# American Institute of Mathematical Sciences

November  2009, 3(4): 625-648. doi: 10.3934/ipi.2009.3.625

## Variational denoising of diffusion weighted MRI

 1 West Virginia University, Morgantown, WV 26506, United States 2 University of Florida, Gainesville, FL 32601, United States, United States, United States 3 National Institutes of Health, Bethesda, MD 20892, United States

Received  October 2008 Revised  August 2009 Published  October 2009

In this paper, we present a novel variational formulation for restoring high angular resolution diffusion imaging (HARDI) data. The restoration formulation involves smoothing signal measurements over the spherical domain and across the 3D image lattice. The regularization across the lattice is achieved using a total variation (TV) norm based scheme, while the finite element method (FEM) was employed to smooth the data on the sphere at each lattice point using first and second order smoothness constraints. Examples are presented to show the performance of the HARDI data restoration scheme and its effect on fiber direction computation on synthetic data, as well as on real data sets collected from excised rat brain and spinal cord.
Citation: Tim McGraw, Baba Vemuri, Evren Özarslan, Yunmei Chen, Thomas Mareci. Variational denoising of diffusion weighted MRI. Inverse Problems & Imaging, 2009, 3 (4) : 625-648. doi: 10.3934/ipi.2009.3.625
 [1] Yuyuan Ouyang, Yunmei Chen, Ying Wu. Total variation and wavelet regularization of orientation distribution functions in diffusion MRI. Inverse Problems & Imaging, 2013, 7 (2) : 565-583. doi: 10.3934/ipi.2013.7.565 [2] Elie Bretin, Imen Mekkaoui, Jérôme Pousin. Assessment of the effect of tissue motion in diffusion MRI: Derivation of new apparent diffusion coefficient formula. Inverse Problems & Imaging, 2018, 12 (1) : 125-152. doi: 10.3934/ipi.2018005 [3] Qiang Liu, Zhichang Guo, Chunpeng Wang. Renormalized solutions to a reaction-diffusion system applied to image denoising. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1839-1858. doi: 10.3934/dcdsb.2016025 [4] Weihong Guo, Jing Qin. A geometry guided image denoising scheme. Inverse Problems & Imaging, 2013, 7 (2) : 499-521. doi: 10.3934/ipi.2013.7.499 [5] Robert D. Sidman, Marie Erie, Henry Chu. A method, with applications, for analyzing co-registered EEG and MRI data. Conference Publications, 2001, 2001 (Special) : 349-356. doi: 10.3934/proc.2001.2001.349 [6] Xianchao Xiu, Lingchen Kong. Rank-one and sparse matrix decomposition for dynamic MRI. Numerical Algebra, Control & Optimization, 2015, 5 (2) : 127-134. doi: 10.3934/naco.2015.5.127 [7] Ryan Compton, Stanley Osher, Louis-S. Bouchard. Hybrid regularization for MRI reconstruction with static field inhomogeneity correction. Inverse Problems & Imaging, 2013, 7 (4) : 1215-1233. doi: 10.3934/ipi.2013.7.1215 [8] Yunho Kim, Paul M. Thompson, Luminita A. Vese. HARDI data denoising using vectorial total variation and logarithmic barrier. Inverse Problems & Imaging, 2010, 4 (2) : 273-310. doi: 10.3934/ipi.2010.4.273 [9] Fangfang Dong, Yunmei Chen. A fractional-order derivative based variational framework for image denoising. Inverse Problems & Imaging, 2016, 10 (1) : 27-50. doi: 10.3934/ipi.2016.10.27 [10] Wei Zhu, Xue-Cheng Tai, Tony Chan. Augmented Lagrangian method for a mean curvature based image denoising model. Inverse Problems & Imaging, 2013, 7 (4) : 1409-1432. doi: 10.3934/ipi.2013.7.1409 [11] Yoon Mo Jung, Taeuk Jeong, Sangwoon Yun. Non-convex TV denoising corrupted by impulse noise. Inverse Problems & Imaging, 2017, 11 (4) : 689-702. doi: 10.3934/ipi.2017032 [12] Feishe Chen, Lixin Shen, Yuesheng Xu, Xueying Zeng. The Moreau envelope approach for the L1/TV image denoising model. Inverse Problems & Imaging, 2014, 8 (1) : 53-77. doi: 10.3934/ipi.2014.8.53 [13] Rongliang Chen, Jizu Huang, Xiao-Chuan Cai. A parallel domain decomposition algorithm for large scale image denoising. Inverse Problems & Imaging, 2019, 13 (6) : 1259-1282. doi: 10.3934/ipi.2019055 [14] Mujibur Rahman Chowdhury, Jun Zhang, Jing Qin, Yifei Lou. Poisson image denoising based on fractional-order total variation. Inverse Problems & Imaging, 2020, 14 (1) : 77-96. doi: 10.3934/ipi.2019064 [15] Juan Carlos De los Reyes, Carola-Bibiane Schönlieb. Image denoising: Learning the noise model via nonsmooth PDE-constrained optimization. Inverse Problems & Imaging, 2013, 7 (4) : 1183-1214. doi: 10.3934/ipi.2013.7.1183 [16] Nils Dabrock, Yves van Gennip. A note on "Anisotropic total variation regularized $L^1$-approximation and denoising/deblurring of 2D bar codes". Inverse Problems & Imaging, 2018, 12 (2) : 525-526. doi: 10.3934/ipi.2018022 [17] Chien Hsun Tseng. Applications of a nonlinear optimization solver and two-stage comprehensive Denoising techniques for optimum underwater wideband sonar echolocation system. Journal of Industrial & Management Optimization, 2013, 9 (1) : 205-225. doi: 10.3934/jimo.2013.9.205 [18] Rustum Choksi, Yves van Gennip, Adam Oberman. Anisotropic total variation regularized $L^1$ approximation and denoising/deblurring of 2D bar codes. Inverse Problems & Imaging, 2011, 5 (3) : 591-617. doi: 10.3934/ipi.2011.5.591 [19] Macarena Boix, Begoña Cantó. Using wavelet denoising and mathematical morphology in the segmentation technique applied to blood cells images. Mathematical Biosciences & Engineering, 2013, 10 (2) : 279-294. doi: 10.3934/mbe.2013.10.279 [20] Wei Zhu. A numerical study of a mean curvature denoising model using a novel augmented Lagrangian method. Inverse Problems & Imaging, 2017, 11 (6) : 975-996. doi: 10.3934/ipi.2017045

2018 Impact Factor: 1.469