Cosine series representation of 3D curves and its application to white matter fiber bundles in diffusion tensor imaging.

We present a novel cosine series representation for encoding fiber bundles consisting of multiple 3D curves. The coordinates of curves are parameterized as coefficients of cosine series expansion. We address the issue of registration, averaging and statistical inference on curves in a unified Hilbert space framework. Unlike traditional splines, the proposed method does not have internal knots and explicitly represents curves as a linear combination of cosine basis. This simplicity in the representation enables us to design statistical models, register curves and perform subsequent analysis in a more unified statistical framework than splines.The proposed representation is applied in characterizing abnormal shape of white matter fiber tracts passing through the splenium of the corpus callosum in autistic subjects. For an arbitrary tract, a 19 degree expansion is usually found to be sufficient to reconstruct the tract with 60 parameters.

[1]  C. Westin,et al.  A method for clustering white matter fiber tracts. , 2006, AJNR. American journal of neuroradiology.

[2]  Claude Brezinski,et al.  Extrapolation Algorithms for Filtering Series of Functions, and Treating the Gibbs Phenomenon , 2004, Numerical Algorithms.

[3]  H. Moser,et al.  Imaging cortical association tracts in the human brain using diffusion‐tensor‐based axonal tracking , 2002, Magnetic resonance in medicine.

[4]  M. Raichle,et al.  Tracking neuronal fiber pathways in the living human brain. , 1999, Proceedings of the National Academy of Sciences of the United States of America.

[5]  Douglas W. Jones,et al.  Shape analysis of brain ventricles using SPHARM , 2001, Proceedings IEEE Workshop on Mathematical Methods in Biomedical Image Analysis (MMBIA 2001).

[6]  T. Bulow.  Spherical diffusion for 3D surface smoothing , 2002, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[7]  A. Yaglom.  Correlation Theory of Stationary and Related Random Functions I: Basic Results , 1987 .

[8]  MakedonFillia,et al.  A surface-based approach for classification of 3D neuroanatomic structures , 2004 .

[9]  Abdul J. Jerri.  The Gibbs Phenomenon in Fourier Analysis, Splines and Wavelet Approximations , 1998 .

[10]  Scott T. Grafton,et al.  Automated image registration: I. General methods and intrasubject, intramodality validation. , 1998, Journal of computer assisted tomography.

[11]  B. Silverman,et al.  Functional Data Analysis , 1997 .

[12]  S. Kwapień,et al.  Random Series and Stochastic Integrals: Single and Multiple , 1992 .

[13]  Anne Gelb,et al.  The resolution of the Gibbs phenomenon for spherical harmonics , 1997, Math. Comput..

[14]  Carl-Fredrik Westin,et al.  Automatic Tractography Segmentation Using a High-Dimensional White Matter Atlas , 2007, IEEE Transactions on Medical Imaging.

[15]  Michael Brady,et al.  Improved Optimization for the Robust and Accurate Linear Registration and Motion Correction of Brain Images , 2002, NeuroImage.

[16]  A. Gruen,et al.  Least squares 3D surface and curve matching , 2005 .

[17]  T. Chan,et al.  Genus zero surface conformal mapping and its application to brain surface mapping. , 2004, IEEE transactions on medical imaging.

[18]  Moo K. Chung,et al.  Large-Scale Modeling of Parametric Surfaces Using Spherical Harmonics , 2006, Third International Symposium on 3D Data Processing, Visualization, and Transmission (3DPVT'06).

[19]  M. Bôcher,et al.  Introduction to the Theory of Fourier's Series , 1906 .

[20]  J. Mazziotta,et al.  Automated image registration , 1993 .

[21]  P. Basser,et al.  In vivo fiber tractography using DT‐MRI data , 2000, Magnetic resonance in medicine.

[22]  P. V. van Zijl,et al.  Three‐dimensional tracking of axonal projections in the brain by magnetic resonance imaging , 1999, Annals of neurology.

[23]  Chi-Wang Shu,et al.  On the Gibbs Phenomenon and Its Resolution , 1997, SIAM Rev..

[24]  Derek K. Jones,et al.  Virtual in Vivo Interactive Dissection of White Matter Fasciculi in the Human Brain , 2002, NeuroImage.

[25]  A. Alexander,et al.  White matter tractography using diffusion tensor deflection , 2003, Human brain mapping.

[26]  J. Foster,et al.  The Gibbs phenomenon for piecewise-linear approximation , 1991 .

[27]  Fillia Makedon,et al.  A surface-based approach for classification of 3D neuroanatomic structures , 2004, Intell. Data Anal..

[28]  Trevor J. Hastie,et al.  3-D curve matching using splines , 1990, J. Field Robotics.

[29]  Moo K. Chung,et al.  Less white matter concentration in autism: 2D voxel-based morphometry , 2004, NeuroImage.

[30]  Mark E. Bastin,et al.  A Probabilistic Model-Based Approach to Consistent White Matter Tract Segmentation , 2007, IEEE Transactions on Medical Imaging.

[31]  P. Basser,et al.  MR diffusion tensor spectroscopy and imaging. , 1994, Biophysical journal.

[32]  Guido Gerig,et al.  Towards a shape model of white matter fiber bundles using diffusion tensor MRI , 2004, 2004 2nd IEEE International Symposium on Biomedical Imaging: Nano to Macro (IEEE Cat No. 04EX821).

[33]  A. Connelly,et al.  Quantification of the shape of fiber tracts , 2006, Magnetic resonance in medicine.

[34]  Nicholas Ayache,et al.  Medical image registration using geometric hashing , 1997 .

[35]  Kim M. Dalton,et al.  Encoding Cortical Surface by Spherical Harmonics , 2008 .

[36]  King-Sun Fu,et al.  Shape Discrimination Using Fourier Descriptors , 1977, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[37]  P. Jezzard,et al.  Correction for geometric distortion in echo planar images from B0 field variations , 1995, Magnetic resonance in medicine.

[38]  Paul A. Yushkevich,et al.  Structure-specific statistical mapping of white matter tracts , 2007, NeuroImage.

[39]  James S. Duncan,et al.  Boundary Finding with Parametrically Deformable Models , 1992, IEEE Trans. Pattern Anal. Mach. Intell..

[40]  Edward R. Dougherty,et al.  Random processes for image and signal processing , 1998, SPIE / IEEE series on imaging science and engineering.

[41]  R. Adler.  An introduction to continuity, extrema, and related topics for general Gaussian processes , 1990 .

[42]  J Sijbers,et al.  Multiscale white matter fiber tract coregistration: A new feature‐based approach to align diffusion tensor data , 2006, Magnetic resonance in medicine.

[43]  Moo K. Chung,et al.  Cortical thickness analysis in autism with heat kernel smoothing , 2005, NeuroImage.