.59	42.87	45.59
	$\mathrm{SSIM}$	0.8751	0.9890	0.9988	0.9996	0.9998

		Cardiac
Models	spokes	12	22	32	42	52
Zerofilled	$\mathrm{SER}$	9.44	12.44	14.80	16.68	18.19
	$\mathrm{SSIM}$	0.7595	0.8332	0.8855	0.9178	0.9386
kt-RPCA	$\mathrm{SER}$	15.50	18.04	19.36	20.31	21.11
	$\mathrm{SSIM}$	0.9206	0.9457	0.9553	0.9620	0.9673
ICTGV	$\mathrm{SER}$	15.93	18.37	19.84	20.86	21.64
	$\mathrm{SSIM}$	0.9229	0.9485	0.9590	0.9653	0.9700
LS	$\mathrm{SER}$	12.01	16.80	19.72	21.18	22.07
	$\mathrm{SSIM}$	0.8343	0.9278	0.9581	0.9674	0.9724
SR-LS	$\mathrm{SER}$	16.48	18.52	19.82	20.81	21.58
	$\mathrm{SSIM}$	0.9313	0.9492	0.9584	0.9647	0.9694
kt-SLR	$\mathrm{SER}$	17.28	19.12	20.50	21.49	22.23
	$\mathrm{SSIM}$	0.9363	0.9519	0.9613	0.9674	0.9719
NNTGV	$\mathrm{SER}$	12.39	16.25	18.94	20.71	21.89
	$\mathrm{SSIM}$	0.8507	0.9213	0.9513	0.9645	0.9716
TV	$\mathrm{SER}$	16.23	18.78	20.39	21.49	22.28
	$\mathrm{SSIM}$	0.9294	0.9529	0.9632	0.9695	0.9738
$\mathbf{WNNTV}$	$\mathrm{SER}$	17.17	19.23	20.72	21.81	22.60
	$\mathrm{SSIM}$	0.9399	0.9560	0.9650	0.9710	0.9752

		Breast
Models	spokes	12	22	32	42	52
Zerofilled	$\mathrm{SER}$	10.03	12.78	14.85	16.43	17.79
	$\mathrm{SSIM}$	0.4077	0.4984	0.5711	0.6266	0.6743
kt-RPCA	$\mathrm{SER}$	20.72	22.19	23.23	24.11	24.90
	$\mathrm{SSIM}$	0.8802	0.9154	0.9354	0.9486	0.9584
ICTGV	$\mathrm{SER}$	19.81	22.44	23.87	24.99	26.02
	$\mathrm{SSIM}$	0.7963	0.8914	0.9261	0.9447	0.9579
LS	$\mathrm{SER}$	13.41	19.75	24.09	25.71	26.56
	$\mathrm{SSIM}$	0.5217	0.7895	0.9584	0.9647	0.9694
SR-LS	$\mathrm{SER}$	21.30	22.85	23.82	24.63	25.37
	$\mathrm{SSIM}$	0.8678	0.9031	0.9187	0.9298	0.9392
kt-SLR	$\mathrm{SER}$	21.47	23.18	24.53	25.25	26.69
	$\mathrm{SSIM}$	0.9206	0.9400	0.9539	0.9539	0.9699
NNTGV	$\mathrm{SER}$	19.95	23.02	24.66	25.95	27.08
	$\mathrm{SSIM}$	0.8842	0.9431	0.9620	0.9720	0.9785
TV	$\mathrm{SER}$	16.96	21.34	24.01	25.82	27.10
	$\mathrm{SSIM}$	0.7514	0.8854	0.9345	0.9557	0.9669
$\mathbf{WNNTV}$	$\mathrm{SER}$	21.76	23.77	25.26	26.58	27.79
	$\mathrm{SSIM}$	0.8835	0.9404	0.9614	0.9727	0.9799

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Table 4. CPU times of different models on reconstructing the Pincat, Cardiac and Breast datasets

	Pincat
Models	kt-RPCA	ICTGV	LS	SR-LS	kt-SLR	NNTGV	TV	$\mathbf{WNNTV}$
Time	106.02 s	289.82 s	30.97 s	34.77 s	373.03 s	184.15 s	57.50 s	62.08 s
	Cardiac
Models	kt-RPCA	ICTGV	LS	SR-LS	kt-SLR	NNTGV	TV	$\mathbf{WNNTV}$
Time	168.62 s	400.26 s	42.56 s	51.07 s	413.34 s	262.50 s	84.17 s	85.79s
	Breast
Models	kt-RPCA	ICTGV	LS	SR-LS	kt-SLR	NNTGV	TV	$\mathbf{WNNTV}$
Time	511.23 s	1270.97 s	260.67 s	171.62 s	2020.51 s	831.49 s	248.91 s	266.79 s

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