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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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

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

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

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

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

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

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

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:

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)

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:

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:

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:

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)

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

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.

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

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.

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

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

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

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

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.

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

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