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Algebraic Geometry in Computer Vision and Robotics

Algebraic Geometry in Computer Vision and Robotics . Xun (Sam) Zhou Multiple Autonomous Robotic Systems (MARS) Lab Dept. of Computer Science and Engineering University of Minnesota. Stewart Mechanism. Introduction. Geometric problems widely appear in computer vision/robotics

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Algebraic Geometry in Computer Vision and Robotics

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  1. Algebraic Geometry inComputer Vision and Robotics Xun (Sam) Zhou Multiple Autonomous Robotic Systems (MARS) Lab Dept. of Computer Science and Engineering University of Minnesota

  2. Stewart Mechanism Introduction • Geometric problems widely appear in computer vision/robotics • Visual Odometry • Map-based localization (image/laser scan) • Manipulators • We need to solve systems of polynomial equations

  3. C p {F} {F} {L} {L} Outline • Visual odometry with directional correspondence • Motion-induced robot-to-robot extrinsic calibration • Optimal motion strategies for leader-follower formations g

  4. Min. No. points Minimize prob. of picking an outlier Motivation • Main challenge: data association • Outlier rejection (RANSAC)  least-squares refinement • Objective: efficient minimal solvers

  5. Related Work • Five points (10 solutions) • [Nister ’04] • Compute null space of a 5x9 matrix • Gauss elimination of a dense 10x20 matrix • Solve a 10th order polynomial  essential matrix • Recover the camera pose from the essential matrix • Three points and one direction (4 solutions) • [Fraundorfer et al. ’10] • Similar to the 5-point algorithm w. fewer unknowns • Solve a 4th order polynomial  essential matrix • [Kalantari et al. ’11] • Tangent half-angle formulae • Singularity at 180 degree rotation • Solve a 6th order polynomial (2 spurious solutions) • Our algorithm • Fast: coefficient of the 4th order polynomial in closed form • Solve for the camera pose directly

  6. 2-DOF in rotation 1-DOF in rotation 2-DOF in translation (scale is unobservable) Problem Formulation • Directional constraint • 3 point matches {1} {2} Objective: determine

  7. Determine 2-DOF in Rotation • Parameterization of R: • Compute

  8. Linear in System of polynomial equations in Determine the Remaining 3-DOF • Problem reformulation

  9. 4 solutions for Determine the Remaining 3-DOF • Problem solution Eliminate Eliminate using Sylvester resultant Back-substitute to solve for Step 1 Step 2 4th order Step 3

  10. Simulation Results • Under image and directional noise • Directional noise (deg): rotate around random axis • Report median errors • Observations • Forward motion out performs sideway • Rotation estimate better than translation [Courtesy of O. Naroditsky, UPenn]

  11. Experimental Results • Setup • Single camera (640x480 pixels, 50 degree FOV) • Record an 825-frame outdoor sequence, total of 430 m trajectory • RANSAC: 200 hypotheses for each image pair • 3p1 has 2 failures, while 5-point has 4 failures Fail to choose inlier set Sample Images [Courtesy of O. Naroditsky, UPenn]

  12. C p Outline • Visual odometry with directional correspondence • Motion-induced robot-to-robot extrinsic calibration • Optimal motion strategies for leader-follower formation g {F} {F} {L} {L}

  13. Multi-robot tracking (MARS) Multi-robot tracking (MARS) Formation Flight (NASA) Talisman L (BAE Systems) Satellite Formation Flight (NASA) Introduction • Motivating applications • Cooperative SLAM • Multi-robot tracking • Formation flight Require global/relative robot pose

  14. Multi-robot tracking (MARS) Formation Flight (NASA) Talisman L (BAE Systems) Talisman L (BAE Systems) Introduction • Motivating applications • Cooperative SLAM • Multi-robot tracking • Formation flight • Determine relative pose using • External references (e.g., GPS, map) • Not always available • Ego motion and robot-to-robot measurements • Distance and/or Bearing • Requires solving systems of nonlinear (polynomial) equations • Contributions • Identified 14 minimal problems using combinations of robot-to-robot measurements (distance and/or bearing) • Provided closed-form or efficient solutions Require global/relative robot pose

  15. Problem Description Goal: Determine relative pose (p, C) for robots moving in 3D • First meet at {1}, {2}, measure subset of {d12, b1, b2 } {2} b2 C p d12 b1 {1}

  16. Problem Description Goal: Determine relative pose (p, C) for robots moving in 3D • First meet at {1}, {2}, measure subset of {d12, b1, b2 } • Then move to {3}, {4}, measure subset of {d34, b3, b4 } {4} {2} 2p4 b4 b2 C p d12 d34 b1 b3 1p3 {1} {3}

  17. Problem Description and Related Work Goal: Determine relative pose (p, C) for robots moving in 3D • First meet at {1}, {2}, measure subset of {d12, b1, b2 } • Then move to {3}, {4}, measure subset of {d34, b3, b4 } • Collect at least 6 scalar measurements for determining the 6-DOF relative pose • Homogeneous (Minimal) • 6 distances • [Wampler ’96], [Lee & Shim ’03] • [Trawny, Zhou, et al. RSS’09] • Homogeneous (Overdetermined) • Distance and/or bearing • [Trawny, Zhou, et al. TRO’10] {4} {2n} {2} 2p4 2p2n b4 b2n b2 d2n-1, 2n C p d12 d34 ... b2n-1 b1 b3 1p2n-1 1p3 {1} {2n-1} {3} Stewart Mechanism

  18. Problem Description and Related Work Goal: Determine relative pose (p, C) for robots moving in 3D • First meet at {1}, {2}, measure subset of {d12, b1, b2 } • Then move to {3}, {4}, measure subset of {d34, b3, b4 } • Collect at least 6 scalar measurements for determining the 6-DOF relative pose • Homogeneous (Minimal) • 6 distances • [Wampler ’96], [Lee & Shim ’03] • [Trawny, Zhou, et al. RSS’09] • Homogeneous (Overdetermined) • Distance and/or bearing • [Trawny, Zhou, et al. TRO’10] • Heterogeneous (Minimal) • (e.g., ) • Our focus {4} {6} {2} 2p4 2p6 b4 d56 C p d34 b1 1p5 1p3 {1} {5} {3}

  19. Combinations of Inter-robot Measurements No. of eqns 1 2 3 4 5 6 scalar  1 equation 3D unit vector  2 equations • All possible combinations up to 6 time steps • 7^6 =117,649 (overdetermined) problems!

  20. Only 14 Minimal Systems No. of eqns 1 2 3 4 5 6 [ICRA ’11] [RSS ’09] [IROS ’10] Sys10 These are formulated as systems of polynomial equations.

  21. C System 10: {4} {8} {2} 2p4 • Relative position known • From the distance • Solve for C from system of equations 2p8 p d12 d34 ... d78 b1 1p7 1p3 {1} {7} {3} 8 solutions solved by multiplication matrix

  22. Methods for Solving Polynomial Equations • Elimination & back-substitution • Multiplication (Action) matrix Original system Triangular system Groebner Basis Resultant Eigendecomp. Symbolic- Numerical method MultiplicationMatrix m solutions

  23. Monomials in the remainder of any polynomial divided by f Multiplication Matrix of a Univariate Polynomial

  24. Extension to Multivariable Case

  25. 8 basis monomials 27 extra monomials Add a linear function: multiply with some monomials Eliminate Solve System 10 by Multiplication Matrix • Represent rotation by Cayley’s parameter • Find the Multiplication matrix via Macaulay Resultant Quadratic in s Arrange polynomials in matrix form: Read off solutions from eigenvectors

  26. C p {F} {F} {L} {L} Outline • Visual odometry with directional correspondence • Motion-induced robot-to-robot extrinsic calibration • Optimal motion strategies for leader-follower formations g

  27. Platooning [tech-faq.com] V formation flight [aerospaceweb.org] Optimal Motion Strateges for Leader-Follower Formations • Vehicles often move in formation X. S. Zhou, K. Zhou, S. I. Roumeliotis, Optimized Motion Strategies for Localization in Leader-Follower Formations, IROS 2011. (To appear)

  28. Optimal Motion Strateges for Leader-Follower Formations • Vehicles often move in formation to improve fuel efficiency • Robot motion affects estimation accuracy • Next-step optimal motion strategies • Finding all critical points that satisfy the KKT optimality conditions distance, or bearing In formation, relative pose unobservable Uncertainty unbounded {F} {L}

  29. Follower Trajectory Average over 50 Monte Carlo trials Average over 50 Monte Carlo trials Simulation Results: Range-only • Leader moves on straight line • Follower desired position • Initial covariance • Measurement noise • MTF: maintaining the formation • CRM: constrained random motion • MME: active control strategy • [Mariottini et al.] • GBS: grid-based search • RAM: our relaxed algebraic method

  30. Visual Odometry Motion-induced Extrinsic Calibration C {F} p Optimal Motion {L} Summary • Algebraic geometric has wide range of applications • Other projects I have also worked on • Multi-robot SLAM • Vision-aided inertial navigation and more …

  31. Algebraic Geometry inComputer Vision and Robotics Xun (Sam) Zhou Multiple Autonomous Robotic Systems (MARS) Lab Dept. of Computer Science and Engineering University of Minnesota

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