Homotopy Continuation Resource Center

Welcome to the HC Resource Center - HCRC, a selection of resources on efficient solving of challenging systems of polynomial equations using the numerical technique of homotopy continuation.

About

This selection is hand-curated by us -- an active team of researchers in computational algebraic geometry with experience in hard problems arising from computer vision applications such as augmented reality and autonomous cars. Many other compute-bound applications can benefit from this information.

It is not our intent to produce a comprehensive list of resources, but, rather an opinionated selection that we personally recommend looking at (including our own work). Our emphasis is on achieving real-time speed for online AR applications, rather than extreme precision, certification or robustness. All references are directly or indirectly aimed towards tackling the real-time scenario.

Feel free to contact us if you need additional information or source code. Please note that this is a dynamic website that is frequently updated.

Authors

Our collaboration began in 2018 at ICERM/Brown Univesity when we designed and optimized the first solver for the problem of reconstructing three images and the camera position from three oriented point correspondences, having algebraic degree of nonlinearity 312. The related paper is

Trifocal Relative Pose from Lines at Points, IEEE Transactions on Pattern Analysis and Machine Intelligence, 2022, CVPR 2020 (online march 23 2019 4:29 UTC), Ricardo Fabbri, Timothy Duff, Hongyi Fan, Margaret Regan, David de Pinho (my former MSc. Student), Elias Tsigaridas, Charles Wampler, Jonathan Hauenstein, Peter Giblin, Benjamin Kimia, Anton Leykin and Tomas Pajdla (pdf | code | datasets | bib)

Please cite this paper when referring to HCRC.

new! Line-point Macaulay2 tutorial in minus/tutorial/linepoint! new!
new! Metric Multiview Geometry -- A Catalogue in Low Dimensions by Duff and Rydel new!

Code

Minus C++ Fast Solver Framework

Macaulay2 packages coming soon!

GPU-based solvers

coming soon!

Datasets

This section lists datasets to evaluate HC. coming soon!

Publications

Our publications

Books

HC for critical points of Dynamical Systems

del Campo, Abraham Martin, and Jose Israel Rodriguez. "Critical points via monodromy and local methods." Journal of Symbolic Computation 79 (2017): 559-574.

Work in Progress!