American Institute of Mathematical Sciences

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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

Yunmei Chen 1, , Weihong Guo 2, , Qingguo Zeng 1, and  Yijun Liu 3,

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.
Keywords: Reconstruction, smoothing, spherical harmonic series., diffusion weighted images.
Mathematics Subject Classification: Primary: 15A29, 94A0.
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:
[1]

D. LeBihan and P. J. Basser, Molecular diffusion and nuclear magnetic resonance, Diffusion and perfusion magnetic resonance imaging, 1995. Google Scholar

[2]

M. E. Moseley, Y. Cohen, J. Mintorovitch, J. L. Chileuitt, D. Norman and P. Weinstein, Evidence of anisotropic self-diffusion in cat brain, Proc. of the 8th ISMRM, (1989), 136-136. Google Scholar

[3]

M. E. Moseley, J. Kucharczyk, H. S. Asgari and D. Norman, Anisotropy in diffusion weighted MRI, Magn. Reson. Med., 19 (1991), 321-326. doi: 10.1002/mrm.1910190222.  Google Scholar

[4]

P. J. Basser and C. Pierpaoli, Microstructual and physiological features of tissues elucidated by quantitative diffusion tensor {MRI}, Magn. Reson. Med., 111(B) 1996, 209-219. Google Scholar

[5]

E. O. Stejskal and J. E. Tanner, Spin diffusion measurements: Spin echoes in the presence of a time-dependent field gradient, Chem. Phys., 42 (1965), 288-292. Google Scholar

[6]

P. J. Basser, J. Mattiello and D. LeBihan, MR diffusion tensor spectroscopy and imaging, Biophys, 66 1994, 259-267. doi: 10.1016/S0006-3495(94)80775-1.  Google Scholar

[7]

D. S. Tuch, R. M. Weisskoff, J. W. Belliveau and V. J. Wedeen, High angular resolution diffusion imaging of the human brain, Proc. of the 7th ISMRM, (1999), 321-321. Google Scholar

[8]

V. J. Wedeen, T. G. Reese, D. S. Tuch, M. R. Weigel, J.-G. Dou, R. M. Weisskoff and D. Chesler, Mapping fiber orientation spectra in cerebral white matter with fourier transform diffusion {MRI}, Proc. of the 8th ISMRM, (2000), 82-82. Google Scholar

[9]

P. J. Basser, J. Mattiello and D. Lebihan, Estimation of the effective self-diffusion tensor from the NMR, Spin Echo. J. Magn. Reson., series B, 103 (1994), 247-254. Google Scholar

[10]

T. L. Chenevert, J. A. Brunberg and J. G. Pipe, Anisotropic diffusion in human white matter: demonstration with MR techniques in vivo, Radiology, 177 (1990), 401-405. Google Scholar

[11]

E. W. Hsu and S. Mori, Analytical expression for the NMR apparent diffusion coefficients in an anisotropy system and a simplified method for determing fiber orientation, Magn. Reson. Med., 34 (1995), 194-200. doi: 10.1002/mrm.1910340210.  Google Scholar

[12]

L. Frank, Characterization of anisotropy in high angular resolution diffusion weighted mri, in "Proc. of 9th ISMRM," Glasgow, Scotland, (2001). Google Scholar

[13]

A. L. Alexander, K. M. Hasan, M. Lazar, J. S. Tsuruda and D. L. Parker, Analysis of partial volume effects in diffusion-tensor MRI, Magn. Reson. Med., 45 (2001), 770-780. doi: 10.1002/mrm.1105.  Google Scholar

[14]

L. Frank, Anisotropy in high angular resolution diffusion-weighted MRI, Magn. Reson. Med., 45 (2001), 935-939. doi: 10.1002/mrm.1125.  Google Scholar

[15]

D. C. Alexander, G. J. Barker and S. R. Arridge, Detection and modeling of non-Gaussian apparent diffusion coefficient profiles in human brain data, Magn. Reson. Med., 48 (2002), 331-340. doi: 10.1002/mrm.10209.  Google Scholar

[16]

Y. Chen, W. Guo, Q. Zeng, X. Yan, F. Huang, H. Zhang, G. He, B. Vemuri and Y. Liu, Estimation, smoothing, and charaterization of apparent diffusion coefficient profiles from high angular resolution DWI, Proc. of CVPR, (2004), 588-593. Google Scholar

[17]

L. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithm, Physica D, 60 (1992), 259-268. doi: 10.1016/0167-2789(92)90242-F.  Google Scholar

[18]

P. Blomgren, T. Chan, P. Mulet and C. K. Wong, Total variation image restoration: Numerical methods and extensions, Proceeding of IEEE Int'l Conference on Image Processing, 3 (1997), 384-387. Google Scholar

[19]

A. Chambolle and P-L.Lions, Image recovery via total variation minimization and related problems, Numerische Mathematik, 76 (1997), 167-188. doi: 10.1007/s002110050258.  Google Scholar

[20]

Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM Journal of AMath., 66 (2006), 1383-1406. doi: 10.1137/050624522.  Google Scholar

[21]

P. Blomgren and T. Chan, Color TV: total variation methods for restoration of vector-valued images, IEEE Trans. on Image Processing, 7 (1998), 304-309. doi: 10.1109/83.661180.  Google Scholar

[22]

Z. Wang, B. C. Vemuri, Y. Chen and T. Mareci, A constrained variational principle for direct estimation and smoothing of the tensor field from complex DWI, IEEE TMI, 23 (2004), 930-939. Google Scholar

[23]

J. Weickert, B. Romeny and M. Viergever, Efficient and reliable schemes for nonlinear diffusion filtering, IEEE Trans. on Img. Proc., 7 (1998), 398-410. Google Scholar

[24]

T. Lu, P. Neittaanm and X. Tai, A parallel splitting up method and its application to Navier-Stokes equations, Applied Mathematics Letters, 4 (1991), 25-29. doi: 10.1016/0893-9659(91)90161-N.  Google Scholar

[25]

T. F. Chan and L. A. Vese, Active contours without edges, IEEE Trans. Image Processing, 10 (2001), 266-277. doi: 10.1109/83.902291.  Google Scholar

[26]

S. D. Conte and C. DeBoor, "Elementary Numerical Analysis," McGraw-Hill, New York, 1972. Google Scholar

show all references

References:
[1]

D. LeBihan and P. J. Basser, Molecular diffusion and nuclear magnetic resonance, Diffusion and perfusion magnetic resonance imaging, 1995. Google Scholar

[2]

M. E. Moseley, Y. Cohen, J. Mintorovitch, J. L. Chileuitt, D. Norman and P. Weinstein, Evidence of anisotropic self-diffusion in cat brain, Proc. of the 8th ISMRM, (1989), 136-136. Google Scholar

[3]

M. E. Moseley, J. Kucharczyk, H. S. Asgari and D. Norman, Anisotropy in diffusion weighted MRI, Magn. Reson. Med., 19 (1991), 321-326. doi: 10.1002/mrm.1910190222.  Google Scholar

[4]

P. J. Basser and C. Pierpaoli, Microstructual and physiological features of tissues elucidated by quantitative diffusion tensor {MRI}, Magn. Reson. Med., 111(B) 1996, 209-219. Google Scholar

[5]

E. O. Stejskal and J. E. Tanner, Spin diffusion measurements: Spin echoes in the presence of a time-dependent field gradient, Chem. Phys., 42 (1965), 288-292. Google Scholar

[6]

P. J. Basser, J. Mattiello and D. LeBihan, MR diffusion tensor spectroscopy and imaging, Biophys, 66 1994, 259-267. doi: 10.1016/S0006-3495(94)80775-1.  Google Scholar

[7]

D. S. Tuch, R. M. Weisskoff, J. W. Belliveau and V. J. Wedeen, High angular resolution diffusion imaging of the human brain, Proc. of the 7th ISMRM, (1999), 321-321. Google Scholar

[8]

V. J. Wedeen, T. G. Reese, D. S. Tuch, M. R. Weigel, J.-G. Dou, R. M. Weisskoff and D. Chesler, Mapping fiber orientation spectra in cerebral white matter with fourier transform diffusion {MRI}, Proc. of the 8th ISMRM, (2000), 82-82. Google Scholar

[9]

P. J. Basser, J. Mattiello and D. Lebihan, Estimation of the effective self-diffusion tensor from the NMR, Spin Echo. J. Magn. Reson., series B, 103 (1994), 247-254. Google Scholar

[10]

T. L. Chenevert, J. A. Brunberg and J. G. Pipe, Anisotropic diffusion in human white matter: demonstration with MR techniques in vivo, Radiology, 177 (1990), 401-405. Google Scholar

[11]

E. W. Hsu and S. Mori, Analytical expression for the NMR apparent diffusion coefficients in an anisotropy system and a simplified method for determing fiber orientation, Magn. Reson. Med., 34 (1995), 194-200. doi: 10.1002/mrm.1910340210.  Google Scholar

[12]

L. Frank, Characterization of anisotropy in high angular resolution diffusion weighted mri, in "Proc. of 9th ISMRM," Glasgow, Scotland, (2001). Google Scholar

[13]

A. L. Alexander, K. M. Hasan, M. Lazar, J. S. Tsuruda and D. L. Parker, Analysis of partial volume effects in diffusion-tensor MRI, Magn. Reson. Med., 45 (2001), 770-780. doi: 10.1002/mrm.1105.  Google Scholar

[14]

L. Frank, Anisotropy in high angular resolution diffusion-weighted MRI, Magn. Reson. Med., 45 (2001), 935-939. doi: 10.1002/mrm.1125.  Google Scholar

[15]

D. C. Alexander, G. J. Barker and S. R. Arridge, Detection and modeling of non-Gaussian apparent diffusion coefficient profiles in human brain data, Magn. Reson. Med., 48 (2002), 331-340. doi: 10.1002/mrm.10209.  Google Scholar

[16]

Y. Chen, W. Guo, Q. Zeng, X. Yan, F. Huang, H. Zhang, G. He, B. Vemuri and Y. Liu, Estimation, smoothing, and charaterization of apparent diffusion coefficient profiles from high angular resolution DWI, Proc. of CVPR, (2004), 588-593. Google Scholar

[17]

L. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithm, Physica D, 60 (1992), 259-268. doi: 10.1016/0167-2789(92)90242-F.  Google Scholar

[18]

P. Blomgren, T. Chan, P. Mulet and C. K. Wong, Total variation image restoration: Numerical methods and extensions, Proceeding of IEEE Int'l Conference on Image Processing, 3 (1997), 384-387. Google Scholar

[19]

A. Chambolle and P-L.Lions, Image recovery via total variation minimization and related problems, Numerische Mathematik, 76 (1997), 167-188. doi: 10.1007/s002110050258.  Google Scholar

[20]

Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM Journal of AMath., 66 (2006), 1383-1406. doi: 10.1137/050624522.  Google Scholar

[21]

P. Blomgren and T. Chan, Color TV: total variation methods for restoration of vector-valued images, IEEE Trans. on Image Processing, 7 (1998), 304-309. doi: 10.1109/83.661180.  Google Scholar

[22]

Z. Wang, B. C. Vemuri, Y. Chen and T. Mareci, A constrained variational principle for direct estimation and smoothing of the tensor field from complex DWI, IEEE TMI, 23 (2004), 930-939. Google Scholar

[23]

J. Weickert, B. Romeny and M. Viergever, Efficient and reliable schemes for nonlinear diffusion filtering, IEEE Trans. on Img. Proc., 7 (1998), 398-410. Google Scholar

[24]

T. Lu, P. Neittaanm and X. Tai, A parallel splitting up method and its application to Navier-Stokes equations, Applied Mathematics Letters, 4 (1991), 25-29. doi: 10.1016/0893-9659(91)90161-N.  Google Scholar

[25]

T. F. Chan and L. A. Vese, Active contours without edges, IEEE Trans. Image Processing, 10 (2001), 266-277. doi: 10.1109/83.902291.  Google Scholar

[26]

S. D. Conte and C. DeBoor, "Elementary Numerical Analysis," McGraw-Hill, New York, 1972. Google Scholar

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