# American Institute of Mathematical Sciences

May  2013, 7(2): 565-583. doi: 10.3934/ipi.2013.7.565

## Total variation and wavelet regularization of orientation distribution functions in diffusion MRI

 1 Department of Mathematics, University of Florida, Gainesville, FL 32611, United States 2 Center for Advanced Imaging, Evanston Hospital, 2650 Ridge Avenue, Evanston, IL 60201, United States

Received  October 2011 Revised  August 2012 Published  May 2013

We introduce a variational model and a numerical method for simultaneous ODF smoothing and reconstruction. The model uses the sparsity of MR images in finite difference domain and wavelet domain as the spatial regularization means in ODF's reconstruction. The model also incorporates angular regularization using Laplace-Beltrami operator on the unit sphere. A primal-dual scheme is applied to solve the model efficiently. The experimental results indicate that with spatial and angular regularization in the process of reconstruction, we can get better directional structures of reconstructed ODFs.
Citation: Yuyuan Ouyang, Yunmei Chen, Ying Wu. Total variation and wavelet regularization of orientation distribution functions in diffusion MRI. Inverse Problems and Imaging, 2013, 7 (2) : 565-583. doi: 10.3934/ipi.2013.7.565
##### References:
 [1] I. Aganj, C. Lenglet, G. Sapiro, E. Yacoub, K. Ugurbil and N. Harel, Reconstruction of the orientation distribution function in single-and multiple-shell q-ball imaging within constant solid angle, Magnetic Resonance in Medicine, 64 (2010), 554-566. doi: 10.1002/mrm.22365. [2] K. Arrow, L, Hurwicz, H. Uzawa and H. Chenery, Studies in linear and non-linear programming, Stanford Mathematical Studies in the Social Sciences, II, Stanford University Press, Stanford, Calif., (1958). [3] H. Assemlal, D. Tschumperlé and L. Brun, Fiber tracking on HARDI data using robust ODF fields, in "IEEE International Conference on Image Processing," Citeseer, (2007), 344-351. doi: 10.1109/ICIP.2007.4379264. [4] P. Basser, J. Mattiello and D. Lebihan, Estimation of the effective self-diffusion tensor from the NMR spin echo, Journal of Magnetic Resonance, Series B, 103 (1994), 247-247. doi: 10.1006/jmrb.1994.1037. [5] P. Basser and C. Pierpaoli, Microstructural and physiological features of tissues elucidated by quantitative-diffusion-tensor MRI, Journal of Magnetic Resonance, Series B, 111 (1996), 209-219. [6] P. Basser, S. Pajevic, C. Pierpaoli, J. Duda and A. Aldroubi, In vivo fiber tractography using DT-MRI data, Magnetic Resonance in Medicine, 44 (2000), 625-632. doi: 10.1002/1522-2594(200010)44:4<625::AID-MRM17>3.0.CO;2-O. [7] A. Chambolle, An algorithm for total variation minimization and applications, Journal of Mathematical Imaging and Vision, 20 (2004), 89-97. doi: 10.1023/B:JMIV.0000011325.36760.1e. [8] A. Chambolle and T. Pock, A first-order primal-dual algorithm for convex problems with applications to imaging, Journal of Mathematical Imaging and Vision, 40 (2011), 120-145. doi: 10.1007/s10851-010-0251-1. [9] Y. Chen, W. Guo, Q. Zeng and Y. Liu, A nonstandard smoothing in reconstruction of apparent diffusion coefficient profiles from diffusion weighted images, Inverse Problems and Imaging, 2 (2008), 205-224. doi: 10.3934/ipi.2008.2.205. [10] O. Christiansen, T. Lee, J. Lie, U. Sinha and T. Chan, Total variation regularization of matrix-valued images, International Journal of Biomedical Imaging, 2007 (2007). doi: 10.1155/2007/27432. [11] M. Descoteaux, E. Angelino, S. Fitzgibbons and R. Deriche, Apparent diffusion coefficients from high angular resolution diffusion imaging: Estimation and applications, Magnetic Resonance in Medicine, 56 (2006), 395-410. doi: 10.1002/mrm.20948. [12] M. Descoteaux, E. Angelino, S. Fitzgibbons and R. Deriche, Regularized, fast, and robust analytical Q-ball imaging, Magnetic Resonance in Medicine, 58 (2007), 497-510. doi: 10.1002/mrm.21277. [13] M. Descoteaux, R. Deriche, T. Knösche and A. Anwander, Deterministic and probabilistic tractography based on complex fibre orientation distributions, IEEE transactions on medical imaging, 28 (2009), 269-286. doi: 10.1109/TMI.2008.2004424. [14] E. Esser, X. Zhang and T. Chan, A general framework for a class of first order primal-dual algorithms for convex optimization in imaging science, SIAM Journal on Imaging Sciences, 3 (2010), 1015-1046. doi: 10.1137/09076934X. [15] L. Frank, Anisotropy in high angular resolution diffusion-weighted MRI, Magnetic Resonance in Medicine, 45 (2001), 935-939. doi: 10.1002/mrm.1125. [16] T. Goldstein and S. Osher, The split Bregman method for L1 regularized problems, SIAM Journal on Imaging Sciences, 2 (2009), 323-343. doi: 10.1137/080725891. [17] L. He, T.-C. Chang, S. Osher, T. Fang and P. Speier, MR image reconstruction by using the iterative renement method and nonlinear inverse scale space methods, UCLA CAM Reports 06-35, (2006). [18] D. Jones, A. Simmons, S. Williams and M. Horsfield, Non-invasive assessment of axonal fiber connectivity in the human brain via diffusion tensor MRI, Magnetic Resonance in Medicine, 42 (1999), 37-41. doi: 10.1002/(SICI)1522-2594(199907)42:1<37::AID-MRM7>3.0.CO;2-O. [19] Q. Li, C. A. Micchelli, L. Shen and Y. Xu, A proximity algorithm accelerated by Gauss-Seidel iterations for L1/TV denoising models, Inverse Problems, 28 (2012). doi: 10.1088/0266-5611/28/9/095003. [20] P. Lions and B. Mercier, Splitting algorithms for the sum of two nonlinear operators, SIAM Journal on Numerical Analysis, 16 (1979), 964-979. doi: 10.1137/0716071. [21] M. Lustig, D. Donoho and J. Pauly, Sparse MRI: The application of compressed sensing for rapid MR imaging, Magnetic Resonance in Medicine, 58 (2007), 1182-1195. doi: 10.1002/mrm.21391. [22] S. Ma, W. Yin, Y. Zhang and A. Chakraborty, An efficient algorithm for compressed MR imaging using total variation and wavelets, in "IEEE Conference on Computer Vision and Pattern Recognition (CVPR 2008)," (2008). doi: 10.1109/CVPR.2008.4587391. [23] A. Ramirez-Manzanares and M. Rivera, Basis tensor decomposition for restoring intra-voxel structure and stochastic walks for inferring brain connectivity in DT-MRI, International Journal of Computer Vision, 69 (2006), 77-92. doi: 10.1007/s11263-006-6855-7. [24] T. McGraw, B. Vemuri, Y. Chen, M. Rao and T. Mareci, DT-MRI denoising and neuronal fiber tracking, Medical Image Analysis, 8 (2004), 95-111. doi: 10.1016/j.media.2003.12.001. [25] T. McGraw, B. Vemuri, E. Ozarslan, Y. Chen and T. Mareci, Variational denoising of diffusion weighted MRI, Inverse Problems and Imaging, 3 (2009), 625-648. doi: 10.3934/ipi.2009.3.625. [26] C. A. Micchelli, L. Shen and Y. Xu, Proximity Algorithms for Image Models: Denoising, Inverse Problems, 27 (2011). doi: 10.1088/0266-5611/27/4/045009. [27] E. Stejskal and J. Tanner, Spin diffusion measurements: Spin echoes in the presence of a time-dependent field gradient, The Journal of Chemical Physics, 42 (1965), 288. doi: 10.1063/1.1695690. [28] D. Tschumperlé and R. Deriche, Variational frameworks for DT-MRI estimation, regularization and visualization, in "Ninth IEEE International Conference on Computer Vision," (2003), 116-121. doi: 10.1109/ICCV.2003.1238323. [29] D. Tuch, R. Weisskoff, J. Belliveau and V. Wedeen, High angular resolution diffusion imaging of the human brain, in "Proceedings of the 7th Annual Meeting of ISMRM," (1999), 321-321. [30] D. Tuch, T. Reese, M. Wiegell and V. J. Wedeen, Diffusion MRI of complex neural architecture, Neuron, 40 (2003), 885-895. doi: 10.1016/S0896-6273(03)00758-X. [31] D. Tuch, Q-ball imaging, Magnetic Resonance in Medicine, 52 (2004), 1358-1372. doi: 10.1002/mrm.20279. [32] A. Tristán-Vega, C. Westin and S. Aja-Fernández, Estimation of fiber orientation probability density functions in high angular resolution diffusion imaging, NeuroImage, 47 (2009), 638-650. doi: 10.1016/j.neuroimage.2009.04.049. [33] Y. Wang, J. Yang, W. Yin and Y. Zhang, A new alternating minimization algorithm for total variation image reconstruction, SIAM Journal on Imaging Sciences, 1 (2008), 248-272. doi: 10.1137/080724265. [34] V. Wedeen, T. Reese, D. Tuch, M. Weigel, J. Dou, R. Weiskoff and D. Chessler, Mapping fiber orientation spectra in cerebral white matter with Fourier-transform diffusion MRI, in "Proc. Intl. Sot. Mag. Reson. Med.," 8 (2000), 82-82. [35] V. Wedeen, P. Hagmann, W. Tseng, T. Reese and R. Weisskoff, Mapping complex tissue architecture with diffusion spectrum magnetic resonance imaging, Magnetic Resonance in Medicine, 54 (2005), 1377-1386. doi: 10.1002/mrm.20642. [36] J. Yang, Y. Zhang and W. Yin, A fast alternating direction method for TVL1-L2 signal reconstruction from partial Fourier data, IEEE Journal of Selected Topics in Signal Processing, 4 (2010), 288-297. [37] M. Zhu and T. Chan, An efficient primal-dual hybrid gradient algorithm for total variation image restoration, UCLA CAM Report 08-34, (2008). [38] M. Zhu, S. Wright and T. Chan, Duality-based algorithms for total-variation-regularized image restoration, Computational Optimization and Applications, 47 (2010), 377-400. doi: 10.1007/s10589-008-9225-2.

show all references

##### References:
 [1] I. Aganj, C. Lenglet, G. Sapiro, E. Yacoub, K. Ugurbil and N. Harel, Reconstruction of the orientation distribution function in single-and multiple-shell q-ball imaging within constant solid angle, Magnetic Resonance in Medicine, 64 (2010), 554-566. doi: 10.1002/mrm.22365. [2] K. Arrow, L, Hurwicz, H. Uzawa and H. Chenery, Studies in linear and non-linear programming, Stanford Mathematical Studies in the Social Sciences, II, Stanford University Press, Stanford, Calif., (1958). [3] H. Assemlal, D. Tschumperlé and L. Brun, Fiber tracking on HARDI data using robust ODF fields, in "IEEE International Conference on Image Processing," Citeseer, (2007), 344-351. doi: 10.1109/ICIP.2007.4379264. [4] P. Basser, J. Mattiello and D. Lebihan, Estimation of the effective self-diffusion tensor from the NMR spin echo, Journal of Magnetic Resonance, Series B, 103 (1994), 247-247. doi: 10.1006/jmrb.1994.1037. [5] P. Basser and C. Pierpaoli, Microstructural and physiological features of tissues elucidated by quantitative-diffusion-tensor MRI, Journal of Magnetic Resonance, Series B, 111 (1996), 209-219. [6] P. Basser, S. Pajevic, C. Pierpaoli, J. Duda and A. Aldroubi, In vivo fiber tractography using DT-MRI data, Magnetic Resonance in Medicine, 44 (2000), 625-632. doi: 10.1002/1522-2594(200010)44:4<625::AID-MRM17>3.0.CO;2-O. [7] A. Chambolle, An algorithm for total variation minimization and applications, Journal of Mathematical Imaging and Vision, 20 (2004), 89-97. doi: 10.1023/B:JMIV.0000011325.36760.1e. [8] A. Chambolle and T. Pock, A first-order primal-dual algorithm for convex problems with applications to imaging, Journal of Mathematical Imaging and Vision, 40 (2011), 120-145. doi: 10.1007/s10851-010-0251-1. [9] Y. Chen, W. Guo, Q. Zeng and Y. Liu, A nonstandard smoothing in reconstruction of apparent diffusion coefficient profiles from diffusion weighted images, Inverse Problems and Imaging, 2 (2008), 205-224. doi: 10.3934/ipi.2008.2.205. [10] O. Christiansen, T. Lee, J. Lie, U. Sinha and T. Chan, Total variation regularization of matrix-valued images, International Journal of Biomedical Imaging, 2007 (2007). doi: 10.1155/2007/27432. [11] M. Descoteaux, E. Angelino, S. Fitzgibbons and R. Deriche, Apparent diffusion coefficients from high angular resolution diffusion imaging: Estimation and applications, Magnetic Resonance in Medicine, 56 (2006), 395-410. doi: 10.1002/mrm.20948. [12] M. Descoteaux, E. Angelino, S. Fitzgibbons and R. Deriche, Regularized, fast, and robust analytical Q-ball imaging, Magnetic Resonance in Medicine, 58 (2007), 497-510. doi: 10.1002/mrm.21277. [13] M. Descoteaux, R. Deriche, T. Knösche and A. Anwander, Deterministic and probabilistic tractography based on complex fibre orientation distributions, IEEE transactions on medical imaging, 28 (2009), 269-286. doi: 10.1109/TMI.2008.2004424. [14] E. Esser, X. Zhang and T. Chan, A general framework for a class of first order primal-dual algorithms for convex optimization in imaging science, SIAM Journal on Imaging Sciences, 3 (2010), 1015-1046. doi: 10.1137/09076934X. [15] L. Frank, Anisotropy in high angular resolution diffusion-weighted MRI, Magnetic Resonance in Medicine, 45 (2001), 935-939. doi: 10.1002/mrm.1125. [16] T. Goldstein and S. Osher, The split Bregman method for L1 regularized problems, SIAM Journal on Imaging Sciences, 2 (2009), 323-343. doi: 10.1137/080725891. [17] L. He, T.-C. Chang, S. Osher, T. Fang and P. Speier, MR image reconstruction by using the iterative renement method and nonlinear inverse scale space methods, UCLA CAM Reports 06-35, (2006). [18] D. Jones, A. Simmons, S. Williams and M. Horsfield, Non-invasive assessment of axonal fiber connectivity in the human brain via diffusion tensor MRI, Magnetic Resonance in Medicine, 42 (1999), 37-41. doi: 10.1002/(SICI)1522-2594(199907)42:1<37::AID-MRM7>3.0.CO;2-O. [19] Q. Li, C. A. Micchelli, L. Shen and Y. Xu, A proximity algorithm accelerated by Gauss-Seidel iterations for L1/TV denoising models, Inverse Problems, 28 (2012). doi: 10.1088/0266-5611/28/9/095003. [20] P. Lions and B. Mercier, Splitting algorithms for the sum of two nonlinear operators, SIAM Journal on Numerical Analysis, 16 (1979), 964-979. doi: 10.1137/0716071. [21] M. Lustig, D. Donoho and J. Pauly, Sparse MRI: The application of compressed sensing for rapid MR imaging, Magnetic Resonance in Medicine, 58 (2007), 1182-1195. doi: 10.1002/mrm.21391. [22] S. Ma, W. Yin, Y. Zhang and A. Chakraborty, An efficient algorithm for compressed MR imaging using total variation and wavelets, in "IEEE Conference on Computer Vision and Pattern Recognition (CVPR 2008)," (2008). doi: 10.1109/CVPR.2008.4587391. [23] A. Ramirez-Manzanares and M. Rivera, Basis tensor decomposition for restoring intra-voxel structure and stochastic walks for inferring brain connectivity in DT-MRI, International Journal of Computer Vision, 69 (2006), 77-92. doi: 10.1007/s11263-006-6855-7. [24] T. McGraw, B. Vemuri, Y. Chen, M. Rao and T. Mareci, DT-MRI denoising and neuronal fiber tracking, Medical Image Analysis, 8 (2004), 95-111. doi: 10.1016/j.media.2003.12.001. [25] T. McGraw, B. Vemuri, E. Ozarslan, Y. Chen and T. Mareci, Variational denoising of diffusion weighted MRI, Inverse Problems and Imaging, 3 (2009), 625-648. doi: 10.3934/ipi.2009.3.625. [26] C. A. Micchelli, L. Shen and Y. Xu, Proximity Algorithms for Image Models: Denoising, Inverse Problems, 27 (2011). doi: 10.1088/0266-5611/27/4/045009. [27] E. Stejskal and J. Tanner, Spin diffusion measurements: Spin echoes in the presence of a time-dependent field gradient, The Journal of Chemical Physics, 42 (1965), 288. doi: 10.1063/1.1695690. [28] D. Tschumperlé and R. Deriche, Variational frameworks for DT-MRI estimation, regularization and visualization, in "Ninth IEEE International Conference on Computer Vision," (2003), 116-121. doi: 10.1109/ICCV.2003.1238323. [29] D. Tuch, R. Weisskoff, J. Belliveau and V. Wedeen, High angular resolution diffusion imaging of the human brain, in "Proceedings of the 7th Annual Meeting of ISMRM," (1999), 321-321. [30] D. Tuch, T. Reese, M. Wiegell and V. J. Wedeen, Diffusion MRI of complex neural architecture, Neuron, 40 (2003), 885-895. doi: 10.1016/S0896-6273(03)00758-X. [31] D. Tuch, Q-ball imaging, Magnetic Resonance in Medicine, 52 (2004), 1358-1372. doi: 10.1002/mrm.20279. [32] A. Tristán-Vega, C. Westin and S. Aja-Fernández, Estimation of fiber orientation probability density functions in high angular resolution diffusion imaging, NeuroImage, 47 (2009), 638-650. doi: 10.1016/j.neuroimage.2009.04.049. [33] Y. Wang, J. Yang, W. Yin and Y. Zhang, A new alternating minimization algorithm for total variation image reconstruction, SIAM Journal on Imaging Sciences, 1 (2008), 248-272. doi: 10.1137/080724265. [34] V. Wedeen, T. Reese, D. Tuch, M. Weigel, J. Dou, R. Weiskoff and D. Chessler, Mapping fiber orientation spectra in cerebral white matter with Fourier-transform diffusion MRI, in "Proc. Intl. Sot. Mag. Reson. Med.," 8 (2000), 82-82. [35] V. Wedeen, P. Hagmann, W. Tseng, T. Reese and R. Weisskoff, Mapping complex tissue architecture with diffusion spectrum magnetic resonance imaging, Magnetic Resonance in Medicine, 54 (2005), 1377-1386. doi: 10.1002/mrm.20642. [36] J. Yang, Y. Zhang and W. Yin, A fast alternating direction method for TVL1-L2 signal reconstruction from partial Fourier data, IEEE Journal of Selected Topics in Signal Processing, 4 (2010), 288-297. [37] M. Zhu and T. Chan, An efficient primal-dual hybrid gradient algorithm for total variation image restoration, UCLA CAM Report 08-34, (2008). [38] M. Zhu, S. Wright and T. Chan, Duality-based algorithms for total-variation-regularized image restoration, Computational Optimization and Applications, 47 (2010), 377-400. doi: 10.1007/s10589-008-9225-2.
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