Total variation regularization for bioluminescence tomography with the split Bregman method

Jinchao Feng; Chenghu Qin; Kebin Jia; Shouping Zhu; Kai Liu; Dong Han; Xin Yang; Quansheng Gao; Jie Tian

doi:10.1364/AO.51.004501

Total variation regularization for bioluminescence tomography with the split Bregman method

Jinchao Feng, Chenghu Qin, Kebin Jia, Shouping Zhu, Kai Liu, Dong Han, Xin Yang, Quansheng Gao, and Jie Tian

Applied Optics
Vol. 51,
Issue 19,
pp. 4501-4512
(2012)
•https://doi.org/10.1364/AO.51.004501

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Jinchao Feng, Chenghu Qin, Kebin Jia, Shouping Zhu, Kai Liu, Dong Han, Xin Yang, Quansheng Gao, and Jie Tian, "Total variation regularization for bioluminescence tomography with the split Bregman method," Appl. Opt. 51, 4501-4512 (2012)

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Abstract

Regularization methods have been broadly applied to bioluminescence tomography (BLT) to obtain stable solutions, including $l_{2}$ and $l_{1}$ regularizations. However, $l_{2}$ regularization can oversmooth reconstructed images and $l_{1}$ regularization may sparsify the source distribution, which degrades image quality. In this paper, the use of total variation (TV) regularization in BLT is investigated. Since a nonnegativity constraint can lead to improved image quality, the nonnegative constraint should be considered in BLT. However, TV regularization with a nonnegativity constraint is extremely difficult to solve due to its nondifferentiability and nonlinearity. The aim of this work is to validate the split Bregman method to minimize the TV regularization problem with a nonnegativity constraint for BLT. The performance of split Bregman-resolved TV (SBRTV) based BLT reconstruction algorithm was verified with numerical and in vivo experiments. Experimental results demonstrate that the SBRTV regularization can provide better regularization quality over $l_{2}$ and $l_{1}$ regularizations.

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Wavelength	600 nm	620 nm
$μ_{a}$ [ ${mm}^{- 1}$ ]	0.0281	0.0109
$μ_{s}^{'}$ [ ${mm}^{- 1}$ ]	1.6667	1.6129

Regularization method	Regularization parameter	Reconstructed position error	Reconstructed density error	MSE
$l_{2}$	$2.15 \times 10^{- 6}$	0.21 mm	77.81%	5.62
$l_{1}$	$4.91 \times 10^{- 5}$	0.89 mm	51.52%	6.39
SBRTV	$9.9 \times 10^{- 8}$	0.21 mm	6.67%	4.05

Regularization method	Regularization parameter	No. source	Reconstructed position error	Reconstructed density error	MSE
$l_{2}$	$4.56 \times 10^{- 5}$	No. 1	0.54 mm	83.77%	14.26
$l_{2}$	$4.56 \times 10^{- 5}$	No. 2	0.15 mm	86.44%	14.26
$l_{1}$	$2.36 \times 10^{- 4}$	No. 1	1.05 mm	39.35%	17.38
$l_{1}$	$2.36 \times 10^{- 4}$	No. 2	0.99 mm	46.26%	17.38
SBRTV	$1.15 \times 10^{- 7}$	No. 1	0.21 mm	25.35%	9.65
SBRTV	$1.15 \times 10^{- 7}$	No. 2	0.15 mm	48.38%	9.65

Regularization method	Regularization parameter	No. source	Reconstructed position error	Reconstructed density error	MSE
$l_{2}$	$4.56 \times 10^{- 5}$	No. 1	0.54 mm	85.81%	21.31
		No. 2	0.14 mm	84.5%
		No. 3	0.07 mm	67.91%
$l_{1}$	$2.36 \times 10^{- 4}$	No. 1	1.05 mm	44.74%	27.61
		No. 2	1.13 mm	67.24%
		No. 3	0.93 mm	13.36%
SBRTV	$1.15 \times 10^{- 7}$	No. 1	0.21 mm	49.65%	18.43
		No. 2	0.15 mm	45.03%
		No. 3	0.21 mm	5.39%

	650–700 nm		700–760 nm
Wavelength	$μ_{a}$	$μ_{s}^{'}$	$μ_{a}$	$μ_{s}^{'}$
Muscle	0.0771	0.4128	0.0396	0.3373
Heart	0.0521	0.9446	0.0282	0.8529
Lung	0.1741	2.1574	0.0899	2.0775
Liver	0.3129	0.6680	0.1646	0.6198
Bone	0.0540	2.4441	0.0280	2.2005

	$l_{2}$	$l_{1}$	SBRTV
Iteration number	19	22063	9
Time (seconds)	113	437	2084
$λ$	$4.7 \times 10^{- 5}$	$1.6 \times 10^{- 3}$	$4.7 \times 10^{- 5}$
Reconstructed error (mm)	1.66	0.84	0.84

	$l_{2}$	$l_{1}$	SBRTV
Iteration number	19	22063	9
Time (seconds)	113	437	2084
$λ$	$4.7 \times 10^{- 5}$	$1.6 \times 10^{- 3}$	$4.7 \times 10^{- 5}$
Reconstructed error (mm)	1.66	0.84	0.84

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