simulating decorative mosaics
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Simulating Decorative Mosaics
Alejo HausnerUniversity of Toronto
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Mosaic Tile Simulation
• Reproduce mosaic tiles– long-lasting (graphics is ephemeral)– realism with very few pixels– pixels have orientation
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Real Tile Mosaics
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Real Mosaics II
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The Problem
• Square tiles cover the plane perfectly
• Variable orientation --> loose packing
• Opposing goals:– non-uniform grid– dense packing
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Previous Work
• Romans: algorithm?
• Relaxation (Haeberli’90) – scatter points on image– voronoi region = tile– tiles not square
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Previous Work
• Escherization (Kaplan’00) – regular tilings– use symmetry groups
• Stippling (Deussen’01)
– voronoi relaxation– round dots
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Voronoi Diagrams
• What are they– sites, and regions closest to each site
• How to compute them?– Divide & conquer (PS)– sweepline (Fortune)– incremental
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Voronoi Diagram
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Centroidal Voronoi Diagrams
• VD sites are not centroids
• Lloyd’s method (k-means)– move site to centroid,– recalculate VD– repeat
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Centroidal Voronoi Diagrams
• VD sites are not centroids
• Lloyd’s method (k-means)– move site to centroid,– recalculate VD– repeat
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Centroidal Voronoi Diagrams
• VD sites are not centroids
• Lloyd’s method (k-means)– move site to centroid,– recalculate VD– repeat
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Centroidal Voronoi Diagram
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CVD uses
• Nature:– honeycombs– giraffe spots
• Sampling– approximates Poisson-disk (low discrepancy)– can bias for filter function
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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
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Key idea
• Cone is distance function– radius = height
• Non-euclidean distance:– different kind of cone– eg square pyramid– can be non-isotropic
(rotate pyramid around Z)
r
h
h=|x-a|+|y-b|
(a,b)
(x,y)h
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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
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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.
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Lloyd Near Convergence
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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
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Orientation Field
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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.
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Tiles Straddle Edge
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Thick Edge: Centroid Repelled
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After 10 Iterations
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Original Voronoi Diagram
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After 20 Iterations
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One Tile per Voronoi Region
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Thinner Tiles
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Round Tiles
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Rhomboidal Tiles
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Tiles are not Pixels4000 pixels 4000 tiles
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More Pictures
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“Painterly” Rendering
Tile = Paint stroke
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Summary
• New method for packing squares on curvilinear grid
• Minimizes sum of particle distances
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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?
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