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Simulating Decorative Mosaics

Simulating Decorative Mosaics. Alejo Hausner University of Toronto. Mosaic Tile Simulation. Reproduce mosaic tiles long-lasting (graphics is ephemeral) realism with very few pixels pixels have orientation. Real Tile Mosaics. Real Mosaics II. The Problem.

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Simulating Decorative Mosaics

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  1. Simulating Decorative Mosaics Alejo Hausner University of Toronto

  2. Mosaic Tile Simulation • Reproduce mosaic tiles • long-lasting (graphics is ephemeral) • realism with very few pixels • pixels have orientation Simulating Decorative Mosaics

  3. Real Tile Mosaics Simulating Decorative Mosaics

  4. Real Mosaics II Simulating Decorative Mosaics

  5. The Problem • Square tiles cover the plane perfectly • Variable orientation --> loose packing • Opposing goals: • non-uniform grid • dense packing Simulating Decorative Mosaics

  6. Previous Work • Romans: algorithm? • Relaxation (Haeberli’90) • scatter points on image • voronoi region = tile • tiles not square Simulating Decorative Mosaics

  7. Previous Work • Escherization (Kaplan’00) • regular tilings • use symmetry groups • Stippling (Deussen’01) • voronoi relaxation • round dots Simulating Decorative Mosaics

  8. Voronoi Diagrams • What are they • sites, and regions closest to each site • How to compute them? • Divide & conquer (PS) • sweepline (Fortune) • incremental Simulating Decorative Mosaics

  9. Voronoi Diagram Simulating Decorative Mosaics

  10. Centroidal Voronoi Diagrams • VD sites are not centroids • Lloyd’s method (k-means) • move site to centroid, • recalculate VD • repeat Simulating Decorative Mosaics

  11. Centroidal Voronoi Diagrams • VD sites are not centroids • Lloyd’s method (k-means) • move site to centroid, • recalculate VD • repeat Simulating Decorative Mosaics

  12. Centroidal Voronoi Diagrams • VD sites are not centroids • Lloyd’s method (k-means) • move site to centroid, • recalculate VD • repeat Simulating Decorative Mosaics

  13. Centroidal Voronoi Diagram Simulating Decorative Mosaics

  14. CVD uses • Nature: • honeycombs • giraffe spots • Sampling • approximates Poisson-disk (low discrepancy) • can bias for filter function Simulating Decorative Mosaics

  15. Hardware-assisted VD’s • SG99: Hoff et al • uses graphics hardware • draw cone at each site • orthogonal view from above • each region is single-coloured • can extend to non-point sites (curves) project Simulating Decorative Mosaics

  16. Key idea r • Cone is distance function • radius = height • Non-euclidean distance: • different kind of cone • eg square pyramid • can be non-isotropic (rotate pyramid around Z) h (a,b) h (x,y) h=|x-a|+|y-b| Simulating Decorative Mosaics

  17. Basic Tiling Algorithm • Compute orientation field (details later) • scatter points on image • use pyramids to get oriented tiles • apply Lloyd’s method to spread sites evenly • draw oriented tile at each site Simulating Decorative Mosaics

  18. Details • Lloyd’s method: • To compute centroid of each Voronoi region: 1: read back pixels, 2: get average (row,col) per colour 3: convert back to object coords. Simulating Decorative Mosaics

  19. Lloyd Near Convergence Simulating Decorative Mosaics

  20. Orientation Field • Choose edges that need emphasis • compute generalized VD for edges (Hoff99) • get gradient vector of distance field • distance = z-buffer distance • gradient orientation = tile orientation • points away from edges Simulating Decorative Mosaics

  21. Orientation Field Simulating Decorative Mosaics

  22. Edge Discrimination • Must line up tiles on edges • But both sides of edge oriented same • square tiles ==> 180o rotational symmetry • Use hardware • draw edge thick, different colour • voronoi regions move away from edges • leaves gap where edge is. Simulating Decorative Mosaics

  23. Tiles Straddle Edge Simulating Decorative Mosaics

  24. Thick Edge: Centroid Repelled Simulating Decorative Mosaics

  25. After 10 Iterations Simulating Decorative Mosaics

  26. Original Voronoi Diagram Simulating Decorative Mosaics

  27. After 20 Iterations Simulating Decorative Mosaics

  28. One Tile per Voronoi Region Simulating Decorative Mosaics

  29. Thinner Tiles Simulating Decorative Mosaics

  30. Round Tiles Simulating Decorative Mosaics

  31. Rhomboidal Tiles Simulating Decorative Mosaics

  32. Tiles are not Pixels 4000 tiles 4000 pixels Simulating Decorative Mosaics

  33. More Pictures Simulating Decorative Mosaics

  34. “Painterly” Rendering Tile = Paint stroke Simulating Decorative Mosaics

  35. Summary • New method for packing squares on curvilinear grid • Minimizes sum of particle distances Simulating Decorative Mosaics

  36. Further Work • Reduce “grout” • final pass: adjust tile shapes • currently don’t use adjacency info • use divergence of orientation field · • Improve colour • real tiles have fixed colour set: Dither? How? Simulating Decorative Mosaics

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