MULTI-CHANNEL EEG COMPRESSION BASED ON 3D DECOMPOSITIONS
Justin Dauwels
⋆
Srinivasan K.
⋆†
Ramasubba Reddy M.
†
Andrzej Cichocki
‡
⋆
Nanyang Technological University, Singapore, E-mail: justin@dauwels.com
†
Indian Institute of Technology Madras, Chennai, India, E-mail: srinivasan.sivam@gmail.com
‡
RIKEN BSI, Laboratory for Advanced Brain Signal Processing, Wako-Shi, Japan
ABSTRACT
Various compression algorithms for multi-channel electroen-
cephalograms (EEG) are proposed and compared. The multi-
channel EEG is represented as a three-way tensor (or 3D
volume) to exploit both spatial and temporal correlations effi-
ciently. A general two-stage coding framework is developed
for multi-channel EEG compression. In the first stage, we
consider (i) wavelet-based volumetric coding; (ii) energy-
based lossless compression of wavelet subbands; (iii) tensor
decomposition based coding. In the second stage, the residual
is quantized and coded. Through such two-stage approach,
one can control the maximum error (worst-case distortion).
Numerical results for a standard EEG data set show that
tensor-based coding achieves lower worst-case error and
comparable average error than the wavelet- and energy-based
schemes.
Index Terms— arithmetic coding, three-way tensor, ten-
sor decomposition, wavelet transform
1. INTRODUCTION
Electroencephalogram (EEG) is a recording of the electrical
activity of the human brain, usually acquired by a number
electrodes placed on the scalp. In the past decade, there has
been tremendous growth in EEG based research activities,
e.g., automated EEG analysis for diagnosis of neurological
diseases, and brain computer interfacing (BCI) [1]. In most
applications, EEG recordings are done for an extended pe-
riod, and long-term recordings often generate massive EEG
data sets. Therefore, EEG compression plays an important
role for efficient storage and transmission. The main chal-
lenges for EEG compression are as follows:
• the number of EEG channels can be large (e.g., 256),
• the sampling rate can be high (several kHz) in order to
capture evoked potentials and high frequency oscilla-
tions.
Many techniques have been developed for compress-
ing EEG (see, e.g., [2] and references therein). However,
those methods often compress individual channel separately.
EEG signals from adjacent channels are often strongly corre-
lated (inter-channel correlation), and each individual channel
has temporal correlations (intra-channel correlation). The
single-channel EEG compression algorithms, when extended
directly to multi-channel EEG, will be inefficient as the inter-
channel correlations are not exploited. For multi-channel
EEG, intra- and inter-channel correlations must be exploited
together for efficient compression.
Multi-channel EEG compression is less intensively stud-
ied, and one could find only few instances in the literature;
we categorize the algorithms into lossless [3, 4] and lossy
[5] methods. All multi-channel compression schemes con-
sider inter- and intra-channel correlation separately, and ex-
ploit them by different techniques. However, intra- and inter-
channel correlations are often not independent and exploiting
them in a singe step may improveefficiency. We explore ways
to arrange the multi-channel EEG in suitable form, particu-
larly, to exploit both types of correlations in a single step.
In our previous work [2], we introduced a pre-processing
technique where single-channel EEG is arranged as a matrix
before compression; this representation improved the Rate-
Distortion (R-D) performance over conventionalcompression
schemes. Extending our previous work, in [9] we explored
several ways to arrange multi-channel EEG as matrices and
tensors, and evaluated several matrix/tensor decomposition
techniques based on their R-D performance. Here we present
a more systematic study of multi-way (or volumetric) rep-
resentations of multi-channel EEG for the purpose of com-
pression; we consider several compression schemes that use
such representations, more specifically, based on 3D wavelet
transforms or tensor decompositions. Moreover, we develop
a two-stage framework for compression: in the first stage we
compress the EEG by means of volumetric coding, tensor
decomposition and energy-based coding of significant sub-
bands; in the second stage, we apply arithmetic coding to the
time residual for the first two algorithms and wavelet-domain
residual for third algorithm, after uniform quantization. Such
two-stage compression scheme allows us to bound the maxi-
mum (worst-case)distortion. We discuss the compression per-
formance of the three compression algorithms by means of an
average and worst-case distortion measure.
This paper is structured as follows. In Section 2 we ex-
plain the multi-way (or volumetric data) representation of
multi-channel EEG. We outline our compression algorithms
(a) t/dt/s volume (b) s/s/t volume
Fig. 1. Formation of 3D volume from multi-channel EEG. (a) The matrices
(I
1
, I
2
, . . . , I
M
) formed from single-channel EEG are stacked as volume
(“t/dt/s volume”). (b) At any time instance, we form matrix from the multi-
channel EEG (A & B denote samples from adjacent electrodes in EEG mon-
tage). N such matrices formed at subsequent time instances are then stacked
along the z-direction to form a volume (“s/s/t volume”).
in Section 3, and present our results in Section 4, followed by
concluding remarks in Section 5.
2. TENSOR/VOLUMETRIC DATA FORMATION
FROM EEG
Spatially adjacent channels of multi-channel EEG are strongly
correlated, and each individual channel is strongly correlated
across time. To exploit both spatial and temporal correlations
simultaneously, we arrange multi-channel EEG as a 3D vol-
ume or three-way tensor. We consider two specific ways to
extract a volumetric data from multi-channel EEG, where the
three axes capture spatial and temporal variations in different
form.
In Fig. 1(a) we illustrate the EEG volume formed accord-
ing to our first method. The k-th slice I
k
of the volume I,
extracted from channel k, can be written as:
I
(k)
t/dt/s
= {I
k
|k = 1, . . . , M } (1)
=
i
k
(1) i
k
(2) · · · i
k
(N)
i
k
(2N ) i
k
(2N − 1) · · · i
k
(N + 1)
.
.
.
.
.
.
.
.
.
.
.
.
· · · · · i
k
(N
2
)
(N×N)
.
From our previous studies [2, 6], we found that such ar-
rangement leads to improved compression performance over
conventional vector-based compression schemes. Next, the
matrices associated with the single-channel EEG signals are
stacked to form 3D volume, as shown in Fig. 1(a). Adjacent
slices in the tensor correspond to adjacent EEG channels. We
refer to this volume as “t/dt/s”, where the x, y, and z direc-
tions reflect temporal (t), delayed (dt) temporal, and spatial
(s) variations respectively.
We also consider an alternative method to form a tensor
from multi-channel EEG. A matrix is formed from the multi-
channel EEG at each time instance. We arrange the matrix
such that its elements follow similar adjacency as the EEG
montage; for the sake of brevity, we omit the details. We
stack the matrices from subsequent time instances to form a
volume, as shown in Fig. 1(b). The x − y plane reflects the
spatial correlations, and the temporal correlations is along the
z direction. We refer to this volumetric data as “s/s/t”. The
k-th slice of the volume may be written as:
I
(k)
s/s/t
={i
(i,j)
(k)|k = 1, . . . , N } (2)
=
i
(1,1)
(k) i
(1,2)
(k) · · · i
(1,N
2
)
(k)
i
(2,1)
(k) i
(2,2)
(k) · · · i
(2,N
2
)
(k)
.
.
.
.
.
.
.
.
.
.
.
.
i
(N
1
,1)
(k) i
(N
1
,2)
(k) · · · i
(N
1
,N
2
)
(k)
(N
1
×N
2
)
,
where i and j refer to the position in the x-y plane, whereas
the slice number k refers to the time index. The dimension of
the x − y plane is limited by the number of channels, and the
slices in the x − y plane may be square or rectangular.
3. COMPRESSION ALGORITHMS
We first perform lossy coding (Stage 1), followed by arith-
metic coding of the quantized residuals (Stage 2). We con-
sider three lossy compression algorithms (Stage 1): (i) 3D
Wavelet volumetric coding, (ii) 3D Wavelet subband specific
arithmetic coding, and (iii) tensor decomposition(PARAFAC)
based coding. In the following we explain our three compres-
sion algorithms.
3.1. Wavelet-based Compression
3.1.1. Volumetric Coding Approach
Fig. 2 shows a diagram of the proposed two-stage coder for
multi-channel EEG signals. We denote the EEG volume by I
(both types of volumes, cf. Fig 1). In the first stage, we
compress I with a scalable wavelet encoder based on succes-
sive bit-plane encoding, resulting in the compressed data I
en
;
we use a bi-orthogonal wavelet transform (5/3 filters) as in
our previous work [2]. The compressed data I
en
is then de-
coded, yielding the reconstructed data I
l
. Next we quan-
tize the residue ε = I − I
l
, resulting in ε
q
, which is com-
pressed by arithmetic coding, leading to ε
q−en
. Both are used
by the decoder to approximate the original data. The com-
pressed data I
en
is first decoded, yielding the lossy recon-
structed data I
l
. The data ε
q−en
is passed through an arith-
metic decoder and then dequantized, resulting in ˆε. The latter
is an approximation of the residual ε. Eventually, the data I
is reconstructed as I
nl
= I
l
+ ˆε. The volume I
nl
is at last
rearranged to yield the reconstructed EEG signal(s). We can
readily confirm the following relations:
I = I
l
+ ε (3)
I
nl
= I
l
+ ˆε. (4)
Therefore, it follows that ||ε− ˆε||
∞
= ||I −I
nl
||
∞
, and hence
||ε − ˆε||
∞
≤ δ is equivalent to ||I − I
nl
||
∞
≤ δ. The residual
ε is uniformly quantized to generate quantization indices ε
q
,
with maximum error no larger than δ:
Formation
of
Volume/Tensor
Wavelet-based
3D Encoder
Wavelet-based
3D Decoder
+
Q(·)
Uniform
Quantizer
Residual
Arithmetic
Coder
−
I
I
l
ε
ε
q
ε
q−en
I
en
Multi-channel
EEG
Fig. 2. Wavelet-based volumetric coding of multi-channel EEG
Table 1. Wavelet-based subband specific coding procedure
Step 1: Initialization
(a) Form the volume from the multi-channel EEG,
I ← t/dt/s or s/s/t volume
(b) Compute the Wavelet transform of the volume.
I
w
= 3D-DWT(I, D)
// D-level decomposition yields 7D + 1 subband cubes
(c) Determine the relative energy density (RED
i
) of the sub-
bands (i = 1, . . . , 7D + 1)
RED(i) =
P
j
I
i
w
(j)
2
N
i
·
P
j
I
w
(j)
2
where N
i
is the number of elements in i
th
subband
(d) Coding order O = descend(RED)
(e) Set the threshold τ (% of total energy)
Step 2: First-pass coding - Coding of significant subbands until τ . Set
relative-energy RE = 0, i = 1.
while (RE < τ )
(a) Bitstream ← AC(I
O(i)
w
) // Code the subband according
to O by Arithmetic coding
(b) RE = RE + RED
O(i)
· N
O(i)
// update relative energy of the coded subband. N (O(i))
is number of elements in the subband O(i)
(c) i ← i + 1
end
Step 3: Second-pass coding - lossy coding of the remaining subbands
(O(i + 1) to O(7D + 1))
for (j = i + 1 : 7D + 1)
(a) I
O(j)
w
= Q(I
O(j)
w
, δ)
// Quantize the wavelet coefficients with quantizer step-
size δ
(b) Bitstream ← AC(I
O(j)
w
)
end
ε
q
=
⌊
ε+δ
2δ+1
⌋, ε > 0
⌊
ε−δ
2δ+1
⌋, ε < 0
, (5)
where ⌊·⌋ denotes the integer part of the argument. At the
decoder end, the residual bitstream ε
q−en
is decoded to yield
ε
q
, followed by a dequantizer defined to guarantee ||ε − ˆε|| ≤
δ:
ˆε = (2δ + 1)ε
q
. (6)
By adding the lossy reconstruction I
l
and the dequantized
residual ˆε, we obtain the final near-lossless reconstruction
I
nl
with guarantee ||I − I
nl
|| ≤ δ. In words, the maxi-
mum distortion is therefore bounded to δ. The pre-processing
step, i.e., formation of tensor from multi-channel EEG, is the
principle difference from the coders used in image compres-
sion [7].
3.1.2. Subband Specific Arithmetic Coding (SAC)
In this approach, we first order the wavelet subbands based on
their relative energy density (RED). We use the same wavelets
as in volumetric coding approach (cf. Section 3.1.1). In first
stage, we compress the most significant wavelet subbands
losslessly, followed by lossy compression of the subbands
with smaller energy concentration. Specifically, in the first
stage, we apply arithmetic coding to the subbands with high-
est RED, until a certain threshold τ (% of total energy) is
reached. The remaining subbands are less significant in terms
of their RED; we first quantize them (cf. (5)), and then ap-
ply arithmetic coding. This two-stage procedure results in
lossy compression of the EEG signals. The pseudo-code of
the algorithm is presented in Table 1. We code each subband
separately using simple arithmetic coding where all the coef-
ficients within the same subband are represented by a single
probability model. It is noteworthy that in the second coding
step, we quantize the wavelet subbands; this may lead to a
substantial error in time domain. In other words, we cannot
control the maximum distortion in time domain through this
approach.
3.2. Tensor-based Compression
We apply parallel factor decomposition (PARAFAC) decom-
position [8] to the three-way tensor I, formed from multi-
channel EEG. In our previous study [9], we have shown
that PARAFAC yielded the best compression performance
among various other matrix and tensor decompositions. The
PARAFAC based decomposition of a three-way tensor is
given by:
I =
r
X
i=1
a
i
◦ b
i
◦ c
i
+ E, (7)
where E represents the residual tensor, and a, b, and c repre-
sent the factors along the three modes, whereas ◦ stands for
the outer-product along the particular mode. These three fac-
tors efficiently capture the major variations along the three
modes. In the first-stage we encode the PARAFAC factors
using a simple bit-plane coding scheme, and in the second
stage, we apply arithmetic coding to the residuals after uni-
form quantization (5).
4. RESULTS
We test the performance of our compression algorithms on
the EEG-Motor Mental Imagery datasets of physiobank
database [10]. This EEG dataset consists of 64-channel
recordings, recorded from healthy subjects at 80Hz sampling
Compression Ratio
PRD(%)
1 2 3 4 5 6 7 8 9 10
0
5
10
15
3D Volumetric
coding
PARAFAC
coding
Subband AC
(a) PRD(%)
Compression Ratio
PSNR(x, ˜x)
1 2 3 4 5 6 7 8 9 10
15
20
25
30
35
40
3D Volumetric
coding
PARAFAC coding
Subband
AC
(b) PSNR(x, ˜x)
Fig. 3. Compression performance of the wavelet-based volumetric coding,
subband specific arithmetic coding and PARAFAC based coding for t/dt/s
volume/three-way tensor.
rate and with 12 bit resolution. We analyze the performance
of the algorithms based on compression ratio:
CR =
L
orig
L
comp
, (8)
where L
orig
and L
comp
are the bit length of original and recon-
structed multi-channel EEG signals respectively. The quality
of the reconstructed signal (˜x) is assessed using percent root-
mean-square distortion (PRD (%)):
PRD (%) =
v
u
u
t
P
N
i=1
(x(i) − ˜x(i))
2
P
N
i=1
x(i)
2
× 100. (9)
We also use an alternative quantitative distortion measure,
based on the maximum absolute difference between x and ˜x:
PSNR(x, ˜x) = 10 log
10
2
Q
− 1
max(|x − ˜x|)
. (10)
We consider segments of 1024 samples from each chan-
nel, arranged in a suitable volume size, specifically, 32 × 32 ×
64 for t/dt/s volume, 8 × 8 × 1024 for s/s/t volume. In all
the three algorithms, we vary the quantization step-size from
0 (lossless) till 19 (lossy), and measured the CR and PRD(%)
for each step-size. The energy threshold (τ) for the subband
specific arithmetic coding is fixed to 50%; we obtained the
best results for that value of the threshold. The results are
summarized in Fig. 3. Since the results for t/dt/s and s/s/t are
similar, we only show results for the t/dt/s volume construc-
tion.
It is clear from Fig. 3 that SAC outperforms both vol-
umetric and PARAFAC coding with respect to PRD(%);
with respect to PSNR, volumetric and PARAFAC coding
perform similarly, but they clearly outperform SAC. The
residual quantization is performed in time-domain for volu-
metric and PARAFAC coding, whereas in SAC, quantization
is performed in wavelet domain. The idea behind SAC is to
quantize the residual wavelet subbands with least energy. As
we are quantizing wavelet coefficients in SAC, the maximum
error in time domain cannot be controlled, and consequently,
it is larger than the other two approaches considered here.
Interestingly, it is promising that the average error (PRD) is
smaller for SAC compared to other two approaches. How-
ever, the distortion may be large in very few samples, which
may be tolerable in some specific applications.
5. CONCLUSION
We have presented novel compression schemes for multi-
channel EEG. The main idea is to exploit the intra- and
inter-channel correlations simultaneously by arranging the
multi-channel EEG as a volume, and to represent that volume
in different ways. Particularly, we considered volumetric
coding, energy-based coding of wavelet subbands, and ten-
sor based coding. Next we compressed the residual, which
allows us to bound the worst-case distortion (in volumetric
and PARAFAC coding). The tensor-based coding scheme
yields smaller worst-case error than both subband specific
coding and volumetric coding, yet the average error is only
slightly larger than in subband specific coding and much
smaller than in volumetric coding. Therefore, tensor-based
coding is an attractive approach for multi-channel EEG com-
pression. If larger worst-case distortion is tolerable, wavelet
subband coding may also be a suitable option. In our future
study, we planned to improve the worst-case error of the pro-
posed wavelet subband specific coding by suitable threshold
selection.
6. REFERENCES
[1] E. Nidermeyer and F. L. D. Silva, Electroencephalography: Basic
Principles, Clinical applications and related fields, 5th ed., Lippincott
Williams and Wilkins, 2005.
[2] K. Srinivasan, J. Dauwels, and M. R. Reddy, “A two-dimensional ap-
proach to lossless EEG compression,” Biomedical signal processing
and control, vol. 6, pp. 387–394, 2011.
[3] G. Antoniol and P. Tonella, “EEG data compression techniques,” IEEE
Transactions on Biomedical engineering, vol. 44, no. 2, pp. 105–114,
Feb. 1997.
[4] Y. Wongsawat, S. Oraintara, T. Tanaka, and K. Rao, “Lossless multi-
channel EEG compression,” in Proceedings of IEEE International
Symposium on Circuits and Systems (ISCAS), sep. 2006, pp. 1611–
1614.
[5] Q. Liu, M. Sun, and R. Sclabassi, “Decorrelation of multichannel EEG
based on hjorth filter and graph theory,” in Proc. 6th International
Conference on Signal Processing, vol. 2, Aug. 2002, pp. 1516–1519.
[6] K. Srinivasan and M. R. Reddy, “Efficient pre-processing technique
for lossless real-time EEG compression,” Electronics Letters, vol. 46,
no. 1, pp. 26–27, Jan. 2010.
[7] S. Yea and W. Pearlman, “A wavelet-based two stage near lossless
coder,” IEEE transactions on Image processing, vol. 15, no. 11, pp.
3488–3500, Nov. 2006.
[8] T. G. Kolda and B. W. Bader, “Tensor decompositions and applica-
tions,” SIAM Review, vol. 51, no. 3, pp. 455–500, September 2009.
[9] J. Dauwels, K. Srinivasan, M. R. Reddy, and A. Chichocki, “Multi-
channel EEG compression based on matrix and tensor decomposi-
tions,” in Proc. International conference on acoustics, speech and sig-
nal processing (ICASSP), 2011, pp. 629–632.
[10] A. L. Goldberger, L. A. N. Amaral, L. Glass, J. M. Hausdorff, P. C.
Ivanov, R. G. Mark, J. E. Mietus, G. B. Moody, C.-K. Peng, and H. E.
Stanley, “PhysioBank, PhysioToolkit, and PhysioNet : Components of
a New Research Resource for Complex Physiologic Signals,” Circu-
lation, vol. 101, no. 23, pp. e215–220, 2000.