CONVOLUTION KERNEL COMPENSATION APPLIED TO 1D

AND 2D BLIND SOURCE SEPARATION

Damjan Zazula, Aleš Holobar, Matjaž Divjak

Faculty of Electrical Engineering and Computer Science, University of Maribor

Smetanova 17, 2000 Maribor, Slovenia

Keywords: Blind source separation, Convolution kernel compensation, Bioelectrical signal decomposition,

Range imaging.

Abstract: Many practical situations can be modelled with multiple-input multiple-output (MIMO) models. If the input

sources are mutually orthogonal, several blind source separation methods can be used to reconstruct the

sources and model transfer channels. In this paper, we derive a new approach of this kind, which is based on

the compensation of the model convolution kernel. It detects the triggering instants of individual sources,

and tolerates their non-orthogonalities and high amount of additive noise, which qualifies the method in

several signal and image analysis applications where other approaches fail.. We explain how to implement

the convolution kernel compensation (CKC) method both in 1D and 2D cases. This unified approach made

us able to demonstrate its performance in two different experiments. A 1D application was introduced to the

decomposition of surface electromyograms (SEMG). Nine healthy males participated in the tests with 5%

and 10% maximum voluntary isometric contractions (MVC) of biceps brachii muscle. We identified 3.4 ±

1.3 (mean ± standard deviation) and 6.2 ± 2.2 motor units (MUs) at 5% and 10% MVC, respectively. At the

same time, we applied the 2D version of CKC to range imaging. Dealing with the Middlebury Stereo Vision

referential set of images, our method found correct matches of 91.3± 12.1% of all pixels, while the obtained

RMS disparity difference was 3.4 ± 2.5 pixels. This results are comparable to other ranging approaches, but

our solution exhibits better robustness and reliability.

1 INTRODUCTION

Blind source separation (BSS) has matured to a very

well established theory which has given a fresh im-

petus to several applications in different research

fields. If a problem can be modelled in multiple-

input multiple-output sense (MIMO) and the input

excitations of such a model can be considered or-

thogonal sources, many BSS techniques are available

to separate those sources. Robust and useful solu-

tions have been reported for telecommunications

(Madhow, 1998), seismic and radar signals (Desodt,

1994), speech processing (Gribonval, 2002), bioelec-

tric signals (Barros, 1999), image processing (Hy-

värinen, 2002), etc.

The majority of BSS-based approaches take ad-

vantage of the sources’ orthogonality. Several obser-

vations, i.e. the output signals of the presumed

MIMO model, are taken into account referring to

their mutual information contents, such as covariance.

The covariance-based techniques build a covariance

matrix which comprises the information on the model

transfer channels, i.e. the model convolution kernel,

and the covariance of sources. Actually, the source

covariance matrix appears to be diagonal, which un-

veils the convolution kernel. The information on the

convolution kernel is, afterwards, used to deconvolve

also the original source signals (Cardoso, 1998), (Be-

louchrani, 1997).

However, there are two major drawbacks that

degrade the success of BSS in certain cases, which is

when the number of observations is lower than the

number of sources and when the sources lack the

orthogonality. The both drawbacks prevent a proper

identification of the convolution kernel, which hin-

ders applications in the biomedical field, for example.

The obtained shapes of source signals and modelled

channel responses are distorted because they are pro-

jected into an orthogonal space, in underdetermined

cases also with lower number of dimensions as

126

Zazula D., Holobar A. and Divjak M. (2006).

CONVOLUTION KERNEL COMPENSATION APPLIED TO 1D AND 2D BLIND SOURCE SEPARATION.

In Proceedings of the International Conference on Signal Processing and Multimedia Applications, pages 126-133

DOI: 10.5220/0001568801260133

Copyright

c

SciTePress

needed. So, the obtained results equal unknown linear

combinations of the original sources and channel

responses.

Recently, a novel approach was proposed which

successfully separates the contributions of sources

even if they are only close-to-orthogonal and if the

number of observations is underdetermined (Holobar,

2004). It is based on the fact that there is variety of

situations where sources produce only a limited num-

ber of finite symbols (source activity). Being sent

through the transfer channel, those symbols are con-

volved with the channel responses and they appear in

the observations as the contributions of symbols. If,

for example, bioelectric signals are considered, elec-

trocardiograms (ECG) can be modelled with fully

orthogonal sources (there is no overlap possible be-

tween different types of heart beats, such as normal

systoles and extrasystoles), while electromyograms

(EMG) lose orthogonality with increasing contraction

forces (motor-unit action potentials exhibit more

overlapping) (De Luca, 1996). On the other hand,

observing certain types of communications, such as

CDMA (Madhow, 1998), orthogonality of sources

may be supposed as well. Moreover, we are not con-

strained to 1D; similar reasoning may be extended to

2D images. If an image is taken as a MIMO output

observation, it can also be considered the result of

some source activities transferred through the model

channels. In this case, sources produce symbols in the

form of 2D regions (subimages that contribute to the

observed image), and are expected to be orthogonal

(subimages do not overlap).

Given a number of observations of some sources,

the contributions produced by the transferred source

symbols may be characterized by their shape and

appearance (triggering) instants. Stationarity is also

supposed both for the sources and for the model

channels. We have developed a method which makes

use of the abovementioned facts and detects the trig-

gering instants of the same symbols as they appear in

the observation. Our method actually compensates

the detected shape of the observed source contribu-

tions. The approach turned out to be Bayesian opti-

mal (Kay, 1993).

The paper continues as follows: Section 2 reveals

the novel method called convolution kernel compen-

sation (CKC) and extends it from 1D to 2D, Section 3

explains its application to surface EMG signals, while

section 4 shows the method’s efficiency when applied

to range imaging in stereo vision. The paper is con-

cluded by Section 5.

2 DETECTION OF SOURCE-

SYMBOL TRIGGERING

INSTANTS

Consider the following data model:

∑∑

=

−

=

=+−=

K

j

L

l

ijiji

Minvlntlcny

1

1

0

,,1);()()()( … (1)

where y

i

(n) stands for the i-th observation, c

ij

(l) cor-

responds to the contribution of length L of the j-th

source symbol in the i-th observation, and t

j

(n-l)

denotes a sequence of triggering instants for this

symbol,

∑

∞

−∞=

−=

l

jj

lTnnt ))(()(

δ

, with unit-sample

pulses placed at T

j

(l) lags, while v

i

(n) is considered

i.i.d. white noise independent from the sources.

It has been shown 0 that Eq. (1) can be trans-

formed into a multiplicative vector form as follows:

1,,0);()()( −=

+

=

Nnnnn

eeee

…vtCy (2)

where subscript e designates extended vectors and

matrices, C

e

contains the observed contributions of

source symbols,:

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

=

MKM

K

e

CC

CC

C

1

111

(3)

with

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

−

−

=

)1()0(0

0)1()0(

Lcc

Lcc

ijij

ijij

ij

C

, (4)

y

e

(n) stands for the extended vector of observations,

and t

e

(n) for the vector of triggering pulses, both at

lag n:

T

eK

Kee

T

eM

Mee

MLnt

ntMLntntn

Mny

nyMnynyn

)]2(

),....,(),....,2(),....,([)(

)]1(

),....,(),....,1(),....,([)(

11

11

+−−

+−−=

+−

+−=

t

y

(5)

Extended noise vector v

e

(n) is considered constructed

in the same way.

M

e

from Eqs. (5) means an extension factor. If it

fulfils the following inequality

)1(

−

+

≥

⋅

ee

MLKMM ,

(6)

then for K different observed symbols of length L and

M observations the matrix C

e

is of full column rank.

This condition warranties a successful elimination of

contributions of C

e

, as we are going to show in the

next subsection.

CONVOLUTION KERNEL COMPENSATION APPLIED TO 1D AND 2D BLIND SOURCE SEPARATION

127

2.1 Convolution Kernel

Compensation

Recall Eq. (2). It has a typical MIMO structure. From

this point of view, C

e

is a convolution kernel convolv-

ing t

e

(n) to observations y

e

(n). Given y

e

, if we can get

rid of C

e

the triggering instants of unknown source

symbols, t

e

, would result. We called this process

“convolution kernel compensation (CKC)”.

Observe the following expression:

)()(

1

nn

e

T

e

e

yRy

y

−

(7)

where

e

y

R

stands for the sample correlation matrix:

ICRCICttCyyR

ty

22

σσ

+=+==

T

ee

T

e

T

eee

T

ee

ee

(8)

with

e

t

R

denoting sample correlation matrix of

source triggering trains of pulses, and the expression

σ

2

I stands for the correlation matrix of noise v

e

.

For easier comprehension of derivation, continue

with the noise-free case. By substituting (8) into (7),

we see that convolution kernel is eliminated:

)()(

)()()()(

1

111

nn

nnnn

e

T

e

eee

T

e

T

e

T

ee

T

e

e

ee

tRt

tCCRCCtyRy

t

ty

−

−−−−

==

(9)

The expression from (7) is known as Mahalanobius

distance, which, as it is clear from Eq. (9), yields only

the information on source triggering instants. Actu-

ally, its value depends on the number of sources ac-

tive in given time instant n. This is why we call it

activity index.

Suppose we deal with orthogonal sources and n

0

indicates the time instant where one of them gener-

ates a symbol (its contribution appears in the observa-

tion). Then vector t

e

(n

0

) is all zero except the element

which belongs to the generated symbol, say the i-th,

and equals 1. Besides, matrix

e

t

R

is diagonal, and so

is

1−

e

t

R

. It is then straightforward that

)()()(

)()()()()(

,,,0,,

1

0

1

0,

0

ntrntntr

nnnnnp

ieiiieieii

e

T

ee

T

ein

ee

=

===

−−

tRtyRy

ty

(10)

where r

i,I

denotes the i-th diagonal element of

1−

e

t

R ,

and t

e,i

(n) stands for the train value at lag n for the i-th

source symbol. Evidently, in Eq. (10) we have ob-

tained a sequence

in ,

0

p

whose values equal the i-th

source-symbol triggering pulse train to a constant

amplitude factor, r

i,i

. So, all repetitions of that symbol

are detected.

The values of activity index indicate those lags n

i

where individual sources contribute their symbols. If

we select such n

i

’s that cover all different source con-

tributions, a thorough decomposition is done and all

source-symbol triggering pulse trains,

t

i

; i∈[1,K], are

separated.

Let’s also briefly discuss nonideal conditions. In

ideal condition with orthogonal source-symbol con-

tributions, no noise and the number of observations

exceeding the number of different source-symbol

contributions, the convolution kernel

C

e

is completely

eliminated. If any of the ideal conditions cannot be

met,

C

e

is not eliminated but only compensated to a

certain extent. Consequently, the resulting decompo-

sition of source-symbol triggering instants, Eq. (10),

move off the ideal binary valued pulse train (sample

values either 0 or r

i,i

). Hence, the ideal Bernoulli dis-

tribution of any

jn

i

,

p

tends to “smear”, so the prob-

ability distribution of “no trigger” values may start

overlapping the distribution of “trigger” values. A

more detailed explanation goes beyond the scope of

this paper, so we only stress here that even in far non-

ideal cases, such as with the signal-to-noise ratio as

low as 0 dB, confidence level for the detection of

source-symbol triggering instants remain above 98 %.

Some additional results are given in the experimental

part, Sections 3 and 4.

2.2 Extension to 2D Cases

As we have pointed out, analogy between source con-

tributions in 1D and 2D observations can be found.

2D observations, i.e. images, can be interpreted as a

compositum of several subimages appearing at differ-

ent image co-ordinates. Thus, an image may be seen

as a convolution of different regions and the corre-

sponding “triggering” unit-samples whose positions

in 2D determine the region placements within the

image frame.

The most obvious way to implement CKC also

in 2D is vectorization of images. Assume we have set

of images

I

k

; k∈[1,M], and that I

k

(i,j) denotes the

value of the k-th image pixel at (i,j) co-ordinates.

Then

y

k

=vec(I

k

) (11)

is a vector whose elements correspond to the con-

catenated rows of image

I

k

, so that

)]1,1(

,),1,1(),0,1(

,,),1,1(,),1,1(),0,1(

,),1,0(,),1,0(),0,0([)(

21

11

2

2

−−

−−

−

−=

NNI

NINI

NIII

NIIIvec

k

kk

kkk

kkkk

…

……

…

0

0I

(12)

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128

where N

1

and N

2

stand for image dimensions. Every

row of pixels is padded by M

e

zeros (denoted by vec-

tor

0), where M

e

means the extension factor from

inequality (6).

The extension of vector

y

k

from Eq. (11) is per-

formed the same way as for 1D case in Eq. (5). Also

the other decomposition steps explained in Eqs. (7) to

(10) can be implemented without modifications. A

selected n

0

now determines location of a certain

subimage region, with its 2D co-ordinates being

transformed into n

0

by vectorization. The resulting

sequence

in ,

0

p

comprises pulses at the positions indi-

cating the repetitions of the subimage from location

n

0

. For optimal decomposition results, the number of

observations, M, meaning different images of the

same scene here, must exceed the number of different

subimages.

It has to be emphasized that in Eq. (12) proposed

image vectorization leads to a one-row vector, which

limits the decomposition to subimages of one-row

width only. At the same time, these subimages can

extend at most across M

e

image columns, because the

CKC extension introduced in Eq. (5) “joins” the in-

formation of M

e

subsequent samples. If the subimage

regions of interest span larger areas, images have to

be vectorized differently. They have to be segmented

in such a way that the number of rows in every seg-

ment corresponds to the vertical dimension of the

regions looked for. Every segment row is then taken

as a separate observation entering the CKC-based

decomposition. Consequently, only a single image

segment is decomposed at a time, with no correlation

to other segments. However, it is also possible for

several image segments to be included into the same

decomposition run. In this case, those segments have

to be padded by M

e

zeros and concatenated.

3 APPLICATION OF CKC TO

THE SURFACE

ELECTROMIOGRAM

DECOMPOSITION

Human body contains different kinds of electrically

excitable tissues, such as nerves and muscle fibres,

which, when active, conduct measurable biopoten-

tials, typically of length of several ms. These biop-

tentials can be detected either by inserting invasive

needle electrodes into the tissue or by placing pick-

up electrodes on the skin surface, above the investi-

gated organ. Although being more selective, the in-

vasive needle electrodes impose several restrictions

to everyday clinical investigations. Firstly, measure-

ments must be taken in a sterile environment and

under supervision of trained physicians. Secondly, in

order to reduce the tissue damage, there is a constant

need for miniaturization of needle electrodes. This

significantly increases the costs of manufacturing.

Finally, the invasive recoding techniques put a lot of

stress on an investigated subject and increase the fear

from preventive clinical investigations (Merletti,

1994).

The aforementioned problems can be avoided by

using less selective surface electrodes, providing

signal processing techniques exists, which are capa-

ble of extracting clinically relevant information out

of recorded data. Unfortunately, this is not a trivial

task. Namely, the supportive tissues separating the

investigated biological sources from the pick-up

electrodes acts as a low pass filter and hinder the

information in the detected signals. In addition, ac-

quiring surface signals, contributions of different

biological sources are detected. When electrical ac-

tivity of skeletal muscles is observed, for example,

we deal with several tens of sources (so called motor

units, MU), simultaneously contributing their biopo-

tentials (so called action potentials, AP) to the de-

tected EMG interference pattern (Merletti, 1994).

The decomposition of the surface EMG into the con-

tributions of different MUs is, hence, a highly com-

plex problem whose solution has been addressed

with a many different methods. Unfortunately, most

methods suffer from a drop of performance in case of

significant superposition of MU action potentials.

Surface EMG signals can always be modeled

by Eq. (1), provided they have been acquired during

an isometric muscle contraction (De Luca, 1996). In

such a model, observations y

i

(n) correspond to

measured surface signals, c

ij

(n) corresponds to the

action potential of the j-th MU, as detected by the i-

th pick-up electrode, while t

j

(n) stands for a pulse

sequence carrying the information about triggering

times of APs. The length of detected APs, L, de-

pends on the sampling frequency, but typically

ranges from 15 to 25 samples when the Nyquist

frequency is made equal to the bandwidth of the

surface signals. At low contraction levels, different

MUs discharge in relatively regular but random time

instants, independently of each other. At higher con-

traction levels, the MUs start exhibiting weak ten-

dency to synchronize, but this synchronization

hardly exceeds the 5 % of its maximal possible

value. As a result, t

j

(n) can be modelled as close-to-

orthogonal random pulse sequences and the theory

of 1D CKC method can be readily applied to the

SEMG signals. This is further demonstrated by the

CONVOLUTION KERNEL COMPENSATION APPLIED TO 1D AND 2D BLIND SOURCE SEPARATION

129

experimental results described in the next subsec-

tion.

3.1 Experimental Protocol

Nine healthy male subjects (age 26.8 ± 2.2 years,

height 179 ± 7 cm, weight of 72.1 ± 8.3 kg) partici-

pated to our experiment. Surface EMG signals were

acquired during isometric, constant-force contractions

of the dominant biceps brachii muscle. In order to

provide sufficient number of measurements, M, a

matrix of 55 pick-up electrodes arranged in five col-

umns and 11 lines (without the four corner electrodes)

was used while all the contractions were performed at

5% and 10% of the maximum voluntary contraction

(MVC) force. The EMG signals were recorded in

longitudinal single differential configuration, ampli-

fied (gain set to 5000), band-pass filtered (3 dB

bandwidth, 10 –500 Hz), and sampled at 2500 Hz by

a 12 bit A/D converter. During signal acquisition, the

noise and movement artefacts were visually con-

trolled and reduced by applying water to the skin sur-

face. Before any further processing, all the measure-

ments were digitally filtered to suppress the power-

line interference. Recorded signals are exemplified in

Fig. 1.

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

1

2

3

Time [s]

Measurements

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

1

2

3

Time [s]

Measurements

Figure 1: Real surface EMG signals recorded during isomet-

ric, low-level (10% MVC) contraction of dominant biceps

brachii muscle.

The measured signals were extended according to

Eq. (5) with extension factor, M

e

, set to 10. In order to

reconstruct the MU triggering pulses (Fig. 2), 1D

CKC decomposition method was applied to the

measured signals. The identified triggering pulses

were then used by spike triggering sliding window

averaging technique (Disselhorst-Klug, 1999) to re-

construct the MU APs as detected by different pick-

up electrodes (Fig. 3 depicts the first decomposed AP

as it contributes to each of 51 electrodes). Finally,

convolving the estimated AP shapes with the identi-

fied sequence of MU triggering pulses, the MU AP

trains were reconstructed and compared to the origi-

nal measurements (Figs. 4.a and 4.b). Rows 1 to 10 in

Fig. 4 correspond to the ten decomposed MU APs

depicted in the time instants when they trigger and

contribute to the measured SEMGs. They are

summed up in row 4.b).

23.6 24.4 25.2 26 26.8 27.6 28.4 29.2 30

1

2

3

4

5

6

7

8

Time [s]

MU

23.6 24.4 25.2 26 26.8 27.6 28.4 29.2 30

1

2

3

4

5

6

7

8

Time [s]

MU

Figure 2: A part of MU triggering pulses (i.e. time instants

in which the contributions of different MUs appeared in

observations) reconstructed by the 1D CKC method from 30

s long real SEMG signals of dominant biceps brachii muscle

(subject 3, 10% MVC measurement).

2 3

Columns of electrodes

41

11

10

9

8

7

6

5

4

3

2

1

Rows of electrodes

52 3

Columns of electrodes

41

11

10

9

8

7

6

5

4

3

2

1

Rows of electrodes

5

Figure 3: APs of MU 1 reconstructed by the spike triggered

sliding window averaging technique (293 averages accord-

ing to the train depicted in Fig. 2, bottom) from given 30 s

long SEMG observations.

On average, 3.4 ± 1.3 (mean ± standard deviation)

and 6.2 ± 2.2 MUs were identified during the contrac-

tions at 5% and 10% MVC, respectively. The exact

number of active MUs is, of course, unknown. Never-

theless, comparing the energies of the identified MU

action potentials with the energy of the original signal

we can approximately estimate the percentage of the

information that was extracted from the surface EMG

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130

signals. The average ratio yielded 71 ± 15%, proving

that the largest SEMG components were identified

(Fig. 4). Most of the identified MUs showed decreas-

ing firing frequency over time (presumably due to

fatigue).

19.05 19.1 19.15 19.2 19.25 19.3 19.35 19.4

1

2

3

4

5

6

7

8

9

10

b)

a)

Time [s]

Motor units

19.05 19.1 19.15 19.2 19.25 19.3 19.35 19.4

1

2

3

4

5

6

7

8

9

10

b)

a)

Time [s]

Motor units

Figure 4: Contributions of different MUs identified by the

1D CKC method from 30 s long real SEMG signals re-

corded during isometric, 10% MVC measurement of the

dominant biceps brachii (subject 8); a) the originally ob-

served SEMG signal, b) sum of the identified MU contribu-

tions.

Finally, the shapes of the MU action potentials as

detected by the different electrodes were stable over

time and indicated anatomical and physiological MU

properties, such as location of innervation zones,

length of the fibres, and muscle fibre conduction ve-

locity (Fig. 3).

4 APPLICATION OF CKC TO

RANGE IMAGING

Human beings depend on stereo vision for observing

their surroundings. Slight displacement of images

enables them to reliably detect range information,

which can be used to their advantage. In computer

vision, the same effect is used to reconstruct the

range or depth image of a scene based on two or

more input images. Range reconstruction can be

formulated as a matching problem between pixels of

the left and right stereo image. In general, the prob-

lem doesn't have a unique solution due to lack of

image texture, occlusions, periodic image structures

and noise (Šara, 2002). Early algorithms avoided

those problems by reconstructing only sparse range

images (Sonka, 1994). Modern applications, such as

image-based modelling, texture mapping of 3D ob-

jects and similar, require dense range images, where

disparity of almost all image pixels is known. To

alleviate this problem, several constraints are com-

monly used. All the surfaces in the scene are sup-

posed to be Lambertian, the geometry of the stereo-

system should be known (calibrated camera) and

range values are expected to change smoothly, with-

out sharp jumps (Gutierrez, 2003).

Using the geometric properties of the stereosys-

tem, it can be shown that the matching space can be

reduced to two epipolar image rows (Jain, 1995).

Each image row can easily be represented as an ob-

servation

y

k

, extended and decomposed by the CKC

method, referring to the extension introduced in Sub-

section 2.2. In order to detect disparity of every

pixel, its position in the left image row is described

by index n

0

and the pulse train

in ,

0

p

is calculated

according to Eq. (10) along the right epipolar image

row. Ideally, the sequence

in ,

0

p

contains only one

sharp impulse. This pulse indicate the most probable

location n

1

of the subimage which best corresponds

in the right stereo image to that selected by index n

0

in the left image (Fig. 5). Disparity of pixel n

0

is,

therefore, calculated as

Disp(n

0

) = n

0

– n

1

. (13)

In order to achieve more reliable and robust re-

sults, the matching is repeated using the right stereo

image as a starting point. Only pixels with consistent

left-to-right and right-to-left matches are assigned the

final disparity value.

Figure 5: An example of a pulse train

in ,

0

p

, as obtained for

a selected row in the right image. The location of the pulse

determines the location of a suitable subimage region.

As we have explained in Subsection 2.2, the

shape of image regions being matched by our CKC

approach is determined by the number of image rows

included in one decomposition run, and by the exten-

sion factor M

e

. The quality of left-right stereo-image

matching depends on the appearance of the same im-

age regions in both left and right images. This ap-

pearance may not be equally good for smaller or lar-

ger sections of an image object. So, it can be expected

that its depth may be misinterpreted owing to inferior

quality of matching. However, if we observe a part of

CONVOLUTION KERNEL COMPENSATION APPLIED TO 1D AND 2D BLIND SOURCE SEPARATION

131

an object in different sizes, so with different size of

regions inserted in the CKC-based matching, the in-

formation of best fit can be compared for different

region sizes. Thus, the most probable disparity can be

estimated, which is the idea followed by our 2D CKC

range imaging, as exemplified in the next subsection.

4.1 Experiments with CKC-based

range imaging

All experiments were performed on the test images

from the Middlebury Stereo Vision Page (Scharstein,

2002). This test set provides reliable reference data

and is very popular in the research community, ena-

bling the comparison of different range reconstruction

techniques. The results of our CKC-based approach

are depicted in Fig. 6.

a)

b)

c)

Figure 6: An example of reconstructed range image on the

SAWTOOTH stereo test set: (a) left input image, (b) refer-

ence range image, and (c) the resulting CKC-based range

image.

In Table 1, they are compared to results of the

standard correlation-based approach using the follow-

ing performance metrics:

o percentage of image pixels with consistent

left-to-right and right-to-left matches,

o percentage of matched pixels whose disparity

differs from the reference value by more than

1 pixel,

o RMS of disparity difference between matched

pixels and the reference image.

Table 1: Comparison of disparity values, obtained the 2D

CKC method and a typical correlation-based approach

(SSD, 5×5 window). Mean values for four test images

(MAP, TSUKUBA, VENUS, SAWTOOTH) are shown.

Algorithm

Matches

found

(%)

Bad

matches

(%)

RMS dis-

parity

difference

(pixels)

CKC

91.3 ±

12.1

1.6 ± 0.4 3.4 ± 2.5

SSD

78.2 ±

14.2

2.0 ± 0.5 7.9 ± 4.5

5 CONCLUSIONS

We derived a novel method for statistical signal

processing which blindly separates source

contributions superimposed in one or more available

observations. It is based on the correlation of

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APPLICATIONS

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observations, so that the inverse of correlation matrix

is used to compensate the convolution kernel

influence. The method tolerates moderate declines of

sources from the orthogonality, and copes with

considerable amount of additive random noise.

1D version of our CKC approach was applied to

the decomposition of real surface EMG signals. The

reported results demonstrate the CKC method is not

sensitive to superimpositions of MU action potentials

and has high potential in clinical applications for the

non-invasive analysis of single MU properties.

In this paper we also derived a 2D version of

CKC. It makes use of all the benefits mentioned

above also for image processing. One of possible

applications is searching equivalent regions in more

images, whereas the matching on a pair of stereo im-

ages directly imposes a new range imaging technique.

We exemplified it by constructing range images for a

set of reference images. The obtained results are

comparable with other known approaches, but be-

cause of the CKC being rather noise resistant, the

new way of range imaging obtains a better robust-

ness.

Recent investigations prove that the CKC per-

formance can be improved by combining it with non-

linear modifications of observations and by non-

linear modelling instead of present MIMO scheme.

Our research continues in this direction.

REFERENCES

Madhow, U., 1998. Blind Adaptive interference suppression

for direct-sequence CDMA. In Proceedings of the

IEEE, 86 (10), 2049-2069.

Desodt G.., Muller D., 1994. Complex independent compo-

nents analysis applied to the separation of radar signals.

In Signal Processing V, Theories and applications, 665-

668.

Gribonval R., 2002. Sparse decomposition of stereo signals

with matching pursuit and application to blind separa-

tion of more than two sources from a stereo mixture. In

Proceedings of ICASSP’02.

Barros A. K., Mansour A., Ohnishi N., 1999. Removing

artefacts from ECG signals using independent compo-

nents analysis. In Neurocomputing, 22, 173-186.

Hyvärinen A., Inki M., 2002. Estimating overcomplete in-

dependent component bases for image windows. In

Journal of Mathematical Imaging and Vision, 17, 139-

152.

Cardoso J. F., 1998. Blind signal separation: statistical prin-

ciples. In Proceeding of the IEEE, Special issue on

blind identification and estimation, 9 (10), 2009-2025.

Belouchrani A., Meraim K. A., Cardoso J. F., Moulines E.,

1997. A blind source separation technique based on

second order statistics. In IEEE Transactions on signal

processing, 45 (2), 434-444.

Holobar A., Zazula D., 2004. Correlation-based approach to

separation of surface electromyograms at low contrac-

tion forces. In Med. & Biol. Eng. & Comput., 42 (4),

487-496.

Merletti R., 1994. Surface electromyography: possibilities

and limitations. In J. of Rehab. Sci., 7 (3), 25-34.

De Luca C. J., Foley P. J., Erim Z., 1996. Motor unit control

properties in constant-force isometric contrac-tions. In

J. of Neurophysiol., 76 (3), 1503-1516.

Disselhorst-Klug C., Rau G.., Schmeer A., Silny J., 1999.

Non-invasive detection of the single motor unit action

potential by averaging the spatial potential distribution

triggered on a spatially filtered motor unit action poten-

tial. In J. of Electromyogr. and Kinesiol, 9, 67-72.

Kay S. M., 1993. Fundamentals of Statistical Signal Proc-

essing, Prentice-Hall International. London.

Šara R., 2002. Finding the largest Unambiguous Component

of Stereo Matching. In European Conference on Com-

puter Vision, LNCS 2352, 900-914.

Sonka M., Hlavac V., Boyle R., 1994. Image Processing,

Analysis and Machine Vision, Chapman & Hall. Lon-

don.

Gutierrez S., Marroquin J. M., 2003. Disparity estimation

and reconstruction in stereo vision. In Technical report,

Centro de Investigacion en Matematicas, Mexico.

Jain R., Kasturi R., Schunck B. G., 1995. Machine Vision,

McGraw-Hill Inc. International Edition.

Scharstein D., Szeliski R., 2002. A Taxonomy and Evalua-

tion of Dense Two-Frame Stereo Correspondence Algo-

rithms. In International Journal of Computer Vision, 47,

7-42.

CONVOLUTION KERNEL COMPENSATION APPLIED TO 1D AND 2D BLIND SOURCE SEPARATION

133