SMOOTH SURFACES AND THEIR OUTLINES

1. SMOOTH SURFACES AND THEIR OUTLINES • Elements of Differential Geometry • What are the Inflections of the Contour? • Koenderink’s Theorem

2. Informations pratiques • Présentations :http://www.di.ens.fr/~ponce/geomvis/lect11.ppt http://www.di.ens.fr/~ponce/geomvis/lect11.pdf • Deux cours de plus en Janvier: Jeudis 5 et 12 Janvier Examen le 13 Janvier • A quand l’examen?

3. П1 Chasles’ absolute conic: x12 + x22 + x32 = 0, x4 = 0. T diag(Id,0) d = | u |2 = 0. The absolute quadratic complex:

4. v u O y – x u x £ y v = xÇy = = = (u;v) The join of two points Note: u . v = 0 y x 

5. Proposition : P*P T¼* Proposition : PP T¼ Triggs (1997); Pollefeys et al. (1998) y y ’ ’ x x ’ ’ P x P x ¼ ¼ p ’ d d d d ’ P P * * ¼ ¼ d d ’ ’ d d p p ’ ’ P P ’ T T ¼ ¼ x x r r P P x x T T c c ¼ ¼ d d d p p ’ ’ XT p ’ ’ ¼ ¼ p p ’ ’ d d d d ’ ’ X X * * ¼ ¼ d d T T d d x x ’ ’ X X ’ ’ ¼ ¼ T T f f X X p p ’ ’ ¼ ¼ p p x x ’ ’ The AQC general equation: d T = 0, with  = X*TX* Proposition:  T’¼û¢û’ f f d d x x ’ ’ X* X* ¼ ¼ Dual perspective projection Perspective projection r r c c x x ’ ’ x x x x

6. Non-linear, 7 images Non-linear, 20 images Non-linear, 196 images Linear, 20 images 2480 points tracked in 196 images

7. Canon XL1 digital camcorder, 480£720 pixel2 (Ponce & McHenry, 2004) Projective structure from motion : Mahamud, Hebert, Omori & Ponce (2001)

8. Quantitative comparaison with Pollefeys et al. (1998, 2002) (synthetic cube with 30cm edges, corrupted by Gaussian noise)

9. Smooth Shapes and their Outlines Can we say anything about a 3D shape from the shape of its contour?

10. What are the contour stable features?? Reprinted from “Computing Exact Aspect Graphs of Curved Objects: Algebraic Surfaces,” by S. Petitjean, J. Ponce, and D.J. Kriegman, the International Journal of Computer Vision, 9(3):231-255 (1992).  1992 Kluwer Academic Publishers. folds cusps T-junctions Shadows are like silhouettes.. Reprinted from “Solid Shape,” by J.J. Koenderink, MIT Press (1990).  1990 by the MIT.

11. Differential geometry: geometry in the small The normal to a curve is perpendicular to the tangent line. A tangent is the limit of a sequence of secants.

12. What can happen to a curve in the vicinity of a point? (a) Regular point; (b) inflection; (c) cusp of the first kind; (d) cusp of the second kind.

13. The Gauss Map • It maps points on a curve onto points on the unit circle. • The direction of traversal of the Gaussian image reverts • at inflections: it folds there.

14. The curvature C • C is the center of curvature; • R = CP is the radius of curvature; •  = lim dq/ds = 1/R is the curvature.

15. Closed curves admit a canonical orientation.. k<0 k> 0

16. Twisted curves are more complicated animals..

17. A smooth surface, its tangent plane and its normal.

18. Normal sections and normal curvatures Principal curvatures: minimum value k maximum value k Gaussian curvature: K = k k 1 1 2 2

19. The local shape of a smooth surface Elliptic point Hyperbolic point K > 0 K < 0 Reprinted from “On Computing Structural Changes in Evolving Surfaces and their Appearance,” By S. Pae and J. Ponce, the International Journal of Computer Vision, 43(2):113-131 (2001).  2001 Kluwer Academic Publishers. K = 0 Parabolic point

20. The Gauss map Reprinted from “On Computing Structural Changes in Evolving Surfaces and their Appearance,” By S. Pae and J. Ponce, the International Journal of Computer Vision, 43(2):113-131 (2001).  2001 Kluwer Academic Publishers. The Gauss map folds at parabolic points.

21. Smooth Shapes and their Outlines Can we say anything about a 3D shape from the shape of its contour?

22. Theorem [Koenderink, 1984]: the inflections of the silhouette are the projections of parabolic points.

23. Koenderink’s Theorem (1984) K = kk r c Note: k > 0. r Corollary: K and k have the same sign! c