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
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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)
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Key idea
r
• Cone is distance function
h
– radius = height
• Non-euclidean distance:
– different kind of cone
– eg square pyramid
– can be non-isotropic
(rotate pyramid around Z)
(a,b)
h
(x,y)
h=|x-a|+|y-b|
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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
o
– square tiles ==> 180 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 Pixels
4000 tiles
4000 pixels
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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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