Penrose-based Importance Sampling with Blue Noise Properties. Victor Ostromoukhov University of Montreal
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1 Penrose-based Importance Sampling with Blue Noise Properties Victor Ostromoukhov University of Montreal
2 Presentation Outline Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 2
3 Problem Statement Given: Importance Density I(x,y) high low Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 3
4 Problem Statement Given: Importance Density I(x,y) Find: Discrete Sample Distribution Locally Proportional to I(x,y) high low Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 4
5 Problem Statement Given: Importance Density I(x,y) Find: Discrete Sample Distribution Locally Proportional to I(x,y With Blue Noise Fourier Spectrum high Fourier Spectrum low Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 5
6 Quality vs. Speed Weighted Lloyd's Relaxation Quality Poor Good Cumulative Density Function + Sobol low-discrepancy sequence Slow Speed Fast Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 6
7 Quality vs. Speed Weighted Lloyd's Relaxation Our Method Quality Poor Good Cumulative Density Function + Sobol low-discrepancy sequence Slow Speed Fast Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 7
8 Penrose Tiling: A Primer Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 8
9 Penrose Tiling: Milestones Circa 1200 AD, Fibonacci (Leonardo of Pisa) Rabbit Sequence, Fibonacci Numbers 1619, Johannes Kepler Harmonice Mundi, 5-fold Tiling Problem 1974, Sir Roger Penrose Pentaplexity, Penrose Tiling 1984, Dan Shechtman et al. Discovery of Quasi-Crystals Diffractio n Pattern Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 9
10 Penrose Tiling: Definition by Production Rules Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 10
11 Penrose Tiling: Definition by Production Rules Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 11
12 Penrose Tiling: Definition by Production Rules Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 12
13 Penrose Tiling: Definition by Production Rules Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 13
14 Penrose Tiling: Definition by Production Rules Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 14
15 Penrose Tiling: Definition by Production Rules Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 15
16 Penrose Tiling: Definition by Production Rules Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 16
17 Penrose Tiling: Definition by Production Rules Iteration N Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 17
18 Penrose Tiling: Definition by Production Rules Iteration N+1 Ref: Tilings and Patterns by B. Grunbaum and G.C. Shephard Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 18
19 Penrose Tiling: Vertices Iteration N Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 19
20 Penrose Tiling: Vertices Iteration N+1 Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 20
21 Sampling System Fundamentals Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 21
22 Sampling System Fundamentals Extension to Penrose Tiling Fibonacci Number System Adaptive Subdivision Corrective Vectors Lookup Table Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 22
23 Extension of Penrose Tiling Original Our Extension Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 23
24 Extension of Penrose Tiling a b c d e f Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 24
25 Extension of Penrose Tiling Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 25
26 Extension of Penrose Tiling Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 26
27 Fibonacci Number System Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 27
28 Fibonacci Number System Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 28
29 Fibonacci Number System Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 29
30 Fibonacci Number System Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 30
31 Fibonacci Number System Ref: The Art of Computer Programming, Vol. 1, by D.E. Knuth Binary Number System: Fibonacci Number System: Fibonacci Numbers: Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 31
32 Fibonacci Number System Binary Number System Fibonacci Number System Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 32
33 Fibonacci Number System Pentagonal Tiles Only Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 33
34 Fibonacci Number System Iteration 4 Iteration 5 Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 34
35 Fibonacci Number System Thresholding F-codes, in range [1..376] Thresholding at value = 50 Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 35
36 Fibonacci Number System Thresholding F-codes, in range [1..376] Thresholding at value = 100 Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 36
37 Fibonacci Number System Thresholding F-codes, in range [1..376] Thresholding at value = 200 Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 37
38 Adaptive Subdivision Importance Density Function Non-adaptive Adaptive Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 38
39 Adaptive Subdivision Importance Density Function Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 39
40 Adaptive Subdivision Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 40
41 Adaptive Subdivision Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 41
42 Corrective Vectors Lookup table Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 42
43 Corrective Vectors Lookup table Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 43
44 Corrective Vectors Lookup table Before Correction After Correction Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 44
45 Offline Lloyd s Relaxation: init pts + basis frames Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 45
46 Offline Lloyd s Relaxation: init pts + basis frames Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 46
47 Offline Lloyd s Relaxation: iter 1 Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 47
48 Offline Lloyd s Relaxation: iter 2 Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 48
49 Offline Lloyd s Relaxation: iter 3 Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 49
50 Offline Lloyd s Relaxation: iter 4 Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 50
51 Offline Lloyd s Relaxation: iter 5 Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 51
52 Offline Lloyd s Relaxation: iter 6 Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 52
53 Offline Lloyd s Relaxation: iter 7 Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 53
54 Offline Lloyd s Relaxation: iter 8 Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 54
55 Offline Lloyd s Relaxation: iter 9 Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 55
56 Offline Lloyd s Relaxation: iter 10 Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 56
57 Offline Lloyd s Relaxation: iter 11 Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 57
58 Offline Lloyd s Relaxation: iter 12 Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 58
59 Corrective Vectors Lookup Table Lloyd s Relaxation: iter 12 Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 59
60 Corrective Vectors Lookup Table Lloyd s Relaxation: iter 12 Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 60
61 Corrective Vectors Lookup Table Structural Indices ( i s ) 6 most-significant bits in F-code Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 61
62 Corrective Vectors Lookup Table Structural Indices ( i s ) F-code: most-significant bits: Decimal value: 5 Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 62
63 Corrective Vectors Lookup Table Structural Indices ( i s ) i s =5 Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 63
64 Corrective Vectors Lookup Table Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 64
65 Corrective Vectors Lookup Table Importance Indices ( i v ) Normalized Reference Subdivision Level Subdivision Levels Importance Value Importance Value 65
66 Corrective Vectors Lookup Table Importance Indices ( i v ) Normalized Reference Subdivision Level 66
67 System Summary Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 67
68 System Summary Sampled density function Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 68
69 System Summary Adaptive subdivision Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 69
70 System Summary Sampling tiles Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 70
71 System Summary Threshold with F-codes Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 71
72 System Summary Threshold with F-codes Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 72
73 System Summary Corrective Vectors Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 73
74 System Summary Final Sampling Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 74
75 Results Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 75
76 Results Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 76
77 Results Image credits: Paul Debev ec Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 77
78 Results Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 78
79 Results Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 79
80 Results Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 80
81 Case Study: Environment Map Sampling Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 81
82 Results Timings Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 82
83 Conclusions Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 83
84 Challenges and Contributions Extend basic Penrose tiling: simple shapes, no fractal boundaries Ordinal numbering of the vertices Bring as close as possible to the Blue Noise Efficient implementation Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 84
85 Contributions Extend basic Penrose tiling: simple shapes, no fractal boundaries Our extension, triangles & pentagons Ordinal numbering of the vertices Bring as close as possible to the Blue Noise Efficient implementation Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 85
86 Contributions Extend basic Penrose tiling: simple shapes, no fractal boundaries Ordinal numbering of the vertices Fibonacci number system Bring as close as possible to the Blue Noise Efficient implementation Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 86
87 Contributions Extend basic Penrose tiling: simple shapes, no fractal boundaries Ordinal numbering of the vertices Bring as close as possible to the Blue Noise Offline Lloyd relaxation Efficient implementation Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 87
88 Contributions Extend basic Penrose tiling: simple shapes, no fractal boundaries Ordinal numbering of the vertices Bring as close as possible to the Blue Noise Efficient implementation Lookup table-based approach, using structural and importance indices Motivations Penrose Tiling: a Primer Sampling System Fundamentals System Summary Results Conclusions 88
89 89
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