Sharpening Fibers in Diffusion Weighted MRI via Erosion

In this chapter erosion is generalized to the space of diffusion weighted MRI data. This is done effectively by solving a Hamilton-Jacobi-Bellman (HJB) system (erosion) on the coupled space of three dimensional positions and orientations, embedded as a quotient in the group of three dimensional rigid body motions. The solution to the HJB equations is given by a well-posed morphological convolution. We present two numerical approaches to solve the HJB equations: analytical kernels, and finite differences. Proof of concept is given by showing improved visibility of major fiber bundles in both artificial and human data. Furthermore, the method is shown to significantly improve the output of a probabilistic tractography algorithm used to extract the optic radiation.

[1]  Jiang-Hua Lu,et al.  Progress in Mathematics , 2013 .

[2]  L. Hörmander.  Hypoelliptic second order differential equations , 1967 .

[3]  Anthony J. Sherbondy,et al.  ConTrack: finding the most likely pathways between brain regions using diffusion tractography. , 2008, Journal of vision.

[4]  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.

[5]  Lmj Luc Florack,et al.  Decomposition of High Angular Resolution Diffusion Images into a Sum of Self-Similar Polynomials on the Sphere , 2008 .

[6]  T.C.J. Dela Haije.  Geometrical methods for enhancement and tractography in diffusion MRI , 2012 .

[7]  Joachim Weickert,et al.  A Generic Neighbourhood Filtering Framework for Matrix Fields , 2008, ECCV.

[8]  Luc Florack,et al.  Codomain scale space and regularization for high angular resolution diffusion imaging , 2008, 2008 IEEE Computer Society Conference on Computer Vision and Pattern Recognition Workshops.

[9]  J. E. Tanner,et al.  Spin diffusion measurements : spin echoes in the presence of a time-dependent field gradient , 1965 .

[10]  R. Newcomb.  VISCOSITY SOLUTIONS OF HAMILTON-JACOBI EQUATIONS , 2010 .

[11]  Essa Yacoub,et al.  A Hough transform global probabilistic approach to multiple-subject diffusion MRI tractography , 2011, Medical Image Anal..

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

[13]  Juan J. Manfredi,et al.  A VERSION OF THE HOPF-LAX FORMULA IN THE HEISENBERG GROUP , 2002 .

[14]  Joachim Weickert,et al.  An Explanation for the Logarithmic Connection between Linear and Morphological Systems , 2003, Scale-Space.

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

[16]  G. Chirikjian,et al.  Engineering Applications of Noncommutative Harmonic Analysis: With Emphasis on Rotation and Motion Groups , 2000 .

[17]  Robert Tibshirani,et al.  An Introduction to the Bootstrap , 1994 .

[18]  Mara Cercignani,et al.  Twenty‐five pitfalls in the analysis of diffusion MRI data , 2010, NMR in biomedicine.

[19]  W. Klingenberg.  A course in differential geometry , 1978 .

[20]  Jan Sijbers,et al.  Probabilistic fiber tracking using the residual bootstrap with constrained spherical deconvolution , 2011, Human brain mapping.

[21]  J. Tsuruda,et al.  Diffusion-weighted MR imaging of anisotropic water diffusion in cat central nervous system. , 1990, Radiology.

[22]  T. Mareci,et al.  Generalized diffusion tensor imaging and analytical relationships between diffusion tensor imaging and high angular resolution diffusion imaging , 2003, Magnetic resonance in medicine.

[23]  Joachim Weickert,et al.  A general structure tensor concept and coherence-enhancing diffusion filtering for matrix fields , 2009 .

[24]  Michael Breuß,et al.  PDE-Driven Adaptive Morphology for Matrix Fields , 2009, SSVM.

[25]  R. Duits,et al.  Morphological and Linear Scale Spaces for Fiber Enhancement in DW-MRI , 2013, J. Math. Imaging Vis..

[26]  R. Deriche,et al.  Regularized, fast, and robust analytical Q‐ball imaging , 2007, Magnetic resonance in medicine.

[27]  K. Fujii,et al.  Visualization for the analysis of fluid motion , 2005, J. Vis..

[28]  Christopher P Benton,et al.  The direction of measured face aftereffects. , 2008, Journal of vision.

[29]  Martin Welk.  Families of Generalised Morphological Scale Spaces , 2003, Scale-Space.

[30]  S. Lowe.  Analysis On Lie Groups With Polynomial Growth , 2016 .

[31]  A. Bellaïche.  The tangent space in sub-riemannian geometry , 1994 .

[32]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[33]  Bernhard Burgeth,et al.  09302 Abstracts Collection - New Developments in the Visualization and Processing of Tensor Fields , 2009, New Developments in the Visualization and Processing of Tensor Fields.

[34]  Avik W. Ghosh,et al.  Diffusion, convection and erosion on R3 x S2 and their application to the enhancement of crossing fibers , 2011 .

[35]  B. M. ter Haar Romeny,et al.  Evaluating Contextual Processing in Diffusion MRI: Application to Optic Radiation Reconstruction for Epilepsy Surgery , 2014, PloS one.

[36]  Arnold W. M. Smeulders,et al.  The Morphological Structure of Images: The Differential Equations of Morphological Scale-Space , 1994, IEEE Trans. Pattern Anal. Mach. Intell..

[37]  S. Schweitzer.  A Course In Differential Geometry , 2016 .

[38]  Knut-Andreas Lie,et al.  Scale Space and Variational Methods in Computer Vision, Second International Conference, SSVM 2009, Voss, Norway, June 1-5, 2009. Proceedings , 2009, SSVM.

[39]  Joachim Weickert,et al.  Visualization and Processing of Tensor Fields: Advances and Perspectives , 2009 .

[40]  Karsten Tabelow,et al.  Diffusion tensor imaging: Structural adaptive smoothing , 2008, NeuroImage.

[41]  E. Franken.  Enhancement of crossing elongated structures in images , 2008 .

[42]  S. Arridge,et al.  Detection and modeling of non‐Gaussian apparent diffusion coefficient profiles in human brain data , 2002, Magnetic resonance in medicine.

[43]  Remco Duits,et al.  Erratum to: Morphological and Linear Scale Spaces for Fiber Enhancement in DW-MRI , 2012, Journal of Mathematical Imaging and Vision.

[44]  Baba C. Vemuri,et al.  Resolution of complex tissue microarchitecture using the diffusion orientation transform (DOT) , 2006, NeuroImage.

[45]  Marco Reisert,et al.  Left-Invariant Diffusion on the Motion Group in terms of the Irreducible Representations of SO(3) , 2012, ArXiv.

[46]  Remco Duits,et al.  Numerical Schemes for Linear and Non-linear Enhancement of DW-MRI , 2011, SSVM.

[47]  B. M. ter Haar Romeny,et al.  Extrapolating fiber crossings from DTI data : can we gain the same information as HARDI? , 2010 .

[48]  Joachim Weickert,et al.  An Explanation for the Logarithmic Connection between Linear and Morphological System Theory , 2005, International Journal of Computer Vision.

[49]  Duan Xu,et al.  Q‐ball reconstruction of multimodal fiber orientations using the spherical harmonic basis , 2006, Magnetic resonance in medicine.

[50]  Alfred M. Bruckstein,et al.  Scale Space and Variational Methods in Computer Vision , 2011, Lecture Notes in Computer Science.

[51]  P. Grenier,et al.  MR imaging of intravoxel incoherent motions: application to diffusion and perfusion in neurologic disorders. , 1986, Radiology.

[52]  Bernhard Burgeth,et al.  Edge-Enhancing Diffusion Filtering for Matrix Fields , 2012 .

[53]  Paul Jackway.  Morphological scale-space , 1992 .

[54]  Susan A. Murphy,et al.  Monographs on statistics and applied probability , 1990 .

[55]  Remco Duits,et al.  Accelerated Diffusion Operators for Enhancing DW-MRI , 2010, VCBM.

[56]  Remco Duits.  Perceptual organization in image analysis : a mathematical approach based on scale, orientation and curvature , 2005 .

[57]  Remco Duits,et al.  Fiber Enhancement in Diffusion-Weighted MRI , 2011, SSVM.

[58]  Jos B. T. M. Roerdink,et al.  Group morphology , 2000, Pattern Recognit..

[59]  Elias M. Stein,et al.  Fundamental solutions and harmonic analysis on nilpotent groups , 1990 .

[60]  Remco Duits,et al.  Left-Invariant Diffusions on the Space of Positions and Orientations and their Application to Crossing-Preserving Smoothing of HARDI images , 2011, International Journal of Computer Vision.

[61]  Wei Li,et al.  Fast magnetic resonance diffusion‐weighted imaging of acute human stroke , 1992, Neurology.