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Simulating Decorative Mosaics Alejo Hausner University of Toronto [SIGGRAPH2001]

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Decorative Mosaics SIGGRAPH 2001 2 Mosaic Tile Simulation Reproduce mosaic tilings –long-lasting (graphics is ephemeral) –realism with very few pixels –pixels have orientation

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Decorative Mosaics SIGGRAPH 2001 3 Real Tile Mosaics

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Decorative Mosaics SIGGRAPH 2001 4 Real Mosaics II

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Decorative Mosaics SIGGRAPH 2001 5 The Problem Square tiles cover the plane perfectly Variable orientation loose packing Opposing goals: –non-uniform grid –dense packing

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Decorative Mosaics SIGGRAPH 2001 6 Formal Statement Given a rectangular region I 2 in the plane R 2, and a vector field (x, y) defined on that region, find N sites P i (x i, y i ) in I 2 and place N squares of side s, one at each P i, oriented with sides approximately parallel to (x i, y i ), such that all squares are disjoint and the area they cover is maximized.

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Decorative Mosaics SIGGRAPH 2001 7 Previous Work Romans: algorithm? Relaxation (Haeberli 90) –scatter points on image –voronoi region = tile –tiles not square

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Decorative Mosaics SIGGRAPH 2001 8 Previous Work Escherization (Kaplan 00) –regular tilings –use symmetry groups Stippling (Deussen 01) –voronoi relaxation –round dots

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Decorative Mosaics SIGGRAPH 2001 9 Voronoi Diagram What is it? –Input: sites (generators) –Result: regions closest to each site

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Decorative Mosaics SIGGRAPH 2001 10 Voronoi Diagram

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Decorative Mosaics SIGGRAPH 2001 11 Centroidal Voronoi Diagrams VD sites centroids Lloyds alg. (k-means): –1: move site to centroid –2: recalculate VD –3: repeat

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Decorative Mosaics SIGGRAPH 2001 12 Centroidal Voronoi Diagrams VD sites centroids Lloyds alg. (k-means): –1: move site to centroid –2: recalculate VD –3: repeat

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Decorative Mosaics SIGGRAPH 2001 13 Centroidal Voronoi Diagrams VD sites centroids Lloyds alg. (k-means): –1: move site to centroid –2: recalculate VD –3: repeat

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Decorative Mosaics SIGGRAPH 2001 14 CVD After Convergence Almost a hexagonal grid

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Decorative Mosaics SIGGRAPH 2001 15 CVD properties Minimum-energy arrangements In Nature: –honeycombs –giraffe spots Point Sampling: –approximates Poisson-disk (low discrepancy) –can bias for filter function

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Decorative Mosaics SIGGRAPH 2001 16 Key Idea 1 CVDs and Tilings –best circle packing = hexagonal tiling –Euclidean metric: CVD =hexagonal tiling –different metric: CVD = ? Mosaics are Tilings –non-periodic –locally-varying orientation

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Decorative Mosaics SIGGRAPH 2001 17 Key Idea 1 (continued) Create mosaic metric –locally Manhattan (L 1 ) –locally varying orientation BUT: –existing algorithms limited –they assume Euclidean metric –if L 1 metric: they assume isotropic

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Decorative Mosaics SIGGRAPH 2001 18 Hardware-assisted VDs Hoff 99 –uses graphics hardware –draw cone at each site –orthogonal view from above –region color = cone color –can extend to non-point sites (curves) project

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Decorative Mosaics SIGGRAPH 2001 19 Key idea 2 Cone is distance function –radius = height Non-euclidean distance: –different kind of cone –eg square pyramid –can be non-isotropic (rotate pyramid on Z axis) r h h=|x-a|+|y-b| (a,b) (x,y) h h 2 =(x-a) 2 +(y-b) 2 (a,b) (x,y)

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Decorative Mosaics SIGGRAPH 2001 20 Basic Tiling Algorithm Compute orientation field (details later) initialize: random points on image –use pyramids to get oriented tiles use Lloyds method –spreads sites evenly draw oriented tile at each site

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Decorative Mosaics SIGGRAPH 2001 21 Details Lloyds method: –to get Voronoi region centroids: 1: read back pixels (frame buffer) 2: get average (row,col) per colour 3: convert back to object coords. –move sites to centroids –repeat until converged

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Decorative Mosaics SIGGRAPH 2001 22 Algorithm 1.S<-list of random points on the image. 2.repeat until converged: a.for each p in S, place a square pyramid with apex at p. b.rotate each pyramid about the z axis to align it with the direction field (p). c.render the pyramids with an orthogonal projection onto the xy plane, producing a voronoi diagram. d.compute the centroid of each voronoi region. e.move each p to the centroid of its voronoi region. 3.draw a tile centred at each p, oriented along.

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Decorative Mosaics SIGGRAPH 2001 23 Lloyd Near Convergence Manhattan metric (isotropic)

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Decorative Mosaics SIGGRAPH 2001 24 Tile Orientations Artisans algorithm –draw region boundaries –line tiles up on boundaries –build concentric rows Tile orientations –emphasize boundary curves –near curve: must follow curve –far from curve: dont care

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Decorative Mosaics SIGGRAPH 2001 25 Orientation Field Vector Field – (x,y) unit vector at each (x,y) –near a curve: = curve direction –far away: less curve influence Align tiles to –rotate Manhattan-metric pyramids –rotate square tiles

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Decorative Mosaics SIGGRAPH 2001 26 Computing (x,y) Choose curves that need emphasis get VD for curves (Hoff 99) –draw generalized cones z(x,y) = eye distance = z-buffer(x,y) –z = distance from curve get gradient, normalize – (x,y) = z | z|

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Decorative Mosaics SIGGRAPH 2001 27 Generalized Cones

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Decorative Mosaics SIGGRAPH 2001 28 Edge Discrimination Problem: Tiles on Curve – same on both sides (mod 180 o ) –rotate tile 180 o no change –Lloyds alg. ignores edges –result: tiles straddle curves

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Decorative Mosaics SIGGRAPH 2001 29 Edge Discrimination Solution: use hardware –draw edge thick, different color –edge overwrites part of tile –centroids move away from edge –apply Lloyds alg. –leaves gap where edge was.

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Decorative Mosaics SIGGRAPH 2001 30 Edge Discrimination –Tiles on edge –new centroids –move sites –repeat Lloyds –gap created –repeat Lloyds –edge is clear

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Decorative Mosaics SIGGRAPH 2001 31 Edge Discrimination –Tiles on edge –new centroids –move sites –repeat Lloyds –gap created –repeat Lloyds –edge is clear

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Decorative Mosaics SIGGRAPH 2001 32 Edge Discrimination –Tiles on edge –new centroids –move sites –repeat Lloyds –gap created –repeat Lloyds –edge is clear

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Decorative Mosaics SIGGRAPH 2001 33 Edge Discrimination –Tiles on edge –new centroids –move sites –repeat Lloyds –gap created –repeat Lloyds –edge is clear

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Decorative Mosaics SIGGRAPH 2001 34 Edge Discrimination –Tiles on edge –new centroids –move sites –repeat Lloyds –gap created –repeat Lloyds –edge is clear

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Decorative Mosaics SIGGRAPH 2001 35 Edge Discrimination –Tiles on edge –new centroids –move sites –repeat Lloyds –gap created –repeat Lloyds –edge is clear

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Decorative Mosaics SIGGRAPH 2001 36 Edge Discrimination –Tiles on edge –new centroids –move sites –repeat Lloyds –gap created –repeat Lloyds –edge is clear

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Decorative Mosaics SIGGRAPH 2001 37 Putting It All Together

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Initial random tiles

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Putting It All Together 20 iterations of Lloyds

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Putting It All Together Edges cleared

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Putting It All Together Final tiling

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Decorative Mosaics SIGGRAPH 2001 42 Tile Attributes We have: –tile position, orientation Can also control: –size –aspect ratio –shape (round, square, diamond) –colour

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Decorative Mosaics SIGGRAPH 2001 43 Colour Each tile colour –the tiles centre point –average over all pixels Point samples work best for images with uniformly-coloured regions Area samples suit continuous-tone images

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Decorative Mosaics SIGGRAPH 2001 44 Size h*w pixel image with n tiles, this yields tiles with sides of d = pixels. – < 1 accounts for packing inefficiencies due to variations in Smaller tiles for detail area z= (|x-x 0 |+|y-y 0 |)

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Decorative Mosaics SIGGRAPH 2001 45 Size Initial tile positions? –uniform slow convergence –rejection sampling An initial tile position will be accepted with probability (s min /s) 2 –s = tile size –s min = smallest tile size in the image

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Decorative Mosaics SIGGRAPH 2001 46 Aspect Ratio Fine detail is needed when –near a curve –any places where the direction field must be strongly emphasized z= |x-x 0 |+(1- )|y-y 0 |

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Lybian Sibyl Original Curves, regions, (x,y)

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Size Variation 2000 equal-size tiles 2000 variable-size tiles

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Tiles Carry More Information 2000 PIXELS 2000 TILES

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50 Other Effects: Painterly Oval over-size overlapping tiles

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51 Other Effects: Regions Voronoi regions with color from image

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52 More Pictures Second Story Sunlight (Hopper 60)

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53 More Pictures Stained-glass photograph

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54 More Pictures Photograph: Seal on Beach

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Decorative Mosaics SIGGRAPH 2001 55 Summary Pack tiles on curvilinear grid –Low-energy particle arrangements Generalized CVD –Voronoi diagram for any metric –Use graphics hardware

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Decorative Mosaics SIGGRAPH 2001 56 Further Work Reduce grout –final pass: adjust tile shapes currently dont use adjacency info Paint strokes –textured tiles

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Decorative Mosaics SIGGRAPH 2001 57 Further Work User-defined –better tile orientations Tile shapes –higher moments: x 2, xy, etc. Dithering –no regular grid

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