CGI2016 manuscript No.
(will be inserted by the editor)

Supplemental Material for“Multi-class anisotropic blue noise samplingfor discrete element pattern generation”
Naoki Kita · Kazunori Miyata

A BuildRMatrix
In the proposed algorithm, RMatrix is constructed according
to the literature [8]. In our case, the normalized RMatrix isreturned by the BuildRMatrix(·) function.

Algorithm function BuildRMatrix({ri }i=0:c−1 )// user specified per-class values; {ri },// number of classes; cfor i = 0 to c − 1r ← ri // initialize diagonal entries
end forsort the c classes into priority group {Pk }k=0:p−1 withdescreasing ri
C ← ∅ // the set of classes already processedD ← 0 // the density of classes already processedfor k = 0 S
to c − 1
C ← C Pk
foreach class i ∈ Pk
D ← D + r12i
end foreachforeach class i ∈ Pk
foreach class j ∈ C
if i 6= jr(i, j) ← r(j, i) ← √1D
end ifend foeachend foreachend forrscale ← r/ min(r) // element-wise division
return rscale

N. Kita
Japan Advanced Institute of Science and Technology,
E-mail: naoki-kt@jaist.ac.jp
K. Miyata
Japan Advanced Institute of Science and Technology,
E-mail: miyata@jaist.ac.jp

B Statistical Analysis for Multi-class
Anisotropic Blue Noise Sampling
We analyzed our algorithm by calculating its power spectrum
and anisotropy according to the literature [6]. Figures 1 to
3 show the analysis results for 3-class, 5-class, and 7-class
distribution respectively. We plotted the total set and eachclass. To verify the anisotropic sampling, a distribution iswarped back from a scaling projection to a unit square. In theoriginal distribution, we employed a simple scaling transformfrom p = (x, y) to q = (u, v), i.e., q = ϕ(p), whereϕ : (u, v) = (x/2, y).

(1)

Therefore, the points can be warped back by applying a Jacobian matrix expressed as"−1J(ϕ(q)) =

∂x
∂u
∂y∂u

∂x
∂v
∂x
∂y#


=

20.
01(2)

The spectrum analysis results are shown in Figs. 1−3.
Here, simple scaling is used for the warp to test its anisotropic
and multi-class sampling properties (left to right: original(anisotropic) samples, warped (isotropic) samples, power spectrum averaged over 10 runs, and the corresponding radialmean and anisotropy plots). The total set contains approximately 3800 samples, and each class contains almost equal
number of samples.
As can be seen, the total set of all cases exhibits blue
noise properties. In Fig. 1, each class also exhibits blue noiseproperties. However, in Figs. 2 and 3, each class is slightly
biased due to a smaller number of points and may also sufferfrom a scaling effect and constrained distribution resulting
from multi-class sampling. However, we aim for multi-classelement distribution rather than a pure sampling application,such as anti-aliasing; thus, this is acceptable because we cangenerate visually appealing results, which are shown in themain paper.
The spectrum analysis for 16-/32-/64-classes are shown in
Fig. 4. As can be seen, our algorithm can also generate distributions with blue noise properties even with a large numberof classes.

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C Comparison of Sampling Approaches

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Naoki Kita, Kazunori Miyata
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We compare the distribution methods in the main paper.
Here, we compare our method (Fig. 1) to anisotropic sampling with random class assignment (Fig. 6). We find that
there are significantly sparser and denser areas of samples ineach class in Fig. 6 than those shown in Fig. 1, which results
Fig. 4 Spectrum results of distribution with more classes.
in fewer blue noise qualities.
Left to right: 16-/32-/64-classes. In these cases, simple scaling
We also generate a discrete element pattern using three
is applied, as well as warp back from a scaling projection tomethods: (a) multi-class isotropic sampling [8], (b) anisotropica unit square for spectrum and anisotropy analysis.
sampling [6] with random class assignment, and (c) our multiclass anisotropic sampling (Fig. 5). As can be seen, the anisotropyof an element is ignored and the elements are not well populated in Fig. 5a. Although Lagae and Dutré employed an-10-15-10-15-20-15

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Supplemental Material for “Multi-class anisotropic blue noise sampling for discrete element pattern generation”(a) Multi-class (isotropic)(b) Anisotropic (random class assigned)3(c) Ours

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Fig. 5 Comparison of three methods: (a) multi-class isotropic sampling [8], (b) anisotropic sampling [6] with random classassignment, and (c) our multi-class anisotropic sampling. The generated patterns are shown in the top row, and the elementswith proxy shapes are shown in the bottom row.

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placement approaches lack the ability to distribute multi-classor anisotropic elements, we compare our approach to discreteelement texture synthesis approaches.
Although the statistical test shows that our method outperforms other methods, the results from these synthesis approaches are highly dependent on the input exemplar. As
mentioned in Section 1 of the main paper, creating a visuallyappealing swatch (pattern), i.e., an exemplar, is difficult. This
indicates that the proposed method might be useful while creating a visually appealing pattern, and the resulting pattern
can be used as input exemplar for discrete texture synthesis
approaches.

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isotropic Poisson disk sampling approach for object distribution [4], Fig. 5 shows that isotropic Poisson disk samplingis unsuitable for anisotropic element distribution. In Fig. 5b,the elements are well populated; however, it does not consider
multi-class distribution as in Fig. 5a. As a result, the sameclass elements are not well uniformly distributed in the wholedomain. As shown in Fig. 5c, ours is well populated and theelements of the same class are uniformly distributed in thedomain.

D All Patterns Shown to Participants
All of the patterns shown to the participants in our experiment are shown in Fig. 7. Since the previous discrete element

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Naoki Kita, Kazunori Miyata

Ours (Clipped)

Ours

BBT06 [1]

HLT09 [2]IMIM08 [3]LGH13 [4]

MWT11 [6]

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 7 All patterns shown to participants in our experiment. Left to right: ours (clipped), ours, BBT06 [1], HLT09 [2],IMIM08 [3], LGH13 [5], and MWT11 [7]. Top to bottom: (a) leaf, (b) snake, (c) balloon, (d) flower, (e) ant, and (f) wheat.
The results other than ours are courtesy Landes et al. [5].

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