Joint reconstruction in low dose multi-energy CT

Jussi Toivanen; Alexander Meaney; Samuli Siltanen; Ville Kolehmainen

doi:10.3934/ipi.2020028

Article Contents

2020, Volume 14, Issue 4: 607-629. Doi: 10.3934/ipi.2020028 open access

This issue Previous Article Image restoration from noisy incomplete frequency data by alternative iteration scheme Next Article Uniqueness results in the inverse spectral Steklov problem

Joint reconstruction in low dose multi-energy CT

1.
Department of Applied Physics, University of Eastern Finland, POB 1627, FI-70211 Kuopio, Finland
2.
Department of Mathematics and Statistics, University of Helsinki, Helsinki, Finland

^* Corresponding author: Alexander.Meaney@helsinki.fi
^* Corresponding author: Alexander.Meaney@helsinki.fi

Received: May 31, 2019

Revised: November 30, 2019

Published: May 2020

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Abstract

Multi-energy CT takes advantage of the non-linearly varying attenuation properties of elemental media with respect to energy, enabling more precise material identification than single-energy CT. The increased precision comes with the cost of a higher radiation dose. A straightforward way to lower the dose is to reduce the number of projections per energy, but this makes tomographic reconstruction more ill-posed. In this paper, we propose how this problem can be overcome with a combination of a regularization method that promotes structural similarity between images at different energies and a suitably selected low-dose data acquisition protocol using non-overlapping projections. The performance of various joint regularization models is assessed with both simulated and experimental data, using the novel low-dose data acquisition protocol. Three of the models are well-established, namely the joint total variation, the linear parallel level sets and the spectral smoothness promoting regularization models. Furthermore, one new joint regularization model is introduced for multi-energy CT: a regularization based on the structure function from the structural similarity index. The findings show that joint regularization outperforms individual channel-by-channel reconstruction. Furthermore, the proposed combination of joint reconstruction and non-overlapping projection geometry enables significant reduction of radiation dose.

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Mathematics Subject Classification: Primary: 92C55, 65N21; Secondary: 65Z05.

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Figure 1. Illustration of a spectrally non-overlapping projection sampling scheme for low dose multi-energy CT imaging. The X-ray source energies $ E_k, k = 1,2,3 $, are denoted by tube voltages $ \mathrm{kV}_1,\mathrm{kV}_2,\mathrm{kV}_3 $

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Figure 2. Left: The X-ray attenuation coefficients of water, soft tissue, and cortical bone as functions of X-ray energy. The raw data was obtained from NIST [43]. Right: Example of a realistic tungsten X-ray source spectrum with 120 kVp voltage and filtering (2.5 mm Al + 0.2 mm Cu), and monochromatic radiation with the same effective energy. The spectra were computed using the SpekCalc software [48]

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Figure 3. Reconstructed images for the simulation test case using 30 projection directions at each energy

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Figure 4. The RMS errors (left) and the mean SSIM indices (right) for the different reconstruction approaches using 30 projection directions at each energy

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Figure 5. Comparison of (TV) and (S+TV) reconstructions. Results labeled METHOD(90) are based on overall of $ 3\times 90 = 270 $ projection images using the same 90 directions for each energy. Results labeled METHOD(30W) are based on overall of 90 projections using the non-overlapping 30+30+30 directions (the same geometry that is used in Figure 3). Each image shows a region of interest of the full reconstruction

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Figure 6. Error metrics for the (TV) and (S+TV) reconstructions shown in Figure 5. Results labeled METHOD(90) are based on overall of $ 3\times 90 = 270 $ projection images using the same 90 directions for each energy. Results labeled METHOD(30W) are based on overall of 90 projections using the non-overlapping 30+30+30 directions

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Figure 7. Reconstructed images for the experimental data test case

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Table 1. Imaging geometry used for collecting the experimental data

Parameter	Value
Focus-center distance	252 mm
Focus-detector distance	420 mm
Magnification	5/2
Detector pixel size	0.200 mm
Effective pixel size	0.120 mm
Projection size	552 $ \times $ 576 pixels
Angular range	360°
#projections	720

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Table 2. Energy-specific settings used for collecting the experimental data

Energy	$ U $ (kV)	Filtration	$ I $ (μA)	Exposure time (ms)	Frame averaging
$ E_1 $	50	None	300	125	4
$ E_2 $	80	1 mm Al	180	125	4
$ E_3 $	120	0.5 mm Cu	120	250	4

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Table 3. Selected $ \alpha $ and $ \gamma_k $ parameters for the different regularization terms for the simulated data test case ($ \alpha_{ \rm{SIM}}, \gamma_{k, \rm{SIM}} $) and for the experimental data test case ($ \alpha_{ \rm{EXP}}, \gamma_{k, \rm{EXP}} $). The three values of $ \gamma_k $ are for the three different tube energies with indices $ k = 1,2,3 $

Prior	Acronym	$ \alpha_{ \rm{SIM}} $	$ \gamma_{k, \rm{SIM}} $	$ \alpha_{ \rm{EXP}} $	$ \gamma_{k, \rm{EXP}} $
No prior	No prior	-	-	-	-
Total variation	TV	-	2.25, 0.8, 0.7	-	1.0, 0.5, 0.25
Joint total variation	JTV	2.25	0	1	0
Linear parallel level sets	LPLS	4000	0	6000	0
First difference	D1	400	0	100	0
Structural	S	4e6	0	1e6	0
First difference + TV	D1+TV	50	1.125, 0.4, 0.35	70	0.5, 0.25, 0.125
Structural + TV	S+TV	0.75e5	1.125, 0.4, 0.35	0.5e6	0.5, 0.25, 0.125

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Table 4. Measurement angles used for energies $ E_1 $, $ E_2 $ and $ E_3 $ in the simulated data case (left) and experimental data case (right)

$ E_1 $	$ E_2 $	$ E_3 $	$ E_1 $	$ E_2 $	$ E_3 $
0	2	4	0	4	8
6	8	10	12	16	20
12	14	16	24	28	32
$ \vdots $	$ \vdots $	$ \vdots $	$ \vdots $	$ \vdots $	$ \vdots $
162	164	166	324	328	332
168	170	172	336	340	344
174	176	178	348	352	356

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