Representing Diffusion MRI in 5-D Simplifies Regularization and Segmentation of White Matter Tracts

We present a new five-dimensional (5-D) space representation of diffusion magnetic resonance imaging (dMRI) of high angular resolution. This 5-D space is basically a non-Euclidean space of position and orientation in which crossing fiber tracts can be clearly disentangled, that cannot be separated in three-dimensional position space. This new representation provides many possibilities for processing and analysis since classical methods for scalar images can be extended to higher dimensions even if the spaces are not Euclidean. In this paper, we show examples of how regularization and segmentation of dMRI is simplified with this new representation. The regularization is used with the purpose of denoising and but also to facilitate the segmentation task by using several scales, each scale representing a different level of resolution. We implement in five dimensions the Chan-Vese method combined with active contours without edges for the segmentation and the total variation functional for the regularization. The purpose of this paper is to explore the possibility of segmenting white matter structures directly as entirely separated bundles in this 5-D space. We will present results from a synthetic model and results on real data of a human brain acquired with diffusion spectrum magnetic resonance imaging (MRI), one of the dMRI of high angular resolution available. These results will lead us to the conclusion that this new high-dimensional representation indeed simplifies the problem of segmentation and regularization.

[1]  Alan Connelly,et al.  Direct estimation of the fiber orientation density function from diffusion-weighted MRI data using spherical deconvolution , 2004, NeuroImage.

[2]  M Cercignani,et al.  Diffusion tensor magnetic resonance imaging in multiple sclerosis , 2001, Neurology.

[3]  Carl-Fredrik Westin,et al.  DTI and MTR abnormalities in schizophrenia: Analysis of white matter integrity , 2005, NeuroImage.

[4]  Daniel C. Alexander,et al.  Persistent Angular Structure: New Insights from Diffusion MRI Data. Dummy Version , 2003, IPMI.

[5]  P. Basser,et al.  Estimation of the effective self-diffusion tensor from the NMR spin echo. , 1994, Journal of magnetic resonance. Series B.

[6]  Pierre Vandergheynst,et al.  Multiresolution segmentation of natural images: from linear to nonlinear scale-space representations , 2004, IEEE Transactions on Image Processing.

[7]  Guillermo Sapiro,et al.  Geodesic Active Contours , 1995, International Journal of Computer Vision.

[8]  X. Bresson.  IMAGE SEGMENTATION WITH VARIATIONAL ACTIVE CONTOURS , 2005 .

[9]  J. Sethian,et al.  Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations , 1988 .

[10]  P. Hagmann.  From diffusion MRI to brain connectomics , 2005 .

[11]  Tony F. Chan,et al.  Active contours without edges , 2001, IEEE Trans. Image Process..

[12]  P. Basser,et al.  Water Diffusion Changes in Wallerian Degeneration and Their Dependence on White Matter Architecture , 2000 .

[13]  P. Olver,et al.  Conformal curvature flows: From phase transitions to active vision , 1996, ICCV 1995.

[14]  V. Caselles,et al.  Minimizing total variation flow , 2000, Differential and Integral Equations.

[15]  Yun-Gang Chen,et al.  Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations , 1989 .

[16]  R. Kimmel,et al.  Minimal surfaces: a geometric three dimensional segmentation approach , 1997 .

[17]  D. Geman.  Random fields and inverse problems in imaging , 1990 .

[18]  Jean-Philippe Thiran,et al.  Hand preference and sex shape the architecture of language networks , 2006, Human brain mapping.

[19]  Erwin Kreyszig,et al.  Differential geometry , 1991 .

[20]  D. Tuch.  Q‐ball imaging , 2004, Magnetic resonance in medicine.

[21]  D. Mumford,et al.  Optimal approximations by piecewise smooth functions and associated variational problems , 1989 .

[22]  R. Stephenson.  A and V , 1962, The British journal of ophthalmology.

[23]  D. Cory,et al.  Measurement of translational displacement probabilities by NMR: An indicator of compartmentation , 1990, Magnetic resonance in medicine.

[24]  R. Kikinis,et al.  Quantitative magnetic resonance imaging of brain development in premature and mature newborns , 1998, Annals of neurology.

[25]  J. Thiran,et al.  Diffusion Spectrum Imaging tractography in complex cerebral white matter: an investigation of the centrum semiovale , 2004 .

[26]  Kalvis M. Jansons,et al.  Persistent angular structure: new insights from diffusion magnetic resonance imaging data , 2003 .

[27]  L. Rudin,et al.  Nonlinear total variation based noise removal algorithms , 1992 .

[28]  Joachim Weickert,et al.  A Review of Nonlinear Diffusion Filtering , 1997, Scale-Space.

[29]  Jean-Philippe Thiran,et al.  Fibertract segmentation in position orientation space from high angular resolution diffusion MRI , 2006, NeuroImage.

[30]  V. Wedeen,et al.  Mapping fiber orientation spectra in cerebral white matter with Fourier-transform diffusion MRI , 2000 .

[31]  James A. Sethian,et al.  Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid , 2012 .

[32]  E. M. Wright,et al.  Adaptive Control Processes: A Guided Tour , 1961, The Mathematical Gazette.

[33]  J. Sethian,et al.  A Fast Level Set Method for Propagating Interfaces , 1995 .

[34]  P. Hagmann,et al.  Mapping complex tissue architecture with diffusion spectrum magnetic resonance imaging , 2005, Magnetic resonance in medicine.

[35]  H. Pfeifer.  Principles of Nuclear Magnetic Resonance Microscopy , 1992 .

[36]  Tony Lindeberg,et al.  Scale-Space Theory in Computer Vision , 1993, Lecture Notes in Computer Science.

[37]  H. Soner,et al.  Level set approach to mean curvature flow in arbitrary codimension , 1996 .

[38]  V. Wedeen,et al.  Diffusion MRI of Complex Neural Architecture , 2003, Neuron.

[39]  J. Thiran,et al.  Understanding diffusion MR imaging techniques: from scalar diffusion-weighted imaging to diffusion tensor imaging and beyond. , 2006, Radiographics : a review publication of the Radiological Society of North America, Inc.

[40]  L. Cohen,et al.  Multi-resolution algorithms for active contour models , 1996 .

[41]  Xavier Bresson,et al.  White matter fiber tract segmentation in DT-MRI using geometric flows , 2005, Medical Image Anal..

[42]  Jean-Philippe Thiran,et al.  Imaging the brain neuronal network with diffusion MRI: a way to understand its global architecture , 2006 .

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