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?