Simulating Decorative Mosaics Alejo Hausner University of Toronto

Simulating Decorative Mosaics2 Mosaic Tile Simulation Reproduce mosaic tiles –long-lasting (graphics is ephemeral) –realism with very few pixels –pixels have orientation

Simulating Decorative Mosaics3 Real Tile Mosaics

Simulating Decorative Mosaics4 Real Mosaics II

Simulating Decorative Mosaics5 The Problem Square tiles cover the plane perfectly Variable orientation --> loose packing Opposing goals: –non-uniform grid –dense packing

Simulating Decorative Mosaics6 Previous Work Romans: algorithm? Relaxation (Haeberli90) –scatter points on image –voronoi region = tile –tiles not square

Simulating Decorative Mosaics7 Previous Work Escherization (Kaplan00) –regular tilings –use symmetry groups Stippling (Deussen01) –voronoi relaxation –round dots

Simulating Decorative Mosaics8 Voronoi Diagrams What are they –sites, and regions closest to each site How to compute them? –Divide & conquer (PS) –sweepline (Fortune) –incremental

Simulating Decorative Mosaics9 Voronoi Diagram

Simulating Decorative Mosaics10 Centroidal Voronoi Diagrams VD sites are not centroids Lloyds method (k-means) –move site to centroid, –recalculate VD –repeat

Simulating Decorative Mosaics11 Centroidal Voronoi Diagrams VD sites are not centroids Lloyds method (k-means) –move site to centroid, –recalculate VD –repeat

Simulating Decorative Mosaics12 Centroidal Voronoi Diagrams VD sites are not centroids Lloyds method (k-means) –move site to centroid, –recalculate VD –repeat

Simulating Decorative Mosaics13 Centroidal Voronoi Diagram

Simulating Decorative Mosaics14 CVD uses Nature: –honeycombs –giraffe spots Sampling –approximates Poisson-disk (low discrepancy) –can bias for filter function

Simulating Decorative Mosaics15 Hardware-assisted VDs 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 Mosaics16 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

Simulating Decorative Mosaics17 Basic Tiling Algorithm Compute orientation field (details later) scatter points on image –use pyramids to get oriented tiles apply Lloyds method to spread sites evenly draw oriented tile at each site

Simulating Decorative Mosaics18 Details Lloyds 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 Mosaics19 Lloyd Near Convergence

Simulating Decorative Mosaics20 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 Mosaics21 Orientation Field

Simulating Decorative Mosaics22 Edge Discrimination Must line up tiles on edges But both sides of edge oriented same –square tiles ==> 180 o rotational symmetry Use hardware –draw edge thick, different colour –voronoi regions move away from edges –leaves gap where edge is.

Simulating Decorative Mosaics23 Tiles Straddle Edge

Simulating Decorative Mosaics24 Thick Edge: Centroid Repelled

Simulating Decorative Mosaics25 After 10 Iterations

Simulating Decorative Mosaics26 Original Voronoi Diagram

Simulating Decorative Mosaics27 After 20 Iterations

Simulating Decorative Mosaics28 One Tile per Voronoi Region

Simulating Decorative Mosaics29 Thinner Tiles

Simulating Decorative Mosaics30 Round Tiles

Simulating Decorative Mosaics31 Rhomboidal Tiles

Simulating Decorative Mosaics32 Tiles are not Pixels 4000 pixels 4000 tiles

Simulating Decorative Mosaics33 More Pictures

Simulating Decorative Mosaics34 Painterly Rendering Tile = Paint stroke

Simulating Decorative Mosaics35 Summary New method for packing squares on curvilinear grid Minimizes sum of particle distances

Simulating Decorative Mosaics36 Further Work Reduce grout –final pass: adjust tile shapes currently dont use adjacency info –use divergence of orientation field · Improve colour –real tiles have fixed colour set: Dither? How?