A three-dimensional extension of the slope chain code: analyzing the tortuosity of the flagellar beat of human sperm

In the realm of 3D image processing, accurately representing the geometric nuances of line curves is crucial. Building upon the foundation set by the slope chain code, which adeptly represents intricate two-dimensional curves using an array capturing the exterior angles at each vertex, this study introduces an innovative 3D encoding method tailored for polygonal curves. This 3D encoding employs parallel slope and torsion chains, ensuring invariance to common transformations like translations, rotations, and uniform scaling, while also demonstrating robustness against mirror imaging and variable starting points. A hallmark feature of this method is its ability to compute tortuosity, a descriptor of curve complexity or winding nature. By applying this technique to biomedical engineering, we delved into the flagellar beat patterns of human sperm. These insights underscore the versatility of our 3D encoding across diverse computer vision applications.