Three dimensional mosaics with variable-sized tiles

1. Three dimensional mosaics with variable-sized tiles Visual Comput 2008 報告者:丁琨桓

2. Introduction • Three dimensional mosaics, or surface mosaics, are a beautiful art form where a sculpture is made from putting together tiles on a given shape.

3. Previous work • In computer graphics 2D mosaics have been fully explored. • centroidal Voronoi diagram

4. Previous work • 3D mosaics are much harder since the tiles have to be positioned on the surface of a non-planar object being decorated. • If the shape is complex, adequate tile positioning is a real challenge.

5. Previous work • [Surface mosaics,2006] addressed the problem of mosaics with tiles of the same size. • Using the same tile size for the whole surface is not the best choice, since this size could be too big for some locations with high curvature.

6. Algorithm • Step1: Tiles are initially distributed randomly over the surface • Higher curvature places with higher density of bigger tiles • Smaller curvature places with fewer bigger tiles • Step2: relaxation procedure • move tiles away from one another, leaving some gap for grout and avoiding collisions among tiles. • Step3: rendering specific effects • achieve a more realistic result

7. Evaluating curvatures using model vertex data • [ Re-tiling polygonal surfaces, SIGGRAPH 1992 ] • The method gives a good approximation of the exact curvature, using only the model’s polygonal data. • For each vertex the method finds an associated curvature of this vertex with respect to all edges connected to it.

8. Evaluating curvatures using model vertex data The radius of curvature r : r = tan(θ)|P-A|/2 Point C bisects the Angle APB In 3D the normal vector at P approximates the line segment PC. The term θis estimated with the dot product between a normalized vector A – P and the normal vector at P. Approximation of the curvature in 2D

9. Evaluating curvatures using model vertex data Red is mapped to vertices of higher curvature whereas blue is mapped to relatively flat regions Radius of curvature (Rc) in the plane

10. Mapping curvatures into tile size A : the total area of the object’s surface 2h: the average tile size and h is half this size N : user-specified number of tiles h : half the tile size r : the radius of the circle

11. Mapping curvatures into tile size Function for mapping curvatures into tile sizes

12. Distributing random points on the surface of a polyhedral model • distributed randomly over the surface • polygon capacity Ai : the area of polygon i rci : the polygon radius of curvature f : the mapping function

13. Distributing random points on the surface of a polyhedral model • Polygons with higher curvature, i.e., smaller radius of curvatures, will receive more tiles. distributed randomly distributed with capacity function

14. Relaxation of points on the surface of the model • move the tiles away from each other, to avoid intersections using a repulsive force • repulsive force is proportional to tile size, such that small tiles will concentrate in strongly curved places, and big tiles will push smaller ones to curved regions • f = Kf * ( 1 – d/(r1 + r2))

15. Relaxation of points on the surface of the model • f = Kf * ( 1 – d/(r1 + r2)) • d is the distance between the particles • r1 and r2 are the radii of the ideal circles around the tile. r : the radius of the circle

16. Relaxation of points on the surface of the model r1 r2 r1 r2 d d f > 0 f < 0 f = Kf * ( 1 – d/(r1 + r2))

17. Adjusting the orientation of the tiles • [ Texture Synthesis on Surfaces, SIGGRAPH 2001 ] • Vector field

18. Rendering • To make the results more visually appealing to the user, the final shape of the tiles may be controlled by four parameters Square tiles, turned into general quadrilateral tiles

19. Rendering Comparison of tiles with and without random variation in the shape. Random variables U1, U2, V1, and V2 with values between 85% and 115% of h

20. Result # of tiles :7000 Tsmin : 0.4h Tsmax : 3.15h Rcmin : 0 Rcmax : 25h

21. Result # of tiles :7000 Tsmin : 0.1h Tsmax : 2.3h Rcmin : 0.5 Rcmax : 20h

22. Result Effect of varying the size of tiles ( number of tiles : 4000)

23. Comparison surface mosaic mosaics with variable-sized tiles Comparison with previous result from [surface mosaic]

24. Conclusion • This paper presented a solution efficiently computes the distribution, placement and rendering of tiles • Author plan to extend this work by allowing tiles of variable shapes, not only squares.