1. Visualizing the Evolutions of Silhouettes Junfei Dai Junho Kim Huayi Zeng Xianfeng Gu

2. Outline Introduction Local Properties of Silhouettes Global Properties of Silhouettes Experimental Results

3. Introduction Silhouettes refer to the locus of points on the surface where the view rays tangentially touch the surface. Projected silhouettes refer to the projection images of silhouettes.

4. Introduction Computer Vision   Projected msilhouettes convey rich geometric information about the original surface Non-Photorealistic Rendering   silhouettes carry the most important shape information

5. Introduction Contributions Visualization of all possible topological changes of a silhouette. Development of a theorem of the relation of geodesic curvature of a silhouette and the view cone angle. Introduction of the concept of the aspect surface, all topological changes happen when the view is on the aspect surface.

6. Local Properties of Silhouettes The local properties of silhouettes have been thoroughly studied in computer vision, singularity theory and catastrophe theory.

7. Local Properties of Silhouettes On a smooth surface with at least C2 continuity, k1,k2 are principal curvatures. All points are classified, 1. elliptic, 0 < k1 · k2, 2. hyperbolic, k1 < 0 < k2, 3. parabolic, k1k2 = 0. A special class of parabolic points are called flat, if both k1 and k2 are zeros

8. Local Properties of Silhouettes An asymptotic direction for a parabolic point is the principal direction with zero principal curvature. Asymptotic directions play important roles in analyzing the local behavior of silhouettes.

9. Local Properties of Silhouettes The topological changes happen when the view is along the asymptotic directions of some parabolic points.

10. Global Properties of Silhouettes Lemma 1 Suppose r(u,v) is a generic smooth surface, with local parameters (u,v). The view point is v, r(s) is a silhouette. Then the tangent direction r ’ is conjugate to the view ray direction r-v. (Two tangent vectors dr1,dr2 are conjugate, if =0, where W is Weigarten deformation.)

11. Global Properties of Silhouettes Lemma 2 Suppose r is a silhouette on a generic smooth surface S with a view point v, which is not on the surface. A point p on silhouette is in one of the three cases: – p is elliptic, K(p) > 0; – p is hyperbolic, K(p) < 0; – p is parabolic, but the view direction p-v is not along the asymptotic direction of p. then in a neighborhood of p, the silhouette r is a one dimensionalmanifold.

12. Global Properties of Silhouettes Lemma 3 Suppose S is a generic smooth surface, view point v crosses the surface along the normal direction through p from outside to inside, then – if p is elliptic, then a closed silhouette will shrink to the point p and disappear. – if p is hyperbolic, two silhouettes will intersect and reconnect, the silhouettes are along the asymptotic directions of p.

13. Global Properties of Silhouettes Elliptic Hyperbolic

14. Global Properties of Silhouettes Aspect Surface   Suppose S is a generic smooth surface, is a parabolic curve, at each point e(s) is the asymptotic direction of.The following surface is called the aspect surface of.   The union of the aspect surfaces of all parabolic curves and the surface S itself is called the aspect surface of S, and denoted as W(S ).

15. Global Properties of Silhouettes Theorem 1 Suppose S is a generic smooth surface. The topological changes of the silhouettes only happens when the view point v is on the aspect surface of S.

16. Global Properties of Silhouettes Definition Suppose S is a generic smooth surface. A view v is at a generic position, if it is not on the aspect surface. Otherwise, it is at a critical position.

17. Global Properties of Silhouettes Definition Suppose the view v is in a generic position for a generic smooth surface. The distance between n and S is finite. A connected component of the silhouette is. The view cone surface is defined

18. Global Properties of Silhouettes view cone surface

19. Global Properties of Silhouettes Theorem 2 SupposeS is a generic smooth surface. The view point is not on the surface S. A closed silhouette is smooth (without cusps on S ), where s is the arc length; the cone angle at the view point is, then

20. Experimental Results a b a b

21. Experimental Results c d c d

22. Experimental Results e f e f

23. Experimental Results g h g h

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