Deep Semantic Hashing of 3D Geometric Features
for Efficient 3D Model Retrieval
Takahiko Furuya

Ryutarou Ohbuchi

University of Yamanashi
4-3-11 Takeda, Kofu-shi
Yamanashi-ken, 400-8511, Japan
takahikof AT yamanashi.ac.jp

University of Yamanashi
4-3-11 Takeda, Kofu-shi
Yamanashi-ken, 400-8511, Japan
ohbuchi AT yamanashi.ac.jp

ABSTRACT1

As the scale of 3D model databases increase, speed, in addition toaccuracy, of its search becomes very important. One way to achieve
fast search is to use a compact 3D shape feature whose cost ofcomparison is very small. Binarization of a real-valued feature
vector, via hashing, is a way to obtain such compact feature vector.
Previous algorithms for producing binarized 3D model features via
hashing consisted of two disconnected stages; handcrafted realvalued 3D shape feature extraction followed by hashing into binary
code. This compartmentalized approach, however, leads to lessthan optimal binary codes, as these two stages are optimized
independently. This paper proposes a deep semantic hashingalgorithm called Binarized Deep Local feature Aggregation
Network (BDLAN) which jointly optimizes real-valued feature
extraction per 3D model and its banarization via hashing. BDLAN
training minimizes quantization error caused by binarization.
However, this constraint alone often maps real-valued features totheir nearest binary codes in Hamming space, which are nonoptimal
local minima. To alleviate the issue, we add a simple regularization
called Probabilistic Bit Inversion (PBI) of binary codes.
Experimental evaluation of the proposed algorithms demonstrates
superior efficiency and competitive accuracy to the existing 3Dmodel retrieval algorithms employing real-valued features.

3D shape model has been rapidly growing in numbers,
prompted in part, by proliferation of low-cost 3D range scanners
and 3D printers. A large number of 3D models are stored in a
database to reuse 3D CAD models for mechanical design, toanalyze 3D structure of molecules for medicine, to diagnosediseases from 3D MRI scans of organs, etc. Effective managementof these 3D models requires a technology of shape-based 3D Model
Retrieval (3DMR) that could handle a large-scale database
containing a large number, e.g., 1M or more, of 3D models.
Various approaches aiming at efficient 3DMR have been
proposed (e.g., [19][22][24]). A popular approach is to “compress”
3D model features, e.g., via dimensionality reduction [13][24].
Dimensionality reduction could yield low-dimensional and salientfeatures. However, dimensionality reduction alone is not sufficient
to attain enough scalability. Resulting feature vector is not compact
enough if a 32bit floating point number is used for each dimension.
In addition, distance computation among features, by using Lpnorm, information divergence, etc., can be costly.
A way to obtain more compact and quick-to-compare feature isby hashing of a real-valued feature vector into a binary-valued
vector (e.g., [19][31]). This approach describes each 3D model bya binary vector, or code, in which each element is either “0” or “1”.
As a binary code, it is memory-efficient. In addition, a distance
between a pair of binary codes in Hamming space can be computed
very efficiently by an Exclusive-OR and a “1” population countoperations found in vector instruction set of a modern CPU.
However, binary codes produced by hashing real-valued vectors
tended to have lower retrieval accuracy than the original. We
observe that a compartmentalized strategy that combines ahandcrafted feature with an independently developed hashingalgorithm (e.g., [19][31]) leads to a binary feature vector
nonoptimal in terms of accuracy.
Another approach to retrieval efficiency is by reduction in
number of features per 3D model. If a 3D model is described by aset of features, which are typically local and low-level, a small
subset of it can be selected to describe a 3D model [22] or the setof features can be aggregated into a feature per 3D model [32].
Aggregation local features has been quite popular. A set of local
features can describe diverse partial shapes of a 3D model [5], andan aggregated feature can be robust against non-rigid deformationof the 3D models [36]. Also, the aggregated feature is much less

CCS CONCEPTS• Information systems → Information retrieval → Searchengine architectures and scalability

KEYWORDS
3D model retrieval, feature hashing, binary codes, deep learning.
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http://dx.doi.org/10.1145/3095140.3095148

INTRODUCTION

Sample SOIs &generate RNGs

Convert tooriented points

neuralnetwork

0.00
0.13
1.78
0.01

…

…

I-blockpool byaveraging

0.05
0.00
0.42
0.88

…

…

oriented
point setneuralnetwork…

…

…

polygonal
3D modelAggregate local features

Encode local featuresneuralnetwork

E-block

-0.98
0.84
0.91
-0.87

real-valued
feature r

aggregatedfeaturehash by
sign

-1
1
1
-1

binarycode bb is close to r.
r reflects semantics(regularized by PBI)
of 3D model.
training objectives

mid-level
local features

RNGs

0.08
0.12
0.70
0.39

Hash aggregated feature

A-block

H-block

Figure 1: BDLAN extracts and then aggregates local 3D geometric features to produce a semantic binary code per 3D model.
During training of BDLAN, several bits of a binary code are stochastically inverted for regularization.
costly to store and compare than the set of local features. Still, anaggregated feature is often very high-dimensional (~100k), realvalued vector (e.g., [13][28]) and is too costly for scalable 3DMR.
Our goal is to obtain compact and discriminative binary featuresof 3D models suitable for efficient and accurate shape-based 3Dmodel retrieval. We approach this goal by using a processing
pipeline derived by using an integrated, not compartmentalized,
optimization approach driven by a labeled set of training 3D models.
The proposed integrated feature extraction and hashing pipeline
is a Deep Neural Network (DNN) called Binarized Deep Localfeature Aggregation Network (BDLAN). BDLAN is designed to
produce an accurate binary code per 3D model that is robust against
3D rotation and non-rigid deformation of the 3D model. BDLAN
performs the following three steps in a single deep architecture;(1) extraction of a set of rotationally invariant, local 3D geometricfeatures, (2) aggregation of the local features to a feature per 3Dmodel, and (3) hashing of the aggregated feature. BDLAN is built
upon the recently proposed Deep Local feature Aggregation
Network (DLAN) [30] that produces real-valued and semantic
features for 3DMR. We inject binary constraints into the output of
DLAN to obtain semantic, salient, binarized, compact feature
vector, or codes. Compact binary codes (e.g., 64 bits per code)
generated by BDLAN enable efficient 3DMR both in terms ofmemory footprint and computational time. To our knowledge,
BDLAN is the first deep learning-based feature extraction andhashing algorithm to produce binary codes for scalable 3DMR.
As with DNN-based hashing algorithms for 2D images (e.g.,[10][12][14]), the objective of BDLAN training is to obtain binary
codes that preserve, as much as possible, distances among realvalued features in their binarized counterparts. Such binary features
could be learned by minimizing error caused by binarization, or
quantization, of the real-valued features ([10][12][14][16]).
However, minimizing quantization error alone could result in a
deep hashing neural network that produces suboptimal binary codes.
In an effort to minimize quantization error, the optimization could
get stuck at local minima at which real-valued vector are close to
subopotimal binary codes.
To alleviate this issue, we introduce Probabilistic Bit Inversion(PBI), a simple regularization technique. During training of
BDLAN, PBI randomly inverts several bits of the binary feature.
PBI is expected to help the training escape from the local minima
2

by stochastically quantizing a real-valued feature onto a certain
stochastic neighborhood, not a point, in Hamming space.
Experiments using 3D model retrieval benchmark databases
show that 3DMR using binary codes generated by BDLAN, trainedby using PBI, retains retrieval accuracy comparable to the original,
real-valued feature vector by DLAN. Using binary codes by
BDLAN, however, makes the processing much less memory
intensive and much faster.
Contributions of this paper can be summarized as follows;Proposition of a deep semantic hashing algorithm called
BDLAN for scalable 3DMR.
Proposition of a regularization algorithm called PBI designed
specifically to learn salient binarized feature, or code.
Evaluation of BDLAN and PBI by using 3DMR benchmarks.

Rest of this paper is organized as follows. We review related
work in the next section. In Section 3, the proposed algorithms aredescribed. Experiments and their results are presented in Section 4,
followed by conclusion and future work in Section 5.

2 RELATED WORK
2.1 Feature Hashing for 3D Model Retrieval
Previous algorithms for hashing 3D model feature combine
handcrafted, real-valued features with hashing algorithms based onshallow architectures ([19][25][29][31]). [19] employs a single
layer neural network to hash handcrafted features extracted from
3D models. The single layer neural network is trained by using
pairwise constraints that define two 3D models to be semanticallysimilar or dissimilar. [31] adopts unsupervised feature hashing toefficiently perform part-based 3DMR where a query is a partial 3D
shape. [31] describes each part of a 3D model by a compact binary
code. For the task of 3DMR queried by a hand-drawn sketch, [25]uses simple thresholding to binarize real-valued features of 2D
images, while [29] employs cross-modal manifold learning thatembeds features of sketches and features of 3D models into theircommon Hamming space.
However, the approaches mentioned above do not necessarily
yield binary codes optimal for 3DMR since they combine
handcrafted features having limited descriptive power withlearning models having inadequate expressive power. Recently,

deep learning has been introduced to 3DMR for extracting accurate,
real-valued feature per 3D model. A variety of DNNs tailored to 3Dmodel have been proposed, accepting a variety of 3D shape
representations including voxels [35], point set [1][30], manifold
mesh [15], and rendered 2D views [11][26]. To our knowledge,
however, no prior work has studied feature hashing by using deeparchitecture for scalable 3DMR.

2.2 Probabilistic Approach to RegularizedTraining of Deep Neural Network
Introducing randomness to training DNN could regularize thetraining to avoid overfitting. Dropout [20] and DropConnect [17]are widely-used algorithms that randomly prune neuronal
activations and neuronal connections, respectively. [9] shows that
adding Gaussian noise to connection weights improves
generalization ability of neural networks for regression andclassification tasks. [21] proposes a denoising autoencoder, whichgains robustness against Gaussian noise or masking noise byadding such a noise to the input of the autoencoder. [4] adds
Gaussian noise to gradient of objective function computed during
back propagation. Noisy gradient prevents the training fromoverfitting and enables effective training of a very deep neuralnetwork. [6] adds noise to activation functions when training getsstuck due to saturated gradient of the activation functions.
The proposed PBI can also be classified into the probabilistic
approach for regularizing DNN training. While we applied PBI tolearn binary codes for 3D models, it is general enough to be applied
to other deep hashing algorithms, e.g., ones for 2D image features.

3

PROPOSED ALGORITHM

3.1 Overview of the Proposed Algorithm
The proposed algorithm consists of two parts, the Binarized
Deep Local feature Aggregation Network (BDLAN) architecture forextracting salient binary codes of 3D models, and a novel
regularization strategy designed specifically for binary-valued
hashing called Probabilistic Bit Inversion (PBI). Figure 1 shows theprocessing pipeline of the proposed algorithm. BDLAN takes as its
input a 3D model represented as an oriented point set and produces
a compact binary code of the 3D model. By using Hamming
distance, which can be computed very efficiently, computing
retrieval ranks of 3D models can be very quick.
Like its predecessor DLAN [30], BDLAN generates 3D modelfeatures that have invariance against similarity transformation, i.e.,a combination of translation, uniform scaling, and rotation in 3D
space. A BDLAN feature is also robust against non-rigid
deformation as it uses a set of local features, each of which has
invariance to rotation and translation, to describe a 3D model.

3.2 Extracting Binary Codes using BDLAN
BDLAN comprises four sub-blocks, that are, I-block to compute
low-level local geometrical feature to be input to BDLAN, E-blockto encode the local features, A-block to aggregate the encoded local
features, and H-block to hash the aggregated features into binary

codes (Figure 1). Structures of the first three blocks of BDLAN, Iblock, E-block, and A-block, are identical those of DLAN. The last,
H-block, is newly designed for feature hashing.
I-block: Given a 3D model represented as a 3D polygonal mesh, it
is first converted into an oriented point set by using the algorithm
by Osada et al. [23]. We randomly and uniformly sample 3k points
on the surface of the 3D model. Orientation of each point
corresponds to the normal vector of the triangle at which the pointis sampled. We then sample 100 Spheres-Of-Interest (SOIs) withrandom position and random scale from the oriented point set. Each
SOI is rotationally normalized by applying Principal Component
Analysis (PCA) to the points enclosed by the SOI. The rotationallynormalized SOI is spatially divided into 3D cells by using a 3D grid.
Oriented points within each cell are described by a 10-dim. POD
feature [27] to form a Rotation Normalized Grid (RNG) for the SOI.
E-block: Each of the 100 RNGs representing the 3D model isindependently fed into the E-block to encode the RNG into a midlevel local 3D geometric feature. E-block has two 3D convolution
layers and subsequent three fully-connected layers with ReLU
activation function [2]. The E-block generates a set of 512-dim.,real-valued local geometrical features of the 3D model.
A-block: The set of 100 mid-level local features from E-block isaggregated to a single feature having 512 dim. per 3D model via
average pooling.
H-block: H-block first dimension reduce the aggregated feature
above to a 256-dim. real-valued feature via a fully-connected layer.
Then, a subsequent hash layer binarizes the dimension reduced
feature. The hash layer is also fully-connected but uses hyperbolic
tangent activation function to limit the activation to a range (−1,+1). The activated real-valued feature (denoted as r) is binarized byits sign to generate a binary code (denoted as b) of the 3D model.
When computing Hamming distance, -1 and +1 of the hashing
result are mapped to binary digits 0 and 1, respectively. The numberof bits Nb for binary codes corresponds to the number of neurons(e.g., 64) in the hash layer.

3.3 Training BDLAN with PBI
To learn accurate semantic binary codes, a BDLAN is trained sothat it could correctly predict object category labels of training 3Dmodels while minimizing quantization error due to binarization of
the real-valued 3D model features produced at the output end of the
BDLAN. Such an objective L can be formalized as follows;
=

+

(1)

where Lc is cross-entropy loss that defines classification error ofcategory labels and= ‖ − ‖ indicates quantization error
between real-valued feature r and binary code b. To compute crossentropy loss, we append a classification layer with softmax function
behind the hash layer during training of a BDLAN. The hyperparameter α balances Lc and Lq. We fix α to 1.0 in the experiments.
PBI regularization tries to prevent the training of binary codes
to fall into poor local minima. Specifically, for each time BDLAN
quantizes the real vector r to the binary code b during training, PBI
randomly inverts up to P % of Nb bits of b. For example, when
Nb = 128 and P = 10%, PBI inverts the maximum of 12 bits

3

randomly chosen from the 128 bit of the code. This operationmeans that r could be quantized not only into its nearest binary code
but also to binary codes within its stochastic proximity. The bitinverted binary code b is used to compute the loss described inequation 1. Note that PBI is performed only in the training phase.
After training, the binary code of the 3D model is computed byusing BDLAN without PBI as described in Section 3.2.
To effectively train BDLAN by using small number (~10k) oflabelled 3D models, we adopt two-stage training of BDLAN. Thatis, the first stage trains only E-block by using a large number of(300k in this paper) labelled RNGs. Each RNG is sampled fromeither of labelled 3D models in the training dataset and categorylabel for the RNG is identical to that for 3D model from which the
RNG is sampled. We append a classification layer at the output endof E-block and train the E-block so that cross-entropy loss could be
minimized. The second stage trains the whole BDLAN by using thelabelled 3D models to minimize the entire loss (equation 1). EEblock inherits the result of the first stage while weights of H-block
are randomly initialized. PBI is used at the second stage training.
For both the first and the second stage training, we use Adam [7]with mini-batch size = 32. Training is iterated for 200 epochs and
20 epochs at the first and second stage training, respectively.

4

EXPERIMENTS AND RESULTS(a) ModelNet40(b) ModelNet10

Figure 2: Examples of 3D models in benchmark datasets.
Hash functions for ITQ and AQBC are learned by using DLAN
features extracted from 3D models in the training set. LSH doesn’t
require learning as it is a data-independent hashing algorithm. To
generate ranking results, DLAN uses Cosine distance while theother three algorithms use Hamming distance. We also compare
accuracy of BDLAN against the existing 3DMR algorithms using
handcrafted [8][28] or deep learning-based [11][26][35], realvalued feature of 3D model.
We use Nb = 64 and P = 30% for BDLAN unless otherwise
stated. We used TensorFlow library [18] to implement BDLAN. All
the experiments were conducted on a PC with an Intel Core i7 6700
CPU, a NVIDIA GeForce GTX 1080 GPU, and 64GB DRAM.

4.2 Experimental Results
Accuracy of BDLAN: Figure 3 plots retrieval accuracies of thefive algorithms against the number of feature dimensions (i.e., Nb
for BDLAN). For the four algorithms except for “DLAN+LSH”,

4.1 Experimental Setup






DLAN [30] extracts real-valued, non-negative feature per 3Dmodel. This is the algorithm on which BDLAN is based..
DLAN+LSH hashes the real feature vectors obtained fromDLAN by using Locality Sensitive Hashing [3].
DLAN+ITQ uses Iterative Quantization [34] for hashing.
DLAN+AQBC uses Angular Quantization-based Binary
Codes [33] which is designed to hash non-negative feature
vectors.

BDLAN (proposed)
DLAN+AQBC
DLAN+LSH

DLAN
DLAN+ITQ

MAP [%]

100
90
80
70
60
50
40
30
20
10
4

8

16
32
64 128 256 512 1,024
Number of feature dimensions(a) ModelNet40

4

8

16
32
64 128 256 512 1,024
Number of feature dimensions(b) ModelNet10

100
90
80
70
60
50
40
30
20
10

MAP [%]

Benchmark datasets: Efficiency and accuracy of the proposed
algorithm are evaluated by using two standard benchmark datasetsfor 3DMR, i.e., ModelNet40 and ModelNet10 [35]. Figure 2 shows
examples of 3D models contained in the datasets. ModelNet40 has
9,843 3D models for training and 2,468 3D models for testing.
These 3D models are classified into 40 semantic object categories
such as airplane, car, plant, sofa, etc. On the other hand,
ModelNet10, which is a subset of ModelNet40, contains 3,991 3D
models for training and 908 3D models for testing. The 3D models
in ModelNet10 are categorized into 10 indoor objects, e.g., chair,table, bed, etc. For both datasets, BDLAN is trained by using 3Dmodels in the training set and retrieval accuracy is evaluated byusing those in the test set. We use Mean Average Precision (MAP)[%] as accuracy index. Since the proposed algorithm is influenced
by randomness caused by initialization of DNN parameters, PBI,
etc., every experiment is performed three times and their average
MAP score will be reported.
Competitors: We compare performance of BDLAN with thefollowing four algorithms;

Figure 3: Feature dimensions and retrieval accuracy.
4

Table 1: Comparison of retrieval accuracy.
Algorithms
LFD [8]SV-DSIFT [28]
3D ShapeNets [35]
MVCNN [11]GIFT [26]DLAN [30]BDLAN

MAP [%]
ModelNet40
ModelNet10
40.9
49.8
58.1
68.0
49.2
68.3
79.5
－
81.9
91.1
89.5
83.3
83.1
89.8

Effectiveness of PBI: We investigate the efficacy of PBI for
learning accurate binary codes of 3D models. Figure 4 shows
retrieval accuracies of BDLAN with PBI (P=30%) or without PBI.
For both ModelNet40 and ModelNet10 datasets, PBI slightly but
consistently improves MAP scores of BDLAN. The gain of
retrieval accuracy would be the result from regularization of binary
code learning. PBI would help the training of BDLAN find asolution having better generalization capability.
Figure 5 shows influence of the hyper-parameter P for PBI on
retrieval accuracy. For the experiments, we fixed the number of bits
for binary codes Nb to 64. In the figure, P = 0% means PBI is not
performed during the training of BDLAN. Interestingly, in Figure
5, we can observe that P doesn’t have a large impact on retrieval
accuracy. Very slight peaks can be found at around P = 30%. WhenP = 30%, at most 19 bits of a 64-bit code could be inverted.

MAP [%]

100
90
80
70
60
50
40
30
20
10

ModelNet10, BDLAN with PBI
ModelNet10, BDLAN without PBI
ModelNet40, BDLAN with PBI
ModelNet40, BDLAN without PBI

4

8
16
32
64 128 256 512 1,024
Number of bits Nb for feature hashing

Figure 4: Effectiveness of Probabilistic Bit Inversion (PBI).
9590
MAP [%]

the number of feature dimensions corresponds to the number ofneurons at the output layer of the DNN. On the other hand, for“DLAN+LSH”, 128-dim. real-valued features are extracted byusing DLAN and N hyperplanes are randomly generated to yield Nbit binary codes. Note that only “DLAN”, denoted by dashed lines
in Figure 3, uses real vectors while the others use binary codes for
retrieval.
For both ModelNet40 (Figure3a) and ModelNet10 (Figure3b),
we can observe that “BDLAN” and “DLAN” yield almost the same
MAP scores if binary codes with more than 8~16 bits are used. This
result indicates that BDLAN generates accurate semantic binary
codes of 3D models with low quantization error due to featurehashing. Also, BDLAN shows retrieval accuracies higher than thethree compartmentalized hashing algorithms, i.e., “DLAN+ITQ”,“DLAN+AQBC”, and “DLAN+LSH”. Joint optimization of
extracting semantic real features and their hashing would have apositive effect to obtain binary codes for accurate 3DMR.
“DLAN+AQBC” performs well for the simple ModelNet10 dataset
but suffers for ModelNet40 dataset having more object categories.
Table 1 compares retrieval accuracies of the six state-of-the-art
3DMR algorithms including the proposed BDLAN. All the existingalgorithms in Table 1 employs real-valued features for retrieval.
Note that accuracies of DLAN in Table 1 differs from the original
paper [30] since we used our own implementation of DLAN for theexperiment. BDLAN yields competitive MAP scores to the otherreal feature-based algorithms. Accurate 3DMR can be performed
also by using binary codes obtained from DNN. Efficiency of
BDLAN will be evaluated in the later subsection.

85
80
ModelNet10
ModelNet40

75
70
0

20
40
60
80
Hyper-parameter P for PBI [%]100

Figure 5: Probability of bit inversion and retrieval accuracy.
Efficiency of BDLAN: This subsection evaluates efficiency of
BDLAN both for training and retrieval. As we described in Section
3.3, BDLAN adopts two-step training. The first step (i.e., trainingof E-block) took about 2~3 days and the second step (i.e., trainingof whole BDLAN) took about 2 hours.
We wanted to evaluate retrieval scalability of BDLAN.
Unfortunately, however, the number of 3D models in the test set ofModelNet40 (i.e., 2,468) is too small to evaluate scalability. We
therefore duplicated the 2,468 features of the 3D models up to
100M features to simulate a large-scale 3D model database. Table
2 summarizes retrieval efficiency of BDLAN and DLAN for thesimulated large-scale database. In the table, “Feat.” indicatescomputation time for extracting a feature from a query 3D model,“Dist.” means time for computing distances among the feature of
the query and the 100M features in the database, and “Memory
footprint” is spatial cost to store 100M features of the 3D models.
We used 64-dim. real-valued features for DLAN and 64 bit binary
codes for BDLAN.
Table 2 clearly shows that BDLAN is quite a bit faster than
DLAN. Computing Hamming distances among binary codes
drastically reduces computational time per retrieval.
BDLAN is memory efficient as well; it requires less than
1 GByte for as many as 100M 3D models. DLAN, on the other hand,requires 24 GBytes as it uses a 32 bit floating-point number perfeature dimension.

5

Table 2: Comparison of retrieval efficiency.
Algorithms
DLAN [30]BDLAN

4Per-query time [s]
Feat.
Dist.
Total
0.08
11.50
11.58
0.08
0.37
0.45

Memoryfootprint [GB]
24.41
0.76

CONCLUSION AND FUTURE WORK

Explosive increase in number of 3D models in recent years
requires scalable algorithms for shape-based 3D model retrieval(3DMR). Nevertheless, few prior work has been studied 3D modelfeature that satisfies both high efficiency and high accuracy. In thispaper, we proposed a deep semantic hashing algorithm intended for
scalable 3DMR. The proposed algorithm called Binarized Deep
Local feature Aggregation Network (BDLAN) extracts compact
semantic binary code per 3D model. Compared to real-valued
feature, binary code is much efficient; binary codes can becompared more quickly by using Hamming distance and theyrequire much less memory. To obtain salient binary codes, BDLAN
embeds features of 3D models into the Hamming space that wellreflects semantics of the 3D models by regularizing the training byusing a novel, binary-code specific regularization algorithm called
Probabilistic Bit Inversion (PBI). The experiments demonstrated
superior efficiency and competitive accuracy of BDLAN comparedto the state-of-the-art 3DMR algorithms.
Future work includes evaluation of BDLAN on larger-scale
databases having more 3D models classified into more diverse
object categories. Also, efficacy of PBI regularization should be
evaluated under different settings and scenarios, e.g., by deephashing of 3D model features using base architectures other than
DLAN (e.g., [11] [26][35]), or by deep hashing of 2D image
features (e.g., [10][12][14]).

ACKNOWLEDGMENTS
This research is supported by JSPS Grant-in-Aid for Young
Scientists (B) #16K16055.

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