Maurer-Cartan Forms for Fields on Surfaces: Application to Heart Fiber Geometry

We study the space of first order models of smooth frame fields using the method of moving frames. By exploiting the Maurer-Cartan matrix of connection forms we develop geometrical embeddings for frame fields which lie on spherical, ellipsoidal and generalized helicoid surfaces. We design methods for optimizing connection forms in local neighborhoods and apply these to a statistical analysis of heart fiber geometry, using diffusion magnetic resonance imaging. This application of moving frames corroborates and extends recent characterizations of muscle fiber orientation in the heart wall, but also provides for a rich geometrical interpretation. In particular, we can now obtain direct local measurements of the variation of the helix and transverse angles, of fiber fanning and twisting, and of the curvatures of the heart wall in which these fibers lie.

[1]  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.

[2]  Sinisa Pajevic,et al.  Color schemes to represent the orientation of anisotropic tissues from diffusion tensor data: Application to white matter fiber tract mapping in the human brain , 1999, Magnetic resonance in medicine.

[3]  Charles S. Peskin,et al.  Mathematical aspects of heart physiology , 1975 .

[4]  Kaleem Siddiqi,et al.  Moving Frames for Heart Fiber Geometry , 2013, IPMI.

[5]  Jan J. Koenderink,et al.  Spatial Derivatives and the Propagation of Noise in Gaussian Scale Space , 1993, J. Vis. Commun. Image Represent..

[6]  Alan P Benson,et al.  Regional localisation of left ventricular sheet structure: integration with current models of cardiac fibre, sheet and band structure. , 2007, European journal of cardio-thoracic surgery : official journal of the European Association for Cardio-thoracic Surgery.

[7]  J. Schnabel,et al.  Nonlinear smoothing for reduction of systematic and random errors in diffusion tensor imaging , 2000, Journal of magnetic resonance imaging : JMRI.

[8]  Dd. Streeter,et al.  Gross morphology and fiber geometry of the heart , 1979 .

[9]  Hervé Delingette,et al.  Human Atlas of the Cardiac Fiber Architecture: Study on a Healthy Population , 2012, IEEE Transactions on Medical Imaging.

[10]  Peter Savadjiev,et al.  Local white matter geometry from diffusion tensor gradients , 2010, NeuroImage.

[11]  Peter Savadjiev,et al.  On the Differential Geometry of 3D Flow Patterns: Generalized Helicoids and Diffusion MRI Analysis , 2007, 2007 IEEE 11th International Conference on Computer Vision.

[12]  A. Anderson,et al.  Reduction of noise in diffusion tensor images using anisotropic smoothing , 2005, Magnetic resonance in medicine.

[13]  M. Powell.  The BOBYQA algorithm for bound constrained optimization without derivatives , 2009 .

[14]  Sheng-Kwei Song,et al.  Regional ventricular wall thickening reflects changes in cardiac fiber and sheet structure during contraction: quantification with diffusion tensor MRI. , 2005, American journal of physiology. Heart and circulatory physiology.

[15]  Peter J. Olver,et al.  Joint Invariant Signatures , 2001, Found. Comput. Math..

[16]  598 HERMANN WEYL [ September CARTAN ON GROUPS AND DIFFERENTIAL GEOMETRY La Théorie des Groupes Finis et Continus et la Géométrie Différentielle traitées par la Méthode du Repère Mobile , 2022 .

[17]  Kaleem Siddiqi,et al.  Atlases of Cardiac Fiber Differential Geometry , 2013, FIMH.

[18]  Pierre-Louis Bazin,et al.  Structure from Motion: A New Look from the Point of View of Invariant Theory , 2004, SIAM J. Appl. Math..

[19]  J. Magnus,et al.  Matrix Differential Calculus with Applications in Statistics and Econometrics , 1991 .

[20]  Bin Chen,et al.  Noise removal in magnetic resonance diffusion tensor imaging , 2005, Magnetic resonance in medicine.

[21]  Carl-Fredrik Westin,et al.  Normalized and differential convolution , 1993, Proceedings of IEEE Conference on Computer Vision and Pattern Recognition.

[22]  Tamar Flash,et al.  Affine differential geometry analysis of human arm movements , 2007, Biological Cybernetics.

[23]  J. E. Glynn,et al.  Numerical Recipes: The Art of Scientific Computing , 1989 .

[24]  Edward W Hsu,et al.  Helical myofiber orientation after myocardial infarction and left ventricular surgical restoration in sheep. , 2005, The Journal of thoracic and cardiovascular surgery.

[25]  Diana Adler,et al.  Differential Forms With Applications To The Physical Sciences , 2016 .

[26]  D. L. Bassett,et al.  An engineering analysis of myocardial fiber orientation in pig's left ventricle in systole , 1966 .

[27]  Ohad Ben-Shahar,et al.  The Perceptual Organization of Texture Flow: A Contextual Inference Approach , 2003, IEEE Trans. Pattern Anal. Mach. Intell..

[28]  Olivier D. Faugeras,et al.  Cartan's Moving Frame Method and Its Application to the Geometry and Evolution of Curves in the Euclidean, Affine and Projective Planes , 1993, Applications of Invariance in Computer Vision.

[29]  John Forder,et al.  Histological validation of myocardial microstructure obtained from diffusion tensor magnetic resonance imaging. , 1998, American journal of physiology. Heart and circulatory physiology.

[30]  P. Hunter,et al.  Laminar structure of the heart: ventricular myocyte arrangement and connective tissue architecture in the dog. , 1995, The American journal of physiology.

[31]  P. Olver,et al.  Maurer–Cartan forms and the structure of Lie pseudo-groups , 2005 .

[32]  Peter J. Olver,et al.  Moving frames , 2003, J. Symb. Comput..

[33]  Tristan Needham,et al.  Visual Complex Analysis , 1997 .

[34]  T. Arts,et al.  Characterization of the normal cardiac myofiber field in goat measured with MR-diffusion tensor imaging. , 2002, American journal of physiology. Heart and circulatory physiology.

[35]  S Sideman,et al.  Geodesics as a mechanically optimal fiber geometry for the left ventricle. , 1993, Basic research in cardiology.

[36]  J. Pouysségur,et al.  Correction for Savadjiev et al., Heart wall myofibers are arranged in minimal surfaces to optimize organ function , 2012, Proceedings of the National Academy of Sciences.

[37]  Will Tribbey,et al.  Numerical Recipes: The Art of Scientific Computing (3rd Edition) is written by William H. Press, Saul A. Teukolsky, William T. Vetterling, and Brian P. Flannery, and published by Cambridge University Press, © 2007, hardback, ISBN 978-0-521-88068-8, 1235 pp. , 1987, SOEN.

[38]  Steven Haker,et al.  Differential and Numerically Invariant Signature Curves Applied to Object Recognition , 1998, International Journal of Computer Vision.

[39]  Kaleem Siddiqi,et al.  Generalized Helicoids for Modeling Hair Geometry , 2011, Comput. Graph. Forum.

[40]  Pierre-Louis Bazin,et al.  Structure from Motion: Theoretical Foundations of a Novel Approach Using Custom Built Invariants , 2002, ArXiv.

[41]  Jean-Philippe Thiran,et al.  Sequential anisotropic multichannel Wiener filtering with Rician bias correction applied to 3D regularization of DWI data , 2009, Medical Image Anal..

[42]  C. Henriquez,et al.  Magnetic resonance myocardial fiber-orientation mapping with direct histological correlation. , 1998, American journal of physiology. Heart and circulatory physiology.