Abstract— In this paper, we consider alignment of visual evoked potentials (EP) in the Discrete Fourier Transform (DFT) domain. Visual EPs have important clues for diagnosing medical problems such as multiple sclerosis and optic neuritis. The amplitude of visual EPs are usually smaller than the amplitude of spontaneous EPs which causes difficulties in reliably finding the latencies and amplitudes of important positive and negative peaks in the evoked responses. Therefore, noise cancellation becomes important for determining the features of interest in these waveforms. A well-known noise cancellation method is averaging multiple evoked potentials. Averaging after alignment of EP waveforms can improve the waveform quality substantially since usually evoked potentials have different characteristics and therefore have different latencies and amplitudes in response to the same visual stimulus. In this paper, we use a time alignment method which simultaneously reduces the spectral differences between all waveforms by minimizing the linearly phase shifted forms of the DFTs of these waveforms. We demonstrate that this method successfully aligns multiple visual EPs and achieves a smooth averaged waveform with reduced noise.

I. INTRODUCTION ISUAL evoked potentials are electrical waveforms produced by the brain in response to visual stimuli such as flash light and checkerboard reversal stimuli on a

video screen [1]. The amplitudes and latencies of these evoked responses in these waveforms contain important information about the condition of the visual pathway from the retina to the occipital cortex. For instance, occurrences of these responses are delayed in patients with multiple sclerosis and optic neuritis. Abnormal latencies in visual EP are also used in diagnosis of glaucoma, ischemic optic neuropathy, and tumors compressing the optic nerve, etc. [2]. Lastly, visual EPs are used for grading acuity in some patients with visual impairments [3].

Visual EPs usually have large noise components due to electroencephalogram (EEG) which is also known as spontaneous EP [1,4]. The noise component can easily have larger amplitudes than the visual EPs. Therefore, the features of interest such as latencies and amplitudes of some peaks and valleys in an visual EP are shadowed and cannot be measured reliably. There has been extensive literature

The work of İsmet Şahin was supported by the US National Science

Foundation grant DMR-0520547. İsmet Şahin was with Department of Biomedical Informatics, University

of Pittsburgh, PA 15260 USA. He is now with the Department of Materials Science, University of Maryland at College Park, MD 20742 USA (e-mail: isahin@ gmail.com).

Nuri Yilmazer is with Department of Electrical Engineering, Texas A&M University-Kingsville, Kingsville, TX 78363 USA (e-mail: nuri.yilmazer@tamuk.edu)

which considers separating the evoked potentials from the spontaneous potential [5-10]. A well-known method involves averaging the visual EPs which has the effect of reducing the noise component [1,11]. Since different evoked potentials with different latencies can be recorded in response to the same stimulus, the averaging after alignment of EP waveforms can further improve waveform quality.

In this paper, we use a similar method used in [12,13] where a cost function is constructed for simultaneous minimization of spectral differences between multiple waveforms. We use a computationally efficient form of this simultaneous minimization method [14] for aligning multiple visual EPs. A similar approach has also been used in estimating fundamental periods of fetal ECG waveforms [15]. The control parameters of the cost function are linear phase shift amounts applied to the DFTs of the waveforms in order to achieve alignment in the frequency domain. As a result, differences between the phase-shifted forms of the DFTs of the waveforms are reduced. By using real visual EPs in the experiments, we demonstrate that this method yields accurate alignment of multiple visual EPs and therefore effectively reduces noise. The resulting averaged visual EP can be used to measure average latencies of important peaks and valleys of visual EPs for medical purposes.

In section II, we briefly present the formulation and simplification of the optimization problem for finding optimal time differences between visual EPs. In section III, we present experimental results. These results are followed by concluding remarks in section IV.

II. PROBLEM FORMULATION

Let 0x be a waveform and

1 2 1, ,.....,

Nx x x

− be the

waveforms which are time delayed forms of the reference waveform

0x . Let τ be the time delay vector given by

1 2 1[ .... ]T

Nτ τ τ τ

−= (1)

where T denotes the transpose operation. The thi entry iτ

represents the delay amount between the reference waveform

0x and the delayed waveform

ix . We minimize

a cost function which measures the differences between phase shifted DFTs of the waveforms. In other words, this cost function attains its minimum when the applied phase shifts in discrete Fourier transform domain correspond to true time delays in time domain:

A Discrete Fourier Transform Method for Alignment of Visual Evoked Potentials

İsmet Şahin, IEEE Member and Nuri Yilmazer, IEEE Member

V

978-1-4244-6766-2/10/ $26.00 © IEEE

1 1

0 0

( )N N

mnm n

J Tτ− −

= =

= ∑∑ (2)

where mnT is given by

2

1

2[ ] [ ] [ ] [ ]

k

mn m m n nk k

T Y k P k Y k P k=

= −∑ (3)

where [ ]mY k , 0,1,...., 1m N= − , denotes the DFT of the

waveform mx and [ ]

mP k denotes the phase shift operator

[ ] k mj

mP k e ω τ= (4)

where 2k

k Nω π= . The quantity mnT measures spectral

differences between the phase shifted DFTs of the waveforms

mx and

nx . In [14], we demonstrated that

minimization of ( )J τ is equivalent to minimization of the

following reduced cost function (̂ )J τ

2 1

0 1

ˆ ˆ( )N N

mnm n m

J Tτ− −

= = +

= −∑ ∑ (5)

where

{ }2

1

* *ˆ Re [ ] [ ] [ ] [ ]k

mn m m n nk k

T Y k P kY k P k=

= ∑ . (6)

where * denotes complex conjugate. We also derived the following gradient of the reduced cost function which can be used in gradient based optimization methods

( )1 2 1

01 1 1

(̂ )N N N

n n m n mnn m n m

J e R e e Rτ τ− − −

= = = +

∇ = − −∑ ∑ ∑ (7)

where ie denotes ( 1)x1N − elementary column vector

whose thi entry is 1 and all other entries are zero and mnR

is given by

{ }2

1

* *Re [ ] [ ] [ ] [ ]k

mn k m m n nk k

R j Y k P kY k P kω=

= ∑ . (8)

Notice that computing mnR can be efficiently done once we

already compute ˆmnT since the formulas of both

mnR and

ˆmnT involve the term

* *[ ] [ ] [ ] [ ]m m n nY k P kY k P k . (9)

III. EXPERIMENTAL RESULTS We evaluate our method with real visual evoked

potentials available in the database [16]. All visual evoked potentials in this database are recorded from an electrode which is placed on the left occipital (O1) location of a healthy individual who is subject to two types of visual stimuli. In the first type, called non-target, the stimulus involves only color reversal of the checks on a checker board. In the second type, called target, the stimulus contains a half check displacement in addition to the color reversal. Three-fourth of all experiments are non-target stimuli which are supposed to be ignored by the subject and the other one-fourth are target stimuli which are counted by the subject. The evoked potentials from the electrode are first passed through a band-pass filter with the cutoff frequencies of 0.1 and 70 Hz and then sampled at 250 Hz. Each recording contains 512 samples. The first half contains spontaneous evoked potential before stimulus onset and the other half contains the visual evoked potential in response to the visual stimulus. Even though 30 waveforms are available in the database, we use 15 waveforms since the other waveforms contain substantial noise due to other activities. Further experimental details can be found in [4].

In order to align the visual EPs after stimuli are presented, we only consider the second halves of evoked potentials, i.e.

0 0.2 0.4 0.6 0.8 1

0

20

40

60

VE

P 1

0 0.2 0.4 0.6 0.8 1

0

20

40

VE

P 2

0 0.2 0.4 0.6 0.8 10

20

40

60

TIME (SEC)

VE

P 3

OriginalAligned

OriginalAligned

Fig. 1. Three visual EP waveforms after visual stimulus onset. Second and third waveforms are aligned with respect to the first waveform which is defined to be the reference waveform. The gray and dark lines are the original visual EPs and the aligned EPs respectively after shifting optimal amounts.

the last 256 samples of each EP. The initial guess of time delays is a vector of all zeros. We implement a quasi-Newton method with the BFGS Hessian update and with the line search requiring that the Powell-Wolfe conditions are satisfied [17]. The quasi-Newton method starts with the initial guess and uses the cost function and the analytic gradient given by (5) and (7) respectively. We use all frequency components in evaluations of the cost functions.

We illustrate first three visual EPs in Fig. 1 where VEP1 is the reference waveform and the other two waveforms are aligned with respect to the reference. The quasi-Newton method finds the optimal shift amounts 21.93 and 8.01 samples. We shift second and third waveforms by 22 and 8 samples respectively and plot the results in Fig. 1 with dark lines. Notice that large peaks and valleys in these waveforms occur in similar time ranges after alignment. Similarly, we align all 15 waveforms simultaneously and then average the aligned waveforms. The averaged waveform will have the largest peaks and valleys at similar times with the reference waveform since all waveforms are aligned with respect to the reference waveform. We average the optimal delay amounts between 14 waveforms and the reference waveform and take into account this value in determining true average latencies. The resultant averaged waveform is demonstrated in Fig. 2. Notice that the important peaks and their latencies are clear in this plot since noise is much reduced by averaging latency-corrected waveforms.

0 0.2 0.4 0.6 0.8 1

0

10

20

30

40

AV

ER

AG

E V

EP

TIME (SEC)

Fig. 2. The averaged visual EP.

IV. CONCLUSION In this paper, we describe a DFT method for alignment of

visual EP waveforms. This method tries to find optimal linear phase shifts which are applied to the DFTs of the visual EP waveforms in the Fourier transform domain in order to reduce spectral differences of these waveforms. In other words, the cost function attains its minimum value when phase-shifted DFTs have least differences. We shift all waveforms in time domain with optimal time delays corresponding to optimal phase shifts and then perform averaging. We use this method with real visual EP waveforms and demonstrate that this method reduces noise and therefore clearly shows the latencies and amplitudes of important evoked responses in the averaged waveform.

ACKNOWLEDGMENT We would like to thank Dr. Florencia McAllister,

postdoctoral fellow at Johns Hopkins University School of Medicine, for her useful comments on the medical background of this work.

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