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Abstract— The continuous wavelet transform (CWT) is

widely used in biomedical signal event detection processes. The Berkner transform is an approximation of a CWT that is based on a hierarchical scheme similar to discrete wavelet transform. As a contribution to this novel approach we propose an original reconstruction formula of the decomposition. We used the Berkner transform to propose an adaptive method for scale range determination in a surface EMG reflex detector of the genioglossal muscle. Scale range determination is based on a numerical criterion using the original reconstruction formula.

Keywords—Continuous wavelet transform, surface

electromyography, sleep apnea syndrome

I. INTRODUCTION Obstructive sleep apnea-hypopnea syndrome (OSAS) is a clinical syndrome characterized by recurring episodes of upper airway narrowing and/or closure during sleep. OSA represents an important public health problem since two to four percent of the adult population is affected. Collapse of the upper airway is usually explained by the unbalance between the action of the pharyngeal dilating muscles that demonstrate inspiratory phasic activity, and the collapsing force of the intra-luminal negative pressure generated by the thoracic muscles. The genioglossus is one of the most important pharyngeal dilating muscles phasically active within the breathing cycle [1]. It is well established that applying a negative pressure stimulus on the pharynx both in animals and humans increases the genioglossus activity [2]. There is a possible relationship between the time latency reflex of the genioglossal muscle activation, the oxygen saturation, and the apnea syndrome severity [3]. Thus the measurement of the surface EMG genioglossal (SGEMG) time latency reflex in response to pharyngeal negative pressure stimuli for both control and apneic subjects are of interest for OSA pathophysiology. Prior studies [4], [5] have shown that the surface technique may reflect adequately the bioelectric activity of the genioglossus. Horner & al. [5] have measured the normal subject time latency reflex on the rectified and integrated post-stimulus SGEMG using a threshold of two background EMG activity standard deviation. This approach referred to as “single-threshold techniques” is classically proposed for the estimation of on-off timing of muscle activity. These methods present some drawbacks and lead to

a bias in the estimation of the activity time onset due to a lack in compensation of the time delay introduced by the filter used to process the EMG envelope. Considering surface EMG activity detectors, Merlo [6] lighted up recently the evolution of the proposed solutions in the direction of a signal local analysis where global properties are replaced by local ones. These approaches naturally lead to a more accurate estimation of the muscle on-off timing that is of interest in time reflex measurement. Merlo [6] has also pointed out the advantages of detectors based on the identification of the Motor Unit Action Potential (MUAP) from surface EMG especially in bad signal-to-noise conditions. More precisely he used a matched continuous wavelet transform (CWT), based on a mother wavelet chosen according to the MUAP shape. The method used the fact that in the context of single differential EMG recording the first Order Hermite-Rodriguez function HR1(t) is well suited to describe the basic MUAP shape [7]. The MUAP are transient signals that exhibit variation in duration. For this reason time-scale methods are more and more used to analyze frequency content of EMG signals. Arikidis [8] has recently developed an interscale wavelet maximum method to assist in the diagnosis of neuromuscular disease. EMG signals are decomposed with a redundant dyadic wavelet transform. The chosen wavelet was the first derivative of the cubic B-spline that is well adapted to detection of gradient change. The method is thus based on the fact that each rising edge of MUAP is represented at each decomposition scale by a wavelet coefficient maximum. More generally multiresolution analysis has been applied to other biomedical signals [9]-[12]. We aim to build a multidimensional SGEMG reflex event detector. In order to insure a good accuracy in the time onset estimation (localization on the finest considered scale) but also a robust detection (pertinent scale range choice) we choose the Berkner Transform (BT) because it offers maxima linkage facility and so preserves the robustness and accuracy of the estimation. In this paper we first make some modifications to the BT and establish an original reconstruction formula. We then draw a parallel between the MUAP matched CWT and the BT. Finally we show how to use the BT and its original reconstruction formula to determine the scale range adapted to a genioglossal reflex detector based on coefficient maxima statistics.

Reconstruction Process of the Berkner Transform : Application to Scale Range Determination in a Genioglossal EMG Reflex Time-

Scale Detector

Pierre-Yves Guméry1, Sylvain Meignen2, Hervé Roux-Buisson1, Elise Aithocine1 and Patrick Lévy3. 1 TIMC Laboratory, UMR-CNRS 5525, Joseph Fourier University, Grenoble, France.

2 LMC Laboratory, Joseph Fourier University, Grenoble, France 3Department of Respiratory Medicine and Sleep Laboratory, CHUR de Grenoble, France.

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II. Berkner transform and reconstruction process

A. The Berkner Transform This transformation is an approximation of the Gaussian derivative wavelet transformation [13]. A simple relation allows to follow the maxima lines in the time-scale plane. As reminded by Berkner [13], the binomial sequence of

order N, denoted γN(k), approximate for ∞→N the Gaussian function :

)))²2//()2/()(2/1(

2

1

2

1)( NNke

Nk

N

−−≈π

γ (1)

Here we focus on the approximation of the first derivative ρN(k) of the Gaussian function. It can be written,

( ) NkC

kN

kNkN

+−−−=

112ρ (2)

with ( )!!

!kNk

NNkC

−=

k represents the discrete time and N the scale. From (2) it’s easy to show that ,

ρN(k)=1/2[ρN-1(k)+ρN-1(k-1)] (3)

Owing to this property it will be possible to follow a maximum from the scale N to the scale N+1.The Berkner expansion of a temporal signal s(k) can be written at the time k and the scale N as,

)()()( jksj

jkcNN

+∑∞

−∞== ρ (4)

from equations (3) and (4) it comes,

cN (k)=1/2[cN-1(k)+cN-1(k-1)] (5)

Using this relation Berkner [13] showed that an extremum at time k for the scale N exists for the scale N+1 at the time k or k-1 or vanishes. So in the time scale plane the extrema lines are nearly perpendicular to the time axis beginning at the finest scale and vanishing for higher values of N (larger scales). The wavelet ρN(k) proposed by Berkner is not time centered but it is symmetrical with respect to the time (N+1)/2. This fact induces a left shift of the maxima lines for increasing N. However this deviation has no meaning with respect to the signal. To avoid this drawback we

propose a modified wavelet )(_

kN

ρ time translated from the

integer part of (N+1)/2 denoted (N+1)/2. Thus we write

( )2/)1(()( ++=−

NkNkNρρ (6)

and the modified Berkner expansion,

)()(_

)(_

jksj

jNkNc +∑∞

−∞== ρ (7)

so the recurrence relation (5) is also modified and becomes,

−+= −− )1()(

21)( 1

_

1

__

kckckc NNN if N+1 is even (8a)

++= −− )1()(

21)( 1

_

1

__

kckckc NNN if N+1 is odd (8b)

Fig(1) illustrates the effects of our modification. Fig. 1. (a) Signal with singularities .(b) BT time-scale representation. : (•) new decomposition sheme. (+) Berkner decomposition sheme with systematic deviation. B. The reconstruction process From (6) and (2) one can demonstrate that,

time

A. U.

a)

2/N

time

b)

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∑∞

=

+

+=0 2

1_

2

1s(k)

l

ll kc (9)

which is the exact form of the reconstruction formula. An approximated value will be obtained by limiting the sum to the N first values of l that gives,

∑=

+

+=N

0 2

1_

2

1(k)as

l

ll kc (10)

Another approximation may be obtained using only the local maxima coefficients. Let us define,

+

+=

+

+2

1_

2

1_C

ll

ll kck

(11)

for l_c maximum and

_Cl =0 elsewhere. So the

approximated reconstructed signal becomes,

∑=

+

+=N

0 2

1_C

2

1(k)aS

l

ll k (12)

Let us define the reconstruction error in each case (10) and (12) respectively by,.

( )( )∑

∑=

2)(

2(k)as-s(k)2

ksl

(13)

( )( )∑

∑=2)(

2Sa(k)-s(k)2

ksml (14)

where the sums are extended to the definition domain of s(k). Fig. 2 illustrates an EMG signal and the reconstruction error versus N in the cases where all decomposition coefficients are considered and where only maxima are considered.

III. BT as a MUAP mached CWT

A. Surface EMG and wavelet considerations Surface EMG is a summation of MUAP trains. Using a prototype function f(t) that describes the MUAP shape, a signal model can be described as follows :

∑

ατ−=

ij

ijjj

tf.k)t(MUAPT (15)

where j indicates a specific MU, kj an amplitude factor and τij the occurrence times of MUAPs of the MU. αj respresents a scaling factor. The EMG signal can be expressed as,

Fig. 2. (a) EMG signal. (b) Reconstruction errors versus N

)t(n)t(MUAPT)t(EMGj

j +=∑ (16)

where n(t) represents additive noises. According to this, one can see that different MUAP have the same shape but different widths and amplitudes. On these assumptions, Merlo enhanced [6] that the CWT may be a useful tool for EMG analysis. For optimal time-scale EMG detection performance the mother wavelet ω(t) has to match f(t) in the following decomposition expression,

dt)abt(*.)t(s

a1CWT )a,b( −ω= ∫

−∞

∞− (17)

In the case of surface EMG signals recorded in single differential configuration, ω(t) can be constructed considering the first order Hermite-Rodriguez (HR) function HR1(t) [6] :

.2/².1.

1,)(

1nt

e

n

tH

nktHR

λλ

−

= (18)

where n is the scale factor, kn,1 is a constant that normalizes the energy. H1(t) is the Hermite plolynomial of order one, and λn determines the function duration.

A. U.

a)

b)

______ 2l .

-------- m2l

time

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n

t

n

tH

λλ=

1

(19)

As HR1(t) is proportional to the first derivative of the

Gaussian function, we can use the BT as an approximation of the matched CWT. B. SGEMG signal description In our case the detection and the localization of the onset SGEMG reflex events are made difficult by the presence of transient signals due to electrodes instability. These transients exhibited variable shapes and durations (scale factors). The SGEMG response to pharyngeal negative pressure stimulus can be represented by : SGEMG(t) = EMGbg(t)+EMGr(t)+ Σ ∆i(t)+ Σ ∆’i(t) (20)

where EMGbg(t) represents the EMG background activity, EMGr(t) the reflex response that exhibited a similar frequency band than the background activity but greater amplitude factors (k). ∆i and ∆’i represent transient artifacts with a frequency content respectively close and lower to the EMG one. The transient distribution corresponds to a Poisson’s law with a temporal mean equal to .5 sec. In order to denoise the EMG signal from the transient ∆’i we developed an adaptive method for EMG pertinent scale determination. C. Adaptative EMG scale determination for reflex detector We built the reflex detector on statistic considerations of the CWT maxima coefficients. The reflex detection threshold was defined on the basis of energy considerations of EMGbg maxima coefficients. Here we only focus on the adaptive determination of the EMG scale range. The method used a numerical criterion based on minimization of the Berkner reconstruction error calculated on maxima coefficients [see Fig. 2(b)]. This minimum indicates the scales on which maxima are relevant to the EMGbg signal. The scale determination was conducted on prestimulus EMGbg intervals of 75 ms duration. Kolmogorov test enables us to choose intervals free of artifacts. Fig. 3 illustrates the vanishing of ∆’i artifacts in reconstructed signal calculated on pertinent scales.

V. CONCLUSION AND PERSPECTIVES

In this paper we presented an adaptive way to determine the pertinent scales of a multidimensional SGEMG reflex detector. We used the Berkner Transform that can be considered as a MUAP matched CWT. In the future, we will build a detector that will take into account all the artifact effects and measure, for comparison, both control and apneic subjects time latency genioglossal reflex.

Fig. 3. (a) Original artifacted SGEMG signal with reflex event. (b) Reconstructed signal calculated with BT maxima and an EMGbg pertinent

scales, one can see the ∆’i artifact suppression. The arrows indicate onset of the reflex response.

REFERENCES

[1] J.E. Remmers, W.J. Degroot, E.K. Sauerland, and A.M. Anch, “

Pathogenesis of upper airway occlusion during sleep,” J.Appl.Physiol., vol. 44, pp. 931-938, 1978.

[2] R.L. Horner, J.A. Innes, H.B. Holden, and A. Guz, “Afferent pathway(s) for pharyngeal dilator reflex to negative airway pressure in man: a study using upper airway anesthesia,” J. Physiol.(Lond), vol. 436, pp. 31-34, 1991.

[3] M. Dematteis, J.L. Pépin, M. Jeanmart, C. Deschaux, A. Labarre-Vila, and P. Lévy, “Systematic association between Charcot-Marie-Tooth Disease 5CMTA and sleep apnea syndrome a family study,” Lancet, vol.357, pp. 267-272, 2001.

[4] E.A. Doble, J.C. Leiter, S.L. Knuth, J.A. Daubenspeck. and D.J.R. Bartlett, “A noninvasive intraoral electromyographic electrode for genioglossus muscle,” J.Appl.Physiol., vol. 58 (4), pp. 1378-1382, 1985.

[5] R.L. Horner, J.A. Innes, K. Murphy, and A. Guz, “Evidence for reflex upper airway dilator muscle activation by sudden negative pressure in man,” J. Physiol.(Lond), vol. 436, pp. 15-29,1991.

[6] A. Merlo, D. Farina and R. Merletti, “A Fast and Reliable Technique for Muscle Activity Detection From Surface EMG Signals,” IEEE Trans. Biomed. Eng., vol. 50, pp. 316-323, 2003.

[7] L. Lo Conte, R. Merletti, and G. V. Sandri, “Hermite expansion of compact support waveforms: applications to myoelectric signals,” IEEE Trans. Biomed. Eng., vol. 41, pp. 1147–1159, Dec. 1994.

[8] N. S. Arikidis, E. W. Abel, and A. Foster, “Interscale Wavelet Maximum-A Fine to Coarse Algorithm for Wavelet Analysis of the EMG Interference Pattern,” IEEE Trans. Biomed. Eng., vol. 49, pp. 337-344, 2002.

[9] C. Li, C. Zheng, and C. Tai, “Detection of ECG characteristic points using wavelet transforms,” IEEE Trans. Biomed. Eng., vol. 42, pp. 21–28, Jan. 1995.

[10] S. Kadambe, R. Murray, and G. F. Boudreaux-Bartels, “Wavelet transform-based QRS complex detector,” IEEE Trans. Biom. Engineer., vol. 46, no. 7, pp. 838–848, July 1999.

[11] S. Kadambe and G. F. Boudreaux-Bartels, “Application of the wavelet transform for pitch detection of speech signals,” IEEE Trans. Inform. Theory, vol. 38, pp. 917–924, Mar. 1992.

[12] M. Khalil and J. Duchene, “Uterine EMG analysis: a dynamic approach for change detection and classification,” IEEE Trans. Biomed. Eng., vol. 46, pp. 748–756, June 2000.

[13] K. Berkner, and R. O. Wells, “A New Hierarchical Scheme for Approximating the Continuous Wavelet Transform with Applications to Edge Detection,” IEEE Signal processing lett., vol. 6, no. 8, pp. 193–195, 1999.

EMGbg interval characterization

a)

b)

∆∆’i

time

time