HUMAN eyes exhibit small movements even in steady environmental

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1 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 51, NO. 11, NOVEMBER Analyzing the Dynamic Wavefront Aberrations in the Human Eye D. Robert Iskander*, Senior Member, IEEE, Michael J. Collins, Mark R. Morelande, and Mingxia Zhu Abstract The optics of the human eye are not static in steady viewing conditions and exhibit microfluctuations. Previous methods used for analyzing dynamic changes in the eye s optics include simple Fourier-transform-based methods, which have been used in studies of the eye s accommodation response. However, dedicated tools for the analysis of dynamic wavefront aberrations have not been reported. We propose a set of signal processing tools, the combination of which uncovers aspects of the dynamics of eye s optical aberrations which were hidden from conventional analysis techniques. The methodology includes extraction of artifacts from potentially significant eye movements, filtering, optimal parametric signal modeling, and frequency and time-frequency representations. The exposition of the techniques and their advantages over traditional techniques is illustrated for real dynamic eye wavefront aberration measurements. Index Terms Aberrations, biomedical signal processing, eye, wavefront sensing. I. INTRODUCTION HUMAN eyes exhibit small movements even in steady environmental conditions. This characteristic is not limited to the whole eye but also occurs in individual components of the eye such as the pupil, crystalline lens, and retina. It is not surprising, therefore, that the optical characteristics of the eye also undergo similar fluctuations. The magnitude of these variations is not large, hence, they are often termed microfluctuations [1], [2]. Accurate measurement and analysis of microfluctuations in the optical characteristics of the eye can provide a better insight into the role of these changes in the visual control system. The main focus of studies on microfluctuations in vision has been placed on the accommodative response of the human eye [1] [5]. These studies were usually performed with an objective refractometer that measures only the lower order optical aberration terms, namely defocus and astigmatism. The results of these Manuscript received November 12, 2003; revised March 31, This work was supported by the QUT Strategic Collaborative Program scheme. Asterisk indicates corresponding author. *D. R. Iskander is with the Contact Lens and Visual Optics Laboratory, School of Optometry, Queensland University of Technology, Victoria Park Rd, Kelvin Grove Q4059, Australia ( d.iskander@qut.edu.au). M. J. Collins is with the Contact Lens and Visual Optics Laboratory, School of Optometry, Queensland University of Technology, Kelvin Grove Q4059, Australia ( m.collins@qut.edu.au). M. R. Morelande is with the CRC for Sensor Signal and Information Processing, Department of Electrical and Electronic Engineering, The University of Melbourne, Parkville VIC 3010, Australia ( m.morelande@ ee.mu.oz.au). M. Zhu is with the Contact Lens and Visual Optics Laboratory, School of Optometry, Queensland University of Technology, Kelvin Grove Q4059, Australia ( m.zhu@qut.edu.au). Digital Object Identifier /TBME studies have consistently shown that during steady viewing conditions the eye constantly changes its focus (accommodation). The magnitude of these changes in accommodation are typically less than D (Diopter) and have a maximal amplitude of about 0.5 D [6]. The temporal frequencies of the accommodation fluctuations are typically less than 2 Hz. Traditionally, two main spectral regions are identified for the accommodative fluctuations: the lower frequency region, Hz, and the higher frequency region, Hz. Simple Fourier-transform-based techniques and the coherence analysis have been used to link changes in the accommodation fluctuations with pulse and respiration signals [2], [7], [8]. Recent advances in wavefront sensor technology have provided vision scientists with instruments for measuring the eyes optical aberrations beyond the defocus and astigmatism [9]. The most popular of these instruments are based on the Hartmann- Shack (HS) lenslet array [10]. An HS sensor system generally consists of a laser, an array of small lenses, and a charge-coupled device (CCD) camera. The monochromatic light produced by the laser is reflected from the retina and passed through an array of lenses to form an image that is captured with a CCD camera. This image is then used to calculate the wavefront aberration of the eye which is subsequently fitted with a suitable parametric function such as a finite series of orthogonal radial polynomials [11] [13]. Although most commercially available instruments are designed to acquire single measurements of the wavefront, it is conceptually straightforward to adapt a wavefront sensor system to continuously measure optical aberrations of an eye [14] [16]. Similarly to the overall accommodative response, the wavefront aberration of the eye exhibits temporal microfluctuations. In [14], the dynamics of wavefront aberration from several subjects have been reported. The origin of changes in the wavefront error has not been identified with certainty. Sources of these microfluctuations could include an interaction between the cardiopulmonary system and the crystalline lens, local changes in the tear film thickness, changes in axial length of the eye. Eye movements and instrument noise contribute only marginally to the measured fluctuations [14]. To date, only the effects of pulse and respiration have been clearly identified in the accommodative signal via coherence analysis [2], [7], [8]. These effects have not been confirmed in the wavefront aberration signal, where the frequency component that could be related to a pulse signal was only occasionally observed [14]. However, we postulate that signatures of pulse and respiration could be traceable in the measured aberration signals, as they are in the overall accommodation response /04$ IEEE

2 1970 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 51, NO. 11, NOVEMBER 2004 Analysis of dynamics in the wavefront error of the eye could simply involve inspection of the temporal variations in each of the wavefront aberration signals (radial polynomial terms) and a subsequent frequency analysis of them using the fast Fourier transform (FFT). These traditional techniques could provide a unique description of the aberration signal characteristics, provided that the signals are stationary. However, many biological signals cannot be modeled as a superposition of scaled sinusoids [17]. The wavefront aberrations of the eye have been found to exhibit both amplitude and frequency modulation signal characteristics [18]. Thus, the techniques used to date for the analysis of the dynamics in aberration terms of the eye provide only a general overview of the measured signals. We propose a methodology for comprehensive analysis of dynamics in wavefront aberrations of the human eye. Several standard, as well as specialized signal analysis techniques are considered, the combination of which has the potential to provide answers to some of the questions regarding the genesis of the microfluctuations. In the next section, we provide the protocol for wavefront aberration data acquisition. This is followed by temporal, frequency, and time-frequency analyses of aberration signals which are covered in Section III. Discussion and conclusions are presented in Section IV. II. MEASURING DYNAMIC WAVEFRONT ABERRATIONS We used the Complete Ophthalmic Analysis System (COAS, WaveFront Sciences, Inc.), which is an HS-sensor-based system capable of acquiring wavefront aberration data from the human eye. The dioptric power range of the system is between D and D. The measurement of the dynamic wavefront error is performed at a preset sampling frequency of about 11.5 Hz in the multibuffer option, in which the images from the HS sensor CCD camera are consecutively stored in computer memory. The system exhibits sampling instabilities (jitter) because it is not driven by dedicated hardware. We have previously analyzed this jitter and found its effect on the signal to be insignificant [19]. Thus, the influence of this jitter can be omitted in our analysis. To estimate the underlying noise level of the wavefront sensor for dynamic wavefront measurements we conducted several pilot experiments. Initially, significant subject movement artifacts appeared in the wavefront measurements because of the table that the wavefront sensor was mounted on. To solve this problem, the wavefront sensor unit was fixed to a heavy and stable bench. Second, a model eye was fixed to the wavefront sensor head-rest while a subject was simultaneously positioned in the head-rest and instructed to breath at a rates up to 2 Hz. The estimated power of the instrument noise for the model eye with subject breathing in the head-rest was found to be at least an order of magnitude less than the power of the wavefront aberration signals from a real eye. This gave us confidence that the fluctuations we observed in the wavefront aberration signals were related to ocular factors and not extraneous vibrations. Fig. 1. A typical HS image. The corneal reflection is visible in the center of the image. The subject s head is positioned in a chin-rest in front of the instrument. For most clinical experiments, no additional restraints such as a head-strap or a bite-bar have been found to be necessary [20]. Normally, the instrument s viewing target is fogged (blurred) to suppress accommodation. However, to study the dynamics of aberrations, an accommodation target can be included by means of a beam splitter or by having the fellow eye fixate a target. The subject is advised to blink before commencing the continuous measurement of the wavefront error. We usually acquire 256 measurements resulting in a recording of about 22.3 s. The acquired raw data consists of a set of Hartmann-Shack sensor images. A typical example of such an image is shown in Fig. 1. These images are then used to estimate the wavefront error. Essentially, each high intensity grid spot in the image can be viewed as a local point spread function. The centroid of the local grid spot is used to estimate the transversal aberrations and, where are the central coordinates of the corresponding lenslet in the array. The transversal aberrations are directly related to the slope of the wavefront error where and denote the wavefront error in Cartesian and polar coordinates, respectively,, and is the numerical aperture [21]. The integration of the wavefront error from the measured wavefront slopes can be performed using zonal, modal or iterative spline estimation techniques [22], [23]. This type of analysis is an integral part of a commercial HS system.

3 ISKANDER et al.: ANALYZING THE DYNAMIC WAVEFRONT ABERRATIONS IN THE HUMAN EYE 1971 Fig. 2. Typical dynamic changes in the Zernike polynomial coefficients for a subject focusing on a target. The defocus term, b (t), is identified by a bold line. The piston b (t) and the prismatic terms b (t) and b (t) are omitted. The estimated wavefront error is usually modeled by a finite series of orthogonal Zernike polynomials [11] where, is the th Zernike polynomial,, is the coefficient associated with is the order, is the normalized distance from the origin, and represents the modeling error. The Zernike polynomials are a product of angular frequency functions and radial polynomials where represents the Zernike radial polynomial of order. The estimates of the coefficients, are found by utilizing a least-squares procedure [12]. The first six coefficients in the series correspond to the so-called lower order aberrations, i.e., piston, horizontal and vertical prism, defocus, and astigmatism (at 45 and 0 ). The remaining coefficients constitute the higher-order aberrations which cannot be corrected with traditional sphero-cylindrical lens corrections. Note that the analysis (1) (2) pupil size should be kept constant in the acquisition of dynamic aberrations. In the continuous measurement of aberrations, the estimated wavefront error becomes a function of time where, is the time-varying Zernike coefficient associated with. The coefficients, can be estimated individually at discrete time instants in the same way as the coefficients. A typical measurement of the Zernike polynomial coefficients is shown in Fig. 2. We use the estimated coefficients,to study the dynamics of wavefront aberrations in the human eye. The first three terms (i.e., piston and prisms) do not affect vision and are usually omitted. However, as it is shown later, the prismatic terms and can be used to identify eye blinks or significant eye movements. In the following, we examine the temporal and frequency characteristics of the estimated aberration signals. III. SIGNAL ANALYSIS A number of factors need to be taken into account when measuring the dynamic changes of the eye s optical aberrations. Small eye movements, microfluctuations of position of the cornea, crystalline lens and retina, changes in the tear film distribution covering the eye, and instrument noise are (3)

4 1972 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 51, NO. 11, NOVEMBER 2004 Fig. 3. Typical dynamic changes in the prismatic, b (t) and b (t), and the defocus, b (t), Zernike polynomial terms. The discontinuities in the signals due to eye blinks and potentially significant changes in the eye position are indicated. potential sources of variations in the amplitude and phase of the measured aberration signal. It is, therefore, important that during aberration data acquisition these factors are controlled as much as possible. However, unlike an artificial vision system, the biological eye cannot be mechanically fixed. Several techniques can be used to minimize the changes in aberration signals. For example, a stationary accommodation target can be included in the measurement protocol to help maintain focus and fixation. Also, the respiratory sinus arrhythmia can be controlled by asking the subject to breathe at a metronome dictated frequency [24]. In a prolonged acquisition of aberration data, a subject should be free to blink at a natural blink rate to maintain the consistency of tear-film distribution. Instability of the tear film directly influences the aberration measurements of the eye [25]. The average blink rate for subjects with healthy eyes is about 12 blinks/min [26]. Thus, it is expected to encounter several eye blinks in the data during dynamic aberration measurements. A. Time Domain Representation In Fig. 3, we show traces of the horizontal prism, vertical prism and defocus Zernike polynomial terms for a data acquisition protocol that included an accommodation target. In this example, the subject was asked to blink naturally during the first half of the measurement and try to consciously suppress blinking in the second half. The discontinuities in the signals, i.e., when the signal is identically zero at around, and s, correspond to those frames where the reflection from the retina was not visible in the image captured on the CCD camera. These are evident eye blinks. However, there are several more discontinuities in the signals that although not obvious in the defocus and higher order Zernike terms (not shown here), can be identified from the signals of the prismatic terms as potentially significant eye movements or incomplete eye blinks that should be taken into account in the analysis. For the purpose of studying dynamic aberrations, the eye blinks and large eye movements are considered as interference and are removed from the measured signal. Traditionally, the removal of the eye blinks has been performed by applying a predetermined threshold to the signal, deleting the intervening data points from the record and replacing them by an interpolation using neighboring samples [2], [27]. These techniques are effective for an accommodation signal because the duration of the eye blink is small compared with the period of typical microfluctuations in accommodative response. However, this may not necessarily be the case for aberration signals. Automatic removal of blink artifacts from the aberration signals is useful when examining the temporal characteristics of the dynamics. The identification of these artifacts can be achieved using a simple procedure based on the statistics of the prismatic terms of the aberration data. Firstly, the obvious eye blinks corresponding to time instances at which signals are identically

5 ISKANDER et al.: ANALYZING THE DYNAMIC WAVEFRONT ABERRATIONS IN THE HUMAN EYE 1973 Fig. 4. The prismatic terms b (t) and b (t) with detected artifacts denoted by circles (top and center), and the corresponding artifact free defocus term b (t) (bottom). The discontinuities in b (t) are left intentionally in the signal. zero are removed and the dc (constant) and root mean square (rms) values of the signals calculated, i.e., and where corresponds to the signal with obvious blink terms, removed, and is the signal duration. The signal statistics are then used to find time instances at which the prismatic terms show impulsive behavior. The aberration signals are then cleaned by removing the samples corresponding to the union. In Fig. 4, we show the prismatic terms with identified artifacts denoted by circles and the corresponding artifact free defocus signal. The process described above is similar to the removal of data that do not fall into a 95% confidence interval. However, the equivalence of the two methodologies only applies in the case of ergodic Gaussian data. The interpolation step mentioned earlier is omitted in our analysis since it does not provide any additional information about the measured dynamics of aberrations. In the past, interpolation was used for the accommodation signal for the frequency domain analyses such as the FFT [2], [27]. However, as it is shown in the next section, the artifacts related to potential eye blinks can be removed by appropriate digital filtering. B. Frequency Domain Representation Finding frequency representation of the data with spectral analysis requires the assumption of stationarity. In general, the aberration signals acquired by an HS system are not stationary. In earlier studies, where we used a purpose built HS system with a sampling rate of 10 Hz but smaller lenslet array than that of the COAS system, we observed that some aberration signals exhibit features characteristic of both amplitude and frequency modulation signals [18]. We will return to this issue in Section III-C. For the time being, we will assume that the frequency characteristics of the aberration signals do not vary with time. Cases of such aberration signals arise for subjects with very stable pulse and respiration frequencies or when breathing is stabilized by a metronome dictated frequency. Alternatively, shorter records than 22.3 s of data can be extracted from most aberration signals, for which the assumption of stationarity is then more plausible. Before spectral analysis is performed, each aberration signal is first centered. This is achieved by removing the dc value from the data. Then, each signal has to be appropriately filtered to remove the artifacts and extract the frequency content that is of interest. Similarly to the case of the accommodation signal, the

6 1974 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 51, NO. 11, NOVEMBER 2004 Fig. 5. Periodograms of the defocus Zernike term, b (t), for raw (top) and interpolated (bottom) data, before (dotted line) and after (solid line) filtering. The cutoff frequencies of the filter are set to f =0:1Hz and f =2Hz. Omitting the interpolation step does not significantly change the spectrum of the filtered aberration signal. upper cutoff frequency is set to 2 Hz. With the original sampling frequency of about 11.5 Hz and a typical record length of 256 samples, a frequency resolution of about 0.04 Hz can be achieved. Thus, we choose to set the lower cutoff frequency at 0.1 Hz. Clearly, the presented filter parameters are exemplary only. The digital filter is based on an equivalent of a Butterworth analog filter. The frequency transformation method is used for the conversion from a standard low-pass filter to a band-pass filter. Then the bilinear transformation is used to obtain the digital representation of the filter [28]. The filter order, responsible for the width of the transition band, is chosen to be larger than 10 but less than a third of the record length. The performance of the suggested filtering procedure is shown in Fig. 5 where we show the periodogram of the defocus term, i.e., smoothed periodogram) has limitations in terms of repeatability, particularly with shorter data records. For this reason we decided to use a parametric method of spectral estimation based on autoregressive (AR) signal modeling [29]. Suppose that a centered aberration term, can be modeled by an AR time series of order where is an independent and identically distributed Gaussian process with zero mean and variance. The parameters and are estimated using the technique of least-squares [29]. To determine the best model order,, one can use a criterion that combines the estimated mean square error (MSE) and a suitable penalty function for a range of model orders. Examples of such a methodology include Akaike s information criterion (AIC) [30] (5) (4) with being the number of discrete samples and denoting the sampling frequency, before (dotted line) and after (solid line) applying the filter, for raw signal (top) and the signal with potential artifacts removed and replaced by interpolation (bottom). We note that interpolation is no longer a necessary step in the frequency analysis of the aberration signals. We found that the traditional analysis of the frequency characteristics with FFT-type spectral estimation (periodogram, Rissanen s minimum description length (MDL) [31], or techniques based on the bootstrap [12], [32]. In most cases the bootstrap-based method outperforms other model order selection techniques [33]. However, in practice the MDL is preferred for its computational efficiency and superiority over the AIC for small data records. The estimated optimal order of an

7 ISKANDER et al.: ANALYZING THE DYNAMIC WAVEFRONT ABERRATIONS IN THE HUMAN EYE 1975 Fig. 6. Power spectral estimates for the filtered defocus Zernike term, b (t), using periodogram (dotted line) and the AR process modeling (solid line). AR approximation to the aberration signals was found to range from to for lower order aberration signals and from to for higher order aberrations. Despite these high estimated orders, no obvious spurious peaks were observed in the spectra. Note also that the optimal order is individually estimated for each aberration term. The spectral estimator of an AR process of order is given by (6) C. Time-Frequency Domain Representation The frequency content of most biological signals varies with time, making traditional spectral analysis unsuitable for their analysis. In such cases, the concept of time-frequency representation provides a better insight into the nature of a biological signal than the frequency domain representation alone [34]. The fundamentals of time-frequency signal analysis and processing can be found in [35]. The generalized time-frequency representation of a centered aberration term is given by where and are the least-square estimators of and, respectively. In Fig. 6, we show the AR-based spectral estimate of the considered earlier defocus term in comparison to an estimate based on the periodogram. The optimal model of the AR fitwas. The evident peaks in the AR spectral modeling could be related to the cardiopulmonary signatures [2]. Our experience with clinical data showed that the proposed AR spectral estimation technique provides a clearer indication of the nature of the variations in aberration signals than the FFT-based techniques such as those reported in [2], [14]. However, as mentioned earlier, many aberration signals cannot be assumed to be stationary. In such cases the spectrum alone does not provide a unique representation of the frequency content. In the following we will consider tools for spectral analysis of nonstationary aberration signals. where is the analytic signal, is the Hilbert transformation, the denotes complex conjugation, and is an appropriate kernel function. Equation (7) can be understood as a Fourier transformation of the convolution of the kernel function and an autocorrelation of the analytic signal. Selection of the kernel function depends to a great extent on the application and the nature of the signal at hand. In most cases it is a matter of choice between frequency and time resolutions and the suppression of the cross-terms in the time-frequency representation [34]. (7)

8 1976 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 51, NO. 11, NOVEMBER 2004 Fig. 7. Time (top), frequency (left), and time-frequency (contour graph) representations of the filtered defocus Zernike term, b (t). After experimenting with a large number of time-frequency representations of the aberration signals measured with an HS sensor, we chose a cone-shaped kernel proposed in [36]. This particular kernel has the property of resolving close spectral peaks in the time-frequency plane and proved to provide the clearest representation for a wide range of aberration signals. The time-frequency representation with the cone-shaped kernel is often referred to as the ZAM distribution [35]. In Fig. 7, we show the time record (top), the AR modelbased power spectrum (left), and the ZAM distribution of the filtered defocus Zernike term,, that was considered earlier. We note that in the low frequency region, Hz, the signal is almost stationary (compared with the highest peak in the spectrum representation). At the same time we observe two major components that appear to be less stationary at frequencies greater than 0.5 Hz. Thus, the considered aberration signal is nonstationary and is multicomponent. Note also that the initial part of the signal (approximately 2 s) is not shown in the time and time-frequency plots due to the utilized digital filtering. More examples of time-frequency representation are given in Section IV. D. The Complete Algorithm To summarize our proposed methodology we now provide an outline of the procedure for the analysis of dynamics in the eye s aberrations. In Fig. 8, we show the flowchart of the proposed methodology. A set of aberration terms,, is estimated at discrete time instances. The removal of potential artifacts due to eye-blinks or significant eye movements is performed only for the time domain representation of signals. Otherwise, a band pass filter is used to extract the frequency content of interest and the removal of the potential artifacts. The band-pass filtering includes a centering step to remove the dc value of the signal. Frequency domain representation is achieved by assuming stationarity of the signal and by modeling it with an AR process of order. Estimation of the AR parameters is performed over a range of possible model orders,, and the MDL criterion is used to determine the optimal model. Time-frequency representation is achieved by calculating an analytic signal, and using the ZAM distribution. Together, the three analytic methods as shown in Fig. 7 provide more insight into the nature of the dynamic changes in the aberrations than the time or frequency representation on their own. IV. DISCUSSION AND CONCLUSION A set of signal processing tools has been proposed for the analysis of dynamics in optical aberrations recorded with an HS sensor system. The combined time-frequency analysis of signals provide the vision researcher with a currently unexplored dimension in which the nature of microfluctuations in wavefront aberrations as well as their sources can be investigated. The utility of our approach is demonstrated by considering a protocol in which the subject is asked to increase their breathing rate during the aberration measurement every 4 5 s. Such a protocol is useful for examining the effects of the

9 ISKANDER et al.: ANALYZING THE DYNAMIC WAVEFRONT ABERRATIONS IN THE HUMAN EYE 1977 Fig. 8. The flow chart of the algorithm for the analysis of dynamic aberrations. cardiopulmonary system on wavefront dynamics. Figs. 9 and 10 show the behavior of the and that correspond to the astigmatism at 45 and vertical coma, respectively. Clearly, the power spectrum on its own does not provide sufficient insight into the nature of the dynamics. In the time-frequency representation, on the other hand, we can identify individual signal components and the frequency content that changes in time, as the breathing rate changes. For the astigmatic term, we can clearly see two frequency components, which increase in frequency at about the eighth second. However, further increase of breathing rate at about the twelfth second reduced the frequency components, an effect which is consistent with previous reports of decoupling of the pulse rate [24]. The last few seconds of the recording correspond to the highest rate of breathing. This part of the time-frequency representation is difficult to interpret because the head movements associated with high breathing rate significantly influenced the measurement. In such a case the use of a bite-bar or a head-strap may be necessary. For the comatic term, on the other hand, we see pseudo-periodic changes in the frequency content. It appears that at every increase of the breathing rate the frequency of the major component of coma initially increases followed by a period of frequency decrease. The above example clearly indicates that the frequency analysis alone, whether it is performed with nonparametric (FFT) or parametric (AR) spectral estimation techniques, provides only partial information on the aberrations and does not reveal the true nature of the analyzed signal. Results of a similar nature to those depicted in Figs. 9 and 10 have been obtained in a number of experiments. This leads us to postulate, with some confidence, that some components of the aberration signal are related to the cardiopulmonary system in a similar way to that reported for the accommodation signal and the fluctuations of the pupil [2], [37], [38]. Confirmation of this is being sought via a series of clinical experiments in which the dynamic aberrations are simultaneously recorded with the pulse and respiration signals. The results of these studies will be reported later. Combined time-frequency representation of aberration signals can be useful in other clinical studies such as determining the quality and performance of tear-film [39] or study the dynamics of accommodation [16].

10 1978 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 51, NO. 11, NOVEMBER 2004 Fig. 9. Time (top), frequency (left), and time-frequency (contour graph) representations of the filtered fifth Zernike term (astigmatism at 45 ), for a subject who increased the breathing rate during the measurement. Fig. 10. Time (top), frequency (left), and time-frequency (contour graph) representations of the filtered seventh Zernike term (vertical coma), for a subject who increased the breathing rate during the measurement.

11 ISKANDER et al.: ANALYZING THE DYNAMIC WAVEFRONT ABERRATIONS IN THE HUMAN EYE 1979 REFERENCES [1] F. W. Campbell, J. G. Robson, and G. Westheimer, Fluctuations of accommodation under steady viewing conditions, J. Physiol., vol. 145, pp , [2] M. J. Collins, B. Davis, and J. Wood, Microfluctuations of steady-state accommodation and the cardiopulmonary system, Vis. Res., vol. 35, no. 17, pp , [3] D. A. Atchison, M. J. Collins, C. F. Wildsoet, J. Christensen, and M. D. Waterworth, Measurement of monochromatic ocular aberrations of human eyes as a function of accommodation by the Howland aberroscope technique, Vis. Res., vol. 35, no. 3, pp , [4] F. W. Campbell, The accommodation response of the human eye, Br. J. Physiological Opt., vol. 16, pp , [5] K. Niwa and T. Tokoro, Influence of spatial distribution with blur on fluctuations in accommodation, Optometry Vis. Scie., vol. 75, no. 3, pp , [6] W. N. Charman and G. Heron, Fluctuations in accommodation: A review, Ophthalmic Physiological Opt., vol. 8, pp , [7] B. Winn, J. R. Pugh, B. Gilmartin, and H. Owens, Arterial pulse modulates steady-state ocular accommodation, Curr. Eye Res., vol. 9, pp , [8] A. S. Eadie, J. R. Pugh, and B. Winn, The use of coherence functions in the study of ocular mechanisms, Ophthalmic Physiological Opt., vol. 15, no. 4, pp , [9] R. A. Applegate, L. N. Thibos, and G. Hilmantel, Optics of aberroscopy and super vision, J. Cataract Refractive Surg., vol. 27, no. 7, pp , [10] J. Liang, B. Grimm, S. Goelz, and J. F. Bille, Objective measurement of wave aberrations of the human eye with the use of a Hartmann-Shack wave-front sensor, J. Opt. Soc. Amer. A, vol. 11, no. 7, pp , [11] R. J. Noll, Zernike polynomials and atmospheric turbulence, J. Opt. Soc. Amer. A, vol. 66, no. 3, pp , [12] D. R. Iskander, M. J. Collins, and B. Davis, Optimal modeling of corneal surfaces with Zernike polynomials, IEEE Trans. Biomed. Eng., vol. 48, pp , Jan [13] D. R. Iskander, M. R. Morelande, M. J. Collins, and B. Davis, Modeling of corneal surfaces with radial polynomials, IEEE Trans. Biomed. Eng., vol. 49, pp , Apr [14] H. Hofer, B. Artal, B. Singer, J. L. Aragón, and D. R. Williams, Dynamics of the eye s wave aberration, J. Opt. Soc. Amer. A, vol. 18, no. 3, pp , [15] D. R. Neal, D. M. Topa, and J. Copland, Temporal analysis of ocular aberrations using a Shack-Hartmann aberrometer, Investigat. Ophthalmol. Vis. Sci., vol. 43, E-Abstract 2054, [16] H. B. Dick and S. Kaiser, Dynamische Aberrometrie während der Akkommodation phaker Augen sowie Augen mit potenziell akkommodativer Intraokularlinse, Der Ophthalmologe, vol. 99, no. 11, pp , [17] M. Akay, Ed., Time Frequency and Wavelets in Biomedical Signal Processing. Piscataway, NJ: IEEE Press, [18] D. R. Iskander, M. R. Morelande, and M. J. Collins, Estimating the dynamics of aberration components in the human eye, in Proc. 11th IEEE Workshop Statistical Signal Processing (SSP 2001), Singapore, Aug. 2001, pp [19] M. R. Morelande and D. R. Iskander, Formulation and comparison of two detectors of independent timing jitter in a complex harmonic, IEEE Trans. Signal Processing, vol. 51, pp , Dec [20] R. A. Applegate, C. S. Ballentine, and A. Roorda, Is a bite-bar needed for Hartmann-Shack (H/S) wavefront sensing?, Investigat. Ophthalmol. Vis. Sci., vol. 42, no. 4, p. S604, [21] V. N. Mahajan, Optical Imaging and Aberrations: Ray Geometrical Optics. Reading, MA: SPIE (Int. Soc. Opt. Eng.), [22] W. H. Southwell, Wave-front estimation from wave-front slope measurements, J. Opt. Soc. Amer., vol. 70, no. 8, pp , [23] S. Groening, B. Sick, K. Donner, J. Pfund, N. Lindlein, and J. Schwider, Wave-front reconstruction with a Shack-Hartmann sensor with an iterative spline fitting method, Appl. Opt., vol. 39, no. 4, pp , [24] A. Angelone and N. A. Coulter, Respiratory sinus arrhythmia: A frequency dependent phenomenon, J. Appl. Physiol., vol. 19, pp , [25] R. Tutt, A. Bradley, C. Begley, and L. N. Thibos, Optical and visual impact of tear break-up in human eyes, Investigat. Ophthalmol. Vis. Sci., vol. 41, no. 12, pp , [26] M. Al-Abdulmunem and S. T. Briggs, Spontaneous blink rate of a normal population sample, Int. Contact Lens Clin., vol. 26, pp , [27] C. W. Adams and C. A. Johnson, Steady-state and dynamic response properties of the Mandelbaum effect, Vis. Res., vol. 31, pp , [28] A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing. Englewood Cliffs, NJ: Prentice-Hall, [29] S. M. Kay, Modern Spectral Estimation: Theory and Application. Englewood Cliffs, N.J.: Prentice-Hall, [30] H. Akaike, A new look at the statistical model identification, IEEE Trans. Automat. Contr., vol. AC-19, pp , [31] J. Rissanen, Modeling by shortest data description, Automatics, vol. 14, pp , [32] J. Shao, Bootstrap model selection, J. Amer. Statist. Soc., vol. 91, pp [33] A. M. Zoubir and D. R. Iskander, Bootstrap Techniques for Signal Processing. Cambridge, U.K.: Cambridge University Press, [34] L. Cohen, Time-Frequency Analysis. Englewood Cliffs, NJ: Prentice- Hall, [35] B. Boashash, Ed., Time-Frequency Signal Analysis and Processing. A Comprehensive Reference: Elsevier, [36] Y. Zhao, L. E. Atlas, and R. J. Marks, The use of cone-shaped kernels for generalized time-frequency representations of nonstationary signals, IEEE Trans. Acoust., Speech, Signal Processing, vol. 38, pp , July [37] K. Yana, F. Okuyama, H. Yoshida, S. Fukushima, and T. Tokoro, An evidence of blood pressure and respiration originated variations in fluctuations of pupil diameter of the human eye, in Proc. Int. Conf. IEEE Engineering in Medicine and Biology Soc., vol. 13, 1991, pp [38] G. Calcagnini, S. Lino, F. Censi, and S. Cerutti, Cardiovascular autonomic rhythms in spontaneous pupil fluctuations, Comput. Cardiol., vol. 24, pp , [39] L. N. Thibos and X. Hong, Clinical applications of the Shack-Hartmann aberrometer, Optometry Vis. Sci., vol. 76, no. 12, pp , D. Robert Iskander (M 98 SM 04) received the Magister Inżynier degree in electronic engineering from the Technical University of Lodz, Lodz, Poland, in 1991, and the Ph.D. degree in signal processing from Queensland University of Technology (QUT), Brisbane, Australia, in From 1996 to 2000, he was a Research Fellow at the Signal Processing Research Centre, the Cooperative Research Centre for Satellite Systems, and the Centre for Eye Research, QUT. In 2001, he joined the School of Engineering, Griffith University, Gold Coast, Australia, as a Senior Lecturer. In July 2003, he returned to the Centre for Health Research (Optometry) as a Principal Research Fellow. Since April 2004, he is with the Institute of Health and Biomedical Innovation, Brisbane, Australia, as a Senior Researcher. He is also an Honorary Fellow at Griffith University. Dr Iskander s current research interests include statistical signal processing, visual optics and optometry. He is a member of the Association for Research in Vision and Ophthalmology. Michael J. Collins received the Dip.App.Sc. (Optom), M.App.Sc., and Ph.D. degrees from Queensland University of Technology, Brisbane, Australia, in 1977, 1988, and 1996, respectively. He is an Associate Professor in the School of Optometry at the Queensland University of Technology. His research laboratory, the Contact Lens and Visual Optics Laboratory, specializes in the visual and optical characteristics of the cornea and contact lenses. Dr. Collins is a member of the Optometrists Association of Australia, Fellow of the American Academy of Optometry and a Fellow of the Contact Lens Society of Australia.

12 1980 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 51, NO. 11, NOVEMBER 2004 Mark R. Morelande received the B.Eng. degree in aerospace avionics from Queensland University of Technology (QUT), Brisbane, Australia, in 1997 and the Ph.D. degree in electrical engineering from Curtin University of Technology, Perth, Australia, in From November 2000 to January 2002, he was a Postdoctoral Fellow with the Centre for Eye Research, QUT. Since January 2002, he has been a Research Fellow with the Cooperative Research Centre for Sensor, Signal, and Information Processing, University of Melbourne, Parkville, Australia. His research interests include nonstationary signal analysis and target tracking with particular emphasis on multiple target tracking and application of sequential Monte Carlo methods to tracking problems. Mingxia Zhu received the B.Sc. degree in physics from Shanghai Teacher s University, Shanghai, China, in 1996 and the M.It. degree from Griffith University, Gold Coast, Australia, in Currently, she is studying in a Ph.D. degree program at the School of Optometry, Queensland University of Technology, Brisbane, Australia. She is a member of the Association for Research in Vision and Ophthalmology. Her research interests include microfluctuations of the wavefront aberrations of the eye.