Wavelet decomposition of ocular motor signals was investigated with a view to its use for noise analysis and filtering. Ocular motor noise may be physiological, depending on brain activities, or experimental, depending on the eye recording machine, head movements and blinks. Experimental noise, such as spikes, must be removed, preserving noise due to neuro-physiological activities. The proposed method uses wavelet multiscale decomposition to remove spikes and optimizes the procedure by means of the covariance of the eye signals. To measure the noise on eye motor control, we used the wavelet entropy. The method was tested on patients with cerebellar disorders and healthy subjects. A significant difference in wavelet entropy was observed, indicating this quantity as a valuable measure of physiological motor noise.

1. Journal of Neuroscience Methods 196 (2011) 318–326
Contents lists available at ScienceDirect
Journal of Neuroscience Methods
journal homepage: www.elsevier.com/locate/jneumeth
Spike removal through multiscale wavelet and entropy analysis of ocular motor
noise: A case study in patients with cerebellar disease
Giacomo Veneria,b,c,∗
, Pamela Federighia,b
, Francesca Rosinib
, Antonio Federicob
, Alessandra Rufaa,b,∗∗
a
Eye tracking & Vision Applications Lab - University of Siena, Italy
b
Department of Neurological Neurosurgical and Behavioral Science - University of Siena, Italy
c
Etruria innovazione Spa, Italy
a r t i c l e i n f o
Article history:
Received 20 September 2010
Received in revised form 2 January 2011
Accepted 4 January 2011
Keywords:
Eye-tracking
Spike
Wavelet
Entropy
Despiking
Cerebellum
a b s t r a c t
Wavelet decomposition of ocular motor signals was investigated with a view to its use for noise analysis
and filtering. Ocular motor noise may be physiological, depending on brain activities, or experimen-
tal, depending on the eye recording machine, head movements and blinks. Experimental noise, such as
spikes, must be removed, preserving noise due to neuro-physiological activities. The proposed method
uses wavelet multiscale decomposition to remove spikes and optimizes the procedure by means of the
covariance of the eye signals. To measure the noise on eye motor control, we used the wavelet entropy.
The method was tested on patients with cerebellar disorders and healthy subjects. A significant differ-
ence in wavelet entropy was observed, indicating this quantity as a valuable measure of physiological
motor noise.
© 2011 Elsevier B.V. All rights reserved.
1. Introduction
Eye movements are arguably the most frequent of all human
movements and an essential part of human vision (Yarbus, 1967):
they drive the fovea and consequently the attention towards
regions of interest in space. This enables the visual system to fixate
and to process an image or its details with high resolution (Leigh
and Zee, 2006): act of fixation. Fixations are samples of points
around a point, called centroid, with long duration. An average
of three eye fixations per second generally occurs during active
looking (300 ms).
These eye fixations are intercalated by rapid eye jumps, called
saccades, that can be defined as rapid movements with velocities
that may be higher than 500 deg/s and durations of about 30 ms;
Fig. 1 illustrates a small portion of gaze during visual exploration
on a psychological task, showing five clusters of data points (fix-
ations) and three saccades. Eye movements may be recorded by
∗ Corresponding author at: Department of Neurological, Neurosurgical and
Behavioral Science - University of Siena, Viale Bracci 2, 53100 Siena, Italy.
Tel.: +39 0577 233136; fax: +39 0577 40327.
∗∗ Corresponding author at: Department of Neurological, Neurosurgical and
Behavioral Science - University of Siena, Viale Bracci 2, 53100 Siena, Italy.
Tel.: +39 0577 233136; fax: +39 0577 40327.
E-mail addresses: g.veneri@unisi.it (G. Veneri), federighi@unisi.it (P. Federighi),
rosini3@student.unisi.it (F. Rosini), federico@unisi.it (A. Federico), rufa@unisi.it
(A. Rufa).
URL: http://www.evalab.unisi.it (A. Rufa).
eye-tracking technologies (see Duchowski, 2002 for a review). Eye-
tracking is the process by which eye movements are measured
during visual exploration of a scene or during the execution of spe-
cific tasks. This measurement, can be obtained either by measuring
the position of an eye in the orbit, relative to the head, or the point
of regard, which extracts the coordinates x, y of the eye in space.
Eye gaze analysis and extraction of gaze features is an exciting
and challenging field of research for neuro physiologists and neuro
scientists, since it offers a good, reproducible method for studying
basic mechanisms of brain motor control (from motor command
to reached position, see Girard and Berthoz, 2005 for a review) and
cognitive behavior. Indeed, the human gaze is direct or indirect evi-
dence of human cognition in terms of memory, attention (Corbetta
et al., 2002), intention and decision (Shimojo et al., 2003).
Generally speaking, the gaze signal is processed by extracting
two main classes of features: motor abilities (Table 1 for a short
summary of the subsystems involved and related features) which
studies the ocular motor control, and task specific execution, which
provide insights into ocular motor control, and task specific exe-
cution, which provides information about the ability of subjects to
perform a command, such as searching for an object, remembering,
forming a sequence, making up the mind.
1.1. Eye movements in clinical applications: measuring the
physiological characteristics of motor control noise
The main advances in the study of ocular motor control have
been obtained by lesion studies in monkeys and recording patients
0165-0270/$ – see front matter © 2011 Elsevier B.V. All rights reserved.
doi:10.1016/j.jneumeth.2011.01.006

2. G. Veneri et al. / Journal of Neuroscience Methods 196 (2011) 318–326 319
Fig. 1. A small portion (1500 ms) of the gaze of a patient with cerebellar disorder on
a visual search task: the highlighted fixations are ordered in the time domain, from
first to last. The main clinical features of cerebellar disorders include incoordination,
imbalance, and troubles with stabilizing eye movements. In the simplified form of
the test, subjects are asked to look for a right oriented “E”. The image appeared on
the screen for 20000 ms.
with selective brain lesions or homogeneous groups of patients
with genetic neuro-degenerative diseases. Recently, however, ocu-
lar motor control has also attracted attention for studying complex
tasks, such as recognition of facial expression (Xu-Wilson et al.,
2009) and gaze contingent applications (Veneri et al., 2010a). Some
authors found that humans can adapt ocular motor characteris-
tics (velocity, latency) according to the behavior in order to reduce
energy consumption or improve efficiency; the hypothesis was ver-
ified by van Beers (2008), through resolution of an optimal control
problem of modulating saccade velocity to minimize variance of
gaze during saccades and fixations. The problem can be explained
intuitively as the human ability to reduce energy consumption
(gaze variability) in order to improve efficiency (precision on the
target) (van Beers, 2009).
In the clinical context, one of the most interesting analyses of
motor control was applied to patients with disorders of the cere-
bellum, especially patients with spinocerebellar cerebellar ataxia
type 2 (SCA2), a rare disease. SCA2 is one of a group of genetic
disorders characterized by slowly progressive impairment of the
ability to control eye movements (slow saccades) (Filla et al., 1996;
Bürk et al., 1999; Ramat et al., 2007). In this respect, Leigh and
Kennard (2004) proposed using saccades as a research tool in the
clinical neuro-sciences and the main sequence (a set of relation-
Table 1
Common eye movement features studied: saccade velocity is typically analyzed in
patients with cerebellar or brainstem lesions. Saccadic trajectories and microsac-
cades are studied in attention and voluntary–involuntary tasks. Fixations and ROI
are studied in free visual search tasks.
Subsystem Computation
Gaze shift
Saccades Distance of target from fovea der Stigchel et al., 2006 (deg),
Velocity Leigh and Zee, 2006 (deg/s), Duration (ms),
Latency after target appearance Rayner et al., 1983 (ms)
Pursuit Target velocity (deg/s)
Vergence Target depth (cm)
Gaze holding
Fixations Visual scanning region of interest (ROI), Distance to
nearest ROI (deg), Time spent in ROI (ms), Tremor (deg2
),
Drifts (deg), Microsaccades Engbert and Kliegl, 2003
Vestibular Rotation, translation of head or body (deg)
Optokinetic Speed direction of full-field image motion (deg/s)
ships between saccade amplitude and peak velocity; Bollen et al.,
1993) as a valid tool to classify the subject. These results indicated
the need to investigate a measure of “physiological characteristics of
gaze noise” (PN), such as tremor, square jerks (Leigh and Zee, 2006),
nystagmus (Abel et al., 2008) and drift. According to van Beers
(2007) PN may be additive with or multiplicative of the eye move-
ment, and is lost in recording noise (RN) due to blinks or signal loss;
generally speaking, we can measure subject noise = PN + RN = SDN
(signal) + ADN + RN where SDN is physiological signal dependent
noise and ADN physiological additive noise.
A natural approach to quantify the degree of order of gaze sig-
nals is to consider their spectral features and spectral entropy.
Inouye et al. (1993) were the first to apply spectral analysis by
a Fourier analysis to the study of brain signals. Recently, some
authors (Quiroga et al., 2001; Kamavuako et al., 2009) proposed
the application of multiresolution wavelet decomposition (Mallat,
1989) to overcome the limitations of the Fourier transform and
to study the ocular motor abilities in infantile syndromes (Abel
et al., 2008). The wavelet method arranges successive wavelet
transforms in a hierarchical scheme, enabling the decomposition
of the signal. In the present study, we describe a method to esti-
mate PN by wavelet entropy and to remove RN by multi-scale
wavelet transform (MWT). In particular, we propose a method to
remove signal artefacts such as spikes. Spikes are unwanted and
uninteresting rapid signal artifacts (Juhola et al., 1987) that may
deceive pattern recognition algorithms and be confused with sac-
cades (Fig. 3; see Fischer and Biscaldi, 1993 for the most common
algorithm of saccades), or induce the fixation search algorithm
to over-segmentate (Salvucci and Goldberg, 2000; Veneri et al.,
2010b). Techniques based on linear or adaptive filters, or spectrum
analysis may affect the morphological characteristics of gaze noise.
It is therefore necessary to use advanced techniques to recognise
spikes and to suppress them (despiking), preserving the natural
characteristics of ocular noise. We used MWT and covariance to
decompose the signal and remove spikes. Implementation of the
algorithm is described in Section 3.2.
After despiking, motor control noise was measured by wavelet
entropy (WS; Coifman and Wickerhauser, 1992): in the time
domain, entropy is a measure of signal uncertainty and in the spec-
tral domain it is a measure of signal complexity. High entropy
is associated with signals generated by disordered, random pro-
cesses, and low entropy with ordered or partially ordered processes
(Zunino et al., 2007). If a process is noisy, its signal wavelet decom-
position can be expected to contribute significantly to total wavelet
energy in all frequency bands. This makes WS a good candidate of
for estimating PN.
We evaluated the method performance in a computer-
generated simulation (Section 5.1), and in a small group of SCA2
patients and a mixed group of patients with genetically undefined
cerebellar ataxia (NDC) (Section 5.2). The results provided evidence
that the proposed technique can discriminate healthy subjects from
patients.
2. Materials and methods
Eight SCA2 patients, a mixed group of three patients with unde-
fined genetic cerebellar ataxia (NDC) and 25 healthy subjects were
enrolled in the study. All were in the age range 25–45 years.
Subjects were seated at viewing distance of 78 cm from a 32 in.
color monitor (51 cm×31 cm). Eye position was recorded using
an ASL 6000 system, which consists of a remote-mounted cam-
era sampling pupil location at 240 Hz. A 9-point calibration and
3-point validation procedure was repeated several times to ensure
all recordings had a mean spatial error of less than 0.3 deg. Data
was controlled by a Pentium4 dual core 3 GHz computer, which

3. 320 G. Veneri et al. / Journal of Neuroscience Methods 196 (2011) 318–326
Table 2
Subjects were seated at viewing distance of 78 cm from a 32 in. color monitor (51 cm×31 cm).
Subjects Type Problem on motor control Taking medicine
GFOD Degenerative cerebellar ataxia Yes and documented Yes
MS Degenerative cerebellar ataxia Yes and documented Yes
CN Degenerative cerebellar ataxia Yes and documented Yes
PP Normal No No
GV Normal No No
EU Normal No No
XM Normal No No
SL Normal No No
acquired signals through a fast UART serial port. Head movements
were restricted using chin rest and bite. The subject was asked
to fixate a central red dot; after 500 ms the dot disappeared and
the subject could explore the scene, which consisted of simple
artificial images (Fig. 1). The fixations were identified by the FDT
algorithm developed by Veneri et al. (2010b) and checked manually
and updated by an expert.
In order to test the despiking algorithm, we selected three
patients and five normal subjects (Table 2).
2.1. Computer-generated signal
Computer-generated time series were produced by remov-
ing high level frequencies by window-average filter (window = 10
samples) from 20 randomly selected signals of healthy sub-
jects and adding white noise having a signal-to-noise ratio
SNR = 20 log 10(Asignal/Anoise) ∈ (20 dB, 40 dB); we call this signal
«. Then, from one to ten spikes were added to the signal « in
random positions; we call this signal «s. The spikes had a dura-
tion of 80 ms (∼=20 samples) and amplitude A+
spike
∈ (0, 600 px),
A−
spike
∈ (−600 px,0). Eq. (1) is the formula for spikes:
f (t) =
A+
spike
triang(t, t0, t1) ∀t ∈ (t0, t1)
A−
spike
triang(t, t1, t2) ∀t ∈ (t1, t2)
(1)
where triang is the triangular function. After despiking («ds) we
calculated the ratio of removed spikes to spikes added to the signal
by Eq. (2). We evaluated Eq. (2) for all t (˘) and for all t in spikes
(˘spike).
˘ =
t
(«s(t))
2
− «(t)
(«ds(t) − «(t))
2
· 100% (2)
Eq. (2) is the ratio of “spikes removed” to “spikes added” expressed
as a percentage.
2.2. Analyzing differences between groups
The ˘ was used to build two small groups of 10 × 2 computer-
generated time series with SNRg1 = 1.1 · SNRg2. We applied the
parametric Holm–Sidak procedure, and we evaluated the F func-
tion (variance between items/variance within items) to compare
patients and subjects and to evaluate the application of WS.
3. Theory
Fourier analysis is a popular technique for processing signal
content. It has the disadvantage of losing time information after
transforming a signal into the frequency domain. To obviate this
difficulty, the windowed Fourier transform (WFT) can be used: a
truncate window, the sine and cosine functions of which fit a win-
dow of a given width. However, a unique window is used for all
frequencies in the WFT. The resolution of the analysis is the same
over the whole domain and is limited by the size of the window. In
wavelet analysis, the window size can be varied.
3.1. Wavelet
The wavelet transform (WT) is a time–frequency representation
of the signal that has two main advantages over conventional meth-
ods: it provides an optimal resolution in the time and frequency
domains, and it eliminates the requirement of signal stationarity
(Mallat and Hwang, 1992). It is defined as the convolution of the
signal x(t) and the wavelet functions a,b:
x(t) a,b(t) dt (3)
where a,b(t) is the scaled and translated mother wavelet given
by Eq. (4):
a,b(t) =
1
√
a
t − b
a
a, b ∈ (4)
where a and b are the scale and translation parameters, respec-
tively. The mother wavelet is a function of finite energy and zero
energy (Eq. (5)): see Fig. 2 for a short summary of mother wavelet:
(t) dt = 0 (5)
Therefore, the WT is usually defined at discrete scales a and
discrete times b by choosing the set of parameters:
a = 2−j
; b = k2−j
j, k ∈ ℵ (6)
Then, by correlating the original signal with wavelet functions
at a discrete size and level a, b we obtain details dj of the signal
at several scales. This procedure can be organized in a hierarchical
way called multiresolution or multilevel wavelet decomposition
0 0.5 1 1.5
−2
0
2
db1
0 1 2 3
−2
0
2
db2
0 5 10
−2
0
2
coif2
0 1 2 3
−2
0
2
sym2
0 0.5 1
−0.5
0
0.5
1
bior1.1 decomposition
0 0.5 1
−2
−1
0
1
bior1.1 reconstruction
Fig. 2. Wavelets of five different wavelet families. Other families: Morlet, Mexican
hat and Meyer.

4. G. Veneri et al. / Journal of Neuroscience Methods 196 (2011) 318–326 321
(MWT; Mallat, 1989). The multilevel decomposition separates the
signal into details at different scales and a coarser representation of
the signal named “approximation”. The MWT at level l of decom-
position extracts the coefficients {d1, . . ., dl+1}, where dl+1 is the
approximated coefficients and represents overall signal trend, and
dj is the detailed coefficient and represents the rapid variations in
the signal: d1, . . ., dl are the decompositions at various levels of
frequency, from low to high, respectively. Due to the hierarchical
characteristic of the MWT, the approximated coefficients can be
reconstructed by {dj, . . ., dl, dl+1} coefficients.
3.2. Spike removal—despiking
Besides applications in these fields, the wavelet transform has
been applied to recognition of signal discontinuities. One of the
most interesting applications involves its ability to recognise spikes
(Nenadic and Burdick, 2005). The basic principle proposed by
Donoho and Johnstone (1994) is based on rejecting or accepting
(if we need to identify the spike) only those wavelet coefficients
that exceed a threshold, followed by the inverse wavelet transform
(inverse hard thresholding).
˘dj =
dj if |dj| ≤ thj
0 otherwise
(7)
The threshold is computed by Eq. (8):
thj = j 2 loge N (8)
where N is the number of samples and j is the standard deviation of
noise. The standard deviation j must be estimated from the coef-
ficients d1: the basic idea is to consider noise to be Gaussian, which
may not hold in general, and the detailed coefficients of the finest
scale (such as d1) are assumed to be essentially noise coefficients
with standard deviations equal to (Ehrentreich and Sümmchen,
2001; Sharma et al., 2010; Kamavuako et al., 2010). Under these
assumptions it can be shown that the median of its absolute devi-
ation j = median( | d1 |)/0.6745 effectively estimates the standard
deviation. It is easy to prove that for a zero mean Gaussian distribu-
tion the median should be ≈ · 0.6745 (Nenadic and Burdick, 2005,
Appendix 1).
Finally, it is important to choose a mother wavelet that is suit-
able for the signal of interest. Quiroga et al. (2004) applied a db1 or
coif2 wavelet at four levels. The choice is motivated by the shape of
the spike which should be approximated by an db1 or coif2 wavelet
at the high level. In the current context, it is impossible to make
any assumptions.
3.3. Wavelet entropy
Given {c1,t, . . ., cl+1,t} = {d1(t), . . ., dl+1(t)} the mean energy is
El,k =
k0+ t
t=k0
c2
l,t
N
(9)
where k0 = 1, 1 + t, 1 + 2 t, . . . and N is the number of wavelet
coefficients in the time window.
Ek =
l
El,k (10)
and the probability distribution for each level can be defined as
pl,k =
El,k
Ek
(11)
from the definition of entropy given by Shannon (1948), the time-
varying wavelet entropy (Blanco et al., 1998)
WSk = −
l
pl,k · log2 pl,k (12)
and the WS is the mean of Eq. (12).
4. Calculations
As shown in Ehrentreich and Sümmchen (2001) and Quiroga
et al. (2004), we applied the method of Eq. (7) to gaze signals of
patients. When we needed to eliminate spikes with high precision
to reconstruct the signal, the despiking worked well, but not as
well as we wanted; in particular, the application of Eq. (7) with
MWT at a low level of decomposition affected saccades, and at a
high level of decomposition the spikes were “spread” over more
detailed coefficients dj.
The key idea was to set the level of decomposition of MWT to
the maximum level lmax
1 admissible by the algorithm and to apply
Eq. (7) for j = 1, . . ., loptim and for j = median( | dj |)/0.6745, where
loptim < lmax must be estimated.
4.1. Choosing the correct optimum level
To find the optimum level, we decomposed the signal at the
maximum level in order to separate saccades and spikes as much
as possible and to find the loptim that maximized the ratio of “spikes
removed” to “signal preserved”.
An interesting characteristic of WT is the ability to maintain the
relationship between signals: if a relation exists between x and y,
the WT(x) has the same relation with WT(y). In eye tracking, we
recorded the position of an eye along x and y. van Beers (2007)
found an interesting correlation between x and y during saccades,
and we confirmed this result: the local covariance between x and
y was high during spike and saccade (see Fig. 3). Thanks to the
physiological characteristics of ocular motor signals, we used the
variance and specifically the covariance between x and y to define
loptim. We choose:
loptim =
∀j ∈ (1,lmax)
argmin(j − 1) :
covw(x, y) − covw(dxj, dyj) ≤ 0
∨
covw(dxj, dyj) − covw(dxj−1, dyj−1) ≤ 0
(13)
where ||◦ | | is the median and
covw(x, y) = |diag1,2 (cov(x(t0, t1), y(t0, t1))) |
t0 = t −
w
2
, t1 = t +
w
2
(14)
and w is larger than spike or saccade duration. Since the mean
saccade duration is ≈ 30 ms, w = 40 ms should be a good value.
In other words, despiking by Eq. (7) was stopped when the
median of covariance of detailed coefficients ˘dx, ˘dy was more than
the median covariance of the signal or less than the covariance
of ˘dx, ˘dy at j − 1. Algorithm A.1 describes the implemented pro-
cedure. Intuitively, the algorithm proceeds to despike from level 1
to the optimum level until the information removed is no longer
significant (condition 1) or there is no meaningful information to
be removed (condition 2).
1
See function wmaxlev of Mathworks (2007).

5. 322 G. Veneri et al. / Journal of Neuroscience Methods 196 (2011) 318–326
Fig. 3. (a) Signal of patient CN (samples 1500–3000). A spike is a rapid change on
the signal that can be confused with a saccade, which represents eye movements.
(b) Covariance x, y of signal of patient CN. (c) Spikes s1 and s2 on the signal are easy
to identify by WT: reconstructing the signal by detail coefficient d1, spikes were
more evident than saccades and hard thresholding was applied; unfortunately, to
remove spikes as much as possible and to preserve saccades, extension to the other
dj s with j > 1 was necessary. (d) Evaluating the ratio between the covariance on
detailed coefficients and the covariance of the signal (graph (b)), the procedures
identified the level j where to apply hard thresholding.
4.2. Using wavelet entropy for noise analysis
A ten order MWT was used to separate the signal by scale, and we
removed the main component of spikes from detailed coefficients
according to the method of Eq. (7). Then we applied the entropy
formula to the detailed coefficients to estimate the motor control
noise: WS was applied to each component x and y of the recorded
signal using bi-orthogonal splines and a time window t = 400 ms
(at least 100 points; Eq. (12)), but we only evaluated the mean
of WSk for linf ≤ lmax + 1 scales, where l = lmax = 10 was the level at
which the signal was decomposed:
WS =
linf
k=1
WSk
linf
(15)
linf was chosen according to
linf =
∀j ∈ (1,lmax)
argmax(j) :
cov( dxj, dyj)
cov(x, y)
≤ 0.05 (16)
where cov(x, y) = |diag1,2(cov(x, y))|. From an intuitive point of
view, we calculated WS only on coefficients from level 1 to level
linf, where linf was the maximum level at which x and y covariance
was less than 5% of complete signal covariance.
5. Results and discussion
5.1. Experiment 1: computer-generated case
We tested the algorithm on a computer-generated time series
built as described in Section 2.1. We used the mother wavelet
shown in Fig. 2. Results did not report any correlation between
performance (Eq. (2)) and the level of noise or amplitude of spikes
(Table 3). This means that the algorithm is sufficiently robust to
noise and spike amplitude. Performance varied from 88% (db1) to
92% (coif2). The algorithm preserved the signal and only removed
3–1% of the original signal «.
Wevelt entropy, WS, defined in Eq. (15), was tested in a small
20 × 2 sample of two groups of computer-generated time series
« made by adding white noise to filtered signals with different
SNR derived from subjects ( SNR = 10% · SNRg2). The implemented
procedure proved to estimate 94.8% of the white noise entropy for
SNR = 20 dB and 91.01% for SNR = 40 dB. The maximum performance
(≈111% of the mean) of Eq. (15) was obtained for linf ∈ (2, 5) and
lmax = 10. It provided evidence that WS may be a good candidate for
estimating motor control noise.
Fig. 4 shows the trend of p-values and F over x and y varying SNR
and the level of approximated coefficients: we found a significant
difference (p < 0.05) between groups when WS was estimated for
detailed coefficients from level 5 to 2 (linf ≈ 5).
We obtained a highly significant difference (p < 0.001) for high
SNR, which can be explained as the native property of WS to mea-
sure the microscopic disorder within the system.
5.2. Experiment 2: patients analysis
We applied the despiking algorithm, implemented by “db1”
ten levels, to the subjects of Table 2. Fig. 5 shows the differences
between signal covariance and detailed coefficients of level 1, 2, 3
and Table 4 reports loptim calculated by Eq. (16).
The signals of subjects EU and PP did not report any spikes, and
the algorithm only applied the hard thresholding to PP, but without
any consequence. In the case of EU, the procedure skipped the hard
thresholding. The procedure also worked well on normal subjects
SL, GV and XM (Fig. 6).
In the case of patients GFOD (Fig. 7), MS and CN (Fig. 8),
the procedure removed spikes, but on grouped spikes, such as
Table 3
Performance of the algorithm with different mother wavelets. The best performance was obtained through coif2 and db1.
Mother wavelet ˘spike ˘ Regression over noise amplitude Regression over spike amplitude
db1 0.835 0.881 p > 0.05 p > 0.05
db2 0.799 0.801 p > 0.05 p > 0.05
coif2 0.901 0.954 p > 0.05 p > 0.05
sym2 0.795 0.911 p > 0.05 p > 0.05

6. G. Veneri et al. / Journal of Neuroscience Methods 196 (2011) 318–326 323
10
−6
10
−4
10
−2
10
0
← SNR = 20dB
p−valuex(logscale)
← SNR = 40dB
10−6
10−4
10−2
100
← SNR = 20dB
p−valuey(logscale)
← SNR = 40dB
12345678910l+1
0
10
20
30
F
x
(1,38)
level
12345678910l+1
0
10
20
30
F
y
(1,38)
level
5%
p−mean
p−max/min
Fig. 4. Ability of wavelet entropy to discriminate two groups of 10 × 2 computer-generated time series having different signal to white noise ratio ( SNR = 10% · SNRg2). We
varied signal to noise ratio and the level of detailed coefficients where WS was calculated. The maximum partition was found for linf ∈ (2, 5).
1 2 3
−10
0
10
30
level
Δ
cov
patient MS
patient CN
patient GFOD
normal PP
normal GV
normal EU
normal XM
normal SL
Fig. 5. Differences of covariance of patients and healthy subjects defined by Eq. (13).
the last part of GFOD’s signal, it was unable to reconstruct the
signal.
When a harder despiking is required, it is possible to apply a
simple median filter on spikes identified by the procedure (see Fig.
8).
Statistical analysis of patients and subjects (Fig. 9) reported sig-
nificant differences on the x and y axes for WS applied to coefficients
from 8 to 1 (Fx = 5.29, Fy = 10.07, px = 0.0003, py = 0.02765) and sig-
nificant differences in coefficients linf to 1 as expected, where linf = 5
was chosen according to Eq. (16).
The best performance was obtained for linf = 4 or linf = 3 (px < 0.01,
py < 0.01): Fig. 10c shows the scatter diagram and probability dis-
tribution for linf = 4 and is a further evidence that the procedure
Table 4
Optimum level of subjects according to Eq. (16).
Subjects loptim Spikes
MS 2 Small sparse
CN 2 Small sparse spikes
GFOD 2 Small sparse and grouped spikes
PP 1 None
GV 1 Sparse spikes
EU 0 None
XM 2 Small sparse spikes only on Y
SL 2 Small sparse and grouped spikes
4 4.1 4.2 4.3 4.4 4.5 4.6
x 10
4
200
400
600
800
normal XM
x(px)
time (ms)
4 4.1 4.2 4.3 4.4 4.5 4.6
x 10
4
200
400
600
y(px)
time (ms)
original
despiked
s1
s2
s4
s5
l1
s3
Fig. 6. Healthy subject XM. Group of spikes s1, s2, s3, a large spikes s4 and s5 are
highlighted in the figure; small signal 11 was loss of pupil. There were no spikes on
x and the signal remained unchanged.
4 4.2 4.4 4.6 4.8 5
x 10
4
200
400
600
800
1000
patient GFOD
x(px)
time (ms)
4 4.2 4.4 4.6 4.8 5
x 10
4
200
400
600
y(px)
time (ms)
original
despiked
s1
s3
s2
s4
s1
s2
s4
s3
Fig. 7. Patient GFOD. The implemented technique removed sparse spikes s1, s2 and
s3 well, but was unable to reconstruct the signal on grouped spikes s4.

7. 324 G. Veneri et al. / Journal of Neuroscience Methods 196 (2011) 318–326
1000 2000 3000 4000 5000 6000 7000 8000
0
500
1000
time (ms)
position(px)
1000 2000 3000 4000 5000 6000 7000 8000
0
500
1000
despiking by median filtering after the recognition of spikes
time (ms)
position(px)
original
despiking by median filtering
Fig. 8. (Top) x data of patient CN after application of the method; the magnified
boxes show two spikes not completely removed by the procedure. (Bottom) x data
of patient CN after application of the median filter to spikes identified by hard thresh-
olding; the same two spikes were completely removed but the filter caused artefacts
on saccades.
145l+1
10
0
p−value(logscale)
145l+1
0
10
20
30
F(1,34)value
12345678910l+1
−1
0
1
2
cov
1,2
level
x
y
x
y
5%
1%
l
inf
Fig. 9. Results of statistical analysis of patients and healthy subjects applying WS
at various levels of wavelet coefficients. The best separation was found from coef-
ficient 4: the approximated coefficient (l + 1) indicated the contribution of visual
search exploration and the highly detailed coefficients were influenced by machine
recording or saccades.
1.2
1.2
yentropyof{d
1
,…,d
l+1
}
x entropy of {d
1
,…,d
l+1
}
EU
GV
PP
SL
XM CN
MSGFOD
CTRL
SCA2
NDC
σ
2σ
σ
(a)
0.8
0.8
yentropyofd
1
x entropy of d
1
EU
GV
PP
SL
CN
MS
GFOD
σ
σ
2σ
(b)
pdfx
pdfy
1.3 1.35 1.4 1.45 1.5 1.55 1.6 1.65 1.7
1.35
1.4
1.45
1.5
1.55
1.6
1.65
1.7
1.75
yentropyof{d1
,…,d4
}
x entropy of {d1
,…,d4
}
EU
GV
PP
SL
XM
CN
MS
GFOD
CTRL
SCA2
NDC
(c)
Fig. 10. Scatter diagram of WS. In order to estimate motor control noise, we applied wavelet entropy (WS). (a and b) t-Test of WS applied to entire signal (using all coefficients)
and d1 coefficients did not find any significant difference between patients and healthy subjects (p > 0.05). (c) t-Test analysis and post hoc Holm–Sidak procedure on WS
applied to coefficients d1, . . ., d4 showed a significant difference on x (px < 0.01) and y (py < 0.01) between patients and healthy subjects. Probability distributions over x (pdfx)
and y (pdfy) were calculated by kernel density (Bowman and Azzalini, 1997). Similar results were found for WS applied to coefficients d1, . . . , dlinf
with linf = 5 (px < 0.05,
py < 0.01).

8. G. Veneri et al. / Journal of Neuroscience Methods 196 (2011) 318–326 325
150 200 250 300 350
120
140
160
180
200
220
240
260
280
300
σy
σ
x
0 0.2 0.4 0.6
−0.2
0
0.2
0.4
0.6
0.8
1
σ
y
onfixations
σ
x
on fixations
Fig. 11. Standard deviation evaluated on the entire signal or only on the fixation.
No significant difference (p > 0.05) was found between groups. Standard deviation
cannot be used to evaluate motor control noise on visual search.
successfully separated normal subjects and patients, and especially
subjects and SCA2. The correlation between entropy over x and y
provided further evidence of the method’s efficacy.
No significant difference was found using the approximated
coefficients of d1, . . ., dl+1 (Fig. 10a) and only at level 1 (Fig. 10b).
We argue that motor control noise had no contributions at high
frequencies or was difficult to estimate, confirming the hypothe-
sis that the coefficients d1, . . ., d2 were influenced by the machine
recording or the saccades; on the contrary, the approximated coef-
ficients d11 = dl+1 and less detailed coefficients d9, d10 only indicated
the variability of visual search exploration.
The group NDC did not have a uniform distribution and post hoc
analysis did not find any significant difference: we found similar
results in a pro-saccade test not yet published. No differences were
found in the variance, particularly within fixations variance, which
could be a good candidate for estimating additive motor control
noise (Fig. 11).We could not make any assumptions about the role of
additive noise and signal-dependent noise. Indeed, SCA2 is known
to affect movement, due to the role of the cerebellum, but not the
process of fixation. We obtained results similar to those of the test
of simulated cases (Fig. 4).
6. Conclusions
We used the multiscale wavelet transform to remove spikes,
verifying that our algorithm, based on the work of Donoho and
Johnstone (1994), can be applied to eye-tracking and signals of
patients. We implemented an easy procedure for “fine despik-
ing” and preserving signal characteristics as well as physiological
motor control noise. When the procedure was tested on computer-
generated signals and those of healthy subjects, despiking proved to
remove the 90% of spikes. It allowed us to investigate the role of the
cerebellum in motor control, avoiding artefacts due to spikes. We
then estimated the motor control noise by wavelet entropy during
visual search: the procedure differentiated patients from controls
on y and x, y when applied to coefficients from 8 to 1 and from 4 to
1, respectively.
The ability to measure motor control noise made it possible to
investigate the role of the cerebellum in free visual search in our
current research. Our preliminary results, not yet published, sug-
gest that humans adapt exploration in order to reduce “saccade
energy”. The present method will be used to verify this hypothesis.
Future research will be addressed to mother wavelet construc-
tion to discover characteristic signals, such as square jerks and
microsaccades: correntropy applied to wavelet transform may be
useful to identify important waveforms in fixations. Correntropy
is defined as a generalization of correlation of random processes,
avoiding any assumption of Gaussian distribution (Liu et al., 2007).
It can be used to identify the correlation between x and y during
fixation and therefore “small synchronization” of motor control,
providing a quantitative measure of physiological noise.
Acknowledgments
The authors would like to thank the editor and reviewers for
improvements and suggestions, and Helen Ampt for the formal
improvements.
Appendix A. Algorithm
Algorithm A.1. This algorithm describes the implemented pro-
cedure. WTDECOMPOSITION and WTRECONSTRUCTION are the
wavelet decomposition and reconstruction. HARDTHRESHOLDING
are the function that implements Eq. (7). COVw is the covariance of
two signals on window w.
covxy ← median(COVw(x, y))
cAx, dx ← WTDECOMPOSITION(x, lmax)
cAy, dy ← WTDECOMPOSITION(y, lmax)
for j = 1 to lmaxdo
covdxdy ← median(COVw(dxdy))
pcovxy ← covdxdy
if covxy − covdx(j)dy(j) < 0 or covdx(j)dy(j)−p covxy < 0then
break
end if
dx(j) ← HARDTHRESHOLDING(dx(j))
dy(j) ← HARDTHRESHOLDING(dy(j))
p covxy ← covdx(j)dy(j)
end for
x ← WTRECONSTRUCTION(cAx, dx, lmax)
y ← WTRECONSTRUCTION(cAy, dy, lmax)
Appendix B. Supplementary data
Supplementary data associated with this article can be found, in
the online version, at doi:10.1016/j.jneumeth.2011.01.006.
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