Multi-Class Blue Noise Sampling

Multi-Class Blue Noise Sampling Li-Yi Wei 魏立一 Microsoft Research

Blue noise distribution • random & uniform • applications • sampling • stippling • meshing • texturing • object placement [Ostromoukhovet al. 2004] [Balzer et al. 2009] [Turk 1992]

Previous work • half-toning • [Ulichney 1986; Wang and Parker 1999; Ostromoukhov 2001; Zhou and Fang 2003; Pang et al. 2008; Chang et al. 2009] • dart throwing • [Cook 1986; Mitchell 1987; McCool and Fiume 1992; Jones 2006; Dunbar and Humphreys 2006; White et al. 2007; Wei 2008; Fu and Zhou 2008; Cline et al. 2009; Gamito and Maddock 2010] • relaxation • [Lloyd 1982; Turk 1992; Balzer et al. 2009; Tung et al. 2010; Liu et al. 2010; Levy and Liu 2010] • tiling • [Cohen et al. 2003; Ostromoukhov et al. 2004; Kopf et al. 2006; Lagae and Dutre 2006; Ostromoukhov 2007]

Prior art mostly for 1 sample class • scenarios with multi-class samples sampling (retina cells) stippling (pointillism) texturing (flowers)

Apply 1 class blue noise to > 1 classUniform per class X O O total set class 0 class 1

Apply 1 class blue noise to > 1 classUniform total set O X X total set class 0 class 1

Multi-class blue noise sampling • uniform & random for each class & their unions O O O total set class 0 class 1

Background of blue noise • random & uniform • controlled by spacing r r r -1 r -1 radial mean power spectrum anisotropy

Dart throwing [Dippe and Wold 1985; Cook 1986] • loop: • random sample • conflict check r

Relaxation[Lloyd 1982] • indirectly specify r through sample count N • given a set of N sample • loop: • Voronoi for each sample • move sample to centroid

Core idea for multi-class blue noise • replace scalar spacing r by a matrixr r11 c0 c1 c2 r00 c0 r00r01r02 r10r11r12 r20r21r22 r01 c1 c2

Generating multi-class blue noise • hard disk sampling • control sample spacing r • (like dart throwing) • soft disk sampling • control sample count N • (like Lloyd relaxation)

Multi-class hard disk sampling • like 1-class dart throwing, but • r matrix for conflict check • consistent fill rate 1:4:16 • 0 1 2 2 1 2 2 0 1 2 2 1 2 2 • may kill existing samples c0 c1 c2 c0 0.400.180.09 0.180.200.09 0.090.090.10 c1 c2

Soft disk: single class • Gaussian blob per sample • minimize max(E) → uniform distribution

Soft disk: multi class • minimize max(E) → uniform distribution R/G/B: E(c0 /c1 /c2)

Multi-class soft disk sampling • ~ best candidate dart throwing [Mitchell 1987] • loop for each trial: • random k samples • pick one with min max(E) • X Lloyd relaxation • stuck in multi-class setting

Build r matrix • diagonal entries {rii}i=0:c-1given • how to compute off-diagonal entries {rij}i≠j? • (symmetry: rij= rji) • see paper r00 r01 r02 r03 r10r11r12 r13 r20 r21 r22r23 r30 r31 r32 r33

Analysis [Lagae and Dutre 2008] • spatial uniformity σ • ideal σ in [0.65 0.85]; our σ in [0.65 0.70] • soft disk sampling has larger σ

total set class 0 class 1 class 2

Analysis [Lagae and Dutre 2008] • spectral analysis • (good quality; radial mean diff from 1-class) - 1-class - multi-class power spectrum radial mean anisotropy

Object placement: uniform

Object placement: more classes

Object placement: adaptive

Color stippling input RGBCMYB dots

Sensor layout zoneplate sin(x2+y2) Bayer mosaic Penrose pixel our method

Discrete layout noisy Bayer mosaic random soft disk

Tradeoff • Hard disk sampling • O control sample spacing • X control sample count • O continuous space • X discrete space • X less uniform • O faster • Soft disk sampling • X control sample spacing • O control sample count • O continuous space • O discrete space • O more uniform • X slower

Future work • applications & extensions • 3D or higher dimensions • surfaces or other non-Euclidean domains • anisotropy • acceleration • tiling • parallelization [Bowers et al. 2010] isotropic aniso [Li et al. 2010]

Acknowledgement • Yin Li • Kun Zhou • Xin Tong • Eric Stollnitz • Jason Fondran • http://www.gif-favicon.com/ • Brandon Lloyd • Bill Baxter • Naga Govindaraju • John Manferdelli • Reviewers • http://store.got3d.com/

Multi-Class Blue Noise Sampling