立石科学技術振興財団

助成研究成果集(第30号) 2021

［ 研究助成 (A) ］

ブラインド信号源分離に適用する非線形混合のマルチ部分空間表現に関する研究
A Study on Multi- Subspace Representation of Nonlinear Mixture
with Application in Blind Source Separation
2201034

研究代表者

慶應義塾大学大学院
理工学部・理工学研究科

特任助教

Wang Luapplications, due to inevitably imperfection of[研 究 の 目 的］potential interactions. For instance, to predict
Development and wide deployment of vari-hand movement through brain responses

ous sensor technologies have led to the gen-recorded by multiple electrodes, it is usuallyeration of large amounts of multi-block datanecessary to consider the object pattern in(i. e., multi-set, multi-relational, or multi-view),

addition to the two patterns of time and space.

which need to be jointly analyzed to extract

This means the brain recording can be natural-meaningful information. The technology called

ly fit into a multi-way array with multiple

tensor refers to a multi-way array of numbers,

modes. However, the mostly applied computing

which encapsulates high-dimensional big data

tools for brain research are oriented for

information in a compact format. Benefiting

one-way or two-way data. Consequently, thefrom the power of multi-linear algebra of theirfacilitation and availability of the multi-way

mathematical backbone, data analysis techni-signal processing are attributed to the ability ofques using tensor decomposition shows great

the model interactions between complex latent

flexibility in the choice of constraints that re-components. Moreover, most methods areflect data diversity. Thus, multi-block tensor

suffered from the fact that the observed record-

decomposition and multi-way analysis allow us

ings usually have a low signal-to-noise ratio

to discover meaningful hidden spatial and tem-

(SNR), caused by the inevitable contaminationporal structures of complex data and performfrom the Electrooculography (EOG) signal,

generalizations by capturing multi-linear andbody movement, and artificial noise during themulti-aspect relationships. As a higher-orderpropagation process. It is a typical problem togeneralization of matrices and vectors, theoreti-

as blind source separation (BSS). Therefore, it

cal tools and numerical algorithms for tensor

is crucial to extract accurate target signals

manipulation have been developed across manyfrom the mixture recordings, which reflects thefields including tensor flow for neural networks,

accuracy of prediction when capturing multiple

and tensor decomposition for big data.

interactions and couplings.

Multi-block data are being explored in many
― 1 ―

立石科学技術振興財団⋯, s] ∈R  represent the N -dimensional[研究の内容，成果］

observations and source signals in the k th
In this topic, we propose to apply the tensor

dataset. The mixing matrix A ∈R   repre-

theory to develop a novel structure for the joint

sents channel effect between signals and

BSS technique that allows us to discover thesensors which is unknown. e ∈R  denotes

multi-set components and meaningful spatio-

the possible additive noise, which is generally

temporal structures of data. In our model, theassumed to be zero mean, temporally white andtensor computing is regularized by a low-rank

uncorrelated with the source signals.

prior to exploring a stack of cross covariance

The purpose of BSS is to find a set of invert-

matrices along with temporal information,ible matricesA , A , ⋯ , A  such that towhich in terms of limited storage, providing theestimate the source signals from the observa-

efficiently

tions in multiple datasets K, where A encapsulate and compress large-

scale data into a compact format. Thus, thedenotes (A )objective function is formulated by taking thecoupling

information

intoaccount,for convenience.

Considering the separation criterion of thewhichjoint BSS that are similar to several exi-

explores not only the statistical independence

stingmethod, two assumptions are satisfied

within multiple datasets but also dependence

regarding the sources in this paper, which isamong the different datasets. Thus, the objec-

illustrate in Fig. 1.

tive function is formulated by taking the(1) The sources are statistical uncorrelated

coupling information into account, which ex-within each dataset k formed asplores not only the statistical independenceE[s(t)s(t+τ) ]=0, 1≤i≠ j≤N ,within multiple datasets but also dependenceamong the different datasets. The approach iswhere τ is the time delay and (∙) denotes thebeneficial to decompose the tensor format into

complex conjugate transpose.

multiple low-rank tensors of matrices, thus(2) Only the corresponding sources are de-providing an efficient way for analyzing big andpendence from two different datasets, formed

complex data. In addition, a practical problem

asfor the mixing matrices is its uniqueness andD = [s (t)(s (t+τ)) ]identifiability. We discuss how our results=diag[ρ(τ), ⋯ , ρ (ρ)],

provide the uniqueness decomposition undersome conditions used for estimating the mixing
matrices.
In this research, a joint BSS model is considered on an ensemble of K datasets. The N
observations in each dataset are generated by
the linear mixture of N source signals. The
mixing process is modeled byx =A s +e ,The assumptions regarding the sources : the sources are statisticalindependence within each dataset, but that are dependence among
the different datasets

k=1, 2,⋯, K ,

where x =[x, ⋯, x] ∈R  and s =[s,― 2 ―Fig. 1Illustration of a joint BSS modelTateisi Science and Technology Foundation

Then, we show the average performances inwhere 1≤k≠k≤K and the diagonal matrix(t) andthe different dataset K, where in Fig. 3 (a)(t+τ). Here, we take into account for themeasures the NMSE between the estimatedcase of Κ =3 datasets as an example. Using thematrices and the mixing matrices, and (b)proposed TJBSS algorithm, the underdeter-

evaluates the PPC between the estimatedmined problem also can be effective to deal withsources and the underlying sources. As we canwhen the number of dataset more than 3.

seen, MCCA algorithm presents the poorestρ(τ) define a covariance between s



s

performance even K is increased. Although
Separation of Synthetically Generated Data

MCCA algorithm performs on the same criteri-

We first cross-validated the time delayon, i. e., second-order statistics that shows$\tau$ on the synthetic data used for learning

poorer performance mainly due to the error

parameter of our model. It is clear that theaccumulation from deflation-based separation

more variable as estimation but the longer toapproach. In addition, the performances are notaccess the model. i. e., the tensor structure is

primarily dependent on number of datasets

stacked by the covariance matrices with a time

when N=10. For IVA, TJBSS, and GGD algo-delay τ. The greater value of τ are used torithms, the NMSE is slightly improved in theconstrain the separation model, the better per-point of K=10. MCCA algorithm achieves

formance can be achieved. However, we can see

reliable separation performance when the pa-

from Fig. 2, the performance of the proposedmodel becomes saturated as the parameter τachieves 3 approximately. To prevent thepotential model over-saturation, τ set to 3 suggested by the trends of convergence in both
NMSE and PCC measurements.

(a) NMSE of the estimated matrices and mixing matrices(a) NMSE of the estimated matrices and mixing matrices(b) PCC of the recovered sources and underlying sources(b) PCC of the recovered sources and underlying sources

Fig. 2 The NMSE and PCC performances on the different τ

Fig. 3 The NMSE and PCC performance on the different K― 3 ―立石科学技術振興財団

rameter K is greater than 15. A large enoughthe different sample sizes showed in Fig. 4,

parameter K can provide reliable separation

where (a) measures the NMSE between theeven if the number of sources is large, e. x., weestimated matrices and the mixing matrices,use N=10 here. The proposed TJBSS con-and (b) evaluates the PPC between thesistently presents more accuracy on NMSE andestimated sources and the underlying sources.

PCC, because the approach exploits not only theIt can be seen that all of the algorithms show

statistical independence within multiple data-

similar and reasonable performance, in whichsets, but also dependence among the different

the larger number of samples leads to thedatasets.

performance increase. The poor performance ofIn general, the performances do not depend

MCCA is due to using gradient search algo-on the sample size, because the target dataset isrithm may converge locally, as shown in thegiven for consistency. However, the enoughsimulation results here and the speech separa-sample size are need to ensure that thetion results. GGD as extended approach fromcovariance matrices used for the proposed

MCCA seems reliably solve this problem thatalgorithms are positive definite. It is sufficientachieves the better performance.

in theoretical, however in some scenarios, asmall number of samples can achieve the good

Separation of Real-world Biological Data
In this section, we demonstrate an application

performance. The averaged performances inof the proposed algorithm, which the separationof electroencephalogram (EEG) and electrooculogram (EOG) signals from the real-world
electrode recordings. The recordings arecollected from six subjects. Fig. 5 shows thetime scheme of the experimental paradigm in
one trial. The subjects sit the chair in front of acomputer screen. For each trial, the experiment
begins at t=0 s when a fixation cross appearson the black screen. The prompt starts at the 2(a) NMSE of the estimated matrices and mixing

seconds, i. e., t=2 s, the cue in the form of anarrow points appear on the screen during 1.25 s,
either to the scenarios of left, right, up or down.
The prompt allows subjects to perform the(b) PCC of the recovered sources and underlying sources

Fig. 4The NMSE and PCC performance on the different
sample size T― 4 ―Fig. 5The Mixture Models of the FECG and MECG
Measurements

Tateisi Science and Technology Foundation

corresponding motor imagery task until t=6 s.
The next trial repeats after a 2 seconds breakwith a black screen. 360 trials for each motor[成果の発表，論文等］[1]

L. Wang, T. Ohtsuki, K. Ohwada, N. Honma, H.

Hayashi, “Joint Multiple Subspace-Based BSS

imagery task are recorded from each subject.

Method for Fetal Heart Rate Extraction from

We exploit all the 22 channels available.

Non-Invasive Recordings,” 42st International Con-

EEG and EOG signals extraction are a typical
BSS problem, because of the collecting EEG
signals are inevitably contaminated by some
propagation inside the body, such as EOG
signals and artifact. In practice, the distributionof channels is different due to the location of thedifferent electrodes. Although channels arecomparable dependent, it is difficult to identify
which channel will more contribute to estimation in the different subjects. In this case, thejoint BSS is considered an attractive approach.

ference of the IEEE Engineering in Medicine andBiology Society (EMBC2020), Montreal, Canada,Jul. 2020.
[2]

L. Wang, T. Ohtsuki, K. Owada, N. Honma, and H.

Hayashi, “Fetal Heart Rate Monitoring from Noninvasive ElectrodeRecordings,” 電子情報通信学会
ソサイエティ大会， オンライン， 2020 年 9 月 16
日．
[3]

A. A. Qureshi, L. Wang, T. Ohtsuki, K. Owada, N.

Honma, and H. Hayashi, “Reconstructing Maternal
ECG from Abdominal ECG Using Autoencoder,” 電子情報通信学会ソサイエティ大会，オンライン，
2020 年 9 月 16 日．
[4]

A. A. Qureshi, L. Wang, and T. Ohtsuki, K.

Owada, N. Honma, and H. Hayashi, “Reconstructing
Maternal ECG from Abdominal ECG Using Auto-[今後の研究の方向，課題］

encoder,” 2020 International Conference on Emerging Technologies for Communications (ICETC

Blind separation of source signals havereceived wide attention in various fields such as2020), Tokyo, Japan, Dec. 2020.
[ 5 ] 新飯田夏帆，大槻知明，王路，大和田一成，本間直樹，林隼人，“心拍波形の一次導関数を用いた

speech enhancement, image recognition, wire-

胎児 R ピーク検出と RR 間隔の分布に応じた複数

less communication, and thus have beenの重み付けによる胎児心拍数推定，”電子情報通信

thoroughly studied in the signal processing
community. Among, it is a non-trivial task to

学会総合大会，オンライン，2021 年 3 月 12 日．
[6]

A. A. Qureshi, L. Wang, and T. Ohtsuki, K.

Owada, N. Honma, and H. Hayashi, “Improved Fetal

obtain an accurate and reliable fetal elec-

Heart Rate Detection Via Maternal ECG Removal

trocardiogram (FECG) in a non-invasive fash-

Using Autoencoder in Frequency Domain,” 電子情ion. Problems develop due to the facts, that theelectrocardiogram (ECG) also contains a maternal electrocardiogram (MECG), which canbe from one-half to one-thousandth the magnitude of the MECG. Moreover, the FECG will

報通信学会総合大会， オンライン，2021 年 3 月 12
日．
[7]

L. Wang, T. Ohtsuki, K. Ohwada, N. Honma, H.

Hayashi, “A Multi-Layer Hybrid Network with Its
Application in Fetal Heart Rate Monitoring,” IEEE
Signal Processing Letters. (Under Review)[8]

L. Wang, T. Ohtsuki, K. Ohwada, N. Honma, H.

occasionally overlap the MECG and make it

Hayashi, “Improved Fetal Heart Rate Detection bynormally impossible to detect. Along with theRemoving Maternal Cycle Using Adaptive Temp-

MECG, extensive electrocardiography (EMG)
noise also interferes with the FECG and it canlate,” IEEE Global Communications Conference.
(Under Review)[9]

A. A. Qureshi, L. Wang, and T. Ohtsuki, K.

Owada,

completely mask the FECG.

N.

Honma,

andH.

Hayashi,“An

Autoencoder-Based Heart Rate Detector or Noninvasive Recordings,” International Conference of
the IEEE Engineering in Medicine and Biology
Society. (Under Review)― 5 ―立石科学技術振興財団[10]

N. Niida, L. Wang, and T. Ohtsuki, K. Owada, N.

Multiple Weighting Functions,” International Con-

Honma, and H. Hayashi, “Fetal Heart Rate Detec-

ference of the IEEEEngineering in Medicine andtion UsingFirst Derivative of ECG Waveformand

Biology Society. (Under Review)― 6 ―