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Globally Optimal Estimates for Geometric Reconstruction Problems

Globally Optimal Estimates for Geometric Reconstruction Problems. Tom Gilat, Adi Lakritz Advanced Topics in Computer Vision Seminar Faculty of Mathematics and Computer Science Weizmann Institute 3 June 2007. outline. Motivation and Introduction Background

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Globally Optimal Estimates for Geometric Reconstruction Problems

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  1. Globally Optimal Estimates for Geometric ReconstructionProblems Tom Gilat, Adi Lakritz Advanced Topics in Computer Vision Seminar Faculty of Mathematics and Computer Science Weizmann Institute 3 June 2007

  2. outline • Motivation and Introduction • Background • Positive SemiDefinite matrices (PSD) • Linear Matrix Inequalities (LMI) • SemiDefinite Programming (SDP) • Relaxations • Sum Of Squares (SOS) relaxation • Linear Matrix Inequalities (LMI) relaxation • Application in vision • Finding optimal structure • Partial relaxation and Schur’s complement

  3. Motivation Geometric Reconstruction Problems Polynomial optimization problems (POPs)

  4. 2-views – exact solution Multi view - optimization Triangulation problem in L2 norm

  5. y z x Triangulation problem in L2 norm perspective camera i err

  6. Triangulation problem in L2 norm minimize reprojection error in all cameras Polynomial minimization problem Non convex

  7. Calculating homography given 3D points on a plane and corresponding image points, calculate homography More computer vision problems • Reconstruction problem: known cameras, known corresponding points find 3D points that minimize the projection error of given image points • Similar to triangulation for many points and cameras • Many more problems

  8. Optimization problems

  9. Introduction to optimization problems

  10. optimization problems NP - complete

  11. optimization problems optimization solutions exist: interior point methods problems: local optimum or high computational cost convex non convex Linear Programming (LP) SemiDefinite Programming (SDP)

  12. non convex optimization Non convex feasible set Many algorithms Get stuck in local minima init Max Min level curves of f

  13. optimization problems optimization solutions exist: interior point methods problems: local optimum or high computational cost convex non convex relaxation of problem LP SDP global optimization – algorithms that converge to optimal solution

  14. outline • Motivation and Introduction • Background • Positive SemiDefinite matrices (PSD) • Linear Matrix Inequalities (LMI) • SemiDefinite Programming (SDP) • Relaxations • Sum Of Squares (SOS) relaxation • Linear Matrix Inequalities (LMI) relaxation • Application in vision • Finding optimal structure • Partial relaxation and Schur’s complement

  15. positive semidefinite (PSD) matrices Definition: a matrix M in Rn×n is PSD if 1. M is symmetric: M=MT 2. for all M can be decomposed as AAT (Cholesky) Proof:

  16. positive semidefinite (PSD) matrices Definition: a matrix M in Rn×n is PSD if 1. M is symmetric: M=MT 2. for all M can be decomposed as AAT (Cholesky)

  17. principal minors The kth order principal minors of an n×n symmetric matrix M are the determinants of the k×k matrices obtained by deleting n - k rows and the corresponding n - k columns of M first order: elements on diagonal second order:

  18. diagonal minors The kth order principal minors of an n×n symmetric matrix M are the determinants of the k×k matrices obtained by deleting n - k rows and the corresponding n - k columns of M first order: elements on diagonal second order:

  19. diagonal minors The kth order principal minors of an n×n symmetric matrix M are the determinants of the k×k matrices obtained by deleting n - k rows and the corresponding n - k columns of M first order: elements on diagonal second order:

  20. diagonal minors The kth order principal minors of an n×n symmetric matrix M are the determinants of the k×k matrices obtained by deleting n - k rows and the corresponding n - k columns of M first order: elements on diagonal second order: third order: det(M)

  21. Set of PSD matrices in 2D

  22. Set of PSD matrices This set is convex Proof:

  23. LMI – linear matrix inequality

  24. LMI example: find the feasible set of the 2D LMI

  25. reminder

  26. LMI example: find the feasible set of the 2D LMI 1st order principal minors

  27. LMI example: find the feasible set of the 2D LMI 2nd order principal minors

  28. LMI example: find the feasible set of the 2D LMI 3rd order principal minors Intersection of all inequalities

  29. Semidefinite Programming (SDP) = LMI an extension of LP

  30. outline • Motivation and Introduction • Background • Positive SemiDefinite matrices (PSD) • Linear Matrix Inequalities (LMI) • SemiDefinite Programming (SDP) • Relaxations • Sum Of Squares (SOS) relaxation • Linear Matrix Inequalities (LMI) relaxation • Application in vision • Finding optimal structure • Partial relaxation and Schur’s complement

  31. Sum Of Squares relaxation (SOS) Unconstrained polynomial optimization problem (POP) means the feasible set is Rn H. Waki, S. Kim, M. Kojima, and M. Muramatsu. SOS and SDP relaxations for POPs with structured sparsity. SIAM J. Optimization, 2006.

  32. Sum Of Squares relaxation (SOS)

  33. SOS relaxation for unconstrained polynomials

  34. SOS relaxation for unconstrained polynomials

  35. monomial basis example

  36. SOS relaxation to SDP SDP

  37. SOS relaxation to SDP example: SDP

  38. SOS for constrained POPs possible to extend this method for constrained POPs by use of generalized Lagrange dual

  39. SOS relaxation summary SOS relaxation SDP POP SOS problem Global estimate So we know how to solve a POP that is a SOS And we have a bound on a POP that is not an SOS H. Waki, S. Kim, M. Kojima, and M. Muramatsu. SOS and SDP relaxations for POPs with structured sparsity. SIAM J. Optimization, 2006.

  40. relaxations SOS: SOS relaxation SDP POP SOS problem Global estimate LMI: LMI relaxation SDP + converge POP linear & LMI problem Global estimate

  41. outline • Motivation and Introduction • Background • Positive SemiDefinite matrices (PSD) • Linear Matrix Inequalities (LMI) • SemiDefinite Programming (SDP) • Relaxations • Sum Of Squares (SOS) relaxation • Linear Matrix Inequalities (LMI) relaxation • Application in vision • Finding optimal structure • Partial relaxation and Schur’s complement

  42. LMI relaxations Constraints are handled Convergence to optimum is guaranteed Applies to all polynomials, not SOS as well

  43. A maximization problem Note that: • Feasible set is non-convex. • Constraints are quadratic Feasible set

  44. LMI – linear matrix inequality, a reminder

  45. Motivation An SDP: Goal SDP with solution close to global optimum of the original problem Polynomial Optimization Problem What is it good for? SDP problems can be solved much more efficiently then general optimization problems.

  46. LMI Relaxations is iterative process LMI: POP Step 1: introduce new variables Apply higher order relaxations Linear + LMI + rank constraints Step 2: relax constraints SDP

  47. LMI relaxation – step 1 (the R2 case) Replace monomials by “lifting variables” Rule: Example:

  48. Introducing lifting variables Lifting

  49. New problem is linear, in particular convex Not equivalent to the original problem. Lifting variables are not independent in the original problem:

  50. Goal, more specifically Linear problem (obtained by lifing) + “relations constraints” on lifting variables Relaxation SDP

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