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Blue noise refers to sample distributions that are random and well-spaced, with a variety of applications in graphics, geometry, and optimization. However, prior blue noise sampling algorithms typically suffer from the curse-of-dimensionality, especially when striving to cover a domain maximally. This hampers their applicability for high dimensional domains. We present a blue noise sampling method that can achieve high quality and performance across different dimensions. Our key idea is spoke-dart sampling, sampling locally from hyper-annuli centered at prior point samples, using lines, planes, or, more generally, hyperplanes. Spoke-dart sampling is more efficient at high dimensions than the state-of-the-art alternatives: global sampling and advancing front point sampling. Spoke-dart sampling achieves good quality as measured by differential domain spectrum and spatial coverage. In particular, it probabilistically guarantees that each coverage gap is small, whereas global sampling can only guarantee that the sum of gaps is not large. We demonstrate advantages of our method through empirical analysis and applications across dimensions 8 to 23 in Delaunay graphs, global optimization, and motion planning.
Spoke Darts for Efficient High Dimensional Blue Noise Sampling
Mohamed S. Ebeida
Sandia National Laboratories
Scott A. Mitchell
Sandia National Laboratories
Muhammad A. Awad
Alexandria University
Chonhyon Park
UNC Chapel Hill
Laura P. Swiler
Sandia National Laboratories
Dinesh Manocha
UNC Chapel Hill
Li-Yi Wei
Univ. Hong Kong
Blue noise refers to sample distributions that are random and well-
spaced, with a variety of applications in graphics, geometry, and
optimization. However, prior blue noise sampling algorithms typi-
cally suffer from the curse-of-dimensionality, especially when striv-
ing to cover a domain maximally. This hampers their applicability
for high dimensional domains.
We present a blue noise sampling method that can achieve high
quality and performance across different dimensions. Our key idea
is spoke-dart sampling, sampling locally from hyper-annuli cen-
tered at prior point samples, using lines, planes, or, more gener-
ally, hyperplanes. Spoke-dart sampling is more efficient at high
dimensions than the state-of-the-art alternatives: global sampling
and advancing front point sampling. Spoke-dart sampling achieves
good quality as measured by differential domain spectrum and spa-
tial coverage. In particular, it probabilistically guarantees that each
coverage gap is small, whereas global sampling can only guarantee
that the sum of gaps is not large. We demonstrate advantages of our
method through empirical analysis and applications across dimen-
sions 8 to 23 in Delaunay graphs, global optimization, and motion
Keywords: blue noise, sampling, high dimension, Delaunay
graph, global optimization, motion planning
1 Introduction
Blue noise refers to sample distributions that are random and well-
spaced. These are desirable properties for sampling in computer
graphics, as randomness avoids aliasing while uniformity reduces
noise. Blue noise sampling has been applied to a variety of graphics
applications, such as rendering [Cook 1986; Sun et al. 2013; Chen
et al. 2013], imaging [Balzer et al. 2009; Fattal 2011; de Goes et al.
2012], geometry [Alliez et al. 2003; ¨
Oztireli et al. 2010], animation
[Schechter and Bridson 2012], visualization [Li et al. 2010], and
numerical computation [Ebeida et al. 2014b].
Both the random and well-spaced properties are agnostic with re-
spect to the dimensionality of the underlying sample domain. Thus,
both high and low dimensional applications can and should benefit
from blue noise. However, most existing applications are limited to
low dimensions, predominantly 2-d, in part because of the difficulty
of producing good blue noise distributions in high dimensions. For
example, blue noise has been deployed to construct meshes for 2-d
and 3-d domains, with questionable practicality for much higher di-
mensions; in global optimization, point placement in higher dimen-
sions becomes a critical issue to efficiently explore the parameter
space; in robotic planning, blue noise has been used for configura-
tion spaces up to 6-d [Park et al. 2013] but not in higher dimensions
for more complex or realistic agents and environments.
A key reason mostl results are low dimensional is the curse of di-
mensionality — prior blue noise algorithms do not scale well to
high dimensions, especially when striving to fill in the domain max-
imally [Ebeida et al. 2011]. No prior algorithm can guarantee local
saturation within tractable runtime. The algorithms closest to ob-
taining this goal are based on advancing-front [Liu 1991; Bridson
2007] or k-d darts [Ebeida et al. 2014b]. Our method combines
these two approaches.
Advancing-front methods generate samples locally from the dis-
tribution boundary and gradually advanced towards the rest of the
domain. Building the geometric boundaries explicitly [Liu 1991]
through the intersections of sample disks results in a combinato-
rial explosion in complexity in high dimensions. Advancing-Front
Point dart-throwing (AFP) [Bridson 2007] avoids building the front
geometry. Each accepted sample has a disk around it that rejects fu-
ture samples. For each sample, AFP does rejection sampling from
an annulus around its disk, and proceeds to the next sample after
a fixed number (30) of consecutive rejections. Its advantage is lo-
cality. The advancing front mitigates the effects of domain size.
It ensures that the saturation properties around the current sam-
ple also apply to future and past samples. Its disadvantage is that
point-rejection yields a volume-fraction saturation guarantee which
decreases exponentially with dimension; worse, this guarantee ap-
plies only within the annulus and there is no known bound on the
void volume outside all annuli.
k-d darts [Ebeida et al. 2014b] selects samples using hyperplanes:
select a random axis-aligned hyperplane, find its uncovered subset,
and select a point from this subset. A rejection occurs only when
the entire hyperplane is covered. Its advantage is that hyperplanes
mitigate the effects of high dimensions, because rejection is much
less likely than for point samples. Its disadvantage is that it does not
provide local saturation, because hyperplanes are selected globally
from the entire domain.
Our method achieves guaranteed local saturation within tractable
runtime. Our key idea is to combine the advantages of the two prior
methods: the local saturation of AFP and the dimension-mitigation
of k-d darts. Specifically, our method replaces the point-sampling
of AFP with hyperplane sampling, especially line sampling.
We call our method spoke-dart sampling. A spoke-dart is a set of
spokes passing through a prior sample point. Each spoke is a hyper-
annulus, such as 1-d line segments or a 2-d planar ring as illustrated
in Figure 2. In contrast to constructing the front explicitly, we trim
each spoke by existing sample disks, and select the next sample
from the remaining regions. Since each spoke has a local scope, the
trimming can be efficiently performed locally instead of globally as
in k-d darts.
The advancing front nature also helps our method saturate the do-
main better than prior dimensionality-agnostic methods such as
brute-force dart throwing [Cook 1986] or AFP. Moreover, global
methods such as dart throwing and k-d darts only provide global
saturation. If parameters are chosen so that a global method and
spoke-dart sampling produce the same total gap volume, spoke-
dart sampling will likely have that volume distributed in small gaps
throughout the domain, whereas a global method might have the
gap volume concentrated in one large component. Thus spoke-dart
sampling ensures that the maximum distance from a domain point
to its nearest sample, rc, is smaller. With high probability, (1 ),
it achieves the user-desired saturation, as measured by the desired
ratio β=rc/rfbetween the coverage radius rcand conflict radius
arXiv:1408.1118v1 [cs.GR] 5 Aug 2014
(a) Delaunay graph (b) global optimization
(c) motion planning
Figure 1: Spoke-dart sampling is an effective method for high-dimensional
sampling with applications in Delaunay graph construction (a), global op-
timization (b), and motion planning (c).
rfof sample disks [Mitchell et al. 2012a].
Spoke-dart sampling has polynomial (instead of exponential)
time complexity with respect to the sample-space dimension
dand number of samples n. The only exponential-in-dde-
pendency is when high saturation is desired. The time is
Od(log )(β1)1dn2;if β= 2 and a fixed epsilon (say
105) is desired, then this reduces to O(dn2).We do not know
of explicit formulas for the run-time needed for dart-throwing and
k-d darts to achieve a given local β, but for large domains it ap-
pears that they scale much worse than spoke-dart sampling. An-
other key advantage of spoke-dart sampling is it requires only lin-
ear memory, O(dn),regardless of saturation β. This is in sharp
contrast to grid-based methods such as Simple MPS [Ebeida et al.
2012], where the memory is exponential, O(2d),when the top level
grid is first refined; and careful management is needed to avoid
the O(((log2(β1))d)memory for naive grid refinement. To
our knowledge, we provide the first feasible method to produce
probabilistically-guaranteed locally-saturated blue-noise point-sets
in high dimensions.
We apply spoke-dart sampling to high dimensional Delaunay
graphs, global optimization, and motion planning; see Figure 1.
We generate an approximate Delaunay graph in high dimensions,
where the exact version is too expensive to generate. For global op-
timization, the well-known DIRECT algorithm [Jones et al. 1993]
for Lipschitzian optimization might not scale well to high dimen-
sions due to its hyperrectangular sample neighborhoods and deter-
ministic patterns. Our method places hyperspheres stochastically,
and significantly improves convergence speed. For motion plan-
ning, a common method is RRT (Rapidly-exploring Random Tree)
[Kuffner and Lavalle 2000]. MPS has been used in RRT for up to
six dimension [Park et al. 2013]. Our method can efficiently handle
configuration spaces with more than 20 dimensions.
The contribution of this paper can be summarized as follows:
The idea of spoke-dart sampling, which combines the advan-
tages of state-of-the-art methods: the locality of advancing-
front and the dimension-mitigation of k-d darts;
Efficient trimming and sampling algorithms for spoke-darts in
different dimensions;
Empirical and theoretical bounds for key measures such as
time and memory complexity, and βfor coverage (a.k.a. sat-
uration, maximality, or well-spacedness);
Applications in high dimensional Delaunay graphs, global op-
timization, and motion planning.
2 Background
Blue noise sampling algorithms and applications have much prior
art. Here we focus on those most relevant to high dimensionality,
maximal sampling, non-point samples, and advancing front.
High dimensionality The curse-of-dimensionality refers to the
difficulty of obtaining efficient data-structures and algorithms in
high dimensions. The goal is to avoid time and memory complexi-
ties that have higher than polynomial growth.
Examining nearby samples (e.g. for conflict) is a key step among
all known blue noise methods. However, the number of poten-
tial neighbors grows exponentially with dimension d, related to the
mathematical “kissing number. Finding them efficiently is also
difficult. Neighborhood queries is an active research topic in com-
putational geometry [Samet 2005; Arya and Mount 2005; Miller
et al. 2013]. In high dimensions, it is hard to be more efficient than
examining every point. The good news is that this is only linear in
the output size.
Maximal sampling A maximal disk sampling is fully saturated
and can receive no more samples. As a consequence the distribution
is well-spaced. Such saturation is important for many applications,
as described in the extensive literature on Maximal Poisson-disk
Sampling (MPS) [Cline et al. 2009; Gamito and Maddock 2009;
Ebeida et al. 2011; Ebeida et al. 2012; Yan and Wonka 2013].
There are practical algorithms that achieve maximality in low di-
mensions, but none do so in high-dimensions. The maximal meth-
ods for low dimensions use data structures (e.g. grids or trees)
which do not scale well beyond six dimensions, as surveyed in
Ebeida et al. [2014b]. Brute force dart throwing [Dipp´
e and Wold
1985; Cook 1986] scales easily to high dimensions, but does not
reach maximality even in low dimensions. The notion of a “relaxed
MPS” [Ebeida et al. 2014b], one that is measurably saturated but
short of maximal, is attractive for higher dimensions. Our method
achieves a form of relaxed MPS and is more efficient than the prior
k-d darts method [Ebeida et al. 2014b].
Two-radii MPS methods [Mitchell et al. 2012b; Ebeida et al. 2014a]
provide a way to adjust and measure the saturation of Poisson-disk
distributions. The key idea is to define two radii around each sam-
ple, rcfor domain coverage and rffor inter-sample distances. In
particular, rcis the maximum distance between a domain point and
its nearest sample, and rfis the minimum distance between two
samples. Their ratio β=rc/rfquantifies saturation, and affects
the randomness and well-spacedness of the distribution. We use β
as a control parameter, as do some prior works. Ebeida et al. [2013]
is a post-processing algorithm that can reduce the number of sample
points while preserving β= 1.
Beyond point samples k-d darts [Ebeida et al. 2014b] is the
state-of-the-art for high-dimensional relaxed Poisson-disk sam-
pling, and uses axis-aligned hyperplanes to find regions of interest.
A hyperplane is especially efficient for dealing with arrangements
of spheres, because its intersection with a sphere is a simple ana-
lytic sphere in the lower dimensional space of the hyperplane. How-
ever, k-d dart hyperplanes extend throughout the entire domain, and
it can be expensive to intersect them with densely sampled sub-
regions. Here we seek to avoid global computation. Our method
is essentially a local, advancing front version of the relaxed MPS
method in k-d darts [Ebeida et al. 2014b].
Sun et al. [2013] samples lines and line-segments for rendering ap-
plications, including 3-d motion blur, 4-d lens blur, and 5-d tempo-
ral light fields. For determining sample positions it relies on sub-
routines that do not scale well to high dimensions.
Advancing front Advancing front methods were initially pro-
posed for meshing [Liu 1991; Li et al. 1999; Liu et al. 2008] and
later adopted for sampling in graphics [Dunbar and Humphreys
2006]. The basic idea is to draw new samples from boundaries
(fronts) of existing sample sets and gradually expand towards the
rest of the domain. These methods are designed mainly for low di-
mensions and can run fast in 2-d and 3-d. However, they maintain
geometric and combinatorial structures for the fronts which do not
scale well to high dimensions. Our method is based on advancing
front, but maintains and samples from an implicit front, to avoid
this combinatorial complexity.
3 Method
Our spoke-dart sampling method builds and improves upon both ad-
vancing front and k-d darts. Similar to advancing front, our method
generates new samples from the current sample set boundary and
gradually expands towards the rest of the domain. The key dif-
ferences between our method and prior methods are (1) how the
front is constructed and (2) how new samples are drawn. In partic-
ular, prior methods compute fronts and samples from intersections
of existing sample disks which can be quite complex, whereas our
method uses spoke-darts with very simple structures.
Similar to k-d darts [Ebeida et al. 2014b], spoke-darts are sets of
k-dimensional hyperplanes. However, unlike k-d dart hyperplanes
which are global and axis-aligned, spokes are local hyper-annuli
and randomly oriented. A spoke-dart passes through an extant sam-
ple. Such locality and randomness properties given spoke-darts
computational advantages over k-d darts.
Below, we first describe the basic representation and operations for
spoke-darts, followed by how we use them for blue noise sampling.
3.1 Representation
A spoke-dart is a set of randomly-oriented hyper-annuli that pass
through a given sample point s. Each such annulus is a spoke with
radius spanning [r, r +w], where wis its size (width) and ris its
starting distance away from the sample. For blue noise we use r
equal to the Poisson disk radius, and for some later applications we
use r= 0.Spokes can have various dimensions, up to the dimen-
sion of the sample domain. For example, a 2-d domain can have 1-d
line-spokes and 2-d plane-spokes, as illustrated in Figure 2. Line-
spokes are line segments starting from a random point on the disk
D(s)of a given sample sand extending in the radial direction for a
distance of w. Similarly, a plane-spoke is an annulus starting from
a random great circle on D(s)and extending in the radial direction
for a distance of w.
Spokes can be degenerate in the sense that they have w= 0. A
degenerate line-spoke reduces to a point, and a degenerate plane-
spoke reduces to a great circle, on D(s).
Spokes with three or higher dimensions can be defined analogously.
In principle spoke-dart sampling works with spokes of any dimen-
sion. Since we have used only line-spokes and plane-spokes in our
current implementation, we describe only those below.
(a) line-spokes
(b) plane-spokes
Figure 2: Spokes in a 2-d domain. In (a), a line-spoke (blue) is a radial
line segment of length wstarting from the disk surface D(s)of a sample s.
Degenerate line-spokes (yellow) have zero length and lie on the disk surface.
In (b), a plane spoke (blue) is a planar annulus. Degenerate plane-spokes
(yellow) are circles on the disk. Here pand qindicate example samples
drawn from degenerate and full spokes.
3.2 Operations
As in Poisson-disk sampling, spoke-darts are used to explore the
gaps or voids, the uncovered space that can accept a new sample.
Specifically, we trim a spoke with existing sample disks, and select
a new sample from the remaining uncovered hyperplane region(s).
The key to our design is that all trimming operations are efficient in
high dimensions, even with many nearby sample disks. In particu-
lar, we leverage the degenerate versions of spokes to avoid wasting
time while trimming their full versions.
Spoke generation A random line-spoke is generated by select-
ing its starting position pon the disk surface D(s)uniformly by
area. A random plane-spoke is generated by selecting a great cir-
cle with uniform random orientation, by selecting two such points
that the plane passes through. To generate p, we generate each of
its dcoordinate independently from a normal (Gaussian) distribu-
tion. Then we linearly scale the vector of coordinates to the disk
radius [Muller 1959]. (For r= 0 spokes, pdefines direction only.)
In our current implementation, wis initialized by a global constant.
Line-spoke trimming Our line-spoke trimming method is sum-
marized in Algorithm 1. Our implementation uses two simplifica-
tions for efficiency, to avoid the problem that trimming a full spoke
directly by nearby disks can involve a lot of wasted effort. There
could be many (kissing number) nearby disks.If we iterate over the
disks, many segments are generated early on, which are completely
covered later. Often there is nothing left, but it took many oper-
ations to discover this. Our first simplification is to check if the
degenerate version of a line-spoke (a point) is covered before trim-
ming its full version. This involves no partitioning, and it is dis-
carded by the first disk we find that covers it. It is possible that
we might miss an uncovered segment of a line-spoke, but over-
all it is more efficient. The second simplification is to keep only
the line segment that touches D(s), and ignore the rest of the par-
tition. Hence each trimming operation shortens the length of the
line-spoke instead of fragmenting it into pieces. This favors insert-
ing nearer samples, but the net effect is small; see Figure 4.
Input: input line spoke `1at sample s
Output: trimmed spoke `0
1: `0
2: for each sample s0near sdo
3: `0
4: `0
1only the segment of `0
1touching D(s)
5: if `0
1is empty then
6: return empty
7: return `0
Algorithm 1: Trimming a line spoke.
Input: input plane spoke `2at sample s
Output: trimmed line spoke `1`2
1: `0
2Degenerate(`2)// circle
2: pRandomSample(`0
3: while phas not traversed a full revolution of `0
4: if pis covered by a disk D(s0)then
5: pLeftIntersection(`0
2, D(s0))
6: else
7: wthickness of `2
8: `1LineSpoke(p, w)
9: return TrimLineSpoke(`1)// Algorithm 1
10: return empty // `0
2was covered
Algorithm 2: Trimming a plane spoke.
Plane spoke trimming We apply similar concepts for trimming
plane-spokes as summarized in Algorithm 2. For efficiency we start
with a degenerate spoke, a circle. We search for an uncovered point
on its circle as follows. Take any point pfrom the circle; a point
we used to generate the plane is a good choice. If pis covered by a
disk, we move p“left” along the circle to the intersection point of
the circle and that disk. We repeat this until pis uncovered; or we
have passed our starting point, in which case the circle is completely
covered. For a full plane-spoke, we now throw a line-spoke passing
through pand trim it as in Algorithm 1.
3.3 Sampling
With the representation and operations of spoke-darts above, our
blue noise generation method works as follows. We initialize the
output set with one random sample and put it into the active pool of
front points. When this pool becomes empty our algorithm termi-
nates. We remove a random sample sfrom the pool and try to gen-
erate new samples s0from random spokes `through s. Accepted
samples are added to the pool. We keep throwing spokes from the
same sample suntil mconsecutive spokes failed to generate an ac-
ceptable sample. Our method is summarized in Algorithm 3.
Input: sample domain
Output: output sample set S
1: sRandomSample(Ω)
2: S ← {s}
3: P ← {s}// active pool
4: while Pnot empty do
5: sRandomSelectAndRemove(P)
6: reject 0
7: while reject mdo
8: `RandomSpoke(s, r, r +w)
9: `Trim(`)// Algorithm 1 or 2
10: if `not empty then
11: s0RandomSample(`)
12: S S S{s0}
13: P P S{s0}
14: reject 0
15: else
16: reject reject + 1
17: return S
Algorithm 3: Sampling with spoke-darts.
3.4 Implementation
Note that for each sample on the front, for each spoke-dart we it-
erate over the nearby samples, at distances less than 3r. It saves
time to collect a sample’s neighbors once before throwing any of its
spokes, if the number of neighbors happens to be less than the total
number of points, N < n. Those wishing to reproduce our output
may simply iterate over all the points and gather these neighbors in
an array. In our implementation, we have found that using a k-d
tree saves time in moderate dimensions.
We maintain a k-d tree of the entire point set. We collect the subtree
of neighbors. We update them as we successfully add new disks.
If the neighbor list is huge, Nn, as can happen when dis
very large, then these trees do not save any time over an array, but
they are not significantly more costly either. (None of our run-time
proofs depend on these trees.)
4 Analysis
Here we compare our method against the state-of-the-art, and ana-
lyze the quality and performance of the variations of our method:
{line, plane}×{full, degenerate}spokes.
To the best of our knowledge, k-d dart [Ebeida et al. 2014b] is the
state-of-the-art method for high dimensional blue noise sampling
with high saturation, so we compare to it. For dimensions below
six, we may compare to MPS output produced by the Simple MPS
algorithm [Ebeida et al. 2012].
4.1 Performance Analysis
To consider both speed and saturation, we measure the accumulated
computation time with respect to the number of generated samples
across different dimensions for all candidate methods. Since the
distributions generated by the different methods are not the same,
especially for degenerate and non-degenerate spokes, equal number
of points does not mean equal saturation.
k-d dart versus spoke-darts As shown in Figure 3, when the
domain is relatively empty, line darts [Ebeida et al. 2014b] are bet-
ter than our methods. However, when the domain is relatively full,
100,000$ 150,000$ 200,000$
(a) d=4
50,000$ 150,000$ 250,000$
(b) d=6
0$ 100,000$ 200,000$
(c) d=8
50,000$ 150,000$ 250,000$
Spokes$ Degen.$
(d) d = 10
Figure 3: Comparison between k-d dart and different variations of our method under different dimensions. Each result is sampled from a domain is a
d-dimensional unit box domain with Poisson disk spacing rf. The goal is to fill in the domains with as many samples as possible under the same amount
of computation time. Note that towards the end game with higher fill-rates, our methods consistently outperform k-d dart, with plane-spokes outperforming
line-spokes, and the degenerate versions outperforming the full versions.
our methods are better. It appears that after a critical level of sat-
uration, line darts nearly stall, while spoke-darts continue to add
points. These thresholds depend on the dimension and radius. Fig-
ure 3 shows that if one desires a highly saturated (low β) distri-
bution, our method is orders of magnitude faster. These empirical
results are consistent with our earlier theoretical observations about
the global versus local nature of k-d darts and spoke-dart sampling.
Variations among spoke-darts Figure 3 also points towards the
general trends that degenerate spokes are more efficient than full
spokes, and plane-spokes are more efficient than line-spokes, at
producing points.
4.2 Quality Analysis
We measure quality via coverage β[Mitchell et al. 2012b; Ebeida
et al. 2014a] and the inter-point distance distribution.
Local saturation Figure 4b shows the βguaranteed by the theory
(βguaranteed) and achieved in practice (βachieved) for different values
of m. We see that line-spokes typically achieve a nearly-maximal
(β1) distribution, and can do so using many fewer spokes than
required in theory. Using d= (βguaranteed 1)/(βachieved 1) for
dimension d, we see 48.Figure 4a shows nby m.
1" 100" 10000"
(a) Total inserted points by m.
0" 200" 400" 600" 800" 1000"
(b) βfor different m
Figure 4: Local saturation (coverage) for line-spokes in theory and
practice, for the number of spoke misses m. Here βguaranteed is
the probabilistically-guaranteed saturation upper-bound in theory, and
βachieved is the βobserved in experiments. In practice βis about 8×
closer to 1 than the theory guarantee. For example, for m= 60 we
have βachieved 1.08, almost maximality, whereas βguaranteed 1.6.Since
rfrin practice, it is rcthat is determining β=rc/rf.
(a) different spoke types
(b) different m
Figure 5: Radial profiles, from differential domain distributions [Wei
and Wang 2011]. (a) Different sampling types. Spokes all use the same
m= 1000. Line-spokes, line darts (k-d darts with lines), and Simple MPS
produce nearly identical peaks, but line darts is farthest from saturation. (b)
Line-spokes with different m. The peak becomes more pronounced as satu-
ration is approached. All distributions are quite flat for larger distances.
Distribution We analyze sample distributions via Differential
Domain Analysis (DDA) [Wei and Wang 2011], which essentially
computes histograms of spatial distances between samples. We use
DDA instead of Fourier spectral analysis [Lagae and Dutr´
e 2008]
because it is faster and easier to compute, especially in high di-
mensions; and the two are equivalent, differing only by the choice
of Gaussian versus sinusoidal kernels [Wei and Wang 2011]. In
Figure 5 we plot the 1-d radial means of the high dimensional dis-
tributions with respect to different spoke types and m. As shown in
Figure 5a, all spoke types exhibit MPS-like characteristics. Degen-
erate spokes tend to have sharper profiles than full spokes, analo-
gous to the sharper profiles of boundary sampling methods [Dunbar
and Humphreys 2006]. Figure 5b shows that higher mwill produce
sharper profiles due to higher saturation as measured by β.
4.3 Parameter Trade-offs
The more spokes we generate, the longer the run-time, but the more
saturated the output. Our main control parameter is m, the number
of successively-failed spokes for a given extant sample disk before
removing it from the front. For line-spokes, our guarantee is that
for a given m, with high probability (1 )the achieved βachieved
is less than β.
How many spoke misses are enough? If the user selects the desired
β, then Equation (1) says how large mmust be. Conversely, the
user may pick mbased on a computational budget, and Equation (1)
describes what the probabilistically-guaranteed βwill be. Note β >
1,and ln  > 0,and m1.
m=l(ln )(β1)1dmβ= 1 + ln
One can also pick mand βand bound the probability that βwas
exceeded:  < exp (m(β1)d1).In Figure 6, we see that
m= 12 for β= 2 and = 105and all d. The bound on min
Equation (1) is quite practical for moderate dimensions and β.
1& 1.2& 1.4& 1.6& 1.8& 2&
(a) full scale
1& 1.2& 1.4& 1.6& 1.8& 2&
4" 8"
(b) zoom in
Figure 6: Illustration of Equation (1) with = 105.Only 12 successive
misses per sample are required to achieve β= 2, regardless of d.
We provide some proof intuition for Equation (1) here; the actual
proofs are in the supplementary material, Appendix A. Let us sup-
pose that the algorithm has terminated and there is a void, some
connected part of the domain whose points are farther than rffrom
all samples. This void is bounded by some sample disks, and we
have thrown at least mspokes from each of these disks. Each of
these spokes must have missed this void, otherwise we would have
inserted a sample and reset our miss count. The chance of getting
msuccessive misses is the chance of one miss to the mth power,
which shows mis dependent on the log of . The chance of a single
spoke missing this void is the area the void shares with the spoke’s
disk, divided by the surface area of the disk. Combining this for
all disks bounding the void shows that the natural log of the chance
that they all missed is (at most) proportional to the area of one disk
divided by the total area of the void boundary. Since the void was
not hit, the area of the void is probabilistically-guaranteed to be
small. A void with small area has a small maximum distance from
its interior to its boundary (this is rcrf), in particular smaller than
the radius of a ball with the same surface area as the void. Thus we
get a bound on rc. The exponential-in-(d1) dependence on βis
precisely the dependence of the surface area of a d-ball on its ra-
dius. For β= 2,we only care about voids with at least the surface
area of one of our disks, and this dependence disappears.
In practice, we achieve a much better βthan the theoretical guar-
antee, for all m; See Section 4.2 for a description. This is expected
because the proof makes several worst-case assumptions, such as
the void being shaped like a ball, and ignores chains of misses less
than m.
To bound the overall run-time, we must account for these small miss
chains. We assign the cost of a small miss chain to the successful
sample disk insertion following it, not the disk generating the chain.
Thus each sample accounts for the (< m 1) misses preceding it,
the spoke that created it, plus its own mfinal successive misses, for
a total of at most 2mspokes. Thus in the entire algorithm we throw
at most 2mspokes. Each line-spoke takes time O(dN)to trim,
where N < n is the number of nearby disks in the pool. Multiply-
ing these gives time O(dmn2) = Od(ln ) (β1)1dn2.
5 Applications
We present applications of our method in Delaunay graph (d=
6–14, Section 5.1), global optimization (d=6–15, Section 5.2),
and motion planning (d > 20, Section 5.3). In particular, we use
our spoke-dart operations (Section 3.2) for constructing approxi-
mate Delaunay graphs from given samples, while global optimiza-
tion and motion planning can benefit from our placement of new
samples in blue noise distributions (Section 3.3). All these applica-
tions rely on the underlying domains being sampled as maximally
as possible, as measured by β.
5.1 High-dimensional Delaunay Graph
There is an increasing demand for high dimensional meshes in
various fields such as uncertainty quantification [Witteveen and
Iaccarino 2012] and computational topology [Gerber et al. 2010].
Many applications rely on knowing the distance- and directionally-
significant neighbors of points. These applications often rely on
Delaunay graphs as a core component. Many methods for con-
structing the exact Delaunay graph, D, suffer from the curse of
dimensionality and their effectiveness deteriorates very rapidly as
the dimension increases. Some recent theoretical papers [Miller
and Sheehy 2013; Miller et al. 2013] have considered approximate
graphs and the problem of dimension from the standpoint of com-
plexity analysis, although no implementations or experimental re-
sults of these algorithms are available.
In this section we apply our spoke-dart method to generate an ap-
proximate Delaunay graph D, which contains with high proba-
bility those edges whose dual Voronoi faces subtend a large solid
angle with respect to the site vertex. We call these edges significant
Delaunay edges, and the corresponding Dasignificant Delau-
nay graph. The significant edges are a subset of the true Delaunay
edges, and the Voronoi cell defined by the significant neighbors ge-
ometrically contains the true Voronoi cell. Many high dimensional
applications, such as the classic approximate nearest neighbor prob-
lem, accept approximate Delaunay graphs. One such application is
high dimensional global optimization, as shown in Section 5.2. To
the best of our knowledge, we present the first practical technique
to find a significant Delaunay graph in high dimensions. As a fur-
ther benefit, for each edge our method produces a witness, a domain
point on its true Voronoi face, which can be used to estimate the ra-
dial extent δof the Voronoi cell. This is demonstrated in our global
optimization application; Section 5.2.
Our basic idea is to throw random line-spokes to tease out the sig-
nificant Delaunay edges from a set of spatial neighbors. This is a
very simple method that scales well across different dimensions. It
is summarized in Algorithm 4 with details as follows. We construct
the graph Dfor each vertex sin turn. We initialize its edge pool
with all vertices that are close enough to possibly share a Delaunay
edge with s. We next identify vertices from this pool who are actual
Delaunay neighbors of swith the following probabilistic method.
Using spoke-darts, we throw mline-spokes. We trim each spoke
`using the separating hyperplane between sand each vertex s0in
the pool. There is one pool vertex swhose hyperplane trims `the
most. (In so-called “degenerate” cases multiple vertices trim the
spoke the most and equally. Then we can pick an arbitrary one for
s.) The far end of the trimmed spoke ωis equidistant from sand
s, and no other vertex is closer. Hence ωis the witness that sand
sshare a Voronoi-face (Delaunay-edge), and ssis added to D.
The reason we tend to find the significant neighbors with high prob-
ability is obvious from the above algorithm description. Spokes
sample the solid angle around each vertex suniformly, so the prob-
ability that a given spoke hits a given Voronoi face is proportional
Input: vertex s, Delaunay graph D, NeighborCandidates M, Re-
cursionFlag R
Output: Dwith sadded
1: N=// approx. Delaunay neighbors of s
2: δ(s) = 0 // approx. cell radius of s
3: for i= 1 to mdo
4: `RandomLineSpoke(s, 0,||)// Section 3.2
5: for each sample s0∈ M do
6: π(s, s0)hyperplane between sand s0
7: trim `with π(s, s0)
8: if `got shorter then
9: ss0
10: D← DSss// set union without duplication
11: N ← N S{s}
12: δ(s) = max (δ(s),length(`))
13: if R=true then
14: // update edges of neighbors, removing some
15: for each sample s0∈ N do
16: M ← Neighbors(s0)S{s}
17: D← D\Edges(s0)// remove all edges
18: Recurse(s0,D,M,false)// restore some
19: return D
Algorithm 4: Adding a vertex to the approximate Delaunay graph via our
method. For a new vertex R=true.
to the solid angle the face subtends at s. As the number of spokes
mincreases we are more likely to also find less significant neigh-
bors, and D→ D. (This is analogous to our blue noise sampling
algorithm in Algorithm 3, where larger mincreases our chance of
finding even the small voids.)
6" 8" 10" 12" 14"
(a) computation time
6" 8" 10" 12" 14"
(b) memory requirement
Figure 7: Comparison of speed (a) and memory (b) between Qhull and
spoke-dart sampling for an approximate Delaunay graph. Qhull becomes
infeasible beyond d= 10 whereas our method scales well.
4# 40# 400# 4000# 40000#
(a) effects of mon % of missing edges
4$ 40$ 400$ 4000$ 40000$
(b) effects of mon time
Figure 8: Effects of mon the approximate Delaunay graph. As min-
creases, fewer Delaunay edges are missed (a) but run-time increases (b).
We demonstrate the efficiency of our approach against Qhull [Bar-
ber et al. 1996], a commonly-used code for convex hulls and Delau-
nay triangulations. As test input, we used Poisson-disk point sets
over the unit-box domain in various dimensions. For each case, we
used Qhull to generate the exact solution Dand our method for the
approximate solution D. As Figure 7 shows, the memory and time
requirements of Qhull grows significantly as dincreases. Qhull re-
quired memory that might not be practical for d11. On the other
hand, our method shows a linear growth for time and memory with
d. We see that our method became competitive for d9. Figure 8
shows the effect of mon the time and number of missed edges.
5.2 Rethinking Lipschitzian Optimization
A variety of disciplines — science, engineering or even economics,
— seek the “absolutely best” answer. This usually involves solv-
ing a global optimization problem, where one explores a high-
dimensional design space to find the optimum of some objective
function under a set of feasibility constraints. Local optimality is
not enough. For simple analytical functions, some algorithms are
guaranteed to find the global minimum. However, no method is
guaranteed to find the global minimum for all functions, or even
come close in finite time; for example, no method is guaranteed to
find the minimum of a function resembling white noise. Heuristic
stochastic techniques are usually the best in practice, and some-
times the only option [Horst et al. 2002]. In particular, we con-
sider the important category of Lipschitzian optimization methods
for complex but well-behaved, high-dimensional functions. We
demonstrate how spoke-dart sampling can improve upon DIRECT,
which for many decades has been a preferred method. We believe
this opens the door to new approaches.
Lipschitzian optimization [Shubert 1972] explores the parame-
ter space and provides convergence based on the Lipschitz constant
of the objective function. Specifically, a function fis Lipschitz con-
tinuous with constant K > 0if
|f(xi)f(xj)| ≤ K|xixj|(2)
for all xi6=xjin the feasible domain of f. One can use this con-
dition to show that a neighborhood around a sample point cannot
contain the best solution, and hence can be discarded. In particular,
if Kis known and the best currently-known answer is f, then a
ball around xiof radius |f(xi)f|/K has values above f. We
only need to search the space outside this ball.
DIRECT [Shubert 1972] has two limitations: poor scaling to high
dimensions; and relying on a global K, whose exact value is often
unknown. The DIRECT algorithm [Jones et al. 1993] generalizes
[Shubert 1972] to higher dimensions and does not require knowl-
edge of the Lipschitz constant. DIRECT partitions the domain into
hyperrectangles. It refines those rectangles that could contain a bet-
ter point than the currently best-known f.This refinement recurses
until reaching the maximum number of iterations, or the remaining
possible improvement is small. In particular, DIRECT determinis-
tically decides to refine the jth rectangle if
ii= 1,2, ..., m, and (3)
Here cjis the center of the rectangle, and δjis the distance from
the rectangle’s center to its (farthest) corner. Also ˜
Kruns over all
positive real numbers, and the index iruns through all cells in the
domain. The best currently-known value is f,and is a small
positive number. Intuitively, DIRECT avoids the need to know K
D" E"
(a) rectangular cells
D" E"
(b) Voronoi cells
Figure 9: Main weakness of DIRECT [Jones et al. 1993]. Rectangular
partitions (a) can give misleading estimates of sample-neighborhood sizes.
Voronoi cell partitions (b) improve these estimates, especially for high di-
mensions. For example, the size of the relevant neighborhood of point A in
(a) is overestimated, since all its corners are actually closer to other sample
points, e.g. pis closer to D, as seen in (b).
via Equation (3), in which we consider whether any ˜
Kcould al-
low the cell to contain the global optimum. For small values of ˜
Equation (4) avoids selecting cells which can lead only to minor
improvements. The set Lof all rectangles satisfying these equa-
tions, including their ˜
Kvalues, can be computed efficiently via the
convex hull [Jones et al. 1993]. Specifically, Lis the lower enve-
lope of the convex-hull of points (representing rectangles) plotted
in the two-dimensional domain δby f. Since DIRECT uses rectan-
gles, many cells have the same δand the data points tend to stack
above one another.
However, we question the efficacy of rectangular cells in DIRECT.
As illustrated in Figure 9, they do not appear to be the best way
to describe or measure local neighborhoods around sample points.
Rectangles give misleading δi, and slow DIRECT’s convergence
rate, especially in high dimensions.
Input: target function fover domain
Output: minimum ffound
1: s=Center(Ω) // any sample point
2: S={s}// sample set
3: F f(s)// function values at samples
4: ff(s)// minimum value found so far
5: {δ}={||} // set of cell size estimates
6: while computational budget is not exhausted do
7: L=LowerHull(F,{δ}, f )
8: sRandomSelect(L)
9: s0OneSpokeSample(s, D(s))
10: S S S{s0}
11: F ← F S{f(s0)}
12: fmin(f, f (s0))
13: (D,{δ})DelaunayAdd(D, s0)// Algorithm 4
14: return f
Algorithm 5: Lipschitzian optimization via our method.
Our method We follow the basic steps in DIRECT but improve
it in two major aspects: using Voronoi regions instead of hyper-
rectangular cells, and placing samples via stochastic blue noise in-
stead of deterministic cell division. In particular, to refine a cell,
we first add a new sample within it via our spoke-dart sampling
algorithm. We set the conflict radius to the cell’s inscribed hy-
persphere radius, to avoid adding a sample point that is too close
to a prior sample. We then divide the cell (and update its neigh-
boring cells) via the approximate Delaunay graph as described in
Section 5.1, and use the computed witnesses to estimate the δval-
ues in Equation (3). These two steps replace the corresponding de-
terministic center-sample and rectangular cell division in DIRECT,
respectively. See Algorithm 5 for a summary.
To our knowledge, our method is the first exact stochastic Lips-
chitzian optimization technique that combines the benefits of guar-
anteed convergence in [Jones et al. 1993] and high dimensional ef-
ficiency in [Spall 2005]. Computing blue noise and Voronoi regions
has been intractable in high dimensions, and this is probably why
this direction has not been explored before.
(a) iteration 0, # samples (1,1) (b) iteration 1, # samples (3, 2)
(c) iteration 2, # samples (5, 3) (d) iteration 3, # samples (7, 4)
(e) # samples (31, 31) (f) # samples (101, 101)
(g) # samples (301, 301) (h) # samples (601, 601)
(i) # samples (999, 999) (j) Smooth Herbie
Figure 10: Comparing DIRECT (left) and our method (right) while ex-
ploring the smooth Herbie function (j). We list the number of samples used
by each: (DIRECT, us).
Demonstration A didactic comparison of DIRECT and our
method is illustrated in Figure 10. It uses the smooth Herbie func-
tion, a 2-d test function popular in the optimization community be-
cause it has four local optima of similar value, located in different
quadrants. Notice how DIRECT partitions the space via determin-
istic rectangles while our method uses blue-noise Voronoi cells. Ta-
ble 1 shows the superior performance of our method over a set of
benchmark high-dimensional functions, where the standard mea-
sure of performance is the number of function evaluations needed,
Benchmark dimension DIRECT Our method Speedup
Easom 6 4987 1912 2.60×
Easom 8 64405 8480 7.59×
Easom 10 816937 19081 42.81×
Bohachevsky 7 10315 2125 4.85 ×
Exponential 10 13481 7807 1.72 ×
Exponential 15 36890 10316 3.57×
Table 1: Performance comparison of DIRECT and our method, measured
by the number of function evaluations needed to find the global minimum
within relative error =ffmin
fmax fmin
= 104. Since our method is
random, results are the averages over 100 runs.
because in real problems those tend to be very expensive and dom-
inate the overall cost.
5.3 High-dimensional Motion Planning
Motion planning algorithms are frequently used in robotics, gam-
ing, CAD/CAM, and animation [Yamane et al. 2004; Overmars
2005; Pan et al. 2010]. The main goal is to compute a collision-free
path for real or virtual robots among obstacles. Furthermore, the
resulting path may need to satisfy additional constraints, including
path smoothness, dynamics constraints, and plausible motion for
gaming or animation. This problem has been extensively studied in
many areas for more than three decades. Two main challenges are:
Speed The computation needs to be fast enough for interactive ap-
plications and dynamic environments.
Dimensionality High Degrees-Of-Freedom (DOF) robots are very
common. For example, the simplest models for humans (or
humanoid robots) have tens of DOF, capable of motions like
walking, sitting, bending or picking objects.
Some of the most popular algorithms for high-DOF robots use
sample-based planning [LaValle and Kuffner 2001]. The main
idea is to generate random collision-free sample points in the high-
dimensional configuration space, and join the nearby points us-
ing local collision-free paths. Connected paths provide a roadmap
or tree for path computation or navigation. In particular, RRT
(Rapidly-exploring Random Tree) [Kuffner and Lavalle 2000] in-
crementally builds a tree from the initial point towards the goal
configuration. RRT is relatively simple to implement and widely
used in many applications.
However, prior RRT methods generate samples via white noise
(a.k.a. Poisson process). These samples are not uniformly spaced
in the configuration space, leading to suboptimal computation. Us-
ing Poisson-disk sampling instead can lead to more efficient explo-
ration of the configuration space, as demonstrated in a recent work
by Park et al. [2013]. As summarized in Algorithm 6, the method
uses a precomputed Poisson-disk sampling to guide the generation
of new points which are not too close to prior points. Adaptive sam-
pling can also be used to generate more samples in tight spaces. The
performance of RRT planning can be further improved by multi-
core GPUs.
Due to the curse-of-dimensionality, Park et al. [2013] has been re-
stricted to relatively low dimensional spaces, d6. Our method
offers help here by simply precomputing the sample set via spoke-
dart sampling. We use three well-known motion planning bench-
mark scenarios from OMPL [S¸ucan et al. 2012] to evaluate the per-
formance of the planning algorithm. These scenarios all have 6
DOF, and vary in their level of difficulty. We also compute the mo-
tion of the HRP-4 robot with 23 DOF; see Figure 1c. The total times
Input: start / goal configurations xinit and xgoal within domain
Input: Poisson-disk sample set Pprecomputed via Algorithm 3
Output: RRT Tree T
1: T.add(xinit)
2: P.add(xgoal)
3: for i= 1 to mdo in parallel // multiple threads
4: while xgoal /Tdo
5: yRandomSample(Ω)
6: TExtend(T,y,P)
7: end for
8: return T
Algorithm 6: Parallel Poisson-RRT with precomputed samples.
Benchmark DOF RRT (1 CPU core) GPU Poisson-RRT Speed-up
Easy 6 0.34 0.03 12.14×
AlphaPuzzle 6 32.76 1.31 24.93×
Apartment 6 191.79 11.88 16.15×
HRP-4 23 6.17 0.32 19.28×
Table 2: Comparison of the performances of our GPU-based Poisson-
RRT planning algorithms and a reference single-core CPU algorithm. We
compared the planning time for different benchmarks using 100 trials.
taken by the planner are shown in Table 2. For sampling time, the
only competition comes from line darts, and we have demonstrated
in Figure 3 that our spoke-dart sampling is more efficient.
6 Conclusions and Future Work
In summary, we have presented spoke-dart sampling as a new algo-
rithm for generating well-spaced blue noise distributions in high
dimensions. The method combines the advantages of state-of-
the-art methods: the locality of advancing-front and simplicity of
k-d darts. We demonstrated the usefulness of our method for a va-
riety of applications.
Our method has several parameters. Usually the user has no choice
over the domain dimension d. If quick run-time is desired, then se-
lect m= 12 consecutive misses. If higher saturation (β < 2) is de-
sired, use Equation (1) to select m, but be prepared to wait in high
dimensions. In any event, memory should be a minor issue. De-
generate spokes are faster than full spokes in terms of the number
of points inserted; this advantage tends to disappear as the dimen-
sion increases beyond six. Plane-spokes are faster than line-spokes;
since they effectively reduce the dimension by (only) one, this ad-
vantage tends to disappear as the dimension increases. Moreover,
both degenerate spokes and plane-spokes may be producing more
points more quickly merely because they are inserting more points
at distance rcand creating a tighter, less-random packing than max-
imal Poisson-disk sampling. As such, there is little to recommend
degenerate spokes. Our overall recommendation is to use full line-
spokes. We use line-spokes of extent twice the Poisson-disk radius,
and select the next sample uniformly from the nearest segment. We
would like to understand the effect of the extent and the selection
criteria on the final output distribution.
We would like to analyze and improve the accuracy of generating
approximate Delaunay graphs. We speculate that our approximate
Delaunay graphs may supplant the use of k-nearest neighbors for
computational topology and manifold learning. The benefit is that
our Delaunay graph considers all directions; in contrast, for a point
near a dense cluster, k-nearest neighbors can miss significant neigh-
bors in directions opposite to the cluster.
Spoke-darts may inspire further research in global optimization.
We presented the approach and demonstrated it on a small set of
benchmarks. In our current implementation for motion planning
we precompute all samples. We are investigating the possibility of
sampling on the fly by exploiting the similarity between our method
and RRT tree growth. A potential application for high dimensional
blue noise sampling is rendering. Beyond sampling, we believe
spoke-darts can also benefit numerical integration as demonstrated
in Ebeida et al. [2014b].
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A Bound Proofs for Section 4.3
Here we provide bounds on m,β, and in terms of d. We consider
line-spokes only. A void is an uncovered region. It is bounded by
some disks. The chance of hitting a void will depend on its surface
area Area(void), the d-1 dimensional volume of its boundary.
A.1 Chance of missing the void from one disk
Let us quantify the chance p1(miss)that a line-spoke from disk
D1missed a void. See Figure 11a. Let R1=Area(void
D1)/Area(D1).Since line-spokes are chosen uniformly from the
surface area of the disk, p1(hit) = R1,and p1(miss)=1R1.
(We may multiply the hit chance by 2 if the extent of the void is
small enough that it does not contain antipodal points of the disk
and we use a line rather than a ray for a spoke.)
The chance of missing the void consecutively mtimes is then
1(miss) = Qm
j=1(1 R1) = (1 R1)m.Using the well-
known inequality ex= exp(x)>1x, we have pm
A.2 Chance of missing the void from all disks
The chance of missing mtimes consecutively from all N
bounding disks is then pm
all (miss) = QN
i=1 pm
exp mPN
i=1 Ri= exp(mR),where all sample disks have
the same radius and size so we can drop their subscripts and
If we wish this miss chance to be less than , then it is sufficient to
have exp(mR)<  or mR > ln()>0.
A.3 Bound in terms of β
Now we bound Rin terms of β. Suppose there is a domain point
vin the void at distance rcfrom all samples. Then a ball at vof
radius rvoid =rcrfis strictly inside the void, and Area(void)>
Area(D(rvoid)); see Figure 11b. Since we are in ddimensions and
= (β1)d1
Hence a sufficient condition is m(β1)d1>ln , or
m=l(ln )(β1)1dmβ= 1 + ln
A.4 Example mValues
Table 3 gives example mvalues using Equation (5).
A.5 Other issues
Boundary caveats The astute reader may have noticed that we
made no mention of the domain boundary. The void disk D(rvoid)
must be inside the domain, and Area(void)>Area(D(rvoid)) is
only guaranteed to hold for non-periodic domains. Here we as-
sumed that the void was bounded by disks only, and not the domain
boundary. For bounded domains, this may be finessed by throwing
spokes on or near the domain boundary to ensure it is covered.
β2 1.5 1.25 1.125
2 12 24 47 93
3 12 47 185 737
4 12 93 737 5900
5 12 185 3.0e3 4.8e4
6 12 369 1.2e4 3.8e5
7 12 737 4.8e4 3.1e6
8 12 1.5e3 1.9e5 2.5e7
9 12 3.0e3 7.6e5 2.0e8
10 12 5.9e3 3.1e6 1.6e9
20 12 6.1e6 3.2e12 1.7e18
30 12 6.2e9 3.4e18 1.8e27
40 12 6.4e12 3.5e24 2.0e36
50 12 6.5e15 3.7e30 2.1e45
100 12 3.4e21 4.7e60 3.0e90
Table 3: Values of mby dand βfor = 105as computed from Equa-
tion (5).
Order independence There is a statistical subtly in Ap-
pendix A.2. It does not matter that the consecutive spokes from
one bounding disk were not consecutive with the spokes from an-
other disk. The misses for each of the remaining boundary pieces
is independent of whether the void was hit and reduced by some
spokes from a later front disk. The important thing is that no spoke
ever hit the boundary of the terminal, remaining void.
(a) Shared area of a void and a disk.
(b) An empty ball in a void has
smaller surface area than the
Figure 11: Hitting a void from a neighboring disk.
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