Topological persistence on a Jordan curve

Topological persistence measures the resilience of extrema of a function to perturbations, and has received increasing attention in computer graphics, visualization and computer vision. While the notion of topological persistence for piece-wise linear functions defined on a simplicial complex has been well studied, the time complexity of all the known algorithms are super-linear (e.g. O(n log n)) in the size n of the complex. We give an O(n) algorithm to compute topological persistence for a function defined on a Jordan curve. To the best of our knowledge, our algorithm is the first to attain linear asymptotic complexity, and is asymptotically optimal. We demonstrate the usefulness of persistence in shape abstraction and compression.