# Software

This is some software I have written (some as fun exercises, some for student tutorials, and some for research). While this code is in MATLAB, my research group is working on collective control-based toolboxes that will be made freely available in Python.

## Network Curvature

• ABOUT: This code illustrates a recent that shows changes in Ricci curvature is positively correlated with a systems robustness. This mathematical result was further shown to differentiate time-varying network. In particular, the code posted here implements a notion of discrete Ricci curvature, due to Yann Ollivier, and computes the results on toy examples. This said, one may use Ollivier-Ricci curvature for graphs and networks for ones own research of dynamical systems.

## Grayson's Theorem

• ABOUT: This code illustrates the beautiful result in geometry due to Grayson Theorem. Specifically, a smooth simple closed curve which undergoes the curve-shortening flow will remain smoothly embedded without self-intersections eventually becoming convex, and once it does so it will remain convex. As the evolution (flow) progresses, all points of the curve will move inwards and the shape of the curve will converge to a circle and shrink to a single point. In short, every simple closed curve shrinks to a “round point” - Wikipedia.

## Distribution Metrics for Segmentation

• ABOUT: We present a new distribution metric for image segmentation that arises as a result in prediction theory. Forming a natural geodesic, our metric quantifies “distance” for two density functionals as the standard deviation of the difference between logarithms of those distributions. Using level set methods, we demonstrate the proposed algorithm on classical images.

## Anisotropic Image Smoothing

• ABOUT: This code illustrates how we can utilize concepts from geometry to effectively smooth an image while preserving important details (edge information). We compare this to classic signal processing smoothing that equates to running the heat equation. This is meant for tutorial purposes.

## Classical Shape Analysis

• ABOUT: Notions of machine learning (now with deep learning) is the current hot topic. A subset of machine learning is manifold learning. I wrote this code to help illustrate students what a notion of a “shape basis” is as well as for the interested persons starting out in the world of learning.