On the Differential Geometry of 3D Flow Patterns: Generalized Helicoids and Diffusion MRI Analysis
Abstract
Configurations of dense locally parallel 3D curves occur in medical imaging, computer vision and graphics. Examples include white matter fibre tracts, textures, fur and hair. We develop a differential geometric characterization of such structures by considering the local behaviour of the associated 3D frame field, leading to the associated tangential, normal and bi-normal curvature functions. Using results from the theory of generalized minimal surfaces we adopt a generalized helicoid model as an osculating object and develop the connection between its parameters and these curvature functions. These developments allow for the construction of parametrized 3D vector fields (sampled osculating objects) to locally approximate these patterns. We apply these results to the analysis of diffusion MRI data via a type of 3D streamline flow. Experimental results on data from a human brain demonstrate the advantages of incorporating the full differential geometry.
Cite
Text
Savadjiev et al. "On the Differential Geometry of 3D Flow Patterns: Generalized Helicoids and Diffusion MRI Analysis." IEEE/CVF International Conference on Computer Vision, 2007. doi:10.1109/ICCV.2007.4409086Markdown
[Savadjiev et al. "On the Differential Geometry of 3D Flow Patterns: Generalized Helicoids and Diffusion MRI Analysis." IEEE/CVF International Conference on Computer Vision, 2007.](https://mlanthology.org/iccv/2007/savadjiev2007iccv-differential/) doi:10.1109/ICCV.2007.4409086BibTeX
@inproceedings{savadjiev2007iccv-differential,
title = {{On the Differential Geometry of 3D Flow Patterns: Generalized Helicoids and Diffusion MRI Analysis}},
author = {Savadjiev, Peter and Zucker, Steven W. and Siddiqi, Kaleem},
booktitle = {IEEE/CVF International Conference on Computer Vision},
year = {2007},
pages = {1-8},
doi = {10.1109/ICCV.2007.4409086},
url = {https://mlanthology.org/iccv/2007/savadjiev2007iccv-differential/}
}