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 and Imaging, 2008, 2 (2) : 205-224. doi: 10.3934/ipi.2008.2.205
References:
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D. LeBihan and P. J. Basser, Molecular diffusion and nuclear magnetic resonance, Diffusion and perfusion magnetic resonance imaging, 1995.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[12]

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

[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.

[14]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[26]

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

show all references

References:
[1]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[12]

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

[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.

[14]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[26]

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

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