Optimal selection of regularization parameter in total variation method for reducing noise in magnetic resonance images of the brain
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.Get Access
In the image processing community total variation (TV) is widely acknowledged as a popular and state-of-the-art technique for noise reduction because of its edge-preserving property. This attractive feature of TV is dependent on optimal selection of regularization parameter. Contributions in literature on TV focus on applications, properties and the different numerical solution methods. Few contributions which address the problem of regularization parameter selection are based on regression methods which pre-exist introduction of TV. They are generic and elegantly formulated, and their operation is in series with TV framework. For these reasons they render TV computationally inefficient and there is significant manual tuning when they are deployed in specific applications.
This paper describes a non-regression approach for selection of regularization parameter. It is based on a new concept, the Variational-Bayesian (VB) cycle. Within the context of VB cycle we derive two important results. First, we confirm the notion held for a long time by researchers, within image processing and computer vision community, that variational and Bayesian techniques are equivalent. Second, the value of regularization parameter is equal to noise variance, and is determined, at no computational cost to TV denoising algorithm, from a mathematical model that describes relationship between Markov random field energy and noise level in magnetic resonance images (MRI) of brain. The second result is similar to one reported in  in which the authors, for special choice of regularization operator in different regression methods, derive value of regularization parameter as equal to noise variance.
Our proposal was evaluated on brain MRI images with different acquisition protocols from two clinical trials study management centers. It was based on visual quality, computation time, convergence and optimality.
The result shows that our proposal is suitable in applications where high level of automation is demanded from image processing software.
- Galatsanos NP, Katsaggelos AK. Methods for choosing the regularization parameter and estimating the noise variance in image restoration and their relation. IEEE T Image Process. 1992; 1(3):322–336. CrossRef
- Zhang Y, Brady M, Smith S. Segmentation of brain mr images through a hidden markov random field model and the expectation-maximization algorithm. IEEE T Med Imaging. 2001; 20(1): 45–57. CrossRef
- Osadebey M. Simulation of realistic head geometry using radial vector representation of magnetic resonance image data. Masters thesis; Tampere University of Technology; Finland; 2009.
- Gold R, Kappos L, Arnold DL, Bar-Or A, Giovannoni G, Selmaj K, Tornatore C, Sweetser MT, Yang M, Sheikh SI, Dawson KT. Placebo-controlled phase 3 study of oral bg-12 for relapsing multiple sclerosis. New Engl J Med. 2012; 367(12):1098–1107. CrossRef
- Thulborn KR, Uttecht SD. Volumetry and topography of the human brain by magnetic resonance. Int J Imag Syst Tech. 2000; 11(3):198–208. CrossRef
- Wiest-Daessle N, Prima S, Coupe P, Morrissey S, Barillot C. Rician noise removal by non-Local Means filtering for low signal-to-noise ratio MRI: applications to DT-MRI. Med Image Comput Comput Assist Interv. 2008; 11(Pt 2):171–179.
- Rudin LI, Osher S, Fatemi E. Nonlinear total variation based noise removal algorithms. Physica D. 1992; 60:259–268. CrossRef
- Ching WK, Ng MK, Sze KN, Yau AC. Superresolution image reconstruction from blurred observations by multisensors. Int J Imag Syst Tech. 2003; 13(3):153–160. CrossRef
- Feng J, Zhang JZ. An adaptive dynamic combined energy minimization model for few-view computed tomography reconstruction. Int J Imag Syst Tech. 2013; 23(1):44–52. CrossRef
- Zhang Y, Zhang WH, Chen H, Yang ML, Li TY, Zhou JL. Fewview image reconstruction combining total variation and a highorder norm. Int J Imag Syst Tech. 2013; 23(3):249–255. CrossRef
- Zhu Y, Shi Y. A fast method for reconstruction of total-variation mr images with a periodic boundary condition. IEEE Signal Process Lett. 2013; 20(4):291–294. CrossRef
- Vogel CR, Oman ME. Fast, robust total variation-based reconstruction of noisy, blurred images. IEEE T Image Process. 1998; 7(6):813–824. CrossRef
- Chan SH, Khoshabeh R, Gibson KB, Gill PE, Nguyen TQ. An augmented lagrangian method for total variation video restoration. IEEE T Image Process. 2011; 20(11):3097–3111. CrossRef
- He L, Marquina A, Osher SJ. Blind deconvolution using tv regularization and bregman iteration. Int J Imag Syst Technol. 2005; 15(1):74–83. CrossRef
- Chan TF, Yip AM, Park FE. Simultaneous total variation image inpainting and blind deconvolution. Int J Imag Syst Technol. 2005; 15(1):92–102. CrossRef
- Guo W, Qiao LH. Inpainting based on total variation. Int Conf Wavelt Anal Pattern Recognit. 2007; 2:939–943. CrossRef
- Strong D, Chan T. Edge-preserving and scale-dependent properties of total variation regularization. Inverse Probl. 2003; 19(6):S165. CrossRef
- Bellettini G, Caselles V, Novaga M. The total variation flow in RN. J Differ Equations. 2002; 184(2):475–525. CrossRef
- Chambolle A, Lions PL. Image recovery via total variation minimization and related problems. Numer Math. 1997; 76(2):167–188. CrossRef
- Dobson DC, Santosa F. Recovery of blocky images from noisy and blurred data. SIAM J Appl Math. 1996; 56(4):1181–1198. CrossRef
- Vogel CR, Oman ME. Iterative methods for total variation denoising. SIAM J Sci Comput. 1996; 17(1):227–238. CrossRef
- Lysaker M, Osher S, Tai XC. Noise removal using smoothed normals and surface fitting. IEEE T Image Process. 2004; 13(10):1345–1357. CrossRef
- Chambolle A. An algorithm for total variation minimization and applications. J Math Imaging Vision. 2004; 20(1–2):89–97.
- Chan TF, Golub GH, Mulet P. A nonlinear primal-dual method for total variation-based image restoration. SIAM J Sci Comput. 2006; 20(6):1964–1977. CrossRef
- Zuo W, Lin Z. A generalized accelerated proximal gradient approach for total-variation-based image restoration. IEEE T Image Process. 2011; 20(10):2748–2759. CrossRef
- Beck A, Teboulle M. Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems. IEEE T Image Process. 2009; 18(11):2419–2434. CrossRef
- Wang Y, Yang J, Yin W, Zhang Y. A new alternating minimization algorithm for total variation image reconstruction. SIAM J Imaging Sci. 2008; 1(3):248–272. CrossRef
- Afonso MV, Bioucas-Dias JM, Figueiredo MA. Fast image recovery using variable splitting and constrained optimization. IEEE T Image Process. 2010; 19(9):2345–2356. CrossRef
- Babacan SD, Molina R, Katsaggelos AK. Parameter estimation in tv image restoration using variational distribution approximation. IEEE T Image Process. 2008; 17(3):326–339. CrossRef
- Liao H, Li F, Ng MK. Selection of regularization parameter in total variation image restoration. J Opt Soc Am A. 2009; 26(11):2311–2320. CrossRef
- Golub GH, Heath M, Wahba G. Generalized cross-validation as a method for choosing a good ridge parameter. Technometrics. 1979; 21(2):215–223. CrossRef
- Geisser S. Predictive inference: an introduction. Monographs on statistics and applied probability; New York, Chapman and Hall; 1993. CrossRef
- Wen YW, Chan R. Parameter selection for total-variation-based image restoration using discrepancy principle. IEEE T Image Process. 2012; 21(4):1770–1781. CrossRef
- Chen A, Huo BM, Wen CW. Adaptive regularization for color image restoration using discrepancy principle. IEEE Int Conf Signal Process Commun Comput. 2013; 1–6.
- Engl HW, Hanke M, Neubauer A. Regularization of Inverse Problems. Mathematics and Its Applications; Springer; 1996. CrossRef
- Lin Y, Wohlberg B, Guo H. UPRE method for total variation parameter selection. Signal Process. 2010; 90(8):2546–2551. CrossRef
- Mallows CL. Some comments on cP. Technometrics. 1973; 15(4):661–675.
- Bertalmio M, Caselles V, Rouge B, Sole A. Tv based image restoration with local constraints. J Sci Comput. 2003; 19(1–3):95–122. CrossRef
- Almansa A, Ballester C, Caselles V, Haro G. A tv based restoration model with local constraints. J Sci Comput. 2008; 34(3):209–236. CrossRef
- Dong Y, Hintermuller M, Rincon-Camacho MM. Automated regularization parameter selection in multi-scale total variation models for image restoration. J Math Imaging Vis. 2011; 40(1):82–104. CrossRef
- Palsson F, Sveinsson JR, Ulfarsson MO, Benediktsson JA. Sar image denoising using total variation based regularization with surebased optimization of the regularization parameter. IEEE Int Geosci Remote Sensing Symposium. 2012; 2160–2163.
- Ramani S, Liu Z, Rosen J, Nielsen JF, Fessler JA. Regularization parameter selection for nonlinear iterative image restoration and mri reconstruction using gcv and sure-based methods. IEEE T Image Process. 2012; 21(8):3659–3672. CrossRef
- Stein CM. Estimation of the mean of a multivariate normal distribution. Ann Stat. 1981; 9(6):1135–1151. CrossRef
- Chan R, Chan T, Yip A. Numerical methods and applications in total variation image restoration. In: Scherzer O, editor. Handbook of Mathematical Methods in Imaging. Springer New York; 2011, p. 1059–1094. CrossRef
- Malgouyres F. Minimizing the total variation under a general convex constraint for image restoration. IEEE T Image Process. 2002; 11(12):1450–1456. CrossRef
- Arlot S, Celisse A. A survey of cross-validation procedures for model selection. Stat Surv. 2010; 4:40–79. CrossRef
- Galatsanos NP, Katsaggelos AK. Cross-validation and other criteria for estimating the regularizing parameter. Int Conf Acoust Speech Signal Process. 1991; 4:3021–3024.
- Chan TF, Esedoglu S. Aspects of total variation regularized l1 function approximation. SIAM J Appl Math 2005; 65(5):1817–1837. CrossRef
- Osadebey M, Bouguila N, Arnold D, the ADNI. The clique potential of markov random field in a random experiment for estimation of noise levels in 2d brain mri. Int J Imag Syst Tech. 2013; 23(4):304–313. CrossRef
- Li SZ. Markov Random Field Modeling in Image Analysis. Springer; 2009.
- Mackay DJC. Information Theory, Inference and learning algorithms. Cambridge: Cambridge University Press; 2003.
- Geman S, Geman D. Stochastic relaxation, gibbs distributions, and the bayesian restoration of images. IEEE T Pattern Anal. 1984; 6(6):721–741. CrossRef
- Coupe P, Manjon JV, Gedamu E, Arnold D, Robles M, Collins DL. Robust rician noise estimation for mr images. Med Image Anal. 2010; 14(4):483–493. CrossRef
- Chopra A, Lian H. Total variation, adaptive total variation and nonconvex smoothly clipped absolute deviation penalty for denoising blocky images. Pattern Recogn. 2010; 43(8):2609–2619. CrossRef
- Marquina A, Osher S. Explicit algorithms for a new time dependent model based on level set motion for nonlinear deblurring and noise removal. SIAM J Sci Comput. 2000; 22(2):387–405. CrossRef
- Chan T, Esedoglu S, Park F, Yip A. Total variation image restoration: Overview and recent developments. In: Paragios N, Chen Y, Faugeras O, editors. Handbook of Mathematical Models in Computer Vision. Springer US; 2006, p. 17–31. CrossRef
- Zeng X, Li S. An efficient adaptive total variation regularization for image denoising. 2013 Seventh Int Conf Image Graph. 2013; 55–59. CrossRef
- Chen Q, Montesinos P, Sun QS, Heng PA, Xia DS. Adaptive total variation denoising based on difference curvature. Image Vision Comput. 2010; 28(3):298–306. CrossRef
- Blomgren P, Chan T, Mulet P, Wong CK. Total variation image restoration: numerical methods and extensions. Proc IEEE Int Conf Image Process. 1997; (3):384–387.
- Optimal selection of regularization parameter in total variation method for reducing noise in magnetic resonance images of the brain
Biomedical Engineering Letters
Volume 4, Issue 1 , pp 80-92
- Cover Date
- Print ISSN
- Online ISSN
- The Korean Society of Medical and Biological Engineering
- Additional Links
- Magnetic resonance imaging (MRI)
- Total variation (TV)
- Regularization parameter
- Markov random field
- Noise level
- Author Affiliations
- 1. Department of Electrical and Computer Engineering, Concordia University, 1515 St. Catherine Street West., Montreal, Quebec, H3G 2W1, Canada
- 2. Concordia Institute for Information Systems Engineering, Concordia University, 1515 St. Catherine Street West., Montreal, Quebec, H3G 2W1, Canada
- 3. NeuroRx Research Inc, 3575 Parc Avenue, Suite # 5322, Montreal, QC, H2X 4B3, Canada