Variant of Lloyd's Method Optimizes Point Distributions

Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG
Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013

Capacity-Constrained Point Distributions
A Variant of Lloyd’s Method

Michel Alves dos Santos
Pós-Graduação em Engenharia de Sistemas e Computação
Universidade Federal do Rio de Janeiro - UFRJ - COPPE
Cidade Universitária - Rio de Janeiro - CEP: 21941-972
Docentes Responsáveis: Prof. Dsc. Ricardo Marroquim & Prof. PhD. Cláudio Esperança

{michel.mas, michel.santos.al}@gmail.com

January, 2013
Michel Alves dos Santos: Laboratório de Computação Gráﬁca - LCG

Pós-Graduação em Engenharia de Sistemas e Computação - PESC


Introduction
Applications of Point Distributions...


Presentation hosted on: http://www.lcg.ufrj.br/Members/malves/index



Introduction

Sampling



Introduction

Sampling

Point-Based Rendering




Introduction

Sampling


Geometric Processing




Introduction

Sampling


Geometric Processing


Halftoning

etc...



Desidered Properties
Desidered Properties for Point Distributions...

Red Noise

White Noise

Blue Noise

Blue noise features;
Similar relative distance between points;
No regular appearance (For most applications);
Adaptation to the provided density functions.



Desidered Properties
Desidered Properties for Point Distributions...

Red Noise

White Noise

Blue noise features;
Similar relative distance between points;
No regular appearance (For most applications);
Adaptation to the provided density functions.

Blue Noise

In this presentation we
will discuss about a
technique for optimal
distribution of points!



Capacity-Constrained Point Distributions
Capacity-Constrained Point Distributions: A Variant of Lloyd’s Method
Michael Balzer

Thomas Schl¨ mer
o
University of Konstanz, Germany

Oliver Deussen

Figure 1: (Left) 1024 points with constant density in a toroidal square and its spectral analysis to the right; (Center) 2048 points with the
2
2
density function ρ = e(−20x −20y ) + 0.2 sin2 (πx) sin2 (πy); (Right) 4096 points with a density function extracted from a grayscale image.

Abstract
New

that point distributions adapt to
density function in
general-purpose method for optimizingpoints in an a givenpoint sets;density.
existingis proportional to the the sense
that the number of
area

We present a new general-purpose method for optimizing existing
point sets. The resulting distributions possess high-quality blue
noise characteristics and adapt precisely to given density functions.
Our method is similar to the commonly used Lloyd’s method while
avoiding its drawbacks. We achieve our results by utilizing the concept of capacity, which for each point is determined by the area of
its Voronoi region weighted with an underlying density function.
We demand that each point has the same capacity. In combination
with a dedicated optimization algorithm, this capacity constraint
enforces that each point obtains equal importance in the distribution. Our method can be used as a drop-in replacement for Lloyd’s
method, and combines enhancement of blue Gráﬁca - LCG
Michel Alves dos Santos: Laboratório de Computaçãonoise characteristics

The iterative method by Lloyd [1982] is a powerful and
Resulting distributions possess high-qualitycommonly noise featuresand ﬂexible
blue used to enhance the spectral properties
technique that is
adapt precisely to given density; of existing distributions of points or similar entities. However, the

results from Lloyd’s method are satisfactory only to a limited ex-

tent. First, if the method is not stopped at a
Similar to the commonly used Lloyd’s Method while develop suitable iteration step,
the resulting point distributions will avoiding its
regularity artifacts, as
shown in Figure 2. A reliable universal termination criterion to
drawbacks;
prevent this behavior is unknown. Second, the adaptation to given
heterogenous density functions is suboptimal, requiring additional
application-dependent optimizations to improve the results.

We present a variant of Lloyd’s method which reliably converges toPós-Graduação em Engenharia de Sistemas e Computação - PESC


Proposed Method
initial point set

Lloyd’s method
α ≈ 0.75

α converged

our method
(converged)

zone plate test function

1024 points and their Fourier amplitude sprectrum

α ≈ 0.53

input sites

initial state

−→

capacity-constrained optimization

−→

final state

output sites

Figure 3: Our method takes an existing site distribution and transfers it to a random discrete assignment in which each site has the same
Figure 5:This initial set of is thenpoints is optimizedVoronoi regions are formed and sites are relocatedarethe centroids of their regions, while
capacity. An assignment 1024 optimized so that by Lloyd’s method. After 40 iterations the points to well distributed with a normalized
radius of α ≈ 0.75 Applications: characteristics. HDR Sampling an equilibriumspectral properties and introduces hexagonal
and good blue noise for each site. The optimization stops deteriorates the state with the final site distribution.
simultaneously maintaining the capacity Stippling, Further optimizationat Radiance/Luminance,2 etc.
structures. In contrast, α ≈ 0.75 proves to be ill-suited for the sampling of the zone plate test function with 512 points as strong artifacts
become apparent. Relying on the convergence of α is also not an option as only marginally fewer artifacts can be observed. In this sampling
scenario, stopping Lloyd’s method after about 10 iterations with α ≈ 0.53 would provide the best sampling results. Our method converges
2. move each site siem Engenharia de Sistemas of Computação - PESC
reliably to an equilibrium with better Voronoi Tessellation
Michel AlvesAlgorithm 1: Capacity-Constrainedproperties in both - LCG
dos Santos: Laboratório de Computação Gráfica scenarios.
Pós-Graduação ∈ S to the center of mass e all points


Proposed Method :: Steps and Details

Density Function → Samples → Generation of Sites → Optimization → Optimized Sites



Lloyd’s Method

Figure: (Left) Random dots (red) and polygons. (Right) Result after running
approximate Lloyd relaxation twice - note the artifacts produced by technique.
Used to enhance the spectral properties of existing point distributions.



Lloyd’s Method

But this method presents regularity in distribution!



Lloyd’s Method

Diﬃculty in stopping criterion!



Lloyd’s Method

And poor adaptation to heterogeneous density functions!



Capacity Constrained Vs. Lloyd’s Method
Capacity-Constrained

Lloyd’s Method

CCPD is a variation of the Lloyd’s Method that converges in a
natural way and that in addition not presents the appearance of
regularity still ﬁts precisely to given density functions.
CCPD Uses:

Complexity

Metrics or Distance Functions;
Lloyd
Centroidal Voronoi Tessellations;
CCPD
The Concept of Capacity;
Minimization of Energy (through a Optimization Method).

Memory



Metrics or Distance Functions

Voronoi Tesselations Using Minkowski Metrics: L1 , L2 , L3 , L4 , L5 , L∞ .



Metrics or Distance Functions :: Manhattan or L1

d (x, y) = L1 (x, y) =

n
i =1

|xi − yi |



Metrics or Distance Functions :: Euclidean or L2

d (x, y) = L2 (x, y) = (

n
i =1

|xi − yi |2 )1/2



Metrics or Distance Functions :: L3

d (x, y) = L3 (x, y) = (

n
i =1

|xi − yi |3 )1/3




d (x, y) = L4 (x, y) = (

n
i =1

|xi − yi |4 )1/4




d (x, y) = L5 (x, y) = (

n
i =1

|xi − yi |5 )1/5



Metrics or Distance Functions :: Chebyshev or L∞

d (x, y) = L∞ (x, y) = limp→∞ (

n
i =1

|xi − yi |p )1/p = maxin (|xi − yi |)
=1


Toroidal Square Distance
1

1

2

3

4
6
1

2

4

5
7

6
1

For the current work we used a metric
based on a toroidal square.



Toroidal Square Distance :: Algorithm
§
1
2
3
4
5
6
7
8
9
10
11
12
13

/∗ ∗
∗ Method r e s p o n s i b l e by d i s t a n c e c a l c u l a t i o n s [ t o r o i d a l s q u a r e ] .
∗ @ v a r i a b l e p1 and p2 : p o i n t s on t o r o i d a l s q u a r e .
∗ @ v a r i a b l e s i z e : keeps the dimensions of the input square .
∗/
d o u b l e TSD( c o n s t P o i n t 2& p1 , c o n s t P o i n t 2& p2 , c o n s t P o i n t 2& s i z e )
{
d o u b l e dx = p1 . x − p2 . x ;
i f ( f a b s ( dx ) > s i z e . x / 2 )
{
i f ( p1 . x < s i z e . x / 2 ) dx = p1 . x − ( p2 . x − s i z e . x ) ;
e l s e dx = p1 . x − ( p2 . x + s i z e . x ) ;
}

¤

14

d o u b l e dy = p1 . y − p2 . y ;
i f ( f a b s ( dy ) > s i z e . y / 2 )
{
i f ( p1 . y < s i z e . y / 2 ) dy = p1 . y − ( p2 . y − s i z e . y ) ;
e l s e dy = p1 . y − ( p2 . y + s i z e . y ) ;
}

15
16
17
18
19
20
21
22
23

}

return

s q r t ( dx ∗ dx + dy ∗ dy ) ;

¦


¥


Toroidal Square Distance :: Example

Optimized Sites

Voronoi Tessellation

Note the regions that lie within the limits.



Centroidal Voronoi Tessellation
Non-Centroidal Voronoi

Centroidal Voronoi

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0

0.2

0.2

0.4

0.4

0.6

0.6

0.8

0.8

1

1

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

1

1

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

CVT is a Voronoi Tesselation with the property that each site
itself coincides with the centroid of their respective Voronoi region.



Centroidal Voronoi Tessellation :: Applications
Optimal quadrature rules;
Covolume and ﬁnite diﬀerence methods for PDE’s;
Optimal representation, quantization, and clustering;
Optimal placement of sensors and actuators;
Optimal distribution of resources;
Cell division;
Finite volume methods for PDE’s;
Territorial behavior of animals;
Data compression;
Image segmentation;
Meshfree methods;
Grid generation;
Point distributions and grid generation on surfaces;
Hypercube point sampling;
Reduced-order modeling;



Centroidal Voronoi Tessellation :: Centroids
The centroid of a Voronoi region is nothing but the center of mass
of a region weighted by the density function deﬁned in V area.

z=

Vi
Vi

xρ(x)dx
ρ(x)dx

zi∗

=

M
i =1 xi ρ(xi )
M ρ(x )
i
i =1

For discrete sets of points we have V = {xi }M in Rn and a density
i =1
function ρ(xi ), i = 1, · · · , M. The center of mass is given by zi∗ .
The importance of centroidal Voronoi tessellation is established by
its relationship with the energy function:
M

F (S, V ) =
i =1 Vi

ρ(x)|x − si |2 dx

S → sites; V → voronoi regions; ρ(x) → density function; F (S, V ) → energy function



Centroidal Voronoi Tessellation :: Using Lloyd
Steps for generating a Centroidal Voronoi Tessellation:
1
2
3

Generate a Voronoi tessellation V (S) in a region Ω;
Move each site si ∈ S to the centroid pi of corresponding Voronoi
region vi ∈ V ;
Repeat the previous steps until the sites reach a convergence
criterion.

Energy Equation:
m

|xi − A(xi )|2

F (X, A) =
i =1

1. The relocation of the sites in the centroid position reduces energy F .
2. The algorithm converges to a local minimum F, where each site
coincides with the centroid of the region.
3. In the discrete case, the limited space Ω with density function ρ is
represented by a set X with m samples. A : X → S takes each point in
X to the nearest site in S.



The Concept of Capacity :: Capacity-Constrained
Let S be a set with n sites which determine a Voronoi tessellation
V (S) in limited space Ω with function density ρ(x ).
Definition: The capacity c(si ) of a site si ∈ S with respect to its
respective Voronoi region Vi ∈ V is defined as:
c(si ) =

ρ(x)dx
Vi

We say that a distribution of sites in S adapts optimally to a
density function if the capacity of each site follows:
ρ(x)dx
c(si ) = c ∗ , where c ∗ is defined as c ∗ =

Ω

n

In other words, the capacity of a site is equivalent to the area of
Voronoi region weighted by the density function.



Minimization of Energy :: Voronoi Tessellation
Steps for Minimization:
1
2

Generate the Voronoi tessellation doing the assignment A : X → S
of m points in X for n sites in S with capacity c ∗ .
Minimize the function F (X , A) by swap between two points in X
that belong to different sites in S, with the condition that the
energy is reduced;
Restrictions to make the swap ensure that capacity is maintained
even after the minimization.

3

Repeat the exchange until a stage of stability is achieved.

input sites

initial state

−→

capacity-constrained optimization

−→

final state

output sites

Figure 3: Our method takes an existing site distribution and transfers it to a random discrete assignment in which each site has the same
capacity. This assignment is then optimized so that Voronoi regions are formed and sites are relocated to the centroids of their regions, while
simultaneously maintaining the capacity for each site. The optimization stops at an equilibrium state with the final site distribution.


Overview of Capacity-Constrained Method
Capacity-Constrained:
The Capacity-Constrained method diﬀers from the usual Voronoi
tessellation because it is generated taking into account the capacity
of each site and only optimizing their locations.
Steps to generate the capacity-constrained distribution:
1

Create a set X with m points weighted by the density function ρ(x );

2

Generate the Voronoi tesselation V (S) with conditioned capacity for
the set of n sites S, where each site si has capacity c(si ) = m/n;

3

Move each site si ∈ S to the center of mass of all points xi ∈ X ;

4

Repeat steps 2 and 3 until the new sites achieve the convergence
criterion.

In possession of present provided theory, we will see some evaluations of results obtained according to Balzer et al. (2009).



Evaluation of Results
The method presented here achieves better results when compared
to the Lloyd’s Method, in some respects:

Blue-noise features;
Number of neighbors;
Stopping criteria;
Measuring the quality of the adaptation.
100 %

percentage

60 %

40 %

20 %

0%

4

5

6

7

8

0.95

1.1

0.85

our method

1.0

0.9

Lloyd’s method
0.7

our method
4

number of neighbors

(a) number of neighbors percentage

5

6

7

number of neighbors

(b) normalized Voronoi region area


8

normalized radius α

our method

normalized Voronoi region area

1.2

Lloyd’s method
80 %

0.75

0.65

0.55

16

64

256

1024

4096

16384

number of sites

(c) normalized radius α



Evaluation of Results :: Blue-noise and Neighbors
1024 optimized points

number of neighbors
4

5

6

7

spectral analysis
8
2

power

1.5
1

Lloyd’s method

0.5
0
0

frequency

fc

+10
anisotropy

+5
0

-5

-10
0

frequency

fc

2

power

1.5
1

0.5

our method

0
0

frequency

fc

+10
anisotropy

+5
0

-5

-10
0

frequency

fc

Lloyd’s method generates point distributions with regular structures.



Evaluation of Results :: Stopping and Quality
Stopping Criteria:
In the Lloyd’s Method is necessary a manual intervention or a
speciﬁc criterion determined by the application.
Quality of the adaptation:
The capacity oﬀers the opportunity to measure the quality of
adaptation by a distribution of sites through the errors of the
capacity given by:

δc =

1
n

i =1



c(si )



n

c∗

−

2
1

In respect of capacity:
Constant Density: Lloyd generates a uniform distribution with small errors.
Non-Constant Density: Lloyd generates distribution of sites with large errors.



Results

Using:

Regular Density Functions
Custom Density Functions
Images as Density Functions
Now we will see some results obtained with the technique...



Results :: Regular Density Functions
1.01
1.005
1.01

1

1.005

0.995

1

0.99

0.995
0.99

f(x,y) = c

f(x,y) = c

{(x,y) | x ∈ R, y ∈ R}

(x,y)-> random choice

All tabled numerical results shown in this presentation
are an average of 15 executions for each set of points.



Results :: Constant Regular Grid

Optimized Sites

f (x, y) = c;


{(x, y)|x ∈ R, y ∈ R}




Results :: Constant Regular Grid

1.01
1.005
1
1.01
0.995

1.005
1

1

0.995

0.99

0.5

0.99
0
-1
-0.5

-0.5

0
0.5
1

-1

Regular Density
f(x,y) --> samples

512 Points

1024 Points

2048 Points

16384 Samples

32768 Samples

65536 Samples

Figure: The ﬁgure above shows the number of points and samples for each set.
In the table below we can view times of generation, times of optimization and
steps until convergence.
Amount of Points
512
1024
2048

Generation Time
(<) 00.01 seconds
(<) 00.01 seconds
(<) 00.01 seconds

Optimization Time
00.03 seconds
00.08 seconds
00.25 seconds

Optimization Steps*
12
13
16

Optimization Steps*: number of iterations until convergence.



Results :: Constant Regular Random

Optimized Sites

f (x, y) = c;


(x, y) −→ random choice




Results :: Constant Regular Random
Lattice test for random numbers

1.01
1.00
0.80
0.60
0.40
0.20
0.00

1.005
1
0.995
0.99

0.00
1.00

0.20
0.80

0.40

0.60
0.60

0.40
0.80

0.20
1.00 0.00

Random Density
f(x,y) --> samples

512 Points

1024 Points

2048 Points

16384 Samples

32768 Samples

65536 Samples

Amount of Points
512
1024
2048

Generation Time
(<) 00.01 seconds
(<) 00.01 seconds
(<) 00.01 seconds

Optimization Time
00.04 seconds
00.09 seconds
00.31 seconds

Optimization Steps*
15
13
19




Results :: Regular Grid Vs. Regular Random
Grid

Random

128 Optimized Sites

128 Optimized Sites

Comparison between distributions obtained.
Each experiment used 65536 samples. Iterations: 66 for grid density and 99 for random density.



Results :: Custom Density Functions

Five Mountains

SinSquare

SinXCosY

Shadow Torus

The Waves

All tabled numerical results shown in this presentation
are an average of 15 executions for each set of points.



Results :: Five Mountains

Optimized Sites

f (x, y) = e(−20x

2

−20y 2 )


+ 0.2 sin2 (πx) sin2 (πy)


Results :: Five Mountains

Five Mountains
f(x,y) --> samples

1024 Points

2048 Points

4096 Points

32768 Samples

65536 Samples

131072 Samples

Amount of Points
1024
2048
4096

Generation Time
15.30 seconds
75.90 seconds
355.60 seconds

Optimization Time
00.12 seconds
00.39 seconds
01.29 seconds

Optimization Steps*
20
13
19




Results :: SinSquare

Optimized Sites


f (x, y) = sin (x 2 y 2 )



Results :: SinSquare

SinSquare
f(x,y) --> samples

1024 Points

2048 Points

4096 Points

32768 Samples

65536 Samples

131072 Samples

Amount of Points
1024
2048
4096

Generation Time
12.30 seconds
58.90 seconds
289.00 seconds

Optimization Time
00.18 seconds
00.65 seconds
02.31 seconds

Optimization Steps*
14
17
17




Results :: Shadow Torus

Optimized Sites


f (x, y) = (0.16 − (0.6 − x 2 + y 2 )2 )1/2



Results :: Shadow Torus

Shadow Torus
f(x,y) --> samples

1024 Points

2048 Points

4096 Points

32768 Samples

65536 Samples

131072 Samples

Amount of Points
1024
2048
4096

Generation Time
05.60 seconds
27.10 seconds
135.20 seconds

Optimization Time
00.11 seconds
00.29 seconds
01.20 seconds

Optimization Steps*
20
13
23




Results :: SinXCosY

Optimized Sites


f (x, y) = 0.2 sin (5x) cos (5y)



Results :: SinXCosY

SinXCosY
f(x,y) --> samples

1024 Points

2048 Points

4096 Points

32768 Samples

65536 Samples

131072 Samples

Amount of Points
1024
2048
4096

Generation Time
34.00 seconds
151.40 seconds
813.20 seconds

Optimization Time
00.12 seconds
00.36 seconds
01.33 seconds

Optimization Steps*
14
14
20




Results :: The Waves

Optimized Sites


f (x, y) = x 3 − 3xy 2



Results :: The Waves

The Waves
f(x,y) --> samples

1024 Points

2048 Points

4096 Points

32768 Samples

65536 Samples

131072 Samples

Amount of Points
1024
2048
4096

Generation Time
07.30 seconds
31.90 seconds
168.60 seconds

Optimization Time
00.15 seconds
00.55 seconds
01.72 seconds

Optimization Steps*
16
28
16




Points and
Optimization Time

Number of Points
and Samples

Results :: Stippling :: Image as Density Function

4096 Points

16384 Points

20000 Points

393216 Samples

786432 Samples

1280000 Samples

4096 Points

8192 Points

12288 Points

02.97 seconds

09.62 seconds

17.57 seconds




Results :: Stippling - Corn Plant/Dracaena

4096 Points

16384 Points

20000 Points

393216 Samples

786432 Samples

1280000 Samples

Figure: Input image with 1000x1000 pixels. The ﬁgure above shows the number
of points and the number of samples for each set. In the table below we can
visualize times of generation, times of optimization and steps until convergence.
Amount of Points
4096
16384
20000

Generation Time
25.50 minutes
101.70 minutes
271.20 minutes

Optimization Time
4.41 seconds
20.45 seconds
42.43 seconds

Optimization Steps*
57
24
35




Results :: Stippling - Madonna’s Face

4096 Points

8192 Points

12288 Points

02.97 seconds

09.62 seconds

17.57 seconds

Figure: Madonna’s Face. Input image with 1000x1000 pixels. The ﬁgure shows
the number of points and the optimization time for each set.
Amount of Points
4096
8192
12288

Generation Time
13.40 minutes
55.37 minutes
124.99 minutes


Optimization Time
02.97 seconds
09.62 seconds
17.57 seconds

Optimization Steps
38
36
38



Conclusions
Some conclusions about the method:

1.01
1.00
0.80
0.60
0.40
0.20
0.00

1.005
1
0.995

Performs distribution points optimally.
It is more stable than Lloyd’s Method
therefore uses the concept of capacity
as a form of optimization.
Improves the characteristics of the blue-noise
and has no apparent regularities in the
arrangement of sites.
Displays ‘precise’ adaptation to arbitrary
distribution functions.
No manual intervention is required and
neither depends on the initial distribution
to generate good quality results.

0.99

0.00

1.00

0.20

0.80

0.40

0.60

0.60

0.40

0.80


0.20
1.00 0.00



Thanks

Thanks for your attention!
Michel Alves dos Santos - michel.mas@gmail.com
Michel Alves dos Santos - (Alves, M.)

MSc Candidate at Federal University of Rio de Janeiro.

E-mail: michel.mas@gmail.com, malves@cos.ufrj.br
Lattes: http://lattes.cnpq.br/7295977425362370
Home: http://www.michelalves.com
Phone: +55 21 2562 8572 (Institutional Phone Number)

http://www.facebook.com/michel.alves.santos
http://www.linkedin.com/proﬁle/view?id=26542507



Samples
Voronoi
Sites
Thank you for your attention!



Bibliography
M. Balzer, T. Schlömer, and O. Deussen.
Capacity-constrained point distributions: A variant of Lloyd’s method.
ACM Transactions on Graphics (Proceedings of SIGGRAPH 2009), 28(3):86:1–8, 2009.
F. de Goes, K. Breeden, V. Ostromoukhov, and M. Desbrun.
Blue noise through optimal transport.
ACM Trans. Graph. (SIGGRAPH Asia), 31, 2012.
R. Fattal.
Blue-noise point sampling using kernel density model.
ACM SIGGRAPH 2011 papers, 28(3):1–10, 2011.
H. Li, D. Nehab, L.-Y. Wei, P. V. Sander, and C.-W. Fu.
Fast capacity constrained voronoi tessellation.
In Proceedings of the 2010 ACM SIGGRAPH Symposium on Interactive 3D Graphics and
Games, I3D ’10, pages 13:1–13:1, New York, NY, USA, 2010. ACM.
A. Secord.
Weighted Voronoi stippling.
In Proceedings of the second international symposium on Non-photorealistic animation and
rendering, pages 37–43. ACM Press, 2002.
R. Ulichney.
Digital Halftoning.
MIT Press, 1987.
ISBN 9780262210096.

I dedicate this presentation to Renata Thomaz Lins
do Nascimento, my love, my life!



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