WAVELET TRANSFORM MOMENTS FOR FEATURE

EXTRACTION FROM TEMPORAL SIGNALS

Ignacio Rodriguez Carreño

Department of Electrical and Electronic Engineering, Public University of Navarre, Arrosadia, Pamplona, Spain

Marko Vuskovic

Department of Computer Science, San Diego State University, San Diego, California, USA

Keywords: Pattern recognition, EMG, feature extraction, wavelets, moments, support vector machines.

Abstract: A new feature extraction metho

d based on five moments applied to three wavelet transform sequences has

been proposed and used in classification of prehensile surface EMG patterns. The new method has essen-

tially extended the Englehart's discrete wavelet transform and wavelet packet transform by introducing more

efficient feature reduction method that also offered better generalization. The approaches were empirically

evaluated on the same set of signals recorded from two real subjects, and by using the same classifier, which

was the Vapnik's support vector machine.

1 INTRODUCTION

The electromyographic signal (EMG), measured at

the surface of the skin, provides valuable informa-

tion about the neuromuscular activity of a muscle

and this has been essential to its application in clini-

cal diagnosis, and as a source for controlling assis-

tive devices, and schemes for functional electrical

stimulation. Its application to control prosthetic

limbs has also presented a great challenge, due to the

complexity of the EMG signals.

An important requirement in this area is to accu-

rately classify d

ifferent EMG patterns for controlling

a prosthetic device. For this reason, effective feature

extraction is a crucial step to improve the accuracy

of pattern classification, therefore many signal rep-

resentations have been suggested.

Various temporal and spectral approaches have

been a

pplied to extract features from these signals. A

comparison of some effective temporal and spectral

approaches is given in (Du & Vuskovic 2004),

where the authors have applied moments to short

time Fourier transform (STFT), and short time

Thompson transform (STTT) on prehensile EMG

patterns.

The wavelet transform-base

d feature extraction

techniques have also been successfully applied with

promising results in EMG pattern recognition by

Englehart and others (1998).

The discrete wavelet transform (DWT) and its

eneralization, the wavelet packet transform (WPT),

were elaborated in (Englehart 1989a). These tech-

niques have shown better performance than the oth-

ers in this area because of its multilevel decomposi-

tion with variable trade-off in time and frequency

resolution. The WPT generates a full decomposition

tree in the transform space in which different wave-

let bases can be considered to represent the signal.

The techniques were applied to feature extraction

from surface EMG signals.

However, these techniques produce a large

ount of coefficients, since the transform space has

very large dimension. This fact suggests the system-

atic application of feature selection or projection

methods and dimensionality reduction techniques to

enable the methodology for real time applications.

Englehart applied feature selection and feature pro-

jection that yielded better classification results and

improved time efficiency. Specifically, the principal

component analysis (PCA) was used due to its abil-

ity to model linear dependencies and to reject irrele-

vant information in the feature set (Englehart etal.

1999).

This paper continues the work described above

taking a different approach to feature reduction.

Extending the idea of spectral moments suggested in

Rodriguez Carreño I. and Vuskovic M. (2005).

WAVELET TRANSFORM MOMENTS FOR FEATURE EXTRACTION FROM TEMPORAL SIGNALS.

In Proceedings of the Second International Conference on Informatics in Control, Automation and Robotics - Signal Processing, Systems Modeling and

Control, pages 71-78

DOI: 10.5220/0001190100710078

 SciTePress

(Du & Vuskovic 2004) the sequences of wavelet

coefficients are further subjected to the calculation

of their temporal moments. The main goal of this

work is to propose and empirically compare two

different novel feature extraction approaches based

on simple two-scale DWT and WPT with the two

best Englehart’s approaches using the DWT and the

WPT in combination with principal component

analysis (PCA).

In this new approach, the first five raw moments

were applied to DWT transformed prehensile EMG

sequences, which has proven to be very advanta-

geous in the classification stage. The methods em-

ployed a simple DWT or WPT with only three trans-

form sequences, instead of the full DWT or WPT

used by Englehart. This has eliminated the tedious

feature reduction procedures and PCA.

The evaluation of the three approaches was car-

ried out on the same set of data, and with an identi-

cal classifier based on Vapnik's support vector ma-

chines (SVM) with a linear kernel.

2 PREHENSILE EMGS

The research presented here was motivated by the

need for classification of prehensile elec-

tromiographic signals (EMG) for control of a multi-

functional prosthetic hand (Vuskovic etal. 1995).

Since the hand-preshaping phase in an average ob-

ject grasp takes about 500 ms, it is important to ac-

complish the feature extraction and classification in

less than 400 ms, preferably in 200 ms. Such a diffi-

cult task requires very strong feature extractor and

classifier.

The mioelectric control of multifingered hand

prostheses was studied in several papers, for exam-

ple (Nishikava 1991), (Uchida 1992), (Farry 1996),

and (Huang 1999). Most of the ideas in these efforts

were inspired by Hudgins (Hudgins etal. 1991). In

this work the concept of preshaping of multifunc-

tional grasps was based on the recognition of a par-

ticular finger joint movement. In an earlier work

done at San Diego State University, the approach

was rather different, based on grasp types, instead of

hand configurations in joint space. Once a grasp type

is recognized from the recorded EMGs, it can be

then synergistically mapped into the desired joint

configuration (Vuskovic 1995) for any hand, with

any number of degrees of freedom. We have consid-

ered four basic grasp types according to the

Schlesinger classification (Schlesinger 1919): cylin-

drical grasp (C), spherical grasp (S), lateral grasp (L)

and precision grasp (P), see figure 1.

3 EXPERIMENTAL SETUP

Four-channel surface EMG signals from two healthy

subjects were recorded at 1000 Hz sampling fre-

quency. The recording was done while the subject

has repeatedly performed the four grasp motions.

There were 216 grasp recordings evenly distributed

across the four grasps types: 60 (subject 1) + 4 (sub-

ject 2) for cylindrical grasp, 30+10 for precision

grasp, 30+10 for lateral grasp and 60 + 12 for

spherical grasp. Three different EMG sequence

lengths were used: 200 ms, 300 ms and 400 ms. The

200 and 300 ms sequences were obtained by truncat-

ing the recordings of 400 ms sequences. (The se-

quences of 300 ms were not presented in this paper.)

Figure 1: Four grasp types.

4 DISCRETE WAVELET

TRANSFORM

The DWT is a transformation of the original tempo-

ral signal into a wavelet basis space. The time-

frequency wavelet representation is performed by

repeatedly filtering the signal with a pair of filters

that cut the frequency domain in the middle.

Specifically, the DWT decomposes a signal into

an approximation signal and a detail signal. The

approximation signal is subsequently divided into

new approximation and detail signals. This process

is carried out iteratively producing a set of approxi-

mation signals at different detail levels (scales) and a

final gross approximation of the signal.

The detail Dj and the approximation Aj at level j

can be obtained by filtering the signal with an L-

sample high pass filter g, and an L-sample low pass

filter h. Both approximation and detail signals are

downsampled by a factor of two.

This can be expressed as follows:

[] [] [] [2 ],

jj j

An A n hkA n k

−

−−

∑

Η −

(1)

ICINCO 2005 - SIGNAL PROCESSING, SYSTEMS MODELING AND CONTROL

[] [] [] [2 ],

jj j

Dn D n gkA n k

−

−−

∑

G −

(2)

where

, n = 0,1,…N-1 is the original temporal

sequence, while H and G represent the convolu-

tion/down sampling operators. Sequences g[n] and

h[n] are associated with wavelet function

[]An

()

t and

the scaling function

through inner products: ()ϕ t

[] (),2 (2 ),gn t t n=ψ ψ −

(3)

[] (),2(2 ).hn t t n=ϕ ϕ −

(4)

Operators H and G can be applied repeatedly in al-

teration, for example:

AA A= HH

, DD =

A= GG ,

AD =

AGH ,

DA =

The A and D sequences obtained as the result of

DWT are still massive in terms of the number of

samples, which contributes to large dimensionality of

feature space. Besides, the sequences have a high

noise component inherited from the original EMG

signal.

A= HG

, etc.

A feature extraction approach based on DWT

applied by Englehart (1998, 1998a) consists of four

differentiated phases:

1. Perform full DWT decomposition of the EMG

signals, until scale j = log2 (N), with the Coiflet

wavelet of order 4 (C4);

2. Square the DWT coefficients;

3. Apply PCA for dimensionality reduction tech-

niques;

4. Determine the optimal number of features per

channel based on the target classifier.

An optimization phase is needed before selecting the

adequate number of PCA features in order to maxi-

mize the performance of the target classifier. The

optimum number of features was 100 DWT coeffi-

cients per channel of the EMG signal used in this

work.

5 WAVELET PACKET

TRANSFORM

The WPT is a generalization of DWT. It generates a

full wavelet basis decomposition tree. In each scale,

not only the approximation signal as in DWT, but

also the detail signals are filtered to obtain another

two low and high frequency signals. Many different

representations of a signal can be obtained by select-

ing different wavelet packet basis. In this regard

WPT is superior to DWT, as the chosen basis can be

optimized with respect to frequency or time resolu-

tion.

Englehart (1999) generated a feature extraction

method based on the WPT for EMG signals. In this

method a previous phase must be applied to the set

of training signals. The underlying idea is to select

the WPT basis that best classifies all classes of sig-

nals. For this purpose, Englehart proposed a modi-

fied version of the local discriminant basis (LDB)

algorithm (Englehart 1998a, 2001), to maximize the

discrimination ability of the WPT by using a class

separability cost function (Saito & Coifman 1995).

Once the best basis for classification is defined (for

different channels and different signal lengths), the

following steps must be performed:

1. Perform the full WPT decomposition until

scale j = log

(N), with the Symlet wavelet of

order 5 (S5);

2. Square the WPT coefficients;

3. Average energy maps within each subband;

4. Select the WPT coefficients from a basis cho-

sen previously for each channel and for differ-

ent signal lengths;

5. Extract the optimal number of features based

on the target classifier;

6. Apply PCA transform to the feature space for

dimensionality reduction (removing the eigen-

vectors whose eigenvalues are zero);

7. Extract the optimum number of features per

channel for the target classifier;

The optimal number of features for Englehart’s

WPT based approach and for the support vector ma-

chine as the target classifier (see section 7) was

found to be three features per channel, per signal

length.

6 DWT AND WPT MOMENTS

The new approach for feature extraction presented

here is based on DWT and WPT, and on the calcula-

tion of their temporal moments. The approach was

first proposed in (Rodriguez & Vuskovic 2005) as an

extension of the idea of spectral moments (Du &

Vuskovic 2004).

Specifically, we used two different wavelets suc-

cessfully applied by Englehart on surface EMG sig-

nals: C4 and S5

DWT and WPT moments should not be confused

with wavelet vanishing moments.

WAVELET TRANSFORM MOMENTS FOR FEATURE EXTRACTION FROM TEMPORAL SIGNALS

In order to reduce the dimensionality and to

smooth out the noise, we applied six moments to

transformed signals (DWT and WPT):

[ ], 0,1, 2,...,5,

MSnm

−

⎛⎞

⎜⎟

⎝⎠

∑

(5)

where

represents sequences A, D, AA, DD, AD

and DA used in algorithms described below, while N

is number of samples at the corresponding level of

decomposition.

[]Sn

The new approach based on DWT consists of the

following steps:

1. Perform two-scale decomposition of the input

signal;

2. Compute moments for three transform se-

quences (D, AA, AD);

3. Apply logarithm transform to each feature,

log(0.1+f);

4. Normalize all features using mean value and

standard deviation computed for each feature

across all samples.

The choice of sequences D, AA and AD was

made empirically; it has given the best results in av-

erage for the given set of data. Similar choice was

made for WPT algorithm.

The WPT-based method has the following steps:

1. Perform two-scale decomposition of the input

signal;

2. Select basis obtained from previous application

of the best basis Coifman algorithm;

3. Compute moments for three transform se-

quences (A, DA, DD);

4. Apply logarithm transform to each feature,

log(0.1+f);

5. Normalize all features using mean value and

standard deviation computed for each feature

across all samples.

The optimal basis selection in this method was

based on a single channel. The same basis thus ob-

tained was subsequently used for single and multiple

channels, and for different sequence lengths.

Log transformation was applied to moments as it

effectively reduces the skewness and the kurtosis of

data, consequently resulting in an estimated probabil-

ity density that appears more like normal distribution

(Vuskovic atal. 1995). The nonlinear transformation

of features has significantly improved the classifier

performance.

7 THE SVM CLASSIFIER

The support vector machines ( Christianini & Shaw-

Taylor 2000) are a family of learning algorithms

based on the work of Vapnik (1998), which have

recently gained a considerable interest in pattern

recognition community. The success of SVM comes

from their good generalization ability, robustness in

high dimensional feature spaces and good computa-

tional efficiency.

In this work, a standard SVM classifier with lin-

ear kernel has been used for dichotomic (binary)

classification (Gunn 1997). The multiclass SVM can

also be considered, but this is out of the scope of this

paper.

The previous work on the classification of pre-

hensile EMG patterns (Vuskovic 1996) has shown

that the most difficult is to discriminate cylindrical

from spherical grasps (C/S), and then lateral from

precision grasps (L/P). Therefore the SVM is applied

to these pairs of grasp types and the feature extrac-

tion methods were evaluated accordingly.

The classification tests were performed with

leave-one-out method, where one sample was re-

moved from the data set and the rest of the samples

were used to train the SVM. The procedure was re-

peated for each sample in the data set, and the aver-

age hit rate was computed afterwards.

8 COMPUTATIONAL

COMPLEXITY

Application of WPT and calculation of J scales,

log

≤

N , where N is the length of the original

temporal signal, results in JN coefficients. Conse-

quently, the computational cost of the full-scale

WPT is in the order of

(Englehart 2001). Similarly, the computational com-

plexity of full-scale DWT is half the computational

complexity of the WPT, i.e.

()≤OJN (log )ON N

()

log 2ON N

. Since

our new approaches use only two-scale DWT or

two-scale WPT decomposition, we can enumerate

all the approaches with respect to their computa-

tional complexity in the increasing order: DWT(new)

< WPT (new) < DWT (Englehart) < WPT (Engle-

hart). The complexities are summarized in table 1.

Table 1: Computational complexity

New approach Englehart

DWT WPT DWT WPT

O(N) O(2N) O(N logN/2) O(N logN)

ICINCO 2005 - SIGNAL PROCESSING, SYSTEMS MODELING AND CONTROL

9 EXPERIMENTAL

EVALUATION

In this section we discuss the methodology for the

experimental evaluation of DWT and WPT ap-

proaches.

9.1 Cluster Visualization

In order to compare the effectiveness of a feature

extraction method there is needed some method to

compare the discrimination of clusters in feature

space, either by 2D or 3D scatter plots, or by some

distance measure between clusters. Both methods

are normally based on the transformation of the fea-

ture space through PCA or Fisher-Rao transform,

which both use the inverse of the cluster covariance

matrices. Unfortunately the dimensionality of the

feature space is often larger than the number of sam-

ples, which makes the methods inapplicable due to

the singularity or ill-conditioning of the covariance

matrices. However, the support vector machines of-

fered new possibilities. SVM maximize the margin

between clusters and the separation hyperplane in the

original or kernel-induced feature space without a

need to use covariance matrices.

We use in this work a projection of the original

feature space onto the line perpendicular to the

maximal-margin separation hyperplane:

(6) ,

pXw=

where X is N×d sample (feature) matrix, w is unit, d-

dimensional normal to the separation hyperplane,

and p is N-vector of projected samples. In order to

get a 2D plot of samples another projection vector is

needed:

(7) .

qXu=

The d-dimensional projection vector u doesn't

have to be orthonormal to w, but has to be unique in

some way. Therefore we used the direction of the

minimal variance of both clusters, which is nearly

laying in the separation hyperplane. The vector co-

incides with the eigenvector that corresponds to the

smallest non-zero eigenvalue of the pooled covari-

ance matrix:

112 2

(1)( 1)

(, ) ,

NSNS

S pool S S

−+−

+−

(8)

where N

= N) and S

are sizes and covari-

ance matrices of the two clusters. An example of

cluster diagrams, plot(p,q), is shown in figure 3,

which will be discussed later.

9.2 Hotelling Distance

A useful quantitative measure of cluster discrimina-

tion in multidimensional space is Hotelling distance

between cluster means (T

statistic). The T

can be

computed for projected clusters:

12 12

()(

(, ),

TccCc

C pool C C

−

=−

−

(9),

where

c and C

are sample means and sample co-

variance matrices of projected clusters respectively.

In order to establish the significance of the distance

under some confidence level, the T

distance needs

to be compared with the corresponding critical value

. The critical value can be obtained if we assume

that the quantity

NNr

NN r

)

−−

+−

has F-distribution with degrees of freedom r and

NNr

+−−, where r = 2 in case of 2D pro-

jections (Seber 1984). The above is true under the

assumption that clusters have normal distributions

with nearly equal sizes and covariance matrices. If

this is not the case, a stronger statistic has to be used.

In this work we used statistic suggested in (Yao

1965), where the cluster distance was computed as:

()

12 11 2 2 12

()/ / (

TccCNCN cc

−

=− + −).

(10)

The degrees of freedom for the F-distribution

were estimated from the data (Seber 1984) (not pre-

sented here due to limited space). The test works for

unequal clusters that can have any bell-shaped dis-

tribution. The

values are shown in tables 2 and 3,

and in the scatter diagrams in figure 3. The critical

values

were all below 11. The value of cluster

distances as a quantitative measure of cluster dis-

crimination is that they can be easily and quickly

computed.

9.3 Number of Moments

Once the classification pairs are determined, the next

step is to determine the optimal number of DWT and

WPT moments, which will be used for feature

reduction. This was done experimentally by

extensive

application of feature extractions and classiffications

to different EMG signal lengths and different number

of channels.

WAVELET TRANSFORM MOMENTS FOR FEATURE EXTRACTION FROM TEMPORAL SIGNALS

Based on the bar graphs the selection of five mo-

ments (

, M

,…,M

) was a clear choice.

10 THE RESULTS

The comparison of four different approaches: the

five-moment DWT and WPT as proposed in this pa-

per, and the DWT and WPT of Englehart (1998a,

1999) have been measured by Hotteling distances and

by the classification hit rates applied to two cluster

pairs (C/S) and (L/P).

The results are presented in tables 2 through 5.

The feature extraction was performed for 200 and 400

ms time sequences recorded from a single channel

and from four simultaneous channels. Each channel

represented one surface EMG electrode attached to

the upper-forehand of the subject. Several different

wavelets were used in experiments, but only the two

most successful ones were shown here: the fourth-

order Coifman wavelets (C4) and the fifth-order sym-

lets (S5). The two tables show a roughly good corre-

lation between the Hotelling distances and the classi-

fication hit-rates. The small differences can be ex-

plained by the fact that the Hotelling distances point

the goodness of clustering, while the hit rates stress

the generalization of the trained SVM.

An example of four different cluster scatter dia-

grams is shown in figure 3.

Figure 2: Hotelling distances versus number of

oments

for WPT: (a) 200 ms, single channel, (b) 200ms, four

channels, (c) 400 ms, single channel, (d) 400 ms, four

channels (C/S grasps – lower bars, L/P grasps upper bar

Figure 3: SVM-projected clusters, 200 ms, and four chan-

nels, WPT: (a) C/S - new approach, (b) L/P - new

ap-

proach, (c) C/S - Englehart, (d) L/P - Englehart

The results suggest clear advantage of our novel

method over the Englehart’s approaches mainly due

to the moments used for dimensionality reduction,

instead of applying PCA. In addition, the application

of log transformation on features has helped consid-

erably. Our WPT novel method seems to behave

better at classifying the 200 ms sequences. This is

due to the WPT basis selection, which better charac-

terizes the frequency structure of the transient sig-

nals.

Table 2: Hotelling distances (C/S)

New approach Englehart

WT WPT

Sig.

length

/chnls

C4 S5 C4 S5

DWT

WPT

200/1 75 61 109 97 49 13

200/4

352 466 424 421 201 73

400/1 92 79 96 79 480 45

400/4 366 570 535 488 295 100

Table 3: Hotelling distances (L/P)

New approach Englehart

DWT WPT

Sig.

length

/chnls

C4 S5 C4 S5

DWT

WPT

200/1 33 65 44 100 362 11

200/4 289 3462 1724 723 756 107

400/1 178 166 233 262 118 60

400/4 560 24680 1472 718 2388 168

ICINCO 2005 - SIGNAL PROCESSING, SYSTEMS MODELING AND CONTROL

Table 4: Classification hit rates in % (C/S)

New approach Englehart

WT WPT

Sig.

length

/chnls

CO4 SY5 CO4 SY5

DWT

WPT

200/1 75.0 76.7 79.2 79.2 60.8 62.5

200/4 90.0 94.2 94.2 95.0 86.7 88.3

400/1 80.8 80.0 80.8 77.5 60.8 59.2

400/4 99.2 96. 7 98.3 97.5 88.3 93.3

Table 5: Classification hit rates in % (L/P)

New approach Englehart

WT WPT

Sig.

length

/chnls

CO4 SY5 CO4 SY5

WT WPT

200/1 73.3 81.7 81.7 83.3 56.7 53.3

200/4 91.7 96.7 90.0 98.3 80.0 93.3

400/1 96.7 95.0 93.3 91.7 56.7 63.3

400/4 99.9 99.9 99.9 99.9 88.3 95.0

11 CONCLUSIONS

A new approach of wavelet-based feature extraction

from temporal signals has been proposed. The ap-

proach extends the Englehart's discrete wavelet trans-

form and wavelet packet transform by subjecting the

two-scale, three-sequence wavelet coefficients to

temporal moment computation. This has helped re-

duce significantly the dimensionality of the resulting

feature vectors without loosing the essential informa-

tion in the original patterns. It was found experimen-

tally that first five raw moments represent a good

compromise. The new methods are applied to pre-

hensile EMG signals of various lengths and various

amounts of input signals (surface EMG channels)

and compared to the best approaches of Englehart, on

the same set of signals. For the comparison are used

two quantitative measures: Hotelling statistic and

classification hit rates. The classifier applied to the

extracted features was linear support vector machine,

which has exceptionally good performance in case of

large feature spaces and fewer training samples. The

results have shown superior performance of the new

approach. A brief complexity analysis also shows

that the new approach is more efficient time wise.

Although the methodology was demonstrated on

EMG signals, we believe the methodology can

equally successfully be applied to other temporal

signals.

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