Minimum visible acuity

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  Minimum visible acuity refers to the smallest object that one can detect.

As early as the 17th century, de Valdez measured the distance at which a row of mustard seeds could no longer be counted, and early astronomers such as Robert Hooke were interested in the size of stars that could be detected and their relation to retinal anatomy. In this context, the minimum visible acuity refers to the smallest target that can be detected. Under ideal conditions, humans can detect a long, dark wire (like the cables of the Golden Gate bridge) against a very bright background (like the sky on a bright sunny day) when they subtend an angle of just 0.5 arc seconds (~0.00014 degrees). It is widely accepted that the minimum visible acuity is so small because the optics of the eye (described further in the chapter) spread the image of the thin line, so that on the retina it is much wider, and the fuzzy retinal image of the wire casts a shadow which reduces the light on a row of cones to a level which is just detectably less than the light on the row of cones on either side. In other words, although we specify the minimum visible acuity in terms of the angular size of the target at the retina, it is actually limited by our ability to discriminate the intensity of the target relative to its background.

Increasing the target size, up to a point, is equivalent to increasing its relative intensity. Hecht and Mintz measured the visual resolution of a black line against a background over a large range of background brightness. They found that the minimum visible acuity varied from about 10 minutes at the lowest background levels, to about 0.5 arc seconds at the highest. The limiting factor in the case of minimum visible acuity is the observers’ sensitivity to small variations in the stimulus intensity (ΔI/I), i.e., their contrast sensitivity (discussed further in the chapter). Indeed, Hecht and Mintz state that the retinal image produced by the 0.5 arc second line represents a “fine fuzz of a shadow extending over several cones,” and they calculated that at the highest intensity levels tested, the foveal cones occupying the center of the shadow suffer a drop in intensity (relative to the neighboring cones) of approximately 1%—just at the limit of intensity discrimination.

Although the minimum visible acuity represents one limit to spatial vision, it is in fact a limit in the ability to discern small changes in contrast, rather than a spatial limit per se, and minimum visible acuity is not used clinically.

Minimum resolvable acuity

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  Minimum resolvable acuity refers to the smallest angular separation between neighboring objects that one can resolve.

Resolving double stars is an example of what is now known as minimal resolvable acuity, which is limited by the eyes’ imperfect optics. There is currently still debate about how best to define and measure resolution. However, today, the minimum resolvable acuity is much more likely to be assessed by determining the finest black and white stripes that can be resolved. Under ideal conditions (e.g., high contrast and luminance), humans with very good vision can resolve black and white stripes when one cycle subtends an angle of approximately 1 minute of arc (0.017 degrees). This minimum resolvable acuity represents one of the fundamental limits of spatial vision: it is the finest high-contrast detail that can be resolved. In foveal vision the limit is determined primarily by the spacing of photoreceptors in the retina. The visual system “samples” the stripes discretely, through the array of receptors at the back of the retina ( Fig. 33.1 ). If the receptors are spaced such that the whitest and blackest parts of the grating fall on separate cones ( Fig. 33.1B ), we should be able to make out the grating. But if the entire cycle falls on a single cone ( Fig. 33.1C ), we will see nothing but a gray field (or we may experience a phenomenon called aliasing , in which we misperceive the width or orientation of the stripes). Cones in the fovea have a center-to-center separation of about 0.5 minutes of arc (0.008 degrees), which fits nicely with the observed acuity limit of 1 minute of arc (0.017 degrees—as we need two cones per cycle to be able to perceive it accurately), and each foveal cone has a “private” line to a ganglion cell. Rods and cones in the retinal periphery are less tightly packed together, and many receptors converge on each ganglion cell. As a result, visual acuity is much poorer in the periphery than in the fovea.

Spatial sampling of a sinusoid whose spatial frequency is ( A ) lower than the sampling limit set by the spacing of the cones, ( B ) equal to the sampling limit set by the spacing of the cones, ( C ) higher than the sampling limit set by the spacing of the cones.

Minimum recognizable acuity

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  Minimum recognizable acuity refers to the angular size of the smallest feature that one can recognize or identify.

Although this method has been used since the 17th century, the approach still used by eye doctors today was introduced more than a century ago by Herman Snellen and his colleagues. Snellen constructed a set of block letters for which the letter as a whole was five times as large as the strokes that formed the letter ( Fig. 33.2 ). The distance of the patient was varied until they could no longer accurately read the letters. In later adaptations of the Snellen test, the viewer was positioned at a constant distance (typically 20 feet [6 meters]), and the size of the letters, rather than the position of the viewer, was altered. Visual acuity, measured in this way was defined as:

Snellen acuity chart.

The distance at which the patient can just identify the letters / The distance at which a person with “normal” vision can just identify the letters

Thus, “normal” vision came to be defined as 20/20 (6/6 in metric units). To relate this back to visual angle, a 20/20 letter is designed to subtend an angle of 5 arc minutes (0.083 degrees) at the eye, and each stroke of a 20/20 letter subtends an angle of 0.017 degrees (1 arc minute). Thus, if one can read a 20/20 letter, one can discern detail that subtends 1 minute of arc. If one has to be at 20 feet to read a letter that someone with normal vision can read at 40 feet, one has 20/40 vision (worse than normal). Although 20/20 is often considered the gold standard, most healthy young adults have an acuity level considerably better than 20/20. Illiterate E and Landolt C charts are based on the same principles as the Snellen chart.

Although Snellen’s notation for visual acuity is commonly used, there are other schemes for specifying acuity ( Table 33.2 ). For example, the minimum angle of resolution (MAR) is the angular size of detail of the visual acuity letter in minutes of arc—it is the Snellen denominator (the number below the line, e.g., 40) over the Snellen numerator (the number above the line, e.g., 20). Thus, for example, a Snellen acuity of 20/20 is equivalent to an MAR of 1′; 20/40 to an MAR of 2′, and 20/100 an MAR of 5′. Another method for specifying acuity is the Snellen fraction (the Snellen numerator divided by the denominator or 1/MAR). Acuity is sometimes specified as log MAR (the logarithm of the minimum angle of resolution) and sometimes as log visual acuity (the logarithm of the Snellen fraction). Snell and Sterling developed a metric for quantifying visual loss due to injury or disease. The Snell-Sterling efficiency scale sets 20/20 (MAR = 1′) as 100% efficiency and reduces the efficiency by a fixed percentage (~84%) for every 1-minute loss of acuity.

There are many stimulus and subject variables that influence the minimum recognizable visual acuity (discussed further).

Minimum discriminable acuity

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  Minimum discriminable acuity refers to the angular size of the smallest change in a feature (e.g., a change in size, position, or orientation) that one can discriminate.

Perhaps the most studied example of minimum discriminable acuity is our ability to discern a difference in the relative positions of two features. Our visual systems are very good at telling where things are. Consider two abutting horizontal lines, one slightly higher than the other ( Fig. 33.3A ). It is very easy to discern that, for example, the right line is higher than the left (i.e., discriminate the relative positions of the two lines) even from a long way away. This is an example of a class of visual tasks that have been given the label “hyperacuity” by Gerald Westheimer. These tasks all have in common that they involve judging the relative position of objects, and Westheimer coined the term hyperacuity because, under ideal conditions, humans can make these judgments with a precision that is finer than the size or spacing of foveal cones.

Two “hyperacuity” configurations: Vernier acuity ( A ) and bisection acuity ( B ). In both cases, the subject’s task is to judge the relative position of the features.

The smallest misalignment that we can reliably discern is known as Vernier acuity—named after the Frenchman, Pierre Vernier, whose scale, developed in the 17th century, was widely used to aid ship’s navigators. The success of the Vernier scale was based on the fact that humans are very adept at judging whether nearby lines are lined up or not. Thus, Vernier alignment is still widely used in precision machines, and even in the dial switches in modern ovens. Under ideal conditions, Vernier acuity may be just three arc seconds (~0.0008 degrees)! This performance is even more remarkable when you consider that it is about 10 times smaller than even the smallest foveal cones. Note that the optics of the eye spread the image of a thin line over a number of retinal cones, and that the eyes are in constant motion, and this performance appears even more remarkable.

Vernier acuity is not the most remarkable form of hyperacuity. Guinness World Records 2005 describes the “highest hyperacuity” as follows: In April 1984, Dr. Dennis M. Levi (the author of this chapter) “…repeatedly identified the relative position of a thin, bright green line within 0.8 seconds of arc (0.00024 deg). This is equivalent to a displacement of some 0.25 inches (6 mm) at a distance of 1 mile (1.6 km).” This “remarkable” position acuity was accomplished with a bisection task ( Fig. 33.3B ), but can actually be understood based on the ability to discern small changes in local contrast (i.e., minimum detectable acuity).

As remarkable as the other hyperacuities (sometimes also called “position” acuities) might seem, they do not defy the laws of physics. Geisler calculated that if one placed a machine (known as an ideal discriminator) at the retina, and this machine knew precisely the pattern of photons absorbed by the retinal photoreceptors when the stripes were aligned and the pattern of photons absorbed when they were misaligned, this machine could actually perform an order of magnitude better than even the best humans. So, the information about the Vernier offset is present in the pattern of photons absorbed by the photoreceptors; however, humans must be able to interpret the information despite the constant motion of the eyes. Thus, hyperacuity must ultimately be limited by neurons in the visual cortex that are able to interpolate positional information with high resolution.

There is a good deal of evidence that different mechanisms limit position judgments for closely spaced (or abutting) targets, and for widely separated targets. In the nearby case, both contrast and contrast polarity are important. For example, in a two-line Vernier target, Vernier acuity is better when the lines are both either bright or dark than if one line is bright and the other dark. For longer-range position judgments where the targets are well separated, neither contrast nor contrast polarity nor the local stimulus details matter very much. For long-range position judgments the visual system must localize each of the features, and then compare the position labels of separate cortical mechanisms. The idea that cortical receptive fields have position labels (in addition to labels for other stimulus dimensions) is consistent with the topographical mapping of visual space in the brain (i.e., each point in space is systematically mapped onto the visual cortex). More than a century ago, Hermann Lotze wrote:

“So long as the opinion is maintained that the space relations of impressions pass as such into the soul, it must of course, in the interest of the soul, be further held that each impression is conveyed to it by a distinct fibre, and that the fibres reach the seat of the soul with their relative situation wholly undisturbed.”

Although we now know much more about the anatomy and physiology of the visual system than was known in the 1880s, Lotze clearly recognized that there must be a topographical representation of the world in the visual nervous system (if not the soul!), and that each “fiber” must carry a label about the position of the “impression” that it carried. For this reason, position labels were called “Lotze’s local signs.” Lotze also concluded that local signs played an important role in directing eye movements toward stimuli in the peripheral field of vision. So how accurate is “local sign” information? It turns out that humans can localize the position of a single peripheral feature to within about 1% to 2% of the eccentricity of the target, a little more precisely than our saccadic eye movements to peripheral targets.

Optical quality of the eye

The optics of the eye spread the retinal image. Consider the retinal image of a distant star. Because stars are very far away, they are considered to be a “point source,” that is, if they were in perfect focus and the eye’s optics were perfect, they would subtend an infinitely small angle at the eye. Point sources are useful if one wants to learn about the quality of the eye’s optics. As it turns out, the eye’s optics are far from perfect; in fact, a camera might have better optical quality. The eye’s optics spread the image, so that a point in space forms a distribution (e.g., a Gaussian) on the retina, as illustrated in Fig. 33.4A . This distribution, not to be confused with the office football pool, is known as the point spread function (if the source is a line, it is called the line spread function). Fig. 33.4B–D shows the retinal light distribution for a pair of nearby stars. As the stars become closer and closer together, their images overlap to a greater and greater degree, so that when they are very close, they look like a single distribution. When the separation between the stars on the retina is less than half the spread of each image, they will appear as a single star. This is known as the Rayleigh limit ( Fig. 33.4D ). The Rayleigh limit is determined mainly by the wavelength of the light and the size of the pupil (we soon discuss why). As noted earlier, the ability to distinguish double stars was one of the earliest measures of visual acuity. In a similar vein, an important ritual of certain wedding ceremonies involves the groom showing the bride the double star pair Mizar and Alcor in the handle of the big dipper. Successful sighting of the nearly invisible Alcor portends a successful marriage, so good optics may be the key to a happy marriage!

( A ) Retinal image of a single point. Although the point is infinitesimally small, its image is spread on the retina. ( B–D ) Retinal image of a pair of nearby points, separated by distances equal to two ( B ), one ( C ), and half ( D ) the spread of each image. The latter case is known as the Rayleigh limit.

Another way to determine the quality of an optical system is to measure its modulation transfer function. This is typically done by passing test patterns of sinusoidal gratings ( Fig. 33.5 ) of known contrast through the optical system and measuring the contrast in the image. Sine waves are characterized by their spatial frequency (i.e., the number of stripes in a given distance, usually specified in cycles per degree), their contrast (i.e., the difference in the luminance of the peaks and troughs divided by the sum of the luminance of the peaks and troughs), and their phase (the position of the peaks). Sine waves are especially useful because even after they are degraded by an optical system, they maintain their characteristic shape (i.e., once a sine wave, always a sine wave) and just become smaller in amplitude (and may shift in phase). The ratio of image contrast to object contrast is a measure of image quality. By measuring the ratio of image contrast to object contrast for a range of object spatial frequencies, one can measure the modulation transfer of any optical system. The red line in Fig. 33.6A shows the average modulation transfer functions of a large group of observers with 3-mm pupils (about the average size of the pupil in a well-lighted room). As the object spatial frequency increases, the modulation transfer declines, falling to near zero at a spatial frequency of around 80 cycles per degree. The spatial frequency at which the modulation transfer function falls to zero is called the cutoff spatial frequency. At this spatial frequency the optics do not transmit the sinusoidal variations in the luminance of the object. The theoretical cutoff spatial frequency (SF; in cycles/degree), like the Rayleigh limit, is determined mainly by the wavelength of the light (λ), and the diameter of the pupil (d):

The modulation transfer function.

The modulation transfer function (MTF) of the human eye. ( A ) The red line is the mean MTF of 12 eyes with a 3-mm pupil with refractive error corrected. The blue line is the MTF of an aberration-free eye (i.e., limited by diffraction only). The green line shows schematically the effect of uncorrected refractive error. (Reprinted with permission from Liang J, Williams DR. Aberrations and retinal image quality of the normal human eye. J Opt Soc Am A Opt Image Sci Vis . 1997;14:2873–2883.) ( B ) The red line is the mean MTF of 14 eyes with a 7.3-mm pupil with refractive error corrected. The blue line is the MTF of an aberration-free eye (i.e., limited by diffraction only). (Modified from Campbell FW, Green DG. Optical and retinal factors affecting visual resolution. J Physiol . 1965;181:576–593.)

where d and λ are both specified in millimeters (mm). For red light (630 nm or 0.000630 mm as used by Liang and Williams ) and a 3-mm pupil (about as small as they get), the predicted cutoff spatial frequency is 83 c/deg.

Photoreceptor size and spacing; aperture size; the “Nyquist” limit; aliasing

Retinal anatomy plays a very important role in setting the limits for foveal resolution acuity. Foveal cones are arranged in a regular arrangement (sometimes called a triangular array) and are densely packed (see Fig. 33.7 ). This dense packing is critical to fine resolution. The reason is simple. The visual world is represented by continuous variations in light intensity. However, our visual system “samples” the world discretely, (i.e., by looking at the light distribution through many small individual photoreceptors). The more closely packed the photoreceptors, the better the light distribution can be represented to the visual nervous system. As illustrated in Fig. 33.1 , there is a sampling limit. To properly represent the peaks and troughs in the intensity profile of a sine wave, there must be at least two cones for each cycle of the grating. This is known as the Nyquist limit and is represented in Fig. 33.1B . More formally, the Nyquist limit occurs when the sine wave has a spatial period (i.e., the peak-to-peak distance) that is two times the cone spacing. When the spatial period is smaller than twice the cone spacing (i.e., when the sine wave spatial frequency is higher than the Nyquist frequency), the phenomenon known as “aliasing” may occur, i.e., rather than the cones signaling a perfect replica of the sine wave on the retina, the signal is distorted—it appears to be a sine wave with a lower spatial frequency than the original ( Fig. 33.8 ). Because cones in the fovea have a center-to-center separation of around 0.5 arc minutes, the cone sampling (or Nyquist) limit is a grating spatial period of about 1 minute (i.e., a grating spatial frequency of 60 cycles per degree. Note that because the cones have triangular packing, geometry dictates that the Nyquist frequency will actually be about 15% higher or ~69 c/deg). This represents a fundamental limitation set by the spacing of the foveal cones, and it is not far off the limitations set by the eye’s optics. Thus, in the fovea, the cone spacing is nicely matched to the eye’s optics. As seen later in this chapter, cone spacing in peripheral vision changes dramatically, whereas the optics change only a little.

Inner segments of a human cone mosaic in a strip extending from the foveal center ( upper left arrow ) along the temporal horizontal meridian. The large cells are cones, and the small cells are rods. The horizontal scale bar indicates 10 µm.

From Hirsch J, Curcio C. The spatial resolution capacity of human foveal retina. Vision Res . 1989;29:1095–1101.

( A ) The foveal cone mosaic is represented as sample points (bright dots) . In the other panels it has been sandwiched with gratings of 40 cpd ( C ), 80 cpd ( B ), and 110 cpd ( D ).

From Williams DR. Topography of the foveal cone mosaic in the living human eye. Vision Res . 1988;28:433–454. Copyright Elsevier 1988.

The notion that there is a “highest” spatial frequency that can be accurately represented by the visual system (the Nyquist frequency) raises the interesting question of whether aliasing actually occurs in human vision. Fortunately for us, our optics “protect” us from the effects of aliasing in the fovea by reducing the contrast of high spatial frequencies. Interestingly, it is possible to create a very high spatial frequency pattern on the retina bypassing the optics of the eye (this can be accomplished using lasers to create interference patterns). Under these conditions, humans report the appearance of aliasing—fine gratings appear coarse (and often appear to change their orientation). Fig. 33.8 illustrates how viewing a grating through too coarse a grid produces the kind of wavy, coarse appearance produced by aliasing.

Cone to ganglion cell convergence

If many cones converged on a single ganglion cell, the advantage of having closely spaced cones would be lost. Recall, however, that in the fovea each cone effectively has a “private line” to the brain via the midget bipolar and ganglion cells. So, the receptive field centers of midget ganglion cells are effectively one cone wide. In peripheral vision resolution is limited by the retinal ganglion cells that are widely spaced (sparsely sampled) and that pool information from many cones.

Eccentricity

In contrast to the tightly packed cones in the fovea, cone density decreases dramatically with retinal eccentricity, and there is substantial convergence of cones to ganglion cells ( Chapter 26 ). Humans have many more rods (~90 million) than cones (about 4–5 million), and they have very different geographical distributions on the retina ( Fig. 33.9 ). The cones are most concentrated in the center of the fovea, and their density drops off dramatically with retinal eccentricity. Rods are missing from the center of the fovea, and their density increases to a peak at about 20 degrees, and then declines again. As can be seen in Fig. 33.7 , the cones in the fovea are much more densely packed than in the peripheral retina, and there are no rods in the section from the foveal center. This “rod-free” area (~300 micrometers on the retina) subtends a visual angle of about 1 degree. Moreover, unlike the fovea, in the retinal periphery there is considerable convergence of cones onto ganglion cells.

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