w42-6989 91 53.00 + 0.00 Copyright C 1991 Pcrgamon Press pk

Yinon Res. Vol. 31. No. 5. pp. 815-831. 1991 Printed in Great Bntlin. All nghts rescrwd

NONLINEAR DISTORTION OF GRATINGS FOVEAL RESOLUTION LIMIT

AT THE

R. WILLIAMS’and ORIN PACKER’

NOB~TOSHI SEKIGUCHI,‘~ DAVID

‘Center for Visual Science. University of Rochester. Rochester. NY 14627 U.S.A. and rOlyrnpus Optical Co. Ltd. 2-3 Kuboyama. Hachioji, Tokyo 192. Japan (Received 22 January 1990, in revised form I7 July 1990) Abstract-Aliasing by the fovea1cone mosaic causes high frquency interferena fringes to look like bright and dark zebra stripes (primary zebra stripes) Williams. Vision Reseurch, 25, I95 (1985); Vision Reseurch. 28.433 (l988)]. Some observers report another type of zebra stripes defined by variations in chromaticity as well as brightness, which we call secondary zebra stripes. The conditions required to see the secondary zebra stripes are almost identical to those required to see the primary zebra stripes, exapt that they are seen at approximately half the spatial frequency. We consider the hypothesis that the secondary xebra stripes arise from aliasing by a particular packing arrangement of the M and L cone submosaics. but present evidena favoring an alternative hypothesis based on a known local nonlinearity in the visual system. Spatial sampling Aliasing Cone mosaic Spatial vision Color vision

Contrast

INTRODUCTION

Abasing occurs when an inadequate number of elements sample an image. Aliasing has been demonstrated in human fovea1 vision: gratings with spatial frequencies higher than half the spatial sampling rate of the cone array produce moire effects (Bergmann. 1858; Byram, 1944; Campbell & Green, 1965; Ohzu, Enoch & O’Hair, 1972; Williams & Collier, 1983; Williams, 198.5, 1986, 1988). Laser interferometry has allowed the study of fovea1 aliasing because gratings can be imaged on the retina without blurring by the optics of the eye. Williams (1985. 1988) has exploited aliasing to characterize the cone mosaic in the human eye, estimating the spacing and packing geometry of cones in the living fovea. Aliasing has also provided psychophysical estimates of extrafovea1 cone spacing based on the apparent orientation of aliases there (Coletta & Williams, 1987) as well as their apparent direction of motion (Coletta. Williams & Tiana, 1990; Tiana. Williams, Coletta & Haake, 1991). In the fovea, aliased interference fringes resemble zebra stripes. They look like shimmering, wavy lines defined by variations in brightness. The spacing between these zebra stripes appears largest when the fringe frequency roughly equals the fovea1 cone sampling 815

Nonlinearity

Distortion product

frequency, typically about I IO-120 c/deg. We will refer to these zebra stripes as primary zebra stripes. This paper examines another kind of zebra stripe pattern quite distinct from these primary zebra stripes. Some observers report that a faint secon&7ry zebra stripe pattern can be seen at about half the cone sampling frequency, near the Nyquist frequency. Thus, for a given fovea1 location, these secondary zebra stripes become prominent at spatial frequencies that are about half those that produce the most prominent primary zebra stripes. Both the primary zebra stripes and the secondary zebra stripes appear as local, irregular variations in brightness. However, the secondary zebra stripes are sometimes described as being defined by a variation in hue as well. For example, when viewing a 632.8 nm interference fringe, some observers report that the dark regions in the secondary zebra stripes appear desaturated green. The analysis of the secondary zebra stripes was originally motivated by the notion that they might provide a clue to the packing arrangement of subclasses of middle (M) and long (I,) wavelength sensitive cones, about which there is currently no information in the human. We will refer to this possibility as the chromaric a&z.ring hypothesis which proposes that the secondary zebra stripes arise from aliasing by the

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NOB~TOSHI SE~CICXCHIet al

submosiacs of ,‘1fand L cones that comprise the fovea1 mosaic. Williams and Collier (1983) and Williams. Collier and Thompson (1983) demonstrated the existence of aliasing by the short (S) wavelength cone submosiac. In addition to these psychophysical results, recent theoretical work (Williams. 1990; Ahumada. 1986: Brainard, Wandell & Poirson, 1989) also raises the possibility that the packing arrangement of the M and L cones in the fovea also produces submosaic aliasing. However. a second hypothesis, which we call the contrast hypothesis. accounts for the secondary zebra stripes without appealing to the packing arrangement of the M and L cones. relying instead on sampling by the mosaic as a whole followed by a local nonlinearity in the visual system. The first two experiments in this paper define more precisely how both primary and secondary zebra stripes depend on stimulus conditions. Expcrimcnt I shows how both phenomena dcpcnd on the spatial frcqucncy and orientation of an interference fringe viewed at the fovea1 canter. Experiment 2 shows how this depcndcncc changes with retinal eccentricity within the fovcal region. Thcsc experiments establish an intimate link between the primary zebra stripes, secondary zebra stripes, and the cone mosaic. Then, we dcscribc two competing hypotheses for the generation of the secondary zebra stripes. Finally, we dcscribr experiments designed to distinguish between these two hypotheses. GENERAL METHODS

All experiments in this paper were performed with a computer-controlled laser interferometer, described in detail by Williams (1985). Subsequent modifications of this interferometer, mainly in the control of fringe spatial frequency and orientation, are described by Coletta and Williams (1987). The light source was a He-Ne laser (632.8 nm). Interference fringes were presented for 500 msec every 2.5 set without changing the space-averaged retinal illuminance of 2000 td. The purpose of the 2 set interval between stimulus presentations was to avoid habituation to the stimulus. Fringe contrast was controlled with the pulse overlap technique described by Williams (1985). The two beams of the interferometer were chopped independently by an acousto-optic modulator at 500 Hz. The temporal overlap of the pulse pairs modified the

contrast of the interference fringe; no temporal overlap produced a spatially uniform field while complete overlap produced a unity contrast fringe of the same space-averaged luminance. Fringe spatial frequency was controlled by varying the separation of a pair of coherent point sources focused near the entrance pupil of the observer’s eye. Fringe orientation was controlled by varying the orientation of the point sources in the entrance pupil. The positions of the point sources that determined spatial frequency and orientation were always symmetric about the Stiles-Crawford maximum in the entrance pupil. This allowed arbitrary changes in spatial frequency and orientation without the need to realign the observer. In order to fix the head position relative to the apparatus, dental impressions were used. The right eye was always studied and was aligned both horizontally and vertically as described by Williams (1985). Experiments involving very high spatial frequencies require large separations between the point sources in the entrance pupil. To avoid occlusion of these point sources by the iris, the pupil was dilated with Mydriacyl (0.5%) for expcrimcnts involving spatial frcquencies greater than about 90c/dog. OBSERVATIONS OF SECONDARY ZEBRA STRIPE..

We have identified three observers with normal color vision who reported the appearance of secondary zebra stripes. The subjective reports of all of these observers agreed closely. These reports share the following features, which we propose as defining criteria for secondary zebra stripes, seen when viewing a unity contrast, 632.8 nm interference fringe. Experimental evidence for some of these features is provided by expts I and 2 described subsequently. (I) The secondary zebra stripes are seen at the fovea1 center at spatial frequencies just above the fovea1 resolution limit, typically in the range of 55-75 c/deg. This distinguishes them from the primary zebra stripes which appear at the fovea1 center for frequencies from 100 to 150 c/deg. Examples of sketches from observer NS of primary and secondary zebra stripes are shown in Fig. l(a) and (b), respectively. The fringe orientation was 60 deg from horizontal for both sketches but the fringe spatial frequency was I 13 c/deg for the primary zebra stripes and 57 c/deg for the secondary zebra stripes. The scale bar represents 30 min of visual angle.

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(a) 113 c/deg

(b) 57 c/deg

Black or dark

Dark greenish

,

Reddish

Fig. I. Author’s drawings of the appearance of (a) primary zebra stripes and (b) secondary zebra stripes. The fringe orientation was 60 deg counterclockwise from horizontal and the spatial frequency was (a) I 13 and (b) 57 c/da. The state bar at the bottom represents a visuai angle of OS deg.

(2) The secondary zebra stripes are seen as a pattern of fine wavy lines at the fovea1 center that are slightly darker and greener than the overall fitid. Between these dark green lines are faint stripes that are slightly more reddish than the field as a whole. These color effects distinguish the secondary from the primary zebra stripes, which are seen as fine, red and black lines. (3) Williams (1985, 1988) described the primary zebra stripes as forming an annulus centered on the line of sight whose radius shrinks with increasing spatial frequency. This same behavior can be seen in the secondary zebra stripes although at roughly half the spatial

frequency. However, the observers agreed that the annulus was not as clearly defined as in the case of the primary zebra stripes. (4) The appearance of the secondary zebra stripes must be time-locked to the fringe presentation, which was demarcated for the observer by a tone. This criterion, which is also useful for identifying primary zebra stripes, ensures that the observer is not confusing spatial noise, such as laser speckle, with the phenomenon of interest. The large depth of field of laser interferometers coupled with optical defects inside and outside the eye produce obvious spatial noise or nonuniformities of various kinds in the stimulus

field. In addition, the secondary zebra stripes can be distinguished from the spatial noise because they shimmer or rapidly scintillate, whereas the spatial noise is relatively stationary. (5) The secondary zebra stripes move with the eyes, remaining centered on the line of sight, as do the primary zebra stripes. In general, the spatial patterns of the primary and secondary zebra stripes are not identical, though they share some features. However, for a given fringe spatial frequency and orientation, both the primary and the secondary zebra stripes have the same appearance from day to day. Observer DW has consistently observed both these phenomena since constructing his first interferometer in 1983. The other observer, OP, reported a percept that met most of the criteria described above except that he was uncertain about the color appearance of the pattern. He reported that the secondary zebra stripes were mainly defined by variations in brightness rather than color. Furthermore. with a red (632.8 nm) fringe, there was no spatial frequency for which he was able to set the pattern clearly at the very center of the fovea, though he could see an annular pattern of secondary zebra sttipcs. We also idcntilicd two observers who reported the primary zebra stripes, but could not see the secondary zebra stripes. We have not examined a large number of subjects, so we do not know what fraction of normal observers see the effect. For those observers who do see the phenomenon, it is subtlc and close to threshold, even conditions. The contrast under optimal thresholds for both types of zebra stripes were measured for two observers (DW and NS) with method of adjustment. The observers first adjusted the spatial frequency of a horizontal interference fringe to find the coarsest zebra stripes and then reduced the contrast until the zebra stripes were just barely seen. Whereas the primary zebra stripes required 2@-30% contrast for detection, the secondary zebra stripes required about 80%, very close to the maximum possible contrast of 100%. Therefore, it may be that the effect lies below threshold for many observers. EXPERIMENT 1: THE EFFECT OF SPATIAL FREQUENCY AND ORIENTATION ON PRIMARY AND SECONDARY ZEBRA STRIPES AT THE FOVEAL CENTER

Williams (1988) characterized the dependence of primary zebra stripes on two parameters of a

unity contrast interference fringe: orientation and spatial frequency. He identified which values of these parameters produced so-called moirP zeroes. Moire zeroesoccur when the zebra stripes appear maximally coarse in a relatively locai region of the fringe parameter space. Williams argued that these moire zeros occur when both the following conditions are satisfied: (I) the period of the interference fringe must match the spacing between rows of fovea1 cones. That is, the fringe period at the fovea1 center must be about 0.5 min, corresponding to a spatial frequency of I IO-120 c/deg; (2) the orientation of the fringe must lie parallel to these same rows of cones. For a fringe whose period is matched to the row spacing and which is imaged on a triangular mosaic, the fringe will come into register with the rows of cones with every 60deg of fringe rotation. Our first experiment introduces a new method that confirms Williams’ original findings on the behavior of primary zebra stripes. The method is then used to characterize the dependence of the secondary zebra stripes on spatial frequency and orientation.

Each of two observers (DW and NS) vicwcd a 2 deg field containing a unity contrast intcrferencc fringe. Observer DW used a cylindrical trial lens to correct for astigmatism ( -2.0 D). The purpose of the correction is to superimpose the two retinal images of the circular field stop, one from each of the point sources in the pupil. Without this correction, a double image of the field stop is seen with a lateral shift between the images that could inform the observer about fringe orientation. The cylindrical lens has a very small effect on fringe orientation and spatial frequency, for which the data were corrected. The observer was provided with two knobs, one that controlled the spatial frequency of the fringe and the other its orientation. He adjusted both knobs iteratively to find the coarsest primary or secondary zebra stripes at the fovea1 center. In order to prevent observers from concentrating their settings at a particular fringe orientation, observers were asked to scan over the entire range of the orientation knob before making a fine adjustment in each experimental session. Measurements were made for each observer in three sessions for the primary zebra stripes and three for the secondary zebra stripes. Each session consisted of about 10 settings.

Nonlinear distortion at resolution timit

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of fovea1 cone spacing. as Williams (1988) has previously argued. In addition, the data for each observer falls predominantly into clusters

Fig. 2. Ohscrvcr’s settings in the two-dimensional rdjustmcnt cxpcrimcnt for the primary mhrit stripes (0) and the secondary zebra stripes (0). for obscrvcrs (a) DW and (b) NS.

Figure 2a and b show individual settings plotted in the two-dimensional spatial frequency plane for each of the two observers. The data show the spatial frequencies and orientations that yield individual moire zero settings. in each plot, zero spatial frequency is represented at the origin and spatial frequency increases with increasing distance along any radius from the origin. The inner and outer circles correspond to spatial frequencies of 60 and 120 c/deg, respectively. Fringe orientation is represented on the plot by angle with respect to the origin. Zero and I80 deg correspond to horizontal gratings; 90 and 270deg correspond to vertical gratings. Note that the plot has 180deg rotational symmetry: individual settings always appear as pairs of data points opposite the origin. These pairs can be thought of as the two first-order delta functions of the fringe Fourier spectrum. Solid circles show the settings for primary zebra stripes. Moire zeroes for the primary zebra stripes occur at a nearly constant spatial frequency for both observers, lying in the range l03-122c/deg for DW and I I@-l34c/deg for NS. This agrees well with anatomical estimates

of settings at roughly every 6Odeg of fringe consistent with the triangular orientation, packing of fovea1 cones. Moire zeroes can also be identified for the secondary zebra stripes, and these settings are shown as open circles in Fig. 2a and b. They lie in the range 52-68 c/deg for observer DW and 53-68c/deg for NS. For both observers, the secondary zebra stripe settings lie at about haif the spatial frequency that produces the coarsest primary zebra stripes. Thus the coarsest secondary zebra stripes are obtained when the fringe spatial frequency is near the Nyquist frequency of the cone mosaic. The ratio of the mean spatial frequency settings for primary and secondary moire zeroes is 1.87 for DW and 2.02 for NS. Note that, like the primary zebra stripe settings, the secondary zebra stripe settings also fall predominantly in clusters at orientations that are roughly 60deg apart. Furthermore, these orientations are roughly the same as those for the primary zebra stripes. EXl’ERlMENT 2: THE EFFECT OF ECCENTRKITY WlTlllN TIIE FOVEAL REClON ON PRIMARY AND SECONDARY ZEBRA STRIPES

In expt 2, we examine how the primary and secondary moire zeroes change with retinal eccentricity over a small range within the fovea1 region. Williams (1988) measured the spatial frequencies producing moire zeroes for the primary zebra stripes at a number of retinal eccentricities within the fovea. He showed that these moire zeroes occurred when the fringe spatial frequency matched anatomical estimates of the cone sampling frequency at each eccentricity. We show a similar behavior for the secondary zebra stripes, but at roughly half the spatial frequency at each eccentricity. The results provide additional evidence for the intimate link between the primary zebra stripes, the secondary zebra stripes, and the geometry of the mosaic. Methods

We apply the same technique used by Williams (1988) to examine both the primary and secondary zebra stripes. The technique takes advantage of the fact that there are spatial frequencies for which either the primary or secondary zebra stripes form an annulus centered on the line of sight.

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A~OBLTOSHI

SEKIGUCHI

The three observers. DW, OP, and NS all had normal color vision. The test field was 4 deg in diameter for observer DW, and 2.5 deg for observers OP and NS. A dim white ring, whose thickness was 2.5 min arc, was centered on the test field. It was produced by an incoherent (tungsten) light source and a photographic transparency. The observer fixated the center of the ring and adjusted the fringe spatial frequency until the coarsest zebra stripes fell on the ring. In this experiment. unlike the previous one. the fringe orientation was fixed by the experimenter and was either horizontal or vertical. The radius of the ring determined the retinal eccentricity for which a moire zero was to be obtained. This procedure was repeated for rings of various radii from 0.25 to 1.0 deg. For measurements at the fovea1 center, the ring was removed and the observer adjusted the fringe spatial frequency until the zebra stripes were coarsest at the point of fixation. For each observer. data for the primary zebra stripes wcrc collected in two sessions, as wcrc data for the secondary zebra stripes. Within each session, the retinal eccentricities wcrc randomized and obscrvcrs made four spatial frcqucncy settings at each ccccntricity; two with horizontal intcrfcrcncc fringes and two with vertical fringes.

Settings could only be made up to 0.75 deg of retinal eccentricity for observers DW and OP and 0.625 dcg for NS. At larger eccentricities, the secondary zebra stripes were no longer visible. Also, observer OP could not identify the secondary zebra stripes at the fovea1 center. All observers agreed that the secondary zebra stripes did not form as clear an annulus as did the primary zebra stripes. Figure 3 shows the mean settings for each of the three observers. Data for horizontal and vertical fringes have been averaged. The lower panel shows how the fringe period producing the coarsest primary and secondary zebra stripes depends on retinal eccentricity. Solid and open symbols represent the mean settings for primary and secondary zebra stripes, respectively. Error bars indicate &I SEM. The results show that the fringe period that produces the coarsest primary zebra stripes increases with retinal eccentricity in a similar manner for all three observers. These data are compared with the anatomical data of Msterberg (1935). shown as the solid line, and Curcio,

Ct al.

Sloan, Kalina and Hendrickson (1990). shown as the broken line (mean of seven human eyes). There is good agreement between our psychophysical data and the anatomical measurements. Similar to the case of the primary zebra stripes, the fringe period that produces the coarsest secondary zebra stripes increases with retinal eccentricity, in a manner that keeps roughly constant the ratio of the spatial frequencies corresponding to the primary and secondary moire zeroes. This ratio is plotted in the upper panel of the Fig. 3. The mean ratio is I X3, 1.65 and 1.76 for observers DW, OP. and NS, respectively, close to though slightly less than the factor of 2 relationship suggested by expt I. Williams (1988) found that. at any fovea1 location. the fringe period producing the coarsest zebra stripes was about 16% higher for vertical than for horizontal fringes. His inference from this observation was that cones were

ECCENlRIClTY (dcg) Fig. 3. Bottom

Nonlinear distortion of gratings at the foveal resolution limit.

Aliasing by the foveal cone mosaic causes high frequency interference fringes to look like bright and dark zebra stripes (primary zebra stripes) [Will...
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