The Community for Technology Leaders
RSS Icon
Issue No.04 - April (2013 vol.35)
pp: 983-995
P. MomayyezSiahkal , Sch. of Comput. Sci., McGill Univ., Montreal, QC, Canada
K. Siddiqi , Sch. of Comput. Sci., McGill Univ., Montreal, QC, Canada
The 2D stochastic completion field algorithm, introduced by Williams and Jacobs [1], [2], uses a directional random walk to model the prior probability of completion curves in the plane. This construct has had a powerful impact in computer vision, where it has been used to compute the shapes of likely completion curves between edge fragments in visual imagery. Motivated by these developments, we extend the algorithm to 3D, using a spherical harmonics basis to achieve a rotation invariant computational solution to the Fokker-Planck equation describing the evolution of the probability density function underlying the model. This provides a principled way to compute 3D completion patterns and to derive connectivity measures for orientation data in 3D, as arises in 3D tracking, motion capture, and medical imaging. We demonstrate the utility of the approach for the particular case of diffusion magnetic resonance imaging, where we derive connectivity maps for synthetic data, on a physical phantom and on an in vivo high angular resolution diffusion image of a human brain.
Mathematical model, Stochastic processes, Magnetic resonance imaging, Equations, Discrete wavelet transforms, Solid modeling, Probabilistic logic,spherical harmonics, 3D directional random walk, Fokker-Planck equation, completion fields, diffusion MRI, probabilistic connectivity
P. MomayyezSiahkal, K. Siddiqi, "3D Stochastic Completion Fields for Mapping Connectivity in Diffusion MRI", IEEE Transactions on Pattern Analysis & Machine Intelligence, vol.35, no. 4, pp. 983-995, April 2013, doi:10.1109/TPAMI.2012.184
[1] L. Williams and D. Jacobs, "Stochastic Completion Fields: A Neural Model of Illusory Contour Shape and Salience," Neural Computation, vol. 9, pp. 837-858, 1997.
[2] L. Williams and D. Jacobs, "Local Parallel Computation of Stochastic Completion Fields," Neural Computation, vol. 9, pp. 859-881, 1997.
[3] G. Kanizsa, Organization in Vision: Essays on Gestalt Perception, series Praeger Special Studies. Praeger, 1979.
[4] S. Petry and G. Meyer, The Perception of Illusory Contours. Springer-Verlag, 1987.
[5] S. Ullman, "Filling-In the Gaps: The Shape of Subjective Contours and a Model for Their Generation," Biological Cybernetics, vol. 25, no. 1, pp. 1-6, 1976.
[6] D. Mumford, "Elastica and Computer Vision," Algebraic Geometry and Its Applications. chapter 31, pp. 491-506, Springer-Verlag, 1994.
[7] B. Kimia, I. Frankel, and A. Popescu, "Euler Spiral for Shape Completion," Int'l J. Computer Vision, vol. 54, pp. 159-182, 2003.
[8] P. Parent and S.W. Zucker, "Trace Inference, Curvature Consistency and Curve Detection," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 11, no. 8, pp. 823-889, Aug. 1989.
[9] G. Guy and G. Medioni, "Inferring Global Perceptual Contours from Local Features," Int'l J. Computer Vision, vol. 20, pp. 113-133, 1993.
[10] A. Shashua and S. Ullman, "Structural Saliency: The Detection of Globally Salient Structures Using a Locally Connected Network," Proc. Second IEEE Int'l Conf. Computer Vision, pp. 321-327. 1988,
[11] J. Zweck and L.R. Williams, "Euclidean Group Invariant Computation of Stochastic Completion Fields Using Shiftable-Twistable Functions," J. Math. Imaging and Vision, vol. 21, pp. 135-154, 2004.
[12] C. Poupon, B. Rieul, I. Kezele, M. Perrin, F. Poupon, and J. Mangin, "New Diffusion Phantoms Dedicated to the Study and Validation of High-Angular-Resolution Diffusion Imaging (HARDI) Models," Magnetic Resonance in Medicine, vol. 60, pp. 1276-1283, 2008.
[13] D. LeBihan, E. Breton, D. Lallemand, P. Grenier, E. Cabanis, and M. Laval-Jeantet, "MR Imaging of Intravoxel Incoherent Motions: Application to Diffusion and Perfusion in Neurologic Disorders," Radiology, vol. 161, pp. 401-407, Nov. 1986.
[14] M. Catani, R. Howard, S. Pajevic, and D. DK, "Virtual in Vivo Interactive Dissection of White Matter Fasciculi in the Human Brain," Neuroimage, vol. 17, pp. 77-94, 2002.
[15] S. Frey, J. Campbell, G. Pike, and M. Petrides, "Dissociating the Human Language Pathways with High Angular Resolution Diffusion Fiber Tractography," J. Neuroscience, vol. 28, pp. 11435-11444, 2008.
[16] M. Catani, D. Jones, and D. ffytche, "Perisylvian Language Networks of the Human Brain," Annals of Neurology, vol. 57, pp. 8-16, 2005.
[17] A. Anwander, M. Tittgemeyer, D. von Cramon, A. Friederici, and T. Knösche, "Connectivity-Based Parcellation of Broca's Area," Cerebral Cortex, vol. 17, pp. 816-825, 2007.
[18] M. Rushworth, T. Behrens, and H. Johansen-Berg, "Connection Patterns Distinguish 3 Regions of Human Parietal Cortex," Cerebral Cortex, vol. 16, pp. 1418-1430, 2006.
[19] A. Venkataraman, Y. Rathi, M. Kubicki, C. Westin, and P. Golland, "Joint Generative Model for fMRI/DWI and Its Application to Population Studies," Proc. 13th Int'l Conf. Medical Image Computing and Computer-Assisted Intervention, pp. 191-199, Sept. 2010.
[20] M. Koch, D. Norris, and M. Hund-Georgiadis, "An Investigation of Functional and Anatomical Connectivity Using Magnetic Resonance Imaging," Neuroimage, vol. 16, pp. 241-250, 2002.
[21] D. LeBihan, R. Turner, and P. Douek, "Is Water Diffusion Restricted in Human Brain White Matter? An Echo-Planar NMR Imaging Study," NeuroReport, vol. 4, pp. 887-890, July 1993.
[22] S. Mori, B. Crain, V. Chacko, and P. van Zijl, "Three Dimensional Tracking of Axonal Projections in the Brain by Magnetic Resonance Imaging," Annals of Neurolology, vol. 45, pp. 265-269, 1999.
[23] P. Basser, S. Pajevic, C. Pierpaoli, J. Duda, and A. Aldroubi, "In Vitro Fiber Tractography Using DT-MRI Data," Magnetic Resonance in Medicine, vol. 44, pp. 625-632, 2000.
[24] G. Parker, C. Wheeler-Kingshott, and G. Barker, "Estimating Distributed Anatomical Connectivity Using Fast Marching Methods and Diffusion Tensor Imaging," Trans. Medical Imaging, vol. 21, pp. 505-512, 2002.
[25] C. Poupon, C. Clark, V. Frouin, J. Regis, I. Bloch, D. LeBihan, and J. Mangin, "Regularization of Diffusion-Based Direction Maps for the Tracking of Brain White Matter Fascicles," NeuroImage, vol. 12, pp. 184-195, 2000.
[26] J. Campbell, K. Siddiqi, V. Rymar, A. Sadikot, and B. Pike, "Flow-Based Fiber Tracking with Diffusion Tensor q-Ball Data: Validation and Comparison to Principal Diffusion Direction Techniques," NeuroImage, vol. 27, pp. 725-736, 2005.
[27] P. Behrens et al., "Characterization and Propagation of Uncertainty in Diffusion-Weighted MR Imaging," Magnetic Resonance in Medicine, vol. 50, pp. 1077-1088, 2003.
[28] D. Jones and C. Pierpaoli, "Confidence Mapping in Diffusion Tensor Magnetic Resonance Imaging Tractography Using a Bootstrap Approach," Magnetic Resonance Medicine, vol. 53, pp. 1143-1149, 2005.
[29] J. Berman, S. Chung, P. Mukherjee, C. Hess, E. Han, and R. Henrya, "Probabilistic Streamline q-Ball Tractography Using the Residual Bootstrap," NeuroImage, vol. 39, pp. 215-222, 2008.
[30] P. Basser, J. Matiello, and D.L. Bihan, "MR Diffusion Tensor Spectroscopy and Imaging," Biophysical J., vol. 66, pp. 259-267, Jan. 1994.
[31] D.S. Tuch, "Q-Ball Imaging," Magnetic Resonance in Medicine, vol. 52, pp. 1358-1372, Dec. 2004.
[32] M. Descoteaux, E. Angelino, S. Fitzgibbons, and R. Deriche, "Regularized, Fast, and Robust Analytical q-Ball Imaging," Magnetic Resonance Medicine, vol. 58, pp. 497-510, 2007.
[33] J. Tournier, F. Calamante, D. Gadian, and A. Connelly, "Direct Estimation of the Fiber Orientation Density Function from Diffusion-Weighted MRI Data Using Spherical Deconvolution," NeuroImage, vol. 23, pp. 1176-1185, Sept. 2004.
[34] S. Mori and P.V. Zijl, "Fiber Tracking: Principles and Strategiesa Technical Review," NMR in Biomedicine, vol. 15, pp. 468-480, 2002.
[35] 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, vol. 42, pp. 37-41, 1999.
[36] P. MomayyezSiahkal, J. Campbell, P. Savadjiev, G. Pike, and K. Siddiqi, "Beyond Crossing Fibres: Probabilistic Tractography of Complex Subvoxel Fibre Geometries," Proc. MICCAI Workshop Diffusion Modelling and the Fibre Cup, pp. 81-92, Sept. 2009.
[37] J.G. Malcolm, M.E. Shenton, and Y. Rathi, "Filtered Multi-Tensor Tractography," IEEE Trans. Medical Imaging, vol. 29, no. 9, pp. 1664-1675, Sept. 2010.
[38] P. Fillard, C. Poupon, and J. Mangin, "A Novel Global Tractography Algorithm Based on an Adaptive Spin Glass Model," Proc. 12th Int'l Conf. Medical Image Computing and Computer-Assisted Intervention, pp. 927-934, Sept. 2009.
[39] S. Jbabdi, P. Bellec, R. Toro, J. Daunizeau, M. Pelegrini-Issac, and H. Benali, "Accurate Anisotropic Fast Marching for Diffusion-Based Geodesic Tractography," Int'l J. Biomedical Imaging, vol. 2008, pp. 1-12, 2008.
[40] P. Momayyez and K. Siddiqi, "Rotation Invariant Completion Fields for Mapping Diffusion-MRI Connectivity," Proc. 22nd Int'l Conf. Information Processing in Medical Imaging, 2011.
[41] P. Batchelor, D. Hill, F. Calamante, and D. Atkinson, "Study of Connectivity in the Brain Using the Full Diffusion Tensor from MRI," Proc. 17th Int'l Conf. Information Processing in Medical Imaging, pp. 121-133, 2001.
[42] L. O'Donnell, S. Haker, and C. Westin, "New Approaches to Estimation of White Matter Connectivity in Diffusion Tensor MRI: Elliptic PDEs and Geodesics in a Tensor-Warped Space," Proc. Fifth Int'l Conf. Medical Image Computing and Computer-Assisted Intervention, pp. 459-466, Sept. 2002.
[43] M. Péchaud, M. Descoteaux, and R. Keriven, "Brain Connectivity Using Geodesics in Hardi," Proc. 12th Int'l Conf. Medical Image Computing and Computer-Assisted Intervention, pp. 482-489, Sept. 2009.
[44] N. Hageman, A. Toga, K. Narr, and D. Shattuck, "A Diffusion Tensor Imaging Tractography Algorithm Based on Navier-Stokes Fluid Mechanics," IEEE Trans. Medical Imaging, vol. 28, no. 3, pp. 348-360, Mar. 2009.
[45] P. Fletcher, R. Tao, W. Jeong, and R. Whitaker, "A Volumetric Approach to Quantifying Region-to-Region White Matter Connectivity in Diffusion Tensor MRI," Proc. 20th Int'l Conf. Information Processing in Medical Imaging, pp. 346-358, 2007.
[46] E. Pichon, C. Westin, and A. Tannenbaum, "A Hamilton-Jacobi-Bellman Approach to High Angular Resolution Diffusion Tractography," Proc. Eighth Int'l Conf. Medical Image Computing and Computer-Assisted Intervention, pp. 180-187, 2005.
[47] P. Momayyez and K. Siddiqi, "3D Stochastic Completion Fields for Fiber Tractography," Proc. IEEE Workshop Math. Methods in Biomedical Image Analysis, pp. 178-185, June 2009.
[48] H. Risken, The Fokker-Planck Equation: Methods of Solution and Applications. Springer, 1996.
[49] P. Savadjiev, J. Campbell, and B.P.K. Siddiqi, "3D Curve Inferencefor Diffusion MRI Regularization and Fibre Tractography," Medical Image Analysis, vol. 10, pp. 799-813, Aug. 2006.
[50] B. Fornberg and D. Merrill, "Comparison of Finite Difference and Pseudospectral Methods for Convective Flow over a Sphere," Geophysical Research Letters, vol. 24, pp. 3245-3248, 1997.
[51] D. Alexander, G. Barker, and S. Arridge, "Detection and Modeling of Non-Gaussian Apparent Diffusion Coefficient Profiles in Human Brain Data," Magnetic Resonance in Medicine, vol. 48, pp. 331-340, 2002.
[52] A. Anderson, "Measurement of Fiber Orientation Distributions Using High Angular Resolution Diffusion Imaging," Magnetic Resonance in Medicine, vol. 54, no. 5, pp. 1194-1206, 2005.
[53] T. Yo, A. Anwander, M. Descoteaux, P. Fillard, C. Poupon, and T. Knösch, "Quantifying Brain Connectivity: A Comparative Tractography Study," Proc. 12th Int'l Conf. Medical Image Computing and Computer-Assisted Intervention, pp. 886-893, Sept. 2009.
[54] B. Jeurissen, A. Leemans, D.K. Jones, J.D. Tournier, and J. Sijbers, "Probabilistic Fiber Tracking Using the Residual Bootstrap with Constrained Spherical Deconvolution," Human Brain Mapping, vol. 32, pp. 461-479, 2011.
[55] O. Ciccarelli, A.T. Tossy, G.J. Parker, C.A. Wheeler-Kingshott, G.J. Barker, and P.A. Boulby, "From Diffusion Tractography to Quantitative White Matter Tract Measures: A Reproducibility Study," NeuroImage, vol. 18, pp. 348-359, 2003.
110 ms
(Ver 2.0)

Marketing Automation Platform Marketing Automation Tool