Lecture 1: Introduction

1. Lecture 1: Introduction Administrivia Why vision? Where are we at? What you need to know Quick review of linear algebra, optimization UCLA Vision Lab

2. Questionnaire: rate your degree of familiarity with (0-10) • Linear spaces, linear independence • Bases, change of basis • Inner product, orthogonality • Transformation groups: composition, inverse • Gram-Schmidt • Range, null, rank • Eigenvalues, eigenvectors • Symmetric matrices, skew-symmetric matrices • Diagonalization of symmetric matrices • Pseudo-inverse (Moore-Penrose) • Singular Value Decomposition (SVD) • Minimization via gradient descent • Newton’s method, Gauss-Newton • Constrained optimization, Lagrange multipliers • Least-squares estimation • Kalman filter UCLA Vision Lab

3. Administrative matters • Class website: check for announcements, handouts, assignments • Policy: grading, collaboration, projects UCLA Vision Lab

4. What is vision? • From the 3-D world to 2-D images: image formation (physics). • Domain of artistic reproduction (synthesis): painting, graphics. • From 2-D images to the 3-D world: image analysis (mathematical modeling, inference). • Domain of vision: biological (eye+brain), computational UCLA Vision Lab

5. IMAGE SYNTHESIS: simulation of the image-formation process • Pinhole (perspective) imaging in most ancient civilizations. • Euclid, perspective projection, 4th century B.C., Alexandria (Egypt) • Pompeii frescos, 1st century A.D. (ubiquitous). • Geometry understood very early on, then forgotten. Image courtesy of C. Taylor UCLA Vision Lab

6. PERSPECTIVE IMAGING (geometry) • Re-discovered and formalized in the Renaissance: • Fillippo Brunelleschi, first Renaissance artist to paint with • correct perspective,1413 • “Della Pictura”, Leon Battista Alberti, 1435, first treatise • Leonardo Da Vinci, stereopsis, shading, color, 1500s • Raphael, 1518 Image courtesy of C. Taylor UCLA Vision Lab

7. IMAGE ANALYSIS: THE INVERSE PROBLEM Input: Images (measurements of LIGHT) Intermediate representation: “Features” (2-D geometry) Output: Camera calibration, 3-D pose, scene structure, surface photometry. IN THIS CLASS: only geometry; for photometry take CS174B UCLA Vision Lab

8. IMAGES AND GEOMETRY – History of “Modern” Geometric Vision • Chasles, formulated the two-view seven-point problem in a class homework • assignment in 1855 • Hesse, solved the above problem, 1863 • Kruppa, solved the two-view five-point problem, 1913 • Longuet-Higgins, the two-view eight-point algorithm, 1981 • Liu and Huang, the three-view trilinear constraints, 1986 • Faugeras, uncalibrated reconstruction, 1992 • Tomasi and Kanade, (orthographic) factorization method, 1992 • iata, iata, iata … • MaSKS: generalized rank conditions, 2003. UCLA Vision Lab

9. APPLICATIONS – 3-D Modeling and Rendering UCLA Vision Lab

10. APPLICATIONS – 3-D Modeling and Rendering Image courtesy of Paul Debevec UCLA Vision Lab

11. APPLICATIONS – Image Morphing, Mosaicing, Alignment Images of CSL, UIUC UCLA Vision Lab

12. APPLICATIONS – Real-Time Sports Coverage First-down line and virtual advertising Image courtesy of Princeton Video Image, Inc. UCLA Vision Lab

13. APPLICATIONS – Real-Time Virtual Object Insertion UCLA Vision Lab UCLA Vision Lab

14. APPLICATIONS – Unmanned Aerial Vehicles (UAVs) Rate: 10Hz Accuracy: 5cm, 4o UCLA Vision Lab Berkeley Aerial Robot (BEAR) Project

15. APPLICATIONS – Autonomous Highway Vehicles Image courtesy of E.D. Dickmanns UCLA Vision Lab

16. Dickmanns’ video UCLA Vision Lab

17. And now let’s get to work UCLA Vision Lab

18. Review of linear algebra (Appendix A) UCLA Vision Lab

19. Change of basis UCLA Vision Lab

20. Change of basis (contd.) UCLA Vision Lab

21. Inner product • What is the expression of the inner product in the new basis? UCLA Vision Lab

22. Inner product (contd.) MEMENTO! (will appear in uncalibrated reconstruction) UCLA Vision Lab

23. Transformation groups UCLA Vision Lab

24. Affine transformation • Not a linear transformation! • Can be made linear in HOMOGENEOUS COORDINATES MEMENTO! will appear everywhere UCLA Vision Lab

25. Affine group (contd.) • Composition of affine transformations. • What is the inverse transformation? UCLA Vision Lab

26. Orthogonal group • What is the set of transformations that preserve the inner product? • Remember inner product under a transformation? • More on this later … UCLA Vision Lab

27. Euclidean group UCLA Vision Lab

28. Gram-Schmidt orthogonalization MEMENTO! will appear in calibration (aka Q-R) Structure of the Parameter matrix UCLA Vision Lab

29. Range, null, rank and all that … UCLA Vision Lab

30. 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) UCLA Vision Lab

31. 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 … UCLA Vision Lab

32. Symmetric matrices UCLA Vision Lab

33. Symmetric matrices (contd.) UCLA Vision Lab

34. Skew-symmetric matrices UCLA Vision Lab

35. Skew-symmetric matrices (contd.) UCLA Vision Lab

36. Preview of coming attractions • Essential matrices UCLA Vision Lab