Manifold-valued image processing with SPD matrices

[1]  S. Helgason.  Differential Geometry, Lie Groups, and Symmetric Spaces , 1978 .

[2]  Xavier Pennec,et al.  Statistical Computing on Manifolds: From Riemannian Geometry to Computational Anatomy , 2009, ETVC.

[3]  Jan Sijbers,et al.  Maximum-likelihood estimation of Rician distribution parameters , 1998, IEEE Transactions on Medical Imaging.

[4]  I. Dryden,et al.  Non-Euclidean statistics for covariance matrices, with applications to diffusion tensor imaging , 2009, 0910.1656.

[5]  Xavier Pennec,et al.  A Riemannian Framework for Tensor Computing , 2005, International Journal of Computer Vision.

[6]  Nicholas Ayache,et al.  A Riemannian Framework for the Processing of Tensor-Valued Images , 2005, DSSCV.

[7]  Ross T. Whitaker,et al.  Rician Noise Removal in Diffusion Tensor MRI , 2006, MICCAI.

[8]  R. Woods,et al.  Cortical change in Alzheimer's disease detected with a disease-specific population-based brain atlas. , 2001, Cerebral cortex.

[9]  K. Nomizu.  Invariant Affine Connections on Homogeneous Spaces , 1954 .

[10]  José Manuel Corcuera,et al.  INTRINSIC ANALYSIS OF STATISTICAL ESTIMATION , 1995 .

[11]  P. Thomas Fletcher,et al.  Riemannian geometry for the statistical analysis of diffusion tensor data , 2007, Signal Process..

[12]  R. Bhatia.  On the exponential metric increasing property , 2003 .

[13]  Philipp Grohs,et al.  Total variation regularization on Riemannian manifolds by iteratively reweighted minimization , 2016 .

[14]  Meng Law,et al.  Clinical applications of diffusion tensor imaging. , 2014, World neurosurgery.

[15]  H. Le.  Locating Fréchet means with application to shape spaces , 2001, Advances in Applied Probability.

[16]  Maher Moakher,et al.  A rigorous framework for diffusion tensor calculus , 2005, Magnetic resonance in medicine.

[17]  Thomas Brox,et al.  Nonlinear structure tensors , 2006, Image Vis. Comput..

[18]  Xavier Pennec,et al.  Statistical Computing on Manifolds for Computational Anatomy , 2006 .

[19]  Mi-Suen Lee,et al.  A Computational Framework for Segmentation and Grouping , 2000 .

[20]  C. R. Rao,et al.  Entropy differential metric, distance and divergence measures in probability spaces: A unified approach , 1982 .

[21]  Nicholas Ayache,et al.  Fast and Simple Calculus on Tensors in the Log-Euclidean Framework , 2005, MICCAI.

[22]  Jitendra Malik,et al.  Scale-Space and Edge Detection Using Anisotropic Diffusion , 1990, IEEE Trans. Pattern Anal. Mach. Intell..

[23]  X. Pennec.  Barycentric subspace analysis on manifolds , 2016, The Annals of Statistics.

[24]  Zhizhou Wang,et al.  DTI segmentation using an information theoretic tensor dissimilarity measure , 2005, IEEE Transactions on Medical Imaging.

[25]  W. Förstner,et al.  A Metric for Covariance Matrices , 2003 .

[26]  P. Fillard,et al.  MR diffusion tensor imaging and fiber tracking in spinal cord compression. , 2005, AJNR. American journal of neuroradiology.

[27]  N. Makris,et al.  High angular resolution diffusion imaging reveals intravoxel white matter fiber heterogeneity , 2002, Magnetic resonance in medicine.

[28]  E. Meijering.  A chronology of interpolation: from ancient astronomy to modern signal and image processing , 2002, Proc. IEEE.

[29]  E. Duchesnay,et al.  A framework to study the cortical folding patterns , 2004, NeuroImage.

[30]  M. Unser,et al.  Interpolation revisited [medical images application] , 2000, IEEE Transactions on Medical Imaging.

[31]  J. M. Oller,et al.  AN EXPLICIT SOLUTION OF INFORMATION GEODESIC EQUATIONS FOR THE MULTIVARIATE NORMAL MODEL , 1991 .

[32]  Y. Ollivier.  A visual introduction to Riemannian curvatures and some discrete generalizations , 2012 .

[33]  Joachim Weickert,et al.  Anisotropic diffusion in image processing , 1996 .

[34]  D. Tschumperlé.  PDE's based regularization of multivalued images and applications , 2002 .

[35]  Christian Jutten,et al.  Multiclass Brain–Computer Interface Classification by Riemannian Geometry , 2012, IEEE Transactions on Biomedical Engineering.

[36]  Rachid Deriche,et al.  Regularizing Flows for Constrained Matrix-Valued Images , 2004, Journal of Mathematical Imaging and Vision.

[37]  H. Knutsson.  Representing Local Structure Using Tensors , 1989 .

[38]  P. George,et al.  Delaunay mesh generation governed by metric specifications. Part II. applications , 1997 .

[39]  Dario Bini,et al.  Computing the Karcher mean of symmetric positive definite matrices , 2013 .

[40]  Jonathan H. Manton,et al.  A globally convergent numerical algorithm for computing the centre of mass on compact Lie groups , 2004, ICARCV 2004 8th Control, Automation, Robotics and Vision Conference, 2004..

[41]  Nicholas Ayache,et al.  Uniform Distribution, Distance and Expectation Problems for Geometric Features Processing , 1998, Journal of Mathematical Imaging and Vision.

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

[43]  Pierre Kornprobst,et al.  Mathematical problems in image processing - partial differential equations and the calculus of variations , 2010, Applied mathematical sciences.

[44]  U. Helmke,et al.  Optimization and Dynamical Systems , 1994, Proceedings of the IEEE.

[45]  D. Kendall,et al.  The Riemannian Structure of Euclidean Shape Spaces: A Novel Environment for Statistics , 1993 .

[46]  Carl-Fredrik Westin,et al.  Processing and visualization for diffusion tensor MRI , 2002, Medical Image Anal..

[47]  Xavier Pennec,et al.  Power Euclidean metrics for covariance matrices with application to diffusion tensor imaging , 2010, 1009.3045.

[48]  Rachid Deriche,et al.  Statistics on the Manifold of Multivariate Normal Distributions: Theory and Application to Diffusion Tensor MRI Processing , 2006, Journal of Mathematical Imaging and Vision.

[49]  Guillermo Sapiro,et al.  Diffusion of General Data on Non-Flat Manifolds via Harmonic Maps Theory: The Direction Diffusion Case , 2000, International Journal of Computer Vision.

[50]  João M. F. Xavier,et al.  Newton Algorithms for Riemannian Distance Related Problems on Connected Locally Symmetric Manifolds , 2013, IEEE Journal of Selected Topics in Signal Processing.

[51]  W. Kendall.  Probability, Convexity, and Harmonic Maps with Small Image I: Uniqueness and Fine Existence , 1990 .

[52]  Paul M. Thompson,et al.  Measuring brain variability by extrapolating sparse tensor fields measured on sulcal lines , 2007, NeuroImage.

[53]  Anuj Srivastava,et al.  Statistical Shape Analysis , 2014, Computer Vision, A Reference Guide.

[54]  Andreas Weinmann,et al.  Total Variation Regularization for Manifold-Valued Data , 2013, SIAM J. Imaging Sci..

[55]  Miroslav Lovric,et al.  Multivariate Normal Distributions Parametrized as a Riemannian Symmetric Space , 2000 .

[56]  N. Ayache,et al.  Log‐Euclidean metrics for fast and simple calculus on diffusion tensors , 2006, Magnetic resonance in medicine.

[57]  Stephen D. Howard,et al.  SENSOR MANAGEMENT FOR RADAR: A TUTORIAL , 2006 .

[58]  Nicholas Ayache,et al.  Geometric Means in a Novel Vector Space Structure on Symmetric Positive-Definite Matrices , 2007, SIAM J. Matrix Anal. Appl..

[59]  Mila Nikolova,et al.  A Nonlocal Denoising Algorithm for Manifold-Valued Images Using Second Order Statistics , 2016, SIAM J. Imaging Sci..

[60]  N. Ayache,et al.  Clinical DT-MRI Estimation, Smoothing, and Fiber Tracking With Log-Euclidean Metrics , 2007 .

[61]  Rachid Deriche,et al.  Constrained Flows of Matrix-Valued Functions: Application to Diffusion Tensor Regularization , 2002, ECCV.

[62]  Zhizhou Wang,et al.  A constrained variational principle for direct estimation and smoothing of the diffusion tensor field from complex DWI , 2004, IEEE Transactions on Medical Imaging.

[63]  P. Thompson,et al.  Evaluating Brain Anatomical Correlations via Canonical Correlation Analysis of Sulcal Lines , 2007 .

[64]  Paul M. Thompson,et al.  Extrapolation of Sparse Tensor Fields: Application to the Modeling of Brain Variability , 2005, IPMI.

[65]  Xavier Pennec.  L'incertitude dans les problèmes de reconnaissance et de recalage - Applications en imagerie médicale et biologie moléculaire , 1996 .

[66]  L. Skovgaard.  A Riemannian geometry of the multivariate normal model , 1984 .

[67]  T. Brox,et al.  Diffusion and regularization of vector- and matrix-valued images , 2002 .

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

[69]  J. Sijbers,et al.  Maximum likelihood estimation of signal amplitude and noise variance from MR data , 2004, Magnetic resonance in medicine.

[70]  Guido Gerig,et al.  Nonlinear anisotropic filtering of MRI data , 1992, IEEE Trans. Medical Imaging.

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

[72]  Maher Moakher,et al.  A Differential Geometric Approach to the Geometric Mean of Symmetric Positive-Definite Matrices , 2005, SIAM J. Matrix Anal. Appl..

[73]  Nicholas Ayache,et al.  Clinical DT-MRI estimation, smoothing and fiber tracking with Log-Euclidean metrics , 2006, 3rd IEEE International Symposium on Biomedical Imaging: Nano to Macro, 2006..

[74]  Gabriele Steidl,et al.  A Second Order Nonsmooth Variational Model for Restoring Manifold-Valued Images , 2015, SIAM J. Sci. Comput..

[75]  N. L. Johnson,et al.  Linear Statistical Inference and Its Applications , 1966 .

[76]  D. Le Bihan,et al.  Diffusion tensor imaging: Concepts and applications , 2001, Journal of magnetic resonance imaging : JMRI.

[77]  Xavier Pennec,et al.  Intrinsic Statistics on Riemannian Manifolds: Basic Tools for Geometric Measurements , 2006, Journal of Mathematical Imaging and Vision.