Effects of ocular aberrations on contrast detection in noise

Abstract

Abstract

Abstract:

Abstract  We use adaptive optics (AO) techniques to manipulate the ocular aberrations and elucidate the effects of these ocular aberrations on contrast detection in a noisy background. The detectability of sine wave gratings at frequencies of 4, 8, and 16 circles per degree (cpd) was measured in a standard two-interval force-choice staircase procedure against backgrounds of various levels of white noise. The observer's ocular aberrations were either corrected with AO or left uncorrected. In low levels of external noise, contrast detection thresholds are always lowered by AO correction, whereas in high levels of external noise, they are generally elevated by AO correction. Higher levels of external noise are required to make this threshold elevation observable when signal spatial frequencies increase from 4 to 16 cpd. The linear-amplifier-model fit shows that mostly sampling efficiency and equivalent noise both decrease with AO correction. Our findings indicate that ocular aberrations could be beneficial for contrast detection in high-level noises. The implications of these findings are discussed.

Introduction

Human visual contrast detection is limited by both optical and neural factors. The optical quality of retinal images is degraded by diffraction, monochromatic and chromatic aberrations, and scatter (Charman, 2010). Degradation by diffraction and monochromatic aberrations is fully characterized by the eye's optical transfer function ([OTF]; Goodman, 1996). Adaptive optics (AO) techniques (Fernández, Iglesias, & Artal, 2001; Hofer et al., 2001b; Liang, Williams, & Miller, 1997) can correct the monochromatic aberrations in retinal images, allowing near diffraction-limited OTFs to be achieved with monochromatic stimuli. AO techniques have been extensively applied in recent vision research; Roorda (2011) provides a current review. 

We used AO methods to study the role of optical aberrations in the detection of spatial contrast masked by spatial noise. Spatial contrast perception has been widely studied both experimentally and theoretically using a “threshold versus noise” (TvN) psychophysical task in which observers are required to discriminate between the presence or absence of a spatial sine wave added to a background of experimenter-controlled noise (Burgess, Wagner, Jennings, & Barlow, 1981; Legge, Kersten, & Burgess, 1987; Lu & Dosher, 1999, 2008; Pelli, 1981, 1990). Sine wave contrast thresholds for a given frequency are determined for backgrounds at several noise power levels. The resulting relationship between noise power and threshold signal contrast allows inferences to be drawn about the factors controlling performance. This is commonly referred to as the equivalent input noise method (Burgess et al., 1981; Nachmias, 1964; Pelli, 1981). 

Models for TvN experiments have generally assumed an observer who discriminates between the presence or absence of a sine wave grating signal on a noisy background in a statistically ideal manner, which is subject to certain constraints that can be construed as internal noises (Burgess et al., 1981; Legge et al., 1987; Pelli, 1981, 1990). The observer's performance is assumed to be limited by various sources of noise beyond the external noise that is added to the stimulus by the experimenter to reproduce the experimentally-determined limits of human performance. Such “noisy observer” models have been analyzed at a general level by Pelli and collaborators (Pelli, 1981, 1990; Pelli & Farell, 1999) in a form we will call the Linear Amplifier Model (LAM). In the present study, the analysis centers on the relationship between the external noise level quantified by a parameter N, which corresponds to noise spectral density, and the threshold signal contrast energy E, which is the total squared contrast of the weakest detectable sine wave. These quantities are assumed to be related by a function of the general form 

 

where k is related to the observer's efficiency for using the stimulus information, and N_eq is the power of a hypothetical “equivalent noise” that adds to the external noise to limit performance. Parameter N_eq can be construed as representing the combined effects of photon noise (random fluctuations in quantum absorption) and neural noise (random noises in visual neural system from photoreceptor to cortex). 

When the signal is known exactly and the output correlation is Gaussian distributed, an ideal observer for whom the equivalent noise is zero will have a signal to noise ratio (SNR) d′ = √E/N (Burgess et al., 1981; Pelli, 1990). For a two-alternative force-choice (2AFC) detection task, d′ is related to the proportion correct P_c, through the cumulative normal distribution function Φ, P_c = Φ(d′/√2) (Macmillan & Creelman, 1991). On the other hand, d′ represents the smallest SNR with the required proportion correct level. 

The sampling efficiency, defined as J = d′²/k, reveals an inefficiency in the human observer relative to the ideal observer (Burgess et al., 1981). Sampling efficiency is essentially consistent with the concepts of “calculation efficiency” (Pelli, 1981) and “linear template” (Abbey & Eckstein, 2009). Notably, for sine wave detection task, the template is generally a spatial filter with a center frequency (usually adjacent to the signal frequency) and a bandwidth. Within the tuning of the template, signal and noise will pass through and obtain gain. On the contrary, stimuli outside the tuning of the template will not affect performance. 

LAM, together with the equivalent input noise method, has been demonstrated to be a valuable tool for understanding the factors that affect the particular aspects of visual function (Allard & Faubert, 2006; Bennett, Sekuler, & Ozin, 1999; Kersten, Hess, & Plant, 1988; Legge et al., 1987; Lu & Dosher, 2008; McAnany & Alexander, 2010; Pardhan, Gilchrist, & Beh, 1993; Pelli, 1981; Pelli & Farell, 1999; Pelli, Levi, & Chung, 2004; Radhakrishnan & Pardhan, 2006). According to the relationship between signal contrast energy and noise spectral density, contrast sensitivity could be decomposed into sampling efficiency and equivalent noise, which have remarkable invariance for particular experimental conditions (Pelli & Farell, 1999). This methodology has been widely used for investigating the impacts of optical dysfunctions on contrast sensitivity. For instance, studies have shown that cataracts (Kersten et al., 1988; Pardhan et al., 1993), positive and negative defocus (Radhakrishnan & Pardhan, 2006), and senescent optical changes (Bennett et al., 1999; Pardhan, 2004) alter the equivalent noise but do not change the sampling efficiency. Moreover, Radhakrishnan and Pardhan (2006) found a significant relationship between the total root mean square (RMS) of ocular aberrations and the equivalent noise. Sampling efficiency was found to be susceptible to factors such as amblyopia and optic neuritis (Kersten et al., 1988; Pardhan et al., 1993). These findings imply that different dysfunctions may affect the contrast sensitivity in fundamentally different ways. 

Although various optical factors have been studied as stated above, there is still no direct evidence that shows how sampling efficiency and equivalent noise are affected by the ocular aberrations. Therefore, the purpose of this work is to use AO techniques to manipulate the ocular aberrations and elucidate the role of these ocular aberrations in contrast detection in noise based on the differences of sampling efficiency, equivalent noise, or both. Furthermore, all conventional studies on the contrast sensitivity benefits of AO correction were performed in a uniform background. Here, our study will provide some insight on how AO correction benefits contrast detection in a noisy background. 

Methods

Equipment

Figure 1 shows the schematic diagram of the adaptive optics visual analysis system ([AOVAS]; Li et al., 2009; Xue et al., 2007) used in this study. The AOVAS consisted of a Hartmann-Shack (HS) wavefront sensor with a microlens array and a charge-coupled device (CCD) camera, a 37-actuator piezo deformable mirror, a control system integrated to a personal computer and a visual target device. An infrared beam from a 905-nm laser diode (LD; Boston Laser, Inc., Binghamton, NY) was collimated into the observer's retina. The irradiance of the LD on the cornea was approximately 5 μW, which was 30 times smaller than the maximum permissible exposure safe level, according to the safety standard of American National Standards Institute (ANSI Z136.1, 1993). The beam was reflected from the retina, passed the ocular optics and optical instrument, and then focused on a CCD camera by the microlens array. Thus, the HS wavefront sensor placed conjugate to the observer's pupil and the deformable mirror, measured the ocular wavefront aberrations at 25 Hz. The deformable mirror was used for correcting the ocular aberrations. A beam splitter in front of the HS wavefront sensor allowed for the insertion of a visual target device channel, which consisted of a video converter and an OLED minidisplay (eMagin Corporation, Bellevue, WA; 800 × 600 pixels, 12 mm × 9 mm, 60 Hz refresh rate, green background luminance). An interference filter (550 nm ± 5 nm) was placed in front of the minidisplay. The minidisplay and interference filter were fixed on a stepping motor for adjusting the chromatic focus shift between the HS infrared light source and the green stimulus. The defocus term on the HS wavefront sensor was set to zero and then the adjustment of the stimulus position for a perfect collimation was performed by two experienced experimenters with all ocular aberrations corrected. Three repeated measurements for each experimenter were done and the position of the stepping motor was set to the average of these measurements. The video converter (Li, Lu, Xu, Jin, & Zhou, 2003) was a special circuit that combined two 8-bit output channels of the video card to produce 14 bits of gray levels. Two personal computers were used: one controlled the adaptive optics system in a closed-loop fashion; the other ran Matlab 6.5 and Psychtool (Mathworks, Natick, MA) (Brainard, 1997; Pelli, 1997) extensions to generate visual stimulus. 

Figure 1

 

Schematic diagram of the adaptive optics visual analysis system. LD, Laser Diode; TL, Trial Lens; DM, Deformable Mirror; HS, Hartmann-Shack wavefront sensor; OLED, Minidisplay; VC, Video Converter; PC, Personal Computer; f1 to f6, Lens; M1 to M3, Reflecting Mirror; and BS, Beam Splitter.

The retina illuminance of the stimulus display was set to 84 trolands (luminance of 6.65 cd/m² in the pupil plane), which was measured with an IL17000 Radiometer (International Light Technologies Inc., Peabody, MA). The observers viewed a test field subtending a visual angle of about 1.5° though a 4-mm artificial pupil. A 0.15° half-Gaussian ramp was added to each stimulus to minimize edge effects. Thus the stimulus subtended an angle of about 1.2°, which is commonly accepted to be within the eye's isoplanatic patch (Dubinina, Cherezovaa, Belyakovb, & Kudryashov, 2008). All experiments were performed in a dim light room. The head of the observer was stabilized by a chin rest. The viewing was monocular. 

Observers

Two observers participated in the experiments (JCZ, aged 20, and JZD, aged 21). The observers had normal vision (JZD) or vision that was optically corrected to normal by a trial lens (JCZ). All observers were well trained for the task and naïve to the purpose of the experiments. 

Stimuli

The signals in the task were sine wave gratings with luminance L(x, y) defined by 

 

where the mean luminance L₀ is 6.65 cd/m²; c is the Michelson contrast level set by the experimenter according to pilot studies; f₀ is the spatial frequency (in cpd), with values of 4, 8, and 16 cpd for our experiments; θ was set to zero, such that the vertical sine wave gratings were used; and φ was a random value from 0 to 2π. 

Thus the signal contrast function c(x, y) can be calculated by 

 

Each stimulus was rendered on a circular area with a diameter of approximately 344 pixels. For the different spatial frequencies, namely, 4, 8, and 16 cpd, the check dimensions were 8 × 8 pixels, 6 × 6 pixels, and 4 × 4 pixels, respectively, which were sufficient to maintain the whiteness of the noise (Rovamo & Kukkonen, 1996). The gray level of each check was sampled from a Gaussian distribution with a mean of 0 and standard deviations of 0, 0.02, 0.04, 0.08, 0.16, 0.28, 0.52, and 0.76. The method of generating the noise was based on the demo of Psychtoolbox (Brainard, 1997; Pelli, 1997). That is, singular gray levels exceeding the range that the video card could effectively display were found and replaced by new values from the Gaussian distribution. This process was repeated until all the singular values were replaced. 

The signal and noise were presented in an asynchronous mode (Figure 2). The signal duration was 16.7 ms (one video frame). Dynamic noise was used, and the duration of each noise check was 33.4 ms (two video frames). The noise was combined with the signal through temporal integration. The total stimulus duration was 150 ms, which was short enough to minimize the potential effect of eye movements. 

Figure 2

 

Schematic illustration of the temporal trial sequence in the 2IFC detection task.

The signal energy E is defined as the integral over space and time of the squared signal contrast function (Legge et al., 1987; Watson, Barlow, & Robson, 1983). 

 

where A_s is the pixel area in deg² and t_s is the signal duration in seconds. Thus, the signal energy has a unit of deg²s. 

The external noises consist of four independent samples, each of whose check duration is twice of the signal duration. Suppose that instead of the stimulus having four separate component noises, a single noise had been added to the signal frame. To keep the signal-to-noise ratio the same for an observer that integrates over the entire stimulus duration, the single frame noise should have a variance c₁², with 

 

where r is the ratio of the noise check duration t_n to the signal duration t_s (i.e., r = 2), n_n is the number of independent samples of external noise (i.e., n_n = 4), and c_n is the component noise standard deviation. We will characterize the stimulus noise level by the noise power density N of this equivalent noise, with 

 

where A_n is the noise pixel area. If the durations are in seconds and the area is in deg², the noise spectral density will have the same units as the signal energy (deg²s). 

Procedure

In our experiments, a temporal two-interval forced-choice ([2IFC]; Lu & Dosher, 1998, 1999) detection task was applied. Figure 2 illustrates a typical temporal trial sequence. Tasks were initialized by the observer with a key press. Each trial started with a 250-ms fixation cross in the center of the display. Starting with a brief beep, the presentation sequence in the first trial was as follows: two independent noise frames, one signal or blank frame, another two independent noise frames, and then a blank frame. The lasting time of each refresh was 33.4, 33.4, 16.7, 33.4, 33.4, and 16.7 ms, respectively. After another 250 ms delay, the second same interval began, starting with another brief beep. Between the two intervals, the sine wave grating was randomly presented as the signal, whereas the other was left blank. The observer had to indicate which interval contained the signal by reporting with a key press. Each key press was then immediately followed by another different auditory feedback. 

A three-down one-up staircase procedure (Levitt, 1971) was used to obtain psychometric functions. In this procedure three consecutive correct responses resulted in a reduction of signal contrast of 10%, and one wrong response increased the signal contrast by 10%. Finally, the performance level converged to 79.3% accuracy (d′ = 1.16). 

For each signal spatial frequency, the same task procedure was conducted with and without AO correction. The sequence of tasks with and without AO correction was randomly arranged without informing the observers. Each task was divided into eight blocks, and the observers could take an optional rest after finishing one block. Within each block of 88 trials, the noise levels were randomized. Before the experiments, all observers received several practice trials. The initial signal contrast of each task was set close to the expected threshold based on the results of the practice trials. 

Calculation

The wavefront aberrations reconstructed from measurements with an HS wavefront sensor were described to include 35 Zernike polynomials (up to the seventh order, excluding tip and tilt), according to Optical Society of America wavefront standards (Thibos, Applegate, Schwiegerling, & Webb, 2000). Vertical sine wave gratings were used in the experiments, so the horizontal modulation transfer function (MTF) was used for further calculations. The MTF was calculated from wavefront aberrations for a 4-mm pupil at a wavelength of 550 nm, based on the standard Fourier optics theories (Goodman, 1996). 

Results

Threshold signal energy versus noise spectral density

The results are summarized in Figure 3, which plots the threshold signal energy as a function of noise spectral density for two observers at three signal spatial frequencies. The leftmost data in each panel represent the threshold signal energy in the absence of external noise. Red and black solid circles correspond to the data with (labeled AO) and without (labeled NO AO) AO correction, respectively. Lines passing through the data are derived from the least square fit of the LAM model (best fitting results are listed in Table 1). 

Figure 3

 

Threshold signal energy as a function of noise spectral density for two observers at three signal spatial frequencies with and without AO correction. Red and black solid circles indicate the results with (labeled AO) and without (labeled NO AO) AO correction, respectively. Three columns indicate three signal spatial frequencies (i.e., 4, 8, and 16 cpd) tested in the experiments. Lines passing through the data are derived from the least square fit of the LAM model. The error bar indicates the ±1 standard deviation. Double logarithmic coordinates are used.

Table 1

 

Best-fitting values of model parameters. Notes: Values = Mean ± 95% CI; μdeg²s = 10⁻⁶deg²s.

SF(cpd)	J (%)		Neq (μdeg2s)
	AO	NO AO	AO	NO AO
JCZ
4	26.58 ± 4.30	40.56 ± 5.37	1.53 ± 0.44	2.98 ± 0.64
8	18.85 ± 2.29	26.41 ± 2.84	2.62 ± 0.49	8.91 ± 1.31
16	13.78 ± 2.92	23.50 ± 2.75	9.26 ± 2.49	23.41 ± 3.22
JZD
4	22.78 ± 2.65	42.19 ± 4.33	0.68 ± 0.16	1.82 ± 0.32
8	22.69 ± 2.93	29.45 ± 4.44	0.76 ± 0.18	3.98 ± 0.88
16	21.44 ± 2.56	14.07 ± 1.35	12.83 ± 1.89	11.75 ± 1.40

The performance of both observers shows a similar pattern across the signal spatial frequencies. For natural aberrations (NO AO), the thresholds are systematically elevated with increasing signal spatial frequencies, similar to the results of Solomon (2000). On the other hand, the “knee” of the TvN curve, which reflects the equivalent noise in the log-log axes, increases monotonically with the signal spatial frequencies. 

For most conditions, the AO TvN curve intersects with the NO AO TvN curve. The threshold signal energy is reduced by AO correction in low noise levels but is elevated when the noise level becomes high. An exception occurs at 16 cpd for JZD, which shows a systematic threshold reduction by AO correction. Moreover, the intersection tends to increase with increasing signal spatial frequencies. 

Results of optical aberrations

Figure 4 shows the wavefront aberration map (natural aberrations and the residual aberrations due to the imperfection of AO correction) for both observers in this work. The wavefront aberration data correspond to the average of 24 repeated measurements throughout the experiment. The wavefront aberration maps of natural aberrations are different across observers (Figure 4A and B) but become similar after AO correction (Figure 4C and D). 

Figure 4

 

Wavefront aberration map of the natural aberrations (NO AO, upper row) and the residual aberrations due to the imperfection of AO correction (AO, bottom row) for observer JCZ (left column) and observer JZD (right column). The color bar indicates the wavefront aberration scale in microns. Contour lines are separated by 0.125 microns.

Results of MTFs and noiseless threshold signal energy ratios

The left column of Figure 5 shows the MTFs calculated from the wavefront aberrations with and without AO correction for the two observers. The diffraction-limited MTF for a 4-mm pupil was also calculated for comparison. For both observers, MTFs with AO correction are close to the diffraction limit. 

Figure 5

 

MTFs calculated from wavefront aberrations measured with (red line) and without (black line) AO correction for observers (A) JCZ and (C) JZD. Blue dashed line indicates the diffraction-limited MTF for a 4-mm pupil. The comparison of the noiseless signal threshold energy ratio (purple diamond) corresponding to the threshold energy reduction with AO correction in the absence of noise and the squared MTF ratio (purple dashed line), defined as the squared MTF without AO correction to that with AO correction, for observers (B) JCZ and (D) JZD. The error bar indicates the ±1 standard deviation.

The noiseless threshold signal energy ratio, as shown in Figure 5B and D, corresponds to the reciprocal squared contrast sensitivity (CS) benefit of AO correction (Elliott et al., 2009; Gracia, Marcos, Mathur, & Atchison, 2011; Yoon & Williams, 2002). The average noiseless CS benefit across the signal spatial frequencies was 1.39 with a standard deviation of 0.33. The mean data of the two observers was used for the calculation. This result is consistent with those reported by investigators (Elliott et al., 2009; Gracia et al., 2011; Yoon & Williams, 2002). The MTF benefit, defined as the MTF improvement by AO correction, generally represents the upper limit of the CS benefit. Purple dashed lines in Figure 5B and D represent the reciprocal squared MTF benefits. 

Estimates of equivalent noise and sampling efficiency

In the LAM model, it is assumed that the experiment was performed in a signal-known-exactly (SKE) task without stimulus uncertainty. However, a random phase was used in our experiments. Jeffress (1964) suggested that the detectability d′ is approximately 0.6 greater for a complete phase uncertainty task than that for an SKE task. Following Bennett et al. (1999), the ideal detect threshold for our experiment was assumed to be E = (d′ + 1/√2)²N. Then we rewrote Equation 1 as 

 

The AO and NO AO data were fitted via the same procedure. Customized software in MATLAB (lsqcurvefit, which is based on a nonlinear least square curve fitting method, and nlparci, which is for calculating the standard errors of the parameters) was used for estimating the sampling efficiency J and the equivalent noise N_eq (mean and standard derivation with the 95% confidence interval, M ± 95% CI) by minimizing the sum of the squared difference between the log threshold prediction from the model E_τ^theory (Equation 7) and the observer E_τ. 

 

The estimated parameters are listed in Table 1. The estimates of the equivalent noise as a function of signal spatial frequencies for observers JCZ and JZD are shown in Figure 6A and B, respectively. Figure 6C and D show the estimates of sampling efficiency as a function of signal spatial frequencies for the two observers, respectively. 

Figure 6

 

Equivalent noise (upper row) and sampling efficiency (bottom row) against signal spatial frequency for observer JCZ (left column) and JZD (right column). Red and black symbols in each panel indicate the estimates for AO correction and NO AO correction, respectively. The error bar indicates the ±1 standard deviation.

For natural aberrations (NO AO), the two observers showed a similar pattern. That is, N_eq rises monotonically with the signal spatial frequency. The average of N_eq across all the signal spatial frequencies was 11.77 μdeg²s for JCZ and 5.85 μdeg²s for JZD, with a standard deviation of 1.18 and 0.56 μdeg²s, respectively. On the other hand, the sampling efficiency tends to decrease with increasing signal spatial frequencies for both observers. This feature corresponds to the rise in the slope with increasing signal spatial frequencies, as shown in Figure 4. In addition, J is almost identical for JZD at 8 and 16 cpd. The average of J across all the signal spatial frequencies was 30.16% for JCZ and 28.58% for JZD with a standard deviation of 2.22% and 2.12%, respectively. Thus, we may conclude that for natural aberrations condition, the elevation of the threshold with increasing signal spatial frequencies from 4 to 16 cpd is due to both the increase of N_eq and the drop of J. 

With AO correction, N_eq (4.47 μdeg²s for JCZ and 4.75 μdeg²s for JZD, averaged across signal spatial frequencies) decreased by a factor of 0.65 and 0.78 for JCZ and JZD, respectively. On the other hand, the sampling efficiency (19.74% for JCZ and 22.30% for JZD, averaged across signal spatial frequencies) decreased by a factor of 0.38 and 0.81 for JCZ and JZD, respectively. For JZD, the N_eq at 16 cpd was almost identical for conditions with and without AO correction. Furthermore, the data points of J at 16 cpd for JZD show an opposite feature compared with 4 and 8 cpd. Thus, the AO and NO AO TvN curves have no intersecting point in Figure 4F. These results imply that in some particular circumstances, the AO correction of ocular aberrations could reduce not only the equivalent noise, but also the sampling efficiency. The implications are discussed below. 

Discussion

We measured the threshold of sine wave detection present in various levels of white noises for two observers. Signals at three spatial frequencies, namely, 4, 8, and 16 cpd, were tested, and the ocular aberrations were either corrected by AO or left uncorrected. In most conditions, we found that the detection thresholds are lowered by AO correction in low noise levels, whereas in high noise levels, they are generally elevated by AO correction. We fitted the data to a LAM-type model and found that most of the equivalent noise and sampling efficiency decrease when ocular aberrations are corrected by AO. 

Most studies concerning the effect of imperfect optics, such as cataracts (Kersten et al., 1988; Pardhan et al., 1993), defocus (Radhakrishnan & Pardhan, 2006), and senescent optical changes (Bennett et al., 1999; Pardhan, 2004) on the contrast detection in noise have reported that optical dysfunctions change equivalent noise. Only one published paper (Radhakrishnan & Pardhan, 2006) has studied the role of ocular aberrations on equivalent noise and found that the equivalent noise is significantly correlated to the total wavefront RMS of the ocular aberrations (p = 0.026). Our results showed that N_eq decreases by a factor of 0.72 (mean data of both observers) when correcting the ocular aberrations. This finding supports the hypothesis that the equivalent noise may be a predictor of ocular aberrations (Radhakrishnan & Pardhan, 2006). 

The level of sampling efficiency was found to vary with different experimental conditions (Kersten, 1984; Legge et al., 1987; McAnany & Alexander, 2010; Pardhan, 2004; Pelli & Farell, 1999). Our results were comparable with the previously reported values that approximately ranged from 1% to 30%. Regardless of the cause of the differences in the absolute values, it must be mentioned that in most conditions, correcting the ocular aberrations produces a reduced sampling efficiency. This finding conflicts with the previous reports (Bennett et al., 1999; Kersten et al., 1988; Pardhan et al., 1993) that sampling efficiency is mainly related to the neural changes, but is not altered by optical factors. To our knowledge, there is no direct evidence proving the impacts of ocular aberrations on sampling efficiency. Our results may provide some new insights on this issue. 

Effects of photon noise

It is likely that the ocular aberrations mediate the performance through three aspects: the signal contrast energy, the external noise spectral density, and the photon noise. Photon noise could be quantified using the spectral density N_photon corresponding to the reciprocal of the photon flux (Pelli, 1981, 1990). One troland corresponds to 1.25 × 10⁶ photons per deg²s at 555 nm (Legge et al., 1987). Therefore, N_photon was 9.52 × 10⁻³ μdeg²s in our work (at a retinal illuminance level of 84 troland; neglecting the difference between 550 nm and 555 nm). Our results showed that minimal N_eq was 0.68 μdeg²s (Table 1, Observer JZD, 4 cpd). This value is approximately two orders of magnitude greater than N_photon. For a certain retinal illuminance, the actual amount of N_photon depends on the absorption of retinal receptors. However, the fraction of absorbed photon noise is not precisely known (Pelli, 1990). Pelli (1990) suggested 1% to 10% absorption, corresponding to 10 N_photon to 100 N_photon at the retina. This may make N_photon comparable with N_eq. Therefore, when the signals were embedded in low-level noises (including the noiseless condition), the improvement of the visual performance by AO correction may be attributed to the enhancement of signal contrast and the reduction of photon noise level. When signals were embedded in high-level noises, the contribution of equivalent noise was negligible. In this case, photon noise will not, or will barely, be a dominating factor that affects the performance. This view is also supported by previous studies (Rovamo, Kukkonen, Tiippana, & Nasanen, 1993) that showed how the detection threshold in high-level noises does not change significantly in various retinal illuminance levels. The photon noise elevation induced by ocular aberrations probably increases the equivalent noise, but does not affect the sampling efficiency. 

Equivalent noise and sampling efficiency

The question is then how the observer's responses to the stimuli (i.e., signal and noise) changes due to AO correction of the ocular aberrations. Ocular aberrations degrade the signal energy and the noise spectral density by a more heavily attenuated MTF than diffraction-limited optics. According to Equation 7, the degradation of noise spectral density will give rise to an overestimation of the equivalent noise. This may be the primary explanation to account for the equivalent noise elevation in observers with imperfect optics. Based on this inference, Pelli and collaborators (Pelli, 1990; Pelli et al., 2004) used the eye's optical MTF to adjust the equivalent noise. 

By contrast, changes in the sampling efficiency require the unequal change of signal energy and external noise spectral density (i.e., signal to noise ratio). Here the analysis will center on the case of the high-level external noises, where the equivalent noise could be neglected. It is important to note that the observer's template selectively tuned to a particular frequency band. For convenience, we defined two factors, α and β, representing the total magnification to signal and noise, respectively. The output of signal and noise through tuning of template are therefore αE and βN. 

 

and 

 

where MTF(f) is the optical modulation transfer function normalized by that of diffraction limited optics, T(f) is the template function, f is the horizontal spatial frequency, and f₀ is the signal spatial frequency. If α > β, the sampling efficiency increases. If α < β, the sampling efficiency decreases. If α = β, the sampling efficiency is kept constant. The MTF could be calculated from the ocular aberrations. Therefore, the template function can be essential to visual performance. 

Without estimating the template profile in the present study, we do not yet fully understand the origin of the drop in sampling efficiency caused by correcting the ocular aberrations. Important evidence (Ahumada, 2002; Ahumada & Beard, 1999) of classification images suggested that the template is narrowband for Gabor signal detection, such as matching the signal, which implies that α may be equal to β. However, this will not explain the drop in sampling efficiency. 

Unlike the results of Ahumada and Beard (1999), some studies (Henning, Hertz, & Hinton, 1981; Losada & Mullen, 1995; Lu & Dosher, 2001; Stromeyer & Julesz, 1972) using the filtered noise masking method reported that the template bandwidth is approximately 1 to 2 octaves. The bandwidth is expressed with octaves (log2-units), so even if the template is symmetric (Stromeyer & Julesz, 1972; Henning et al., 1981), the side with the higher frequency will contain more spatial frequencies in linear units. Furthermore, there is also previous evidence (Losada & Mullen, 1995; Lu & Dosher, 2001) that showed an asymmetric template where the side with lower frequency was declining more steeply than the side with higher frequency. This result implies that the template may have more weight in the higher side. Losada and Mullen (1995) estimated the template for signal at 0.5 and 2 cpd in a detection task and found that the template bandwidths are proportional to their center frequency (constant in octaves). If this finding holds true for higher frequencies, it is likely that the bandwidth fraction (linear units) in the higher side may be larger in our experiment. On the other hand, Figure 5 shows that MTF(f) tends to monotonously decline, at least up to 20 cpd. Thus, all the above evidence supports the hypothesis that the template may be more heavily weighted in terms of bandwidth for the higher frequency side, where there is worse optical quality. Consequently, the ocular aberrations will produce more attenuation to noise spectral density than to the signal energy, which will help predict the sampling efficiency decline caused by AO correction of the ocular aberrations. 

The template, however, may vary with the structure of external noise (Abbey & Eckstein, 2007, 2009; Burgess, Li, & Abbey, 1997; Solomon, 2000). Solomon (2000) suggested that the observer may select the higher frequency channel when the sine wave signal is embedded in high levels of low-pass noise, which will produce a greater signal-to-noise ratio. Using the classification image method, Abbey and Eckstein (2007) found that the observers give more weight to higher frequencies for Gaussian bump detection in low-pass noise, because the low-pass noises may contain less information. These studies provide evidence that for some particular noise structure, the observer may have a template with the gain of the side with higher frequency heavily weighted, although this may not support the linear template assumption. A more convincing interpretation for our results will require directly estimating the template with methods like classification image or filtered noise masking. 

The performance varies across signal frequencies. For natural aberrations, the features of TvN curves of different signal frequencies are similar with those reported by Solomon (2000). The LAM fit shows that the estimates of equivalent noise increase with increasing signal frequency. At the same time, the intersection of the TvN curves for AO and NO AO tends to laterally shift toward the high noise levels with increasing signal frequency. From these results, we conclude that the threshold elevation caused by AO correction at high-level noises might occur in the following two necessary conditions: (1) the magnitude of external noise is significantly higher than the equivalent noise; and (2) the ocular aberrations produce more attenuation to noise magnitude than to the signal energy within the tuning of template. 

Contrast gain control

Although the explanation within the framework of linear template seems plausible, the main limitation is that masking by high-level external noise cannot be regarded as just the result of the noise passing through the template. Physiological and psychophysical evidence (Baker, Meese, Georgeson, & Hess, 2011; Ding & Sperling, 2006; Foley, 1994; Geisler & Albrecht, 1992; Heeger, 1992; Legge & Foley, 1980; Meese & Holmes, 2002; Ohzawa, Sclar, & Freeman, 1982; Tolhurst & Heeger, 1997; Tolhurst & Dean, 1987; Watson & Solomon, 1997) both suggested that the observer may involve additional processes such as suppression from contrast gain control. Therefore, an extra explanation of our results may be that it is the result of ocular aberrations providing relief from the cross-channel suppression with noise masking. 

For pattern-masking situations (Foley, 1994; Legge & Foley, 1980; Watson & Solomon, 1997), a model incorporating a divisive gain control can produce both the facilitation (the dipper) in low-contrast masking and the Weber's law behavior in strong masking of high contrast. Differently, the threshold elevation due to the intensive white noise masking is likely to be interpreted as the result of the increased variance to the decision variable (so-called noise masking effects; Legge et al., 1987; Pelli, 1981; Watson & Solomon, 1997). Besides, almost all but two studies (Blackwell, 1998; Goris, Zaenen, & Wagemans, 2008) reported monotonically increasing TvN functions. Blackwell (1998) and Goris et al. (2008) both found the facilitation (the dipper) of contrast detection in weak levels of noise, and Goris et al. (2008) provided a quantitative model involving a within-channel suppression mechanism. The LAM model has been demonstrated to do fairly well in interpreting the TvN functions (Burgess et al., 1981; Legge et al., 1987; Pelli, 1981, 1990; Pelli & Farell, 1999), but the representation of the slope as sampling efficiency is seriously deficient. For example, the LAM model predicts that for high-level external noise, the observer will always give the same response to repeated presentations of the same noise. However, the observer inconsistency was found by Burgess and Colborne (1988), who suggested that an induced internal noise with a magnitude equal to approximately 0.65 to 0.85 times the external noise is necessary to account for the increased slop. Similarly, the stimulus-dependent internal noise (e.g., contrast-gain-control-induced noise or multiplicative noise) was also included in other noisy observer models (Eckstein, Ahumada, & Watson, 1997; Lu & Dosher, 1999) to account for the contrast gain control mechanism. Hence, the threshold elevation by AO correction in high-level external noises may be due in part to the increase of the stimulus-dependent internal noise. If this noise is white and proportional to the external noise, then the random variations dominate the sources of the performance degradation, and thereby the explanation within the framework of linear template remains viable. However, this assumption would be violated if the template varies with the change of external noise magnitude (Abbey & Eckstein, 2007, 2009; Burgess et al., 1997; Eckstein et al., 1997). 

More recently, Baker et al. (2011) argued that contrast gain control should be involved in the interpretation of equivalent noise masking experiments. On the other hand, Watson and Solomon (1997) suggested that “noise masking effects” may also account for the threshold elevation of pattern masking. These results indicate that contrast detection in noise should have a similar mechanism to that with pattern masking because of the same neural substrate for contrast processing (DeValois & DeValois, 1988). Our analysis, however, cannot distinguish whether the ocular aberrations reduce the random variations in white noises (within the tuning of template) or whether they provide relief from the cross-channel suppression. Nevertheless, ocular aberrations lead to an overall degradation of noise masking. At the same time, the signal energy is also degraded by these ocular aberrations. If the degradation to the noise exceeds that to the signal, AO correction of these ocular aberrations will produce the threshold elevation. 

Performance of AO correction

An inspection of Figure 5 shows that the optical calculation (the MTF² ratio) does not perfectly match the subjective performance (the noiseless threshold signal energy ratio). Similar inconsistencies were also observed in previous reports (Elliott et al., 2009; Gracia et al., 2011; Yoon & Williams, 2002). It is well-accepted that at high signal spatial frequencies the CS benefits might be limited by the neural contrast sensitivity and thereby are shown to be lower than the MTF benefits (Elliott et al., 2009; Gracia et al., 2011; Liang et al., 1997; Roorda, 2011; Yoon & Williams, 2002). However, for the low and middle frequencies as used in our experiments, the comparisons of the CS and MTF benefits gave mixed results. Gabor pattern sensitivity for spatial frequencies ranging from 1 to 18 cpd was tested by Elliott et al. (2009), who found that for 6-mm pupil the CS benefits are ranging from 1 to 3 times, which fairly match the MTF benefits, and for 3-mm pupil the CS benefits, ranging from 1 to 2 times, are shown to be slightly higher than the MTF benefits. Gracia et al. (2011) showed that for 5-mm pupil the CS benefits are close to, or slightly lower than, 1 time for spatial frequencies ranging from 1.9 to 7.6 cpd, whereas the MTF benefits are ranging from 1 to 3 times. One possible explanation for these mixed results is that the relatively minor CS benefits at the low and middle frequencies may be overshadowed by variation such as that caused by the micro-fluctuations of ocular aberrations over time (Hofer, Artal, Singer, Aragon, & Williams, 2001a). Besides, prior studies (Dalimier & Dainty, 2008; Elliott et al., 2009; Yoon & Williams, 2002) suggested that the residual aberrations due to the imprecision of AO correction may dilute the CS benefit. On the other hand, although the averaged CS benefits of multiple subjects showed an increasing trend with increasing spatial frequency up to the middle frequency (e.g., 20 cpd), the individual data showed significant variability across the signal spatial frequency (Elliott et al., 2009; Gracia et al., 2011), as we also show in present study. Furthermore, previous evidence (Artal et al., 2004; Chen, Artal, Gutierrez, & Williams, 2007; Rouger, Benard, Gatinel, & Legras, 2010; Sabesan & Yoon, 2009, 2010) revealed that the neural system may adapt to the eye's native aberrations, which could also moderate the benefits of AO correction. Much further investigation is necessary to distinguish the source of these inconsistencies. 

Conclusion

We have used adaptive optics techniques to manipulate the ocular aberrations and elucidate the effects of these ocular aberrations on contrast detection in various levels of white-noise background. We find that ocular aberrations always harm contrast detection in weak levels of noise, whereas they could be beneficial for contrast detection in high levels of noise. For weak levels of noise, it can be analogous to the case of contrast detection in a uniform background, where the signal dominates the performance. For high levels of noise, if ocular aberrations produce more degradation to the noise masking than to the signal, they will benefit the detection. Furthermore, the degradation of noise masking due to ocular aberrations can be from two possible sources: (1) reduction of the random variations (within the template) or (2) relief from the cross-channel suppression (contrast gain control). Our findings expand the knowledge on how ocular aberrations affect the contrast detection in a noisy background, while also improving the understanding of visual processing. 

Acknowledgments

This research was supported by National Science Foundation of China (NSFC): 60808031 (Y.D.) and 30630027 (Y.Z.), Instrument Developing project of the Chinese Academy of Sciences, Grant No. 2010028 (Y.D.). Parts of the results were previously presented in the 8^th International Workshop on Adaptive Optics for Industry and Medicine (2011). The authors would like to thank Albert Ahumada and Jack Yellott for their critical comments and thoughtful suggestions. We also thank an anonymous reviewer for the helpful comments. 

Commercial relationships: none. 

Corresponding author: Bo Liang 

Email: [e-mail]. 

Address: Institute of Optics and Electronics, Chinese Academy of Sciences, Chengdu, Sichuan, China.