CVPR 2024 Tutorial

Event: CVPR 2024 · Duration: 198 min · ▶ Watch on YouTube

Abstract

This segment provides an in-depth look into solving polynomial systems within computer vision. The initial speaker sets the stage by illustrating how various multiview geometry problems, such as relative pose estimation and multiview triangulation, naturally lead to complex polynomial systems. The subsequent speaker then conducts a comprehensive tour of existing solver techniques, including Newton’s method, resultant, Groebner basis, and elimination templates, evaluating their performance against criteria for an ideal solver. The final speaker introduces homotopy continuation as a robust and general method, detailing its theoretical underpinnings, the practical implementation of path-tracking using predictor-corrector steps, and the concept of monodromy for finding all solutions. This segment delves into advanced aspects of homotopy continuation, beginning with Timothy Duff explaining monodromy for finding all solutions to algebraic problems, its application in exploiting symmetries, and practical considerations for parameter homotopy. He demonstrates a 5-point essential matrix estimation using Macaulay2Web and discusses applications in 3-view autocalibration and absolute pose estimation with points and lines. Ricardo Fabbri then takes over, introducing design principles and code-level optimizations for building fast homotopy continuation solvers, focusing on predictor-corrector methods and addressing challenges like singularities and numerical stability. This segment features two presentations on Homotopy Continuation (HC). The first speaker, Hongyi Fan from Cognex, discusses accelerating HC using GPUs by parallelizing path tracking and polynomial evaluation, demonstrating significant performance improvements over CPU-based methods for various computer vision problems. The second speaker, Ricardo Fabbri from Rio de Janeiro State University, introduces the MiNuS fast continuation solver framework, guiding the audience through building a fast solver from a Macaulay prototype to an optimized C++ implementation, complete with a hands-on demonstration of the workflow and code generation.

Speakers

  • Ben
  • Hongyi Fan — Cognex
  • Tim Duff — U. Washington -> U. Missouri
  • Timothy Duff
  • Ricardo Fabbri — Rio de Janeiro State University
  • Chiang-Heng Chien

Talks (7)

  • 00:00:00 — Ben: Introduction to Polynomial Systems in Computer Vision
    • Introduces polynomial systems, their applications in various fields, and specifically in multiview geometry problems like relative pose estimation, PnP, trifocal relative pose, generalized three-view camera pose estimation, and multiview triangulation, highlighting the challenge of solving large polynomial systems.
  • 00:16:00 — Hongyi Fan: Tour of solving polynomial systems in Computer Vision
    • Provides an overview of existing techniques for solving polynomial systems, discussing the requirements of an ideal solver (stable, accurate, fast, easy to access, easy to initialize, scalable). He evaluates Newton’s method, general solvers (resultant and Groebner basis), and specific solvers (elimination template), highlighting their pros and cons.
  • 00:40:25 — Tim Duff: Introduction to homotopy continuation (with a view towards faster solvers)
    • Introduces homotopy continuation as a method for solving polynomial systems, explaining its core idea of path-tracking solutions from an easier system to a target system. He details the numerical integration process via predictor/corrector steps and explains how to find initial solutions via monodromy.
  • 01:05:50 — Timothy Duff: Monodromy and Parameter Homotopy: Theory, Demo, and Applications
    • This talk segment covers the theoretical foundations of monodromy and parameter homotopy, including practical considerations and software implementations, followed by a live demo and discussions on applications in 3-view autocalibration and absolute pose estimation.
  • 01:37:30 — Ricardo Fabbri: Building fast numerical solvers
    • This talk introduces the design principles and code-level optimizations for building fast homotopy continuation solvers, focusing on predictor-corrector methods and addressing challenges like singularities and numerical stability.
  • 02:11:40 — Hongyi Fan: Homotopy Continuation Meets GPU
    • This talk discusses accelerating Homotopy Continuation (HC) using GPUs by parallelizing path tracking and polynomial evaluation, highlighting significant speedups compared to CPU-based methods.
  • 02:23:40 — Ricardo Fabbri: Building your own fast solver: Homotopy Continuation Tutorial
    • This part of the tutorial introduces MiNuS, a fast continuation solver framework, detailing how to build a solver from a Macaulay prototype to an optimized C++ solver, including hands-on steps and a live demo.

Key Takeaways

  • Many computer vision problems, particularly in multiview geometry, can be formulated as polynomial systems, but solving these systems, especially large ones, presents significant computational challenges.
  • Traditional solvers like Newton’s method are fast but sensitive to initial guesses, while general algebraic solvers (resultant, Groebner basis) can be computationally intensive, unstable, and difficult to scale for complex problems.
  • Homotopy continuation offers a powerful and general approach to solving polynomial systems by continuously deforming a known, simpler system into the target system, allowing for the tracking of all solutions.
  • The practical implementation of homotopy continuation involves numerical path-tracking using predictor-corrector steps, and the concept of monodromy can be leveraged to efficiently find all possible solutions to a given polynomial system.
  • Monodromy is a powerful offline method for finding all solutions to algebraic problems by tracking paths in parameter space, especially useful for exploiting problem symmetries.
  • Homotopy continuation can be applied to complex computer vision problems like 5-point essential matrix estimation, 3-view autocalibration, and absolute pose with points and lines, often requiring specialized solvers for efficiency.
  • The design of fast homotopy continuation solvers involves careful consideration of numerical stability, predictor-corrector strategies, and code-level optimizations to handle challenges like singularities and high-dimensional solution spaces.
  • Understanding the underlying mathematical properties and symmetries of a problem, often revealed through monodromy analysis, can significantly reduce the computational complexity of finding solutions.
  • GPU acceleration significantly improves the efficiency of Homotopy Continuation, making it viable for real-time applications in computer vision.
  • Parallelization strategies for HC involve both outer-loop path tracking and inner-loop polynomial/Jacobian evaluation, leveraging GPU architecture for speed.
  • The MiNuS framework provides a structured approach to building custom, fast HC solvers by starting with a Macaulay prototype and progressively optimizing it into C++.
  • Effective GPU utilization requires careful consideration of thread management (warps), memory access patterns, and specialized linear system solvers to minimize overhead and maximize throughput.

Methods / Models / Datasets Mentioned

  • Bertini
  • Bezout-Cayley-Dixon method
  • Buchberger Algorithm
  • Demazure constraints
  • Eigen
  • Elimination Template
  • Epipolar constraints
  • Euler's method
  • F4 Algorithm
  • Fused Kernel
  • GPU-HC
  • Gauss-Jordan elimination
  • Groebner basis
  • Homotopy Continuation
  • Homotopy Continuation (HC)
  • LU decomposition
  • MAGMA
  • Macaulay2
  • MiNuS
  • Monodromy
  • Newton's method
  • Newton-Raphson
  • ODE Integration
  • P1P2L
  • P2P1L
  • P3L
  • P3P
  • PHCpack
  • Predictor-corrector design
  • Predictor/Corrector method
  • Quadrifocal tensor
  • RANSAC
  • RK4
  • Radial camera
  • Resultant
  • Runge-Kutta
  • Runge-Kutta method
  • cuBLAS

Topics

Absolute Pose · Autocalibration · C++ Solver · Computer Vision · Computer vision · Essential Matrix Estimation · GPU Acceleration · Homotopy Continuation · Homotopy continuation · Macaulay2 · MiNuS Framework · Minimal problems · Monodromy · Multiview geometry · Numerical Solvers · Numerical Stability · Numerical methods · Parallelization · Parameter Homotopy · Path-tracking · Polynomial Systems · Polynomial system solvers · Polynomial systems · Predictor-Corrector Methods · Radial Cameras

Notes

Open for commentary — connections to other work, critiques, follow-up reading.