# American Institute of Mathematical Sciences

May  2008, 2(2): 205-224. doi: 10.3934/ipi.2008.2.205

## A nonstandard smoothing in reconstruction of apparent diffusion coefficient profiles from diffusion weighted images

 1 Department of Mathematics, University of Florida, Gainesville, FL 32611, United States, United States 2 Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487, United States 3 Department of Psychiatry & Neuroscience, University of Florida, Gainesville, FL 32653, United States

Received  May 2007 Revised  December 2007 Published  April 2008

We present a new variational framework for simultaneous smoothing and estimation of apparent diffusion coefficient (ADC) profiles from High Angular Resolution Diffusion-weighted MRI. The model approximates the ADC profiles at each voxel by a 4th order spherical harmonic series (SHS). The coefficients in SHS are obtained by solving a constrained minimization problem. The smoothing with feature preserved is achieved by minimizing a variable exponent, linear growth functional, and the data constraint is determined by the original Stejskal-Tanner equation. The antipodal symmetry and positiveness of the ADC are accommodated in the model. We use these coefficients and variance of the ADC profiles from its mean to classify the diffusion in each voxel as isotropic, anisotropic with single fiber orientation, or two fiber orientations. The proposed model has been applied to both simulated data and HARD MRI human brain data . The experiments demonstrated the effectiveness of our method in estimation and smoothing of ADC profiles and in enhancement of diffusion anisotropy. Further characterization of non-Gaussian diffusion based on the proposed model showed a consistency between our results and known neuroanatomy.
Citation: Yunmei Chen, Weihong Guo, Qingguo Zeng, Yijun Liu. A nonstandard smoothing in reconstruction of apparent diffusion coefficient profiles from diffusion weighted images. Inverse Problems & Imaging, 2008, 2 (2) : 205-224. doi: 10.3934/ipi.2008.2.205
##### References:

show all references

##### References:
 [1] Matti Viikinkoski, Mikko Kaasalainen. Shape reconstruction from images: Pixel fields and Fourier transform. Inverse Problems & Imaging, 2014, 8 (3) : 885-900. doi: 10.3934/ipi.2014.8.885 [2] Leonid Kunyansky. Fast reconstruction algorithms for the thermoacoustic tomography in certain domains with cylindrical or spherical symmetries. Inverse Problems & Imaging, 2012, 6 (1) : 111-131. doi: 10.3934/ipi.2012.6.111 [3] Zhichang Guo, Wenjuan Yao, Jiebao Sun, Boying Wu. Nonlinear fractional diffusion model for deblurring images with textures. Inverse Problems & Imaging, 2019, 13 (6) : 1161-1188. doi: 10.3934/ipi.2019052 [4] Guillaume Bal, Jiaming Chen, Anthony B. Davis. Reconstruction of cloud geometry from high-resolution multi-angle images. Inverse Problems & Imaging, 2018, 12 (2) : 261-280. doi: 10.3934/ipi.2018011 [5] Braxton Osting, Dong Wang. Diffusion generated methods for denoising target-valued images. Inverse Problems & Imaging, 2020, 14 (2) : 205-232. doi: 10.3934/ipi.2020010 [6] 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 [7] Alfred K. Louis. Diffusion reconstruction from very noisy tomographic data. Inverse Problems & Imaging, 2010, 4 (4) : 675-683. doi: 10.3934/ipi.2010.4.675 [8] Marc C. Robini, Yuemin Zhu, Jianhua Luo. Edge-preserving reconstruction with contour-line smoothing and non-quadratic data-fidelity. Inverse Problems & Imaging, 2013, 7 (4) : 1331-1366. doi: 10.3934/ipi.2013.7.1331 [9] Cheng Peng, Zhaohui Tang, Weihua Gui, Qing Chen, Jing He. A bidirectional weighted boundary distance algorithm for time series similarity computation based on optimized sliding window size. Journal of Industrial & Management Optimization, 2021, 17 (1) : 205-220. doi: 10.3934/jimo.2019107 [10] Yonggui Zhu, Yuying Shi, Bin Zhang, Xinyan Yu. Weighted-average alternating minimization method for magnetic resonance image reconstruction based on compressive sensing. Inverse Problems & Imaging, 2014, 8 (3) : 925-937. doi: 10.3934/ipi.2014.8.925 [11] Chan Liu, Jin Wen, Zhidong Zhang. Reconstruction of the time-dependent source term in a stochastic fractional diffusion equation. Inverse Problems & Imaging, 2020, 14 (6) : 1001-1024. doi: 10.3934/ipi.2020053 [12] Matteo Bonforte, Jean Dolbeault, Matteo Muratori, Bruno Nazaret. Weighted fast diffusion equations (Part Ⅱ): Sharp asymptotic rates of convergence in relative error by entropy methods. Kinetic & Related Models, 2017, 10 (1) : 61-91. doi: 10.3934/krm.2017003 [13] Jesus Ildefonso Díaz, David Gómez-Castro, Jean Michel Rakotoson, Roger Temam. Linear diffusion with singular absorption potential and/or unbounded convective flow: The weighted space approach. Discrete & Continuous Dynamical Systems, 2018, 38 (2) : 509-546. doi: 10.3934/dcds.2018023 [14] Alexander Barg, Oleg R. Musin. Codes in spherical caps. Advances in Mathematics of Communications, 2007, 1 (1) : 131-149. doi: 10.3934/amc.2007.1.131 [15] Ronen Peretz, Nguyen Van Chau, L. Andrew Campbell, Carlos Gutierrez. Iterated images and the plane Jacobian conjecture. Discrete & Continuous Dynamical Systems, 2006, 16 (2) : 455-461. doi: 10.3934/dcds.2006.16.455 [16] Armengol Gasull, Francesc Mañosas. Subseries and signed series. Communications on Pure & Applied Analysis, 2019, 18 (1) : 479-492. doi: 10.3934/cpaa.2019024 [17] Matteo Bonforte, Jean Dolbeault, Matteo Muratori, Bruno Nazaret. Weighted fast diffusion equations (Part Ⅰ): Sharp asymptotic rates without symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities. Kinetic & Related Models, 2017, 10 (1) : 33-59. doi: 10.3934/krm.2017002 [18] Beom-Seok Han, Kyeong-Hun Kim, Daehan Park. A weighted Sobolev space theory for the diffusion-wave equations with time-fractional derivatives on $C^{1}$ domains. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3415-3445. doi: 10.3934/dcds.2021002 [19] Serena Dipierro, Enrico Valdinoci. On a fractional harmonic replacement. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3377-3392. doi: 10.3934/dcds.2015.35.3377 [20] Wenxiong Chen, Congming Li. Harmonic maps on complete manifolds. Discrete & Continuous Dynamical Systems, 1999, 5 (4) : 799-804. doi: 10.3934/dcds.1999.5.799

2019 Impact Factor: 1.373