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From previous lectures …

From previous lectures …. Orthogonal group. What is the set of transformations that preserve the inner product? Remember inner product under a transformation? More on this later …. Gram-Schmidt orthogonalization. MEMENTO! will appear in calibration (aka Q-R) Structure of the

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From previous lectures …

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  1. From previous lectures … An Introduction to 3-D Vision

  2. Orthogonal group • What is the set of transformations that preserve the inner product? • Remember inner product under a transformation? • More on this later … An Introduction to 3-D Vision

  3. Gram-Schmidt orthogonalization MEMENTO! will appear in calibration (aka Q-R) Structure of the Parameter matrix An Introduction to 3-D Vision

  4. Structure induced by a linear map A X X’ Ra(A) T T Ra(A ) Nu(A) T T Nu(A ) Ra(A) Nu(A) An Introduction to 3-D Vision

  5. Eigenvalues and eigenvectors • Eigenvalues and eigenvectors encode the “essence” of the linear map represented by A: the range space, the null space, the rank, the norm etc. • How do the notions of eigenvalues and eigenvectors generalize to NON-SQUARE matrices? • SVD, later … An Introduction to 3-D Vision

  6. Symmetric matrices An Introduction to 3-D Vision

  7. Symmetric matrices (contd.) An Introduction to 3-D Vision

  8. Lecture 2: some useful tools from linear algebra The Singular Value Decomposition Least-squares solution of linear systems Basic concepts from optimization Lagrange multipliers An Introduction to 3-D Vision

  9. The singular value decomposition An Introduction to 3-D Vision

  10. The SVD (contd.) An Introduction to 3-D Vision

  11. The SVD: geometric interpretation A An Introduction to 3-D Vision

  12. Pseudo-inverse and linear systems An Introduction to 3-D Vision

  13. Fixed-rank approximation • Useful for matrix factorization • MEMENTO! An Introduction to 3-D Vision

  14. Preview of coming attractions • Characterization of the essential matrix • Least-squares solution of Ax=b • Computation of null-space • In general, orthogonal projections An Introduction to 3-D Vision

  15. Unconstrained optimization An Introduction to 3-D Vision

  16. Unconstrained optimization (contd.) An Introduction to 3-D Vision

  17. Iterative minimization (local) • Steepest descent: • Newton’s method: • More in general: An Introduction to 3-D Vision

  18. Gauss-Newton, Levemberg-Marquardt • Quadratic cost function • No second derivatives An Introduction to 3-D Vision

  19. Constrained optimization An Introduction to 3-D Vision

  20. Lagrangian function and multipliers An Introduction to 3-D Vision

  21. Preview of coming attractions • Optimal triangulation An Introduction to 3-D Vision

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