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
- Benjamin Kimia: professor at Brown University
- Timothy Duff: professor at University of Misouri
- Ricardo Fabbri: professor at IPRJ/UERJ, Rio de Janeiro State University
- Hongyi Fan: Cognex corporation
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)
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
- A framework for square problems
- Github
- Highly optimized beyond Eigen
- Fast CPU-oriented solvers
Macaulay2 packages coming soon!
- MonodromySolver package
GPU-based solvers
coming soon!Datasets
This section lists datasets to evaluate HC. coming soon!
Publications
Our publications
- Trifocal Relative Pose from Lines at Points and its Efficient Solution, Arxiv, march 23 2019 4:29 UTC, R. Fabbri, T. Duff, H. Fan, M. Regan, D. de Pinho, E. Tsigaridas, C. Wrampler, J. Hauenstein, P. J. Giblin, B. Kimia, A. Leykin & T. Pajdla (pdf | code | datasets) new!
Books
- Numerically solving polynomial systems with BertiniNumerically solving polynomial systems with {B}ertini, Bates, Hauenstein, Sommese, and Wampler (code)