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

• Published in: IEEE transactions on pattern analysis and machine intelligence Vol. 37; no. 12; pp. 2492 - 2504
• Main Authors: Piuze, Emmanuel, Sporring, Jon, Siddiqi, Kaleem
• Format: Journal Article
• Language: English
• Published: United States IEEE 01.12.2015
• Subjects: Algorithms; Animals; Approximation methods; Differential geometry; Diffusion Magnetic Resonance Imaging - methods; Diffusion MRI; Geometry; Heart Wall Myofibers; Image Enhancement - methods; Image Interpretation, Computer-Assisted - methods; Imaging, Three-Dimensional - methods; Maurer-Cartan Form; Moving Frames; Myocardium - cytology; Myocytes, Cardiac - cytology; Numerical models; Pattern Recognition, Automated - methods; Rats; Reproducibility of Results; Sensitivity and Specificity; Signal Processing, Computer-Assisted; Subtraction Technique; Transmission line matrix methods; diffusion MRI; moving frames; heart wall myofibers; Differential geometry; Maurer-Cartan form
• ISSN: 0162-8828; 1939-3539; 2160-9292; 1939-3539
• DOI: 10.1109/TPAMI.2015.2408352

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.