VA corresponds to the inverse of the minimum angular size of the detail within the just-resolvable configuration expressed as min arc [minimum angle of resolution (MAR)]: the lower the MAR, the higher VA. Since a logarithmic scale is more appropriate as it makes the decrement in size of the optotypes uniform across the range of measurement, VA is generally measured as the logarithm of the MAR (logMAR).

    a “C” or an “E”: detecting the details of a configuration, in fact, means resolving the configuration into its high-frequency components. The type of threshold measured by a resolution task is a detection threshold. It is worth recalling that the detection threshold measures the ability to detect the transition from a state of no stimulation to a state of stimulation. This takes place, for example, in perimetry (the procedure aimed at assessing the integrity of the visual field), when the observer is asked to report the sudden onset of a spot of light presented on a background (1). Incidentally, a second type of threshold is discrimination threshold. Discrimination threshold measures the ability to differentiate a state of stimulation from another state of stimulation (1). Discrimination threshold is therefore the just noticeable difference between two different states of stimulation. For example, given as the independent variable the width of the gap of a Landolt “C”, discrimination threshold is the just noticeable difference between “Cs” with slightly different gap width (the other characteristics of the stimulus are kept constant). In sum, in a discrimination task a comparison is made between different states of stimulation, in a resolution task this is not the case. Signal detection theory is a valid approach to these issues. According to signal detection theory (SDT: Green & Swets, 1966), discrimination (and detection) thresholds results from the capacity of the visual system to extract the signal from noise (2). In other terms, detection and discrimination thresholds depend on the sensory evidence of a signal (the stimulus), that is given by the signal and by a certain amount of noise intrinsically intermingled to it, and a subjective criterion (SC), unconsciously set by the observer. The subjective criterion can be: more “liberal”, when the observer is more inclined to answer “yes, I see it” or “yes, I discriminate it” for signals producing a relatively low degree of sensory evidence; more “conservative”, when he or she is more inclined to answer “no, I do not see it” or “no, I don’t discriminate it” for signals producing a relatively high degree of sensory evidence. The importance of this approach is that, in computing a threshold (be it detection or recognition), SDT assumes that the strength of sensory evidence for a signal as well as the subjective criterion of the observer may change from trial to trial and across individuals.

    Based on the definition of detection threshold of Aleci (1), the resolution task would measure a detection threshold, as it relies on the ability to notice (detect) the difference between a state of non-stimulation (absence of details like a gap along a ring or of three stacked bars) and a state of stimulation (presence of details like a gap along a ring or three stacked bars). Evidently, detecting the details of a configuration means resolving the characteristics of the signal (Figure 1).

To better clarify this statement, we suggest that visual acuity relies on a two-level process of detection of spatial frequencies. Given a serial pattern (e.g., a grating of alternating black and white bars), spatial frequency corresponds to the number of repetitions per space unit (cycles/degree, where 1 cycle is made up of a black bar plus a white bar). More generally, spatial frequency is the number of cycles/deg of a grating that can be imagined to resolve any structured object. For example, a confusing letter, perceived as a blurry spot, has a spatial frequency of a grating whose width of the bars is equal to the rough width of the spot. In a sharp letter, high-frequency components have the spatial frequency of a grating with the width of the bars equal to the width of the stroke of the letter.
When a ring with a gap is presented, the first step for its resolution is noticing the transition from a state of non-stimulation (no stimulus) to a state of stimulation (a

blurry spot). This means the lowest frequency component of the stimulus is detected. The second step is noticing the transition, within the configuration, from a state of no-stimulation (no details in the blurry spot) to a state of stimulation (the spot is a ring with a gap). This means the high frequencies of the stimulus are detected.

    In sum, a detection task evaluates the detection threshold of a global configuration (the lowest spatial frequency of the stimulus) and a resolution task measures the detection threshold of the high spatial frequency components of a configuration, which are its details. Resolution is therefore expressed as the minimum distance of the details of the signal that allows their identification (minimum separable or resolving power).

    For example, consider a high contrast annular stimulus of diameter d and thickness t. The observer is required to determine if there is a break a in the ring. The variable under examination is, therefore, the angular amplitude a. The target stimuli correspond to rings in which a changes by a predetermined step size; non-interrupted rings (null stimuli) are added to check for false-positive responses.

    The task is reporting whenever the gap of the ring, namely the variable a that characterizes the signal (its high-frequency component), is detected: this way, the threshold ae below which a is no longer detected is obtained. If the subject does not detect a, he/she cannot resolve the signal.

    In clinical practice, except for illiterate patients or pre-school children, visual acuity is generally measured through a recognition task. Unlike the simpler resolution task, recognition implies the identification of signs with semantic content, like letters or digits. Resolution is necessary but not sufficient for the recognition task, in other terms for identifying a letter or a digit as that letter or that digit. The recognition of the sign “V” (two upward diverging strokes) as the letter “V” requires not only resolving its elementary constituents (resolution task) but also matching the sign with the corresponding letter stored in the semantic memory. A recognition task relies therefore on additional cognitive functions that are not necessary to the resolution task.

    Within the research setting, visual acuity can be investigated with non-adaptive or adaptive procedures. The method of constant stimuli (n sets of stimuli presented in a randomized order) belongs to the first class. In each set, the size of the stimuli is constant and the only difference lies in their features, like, for example, the width (and position) of the gap in “C” stimuli. The method of constant stimuli is the most reliable way to compute the psychometric function. The psychometric function is a sigmoid curve generated by the psychophysical response to a stimulation. This function expresses the relationship between the probability to perceive a stimulus and its intensity. As the intensity increases, the subject is more likely to notice the stimulus and the threshold is computed from the level that corresponds to a probability equal to or higher than 50% (target probability). In the visual domain the function is called frequency-of-seeing curve (FOSC). Furthermore, the method of constant stimuli is the most straightforward way to provide information about the dispersion of the data around the threshold and to reduce non-stationary responses (trials, in fact, are randomized). Yet, since it requires a high number of trials, it is time-consuming, thereby unsuitable for clinical purposes. A faster alternative is the method of limits, where series of stimuli with decreasing, then increasing intensity are administered, and the threshold is computed as the intensity at which a reversal in the trend of the responses (from correct to incorrect and vice versa) is observed. Arguably, the drawback of the non-adaptive procedures is low efficiency due to the high number of trials. Adaptive nonparametric procedures like the staircase methods or the modified binary search (MOBS), and, even more, adaptive parametric strategies like Best PEST, QUEST, or ZEST (where the next stimulus intensity is determined from all the previous observer’s responses and the parameters, i.e., shape and slope of the psychometric function, are assumed to be known.) overcome this problem. Parametric adaptive strategies tend to converge faster onto the threshold from the very beginning of the examination, greatly reducing the number of trials. Even if this leads to high efficiency, patients, who are generally inexperienced observers, may have difficulty familiarizing themselves with the procedure [see (1) for a review].

    The procedure can be coupled with a yes/no or an alternative forced-choice (n-AFC) design: in the first case, the observer is asked to report if he can resolve the stimulus and the target probability φ (the probability of perceiving ae) should be equal to or higher than 50%.

    In the second case consider the example of a ring that contains the break (target) and one or more uninterrupted rings (null stimuli). The observer is forced to identify the target among the n-alternatives. The intervals can be administered simultaneously (spatial paradigm) or sequentially (temporal paradigm). The target probability will be greater than 0.5 and will depend on n and the adopted psychophysical procedure.

    The reliability and repeatability of visual acuity measurements depend on many factors, namely:

    (I) The luminance of the background;

    (II) The amount of contrast of the signal, i.e., the difference in luminance between signal and background [the term signal, borrowed from the signal detection theory (SDT), identifies a stimulus characterized by an informational content];

Landolt’s charts (Landolt’s C)

Landolt’s Cs are annular stimuli with an interval placed in one of the four cardinal positions.

    The Landolt’s charts, suitable to estimate visual acuity in illiterate or non-cooperating subjects, represent rows of stimuli, each made up of rings the same size. Starting from the upper portion of the chart, each line differs from the previous one for the smaller size of the rings. The angular size of the interval is proportional to the size of the ring (diameter and thickness of the stroke) so that the ratio between the width of the gap and the size of the ring is constant. The minimum angle of resolution of the gap provides the VA value.

    The procedure is compatible with the method of limits associated with a 4-AFC response model, implicit variant. Aleci (1), indeed, distinguished between standard n-AFC, in which the alternatives are presented and the subject is forced to judge the correct one, and implicit n-AFC. In the implicit procedure, the observer is informed about the possible alternatives (in the case of the Landolt’s C the interruption of the ring can be up, down, to the left, or to the right: 4-AFC) and a stimulus (e.g., a ring with the gap on the superior part) is administered at each trial. The subject must choose which one of the mentioned alternatives matches the displayed stimulus. The procedure differs from the classical method of limits since a block of n presentations per signal level (and not a single stimulus) is presented. The observer, informed about the four possible positions of the interval, is forced to report verbally the position of the gap of each symbol displaced along each line (starting from the upper one). The acuity threshold corresponds to the line at which the observer recognizes 62.5% of presentations.

    A version of Landolt’s chart with 8 orientations is available. The response method is therefore 8-AFC and the probability to respond correctly by chance is 12.5%. The target probability should therefore be at least (100%+12.5%)/2=56.25%.

    Although the assessment of VA with Landolt’s charts resembles the example reported in the introduction, there is a substantial difference: in the latter the diameter of the ring is constant. It is therefore questionable whether the acuity measured in the two test conditions is the same. It is arguable that if the diameter of the ring remains constant and the only variable is the width of the gap, the estimate operates only on a local scale, while there is no global processing: both instead occur in the clinical practice, in which the global (the whole stimulus) and local configuration (the gap) covary (Figure 2).

The tumbling E charts (“illiterate” E)

The tumbling E for the assessment of VA in illiterates recruits a perceptual task similar to that of Landolt’s charts. The observer is asked to report the cardinal orientation (above, below, left, or right) of an “E” in which the thickness of the stroke and the width that separates the horizontal bars are 1/5 the size of the symbol. The minimum separable concerns the three horizontal bars: if the three horizontal bars are not resolved, their orientation with the vertical bar cannot be established (Figure 3).

    The ability to report a difference between a null stimulus and the target (detection task) is tested: in the Landolt’s charts, the null stimulus corresponds to a ring with no interruption, in the tumbling E it is comparable to a stimulus with a spatial frequency lower than that of the bars (see Figure 1). How much lower, i.e. how much lower is the VA, in absence of ophthalmological alterations arguably depends on the refractive state, therefore on the blur ratio (BR) of the observer (4,5). A blurred configuration like an E is perceived as a series of contiguous superimposed circles of confusion formed by the strokes. Given that the acuity threshold depends on the relative size of these circles, the BR corresponds to the ratio between the diameter of the

circle and the size of the retinal image of the E. The higher the circle’s size compared to the retinal image, the higher BR, therefore the less resolvable is the stimulus. Optotypes can be recognized for a BR up to 0.5 (4). Based on these optical criteria, the configuration cannot be resolved if the perceived size of the blurred stroke is more than ? the size of the whole symbol.

    The identification of a symbol like “E” depends not only on the BR but also on the cutoff object frequencies. Each symbol, in fact, contains a band of spatial frequencies (designated in cycles per letter) that is particularly effective for its identification (6). These object frequencies mediate letter recognition and are indicative of the acuity threshold. In this respect, a symbol contains a broad spectrum of different spatial frequencies and the visual system can split the information into these frequencies, rather than encoding the proper elementary features. These components are processed by selective neuronal channels via a Fourier-like analysis and are integrated into the cortex to reconstruct the whole picture (7). The object frequencies used for letter recognition are within the range of 1–3 cycles per letter (8,9). They depend both on the adopted symbol [e.g., Landolt C or tumbling “E” (10)] and on the angular dimension, so the identification of small letters should rely mainly on low frequencies (i.e., their gross components) while the identification of large symbols on high-frequency components [i.e., their edges (11,12)].

    As explained in the introduction, the process is made of two steps: the first step consists of the detection of a stimulation: the observer notices the interruption of the no-stimulation state, that is to say, the presence of a stimulus. In this phase, even if he or she becomes aware of the presence of a signal, he is unable to resolve it. The second step is the detection of the details of the signal, in other terms it is the resolution of the stimulus: the stimulus is recognized as a structured pattern. The first phase corresponds to the detection of the low spatial frequency component of the stimulus (“I see that there is something but I don’t know what it is”), the second phase, of interest in VA assessment, involves the detection of the spatial cut off frequency that allows the resolution of the configuration. For sake of

precision, as recalled by an anonymous reviewer, if the lowest frequencies are in the falling left hand part of the contrast sensitivity function, some intermediate frequency, closer to the peak, would be more detectable.

    As for Landolt’s chart, measuring VA with the “illiterate” E makes use of a combination of the method of limits and the method of constant stimuli coupled with a 4-AFC response method, implicit variant. The intensity of the signal (its size) is progressively decreased according to the method of limits, but instead of a single target per intensity, blocks of n presentations with the same signal level are displayed in each line, as it occurs in the method of constant stimuli. The threshold corresponds to the line at which the observer correctly recognizes 62.5% of presentations.

Alphanumeric charts

The most common method for the assessment of VA in literate subjects makes use of charts with alphanumeric symbols. In this case, the task is not limited to the detection and resolution of the target but relies on a subsequent step: the recognition of letters or numbers. Despite the recognition task with alphanumeric symbols is expected to be more complex than with Landolt Cs and tumbling “E” because it implies the mental association of the presented configuration with a known symbol, Sloan showed that letters acuity is substantially equivalent to Landolt’s C measurement (13). More recently, visual acuity measured with Landolt’s Cs was found to be slightly lower than with letters (14,15): maybe the cognitive component that comes into play with alphanumeric symbols helps identification in the presence of mild perceptual deterioration. In fact, the recognition of a letter can take place by approximation or via a Bayesian-like probabilistic mechanism in blurred conditions or when part of the signal is missing (Figure 4). Evidently, this top-down strategy cannot be employed when VA is measured with Landolt’s C (16).

    Visual acuity measured with Landolt’s Cs is lower than acuity estimated with the E charts, too (17-20).
   The difference depends on the spatial characteristics of the stimuli. Landolt rings contain a single gap whereas “Es” have two gaps with a lattice-like spatial configuration: in case of signal deterioration, the Landolt’s C appears as a low spatial frequency circular spot so that it is not possible to localize the position of the gap (null stimulus, see Figure 1). On the contrary, in the tumbling E, the horizontal or vertical arrangement of the bars can still be determined with approximation, increasing from 25% to 50% the chance of guessing (21). For this reason, the tumbling Es should be scaled by 15% compared to the Landolt’s Cs to provide similar estimates (22).

    Reich and Ekabutr found that the two paradigms differ not only for the threshold but also for the slope of the psychometric function, steeper for the E charts compared to the Landolt’s C (18). The slope, indeed, is the dispersion parameter, so a steeper slope indicates less dispersion of acuity values as a function of the proportion of correct responses: it follows that E charts are more accurate than Landolt’s rings.

    When visual acuity is measured with alphanumeric charts, three possible scenarios are expected:
    (I) The detection of the features of the stimulus and the association with the symbols learned during childhood and stored in memory are effective: the stimulus will be recognized. It is worth recalling that some letters, namely those with no edges like “O” or “C”, are more difficult to identify (14,23);

    (II) The detection of the features of the stimulus is defective: the stimulus will be seen blurred, thereby confused with other stimuli;

    (III) The detection of the features of the stimulus is effective but the associative process is defective for central dysfunctions (e.g., agnosia): even if the subject recognizes the symbol, he/she will not be able to name it. In this case, objective methods such as the Oktotype (see below) may be helpful.
    The psychophysical procedure commonly used is a mix of the method of limits and constant stimuli, with progressively smaller rows of letters that must be inspected one at a time. Since the subject is forced to name the symbol even when he/she is unable to recognize it, the response model is n-AFC, implicit version, where n is the number of choices. The Latin alphabet has 26 letters: the response model is, therefore, a 26-AFC and the probability to guess by chance is 3.8%. Therefore, the target probability should be equal or higher than (100%+3.8%)/2=51.9%. To be noted that N-AFC, in its implicit version, assumes that the observer knows what are the possible alternatives so to decide which one of these alternatives matches the displayed stimulus. Even if the possible subset of choices in the Sloan and Snellen chart is 9 and 10, respectively, the observer is not informed by the operator that the presented stimuli will not be selected from the whole pool of 26 letters of the alphabet. So, the alternatives for the observer remain 26.

    For digits [0–9], the response model is 10-AFC: the probability to guess by chance is 10%, then the target probability should be equal or higher than (100% +

10%)/2=55%. The threshold is generally taken as the line whose recognized symbols are more than 50%.

Standardization criteria according to the Consilium Ophthalmologicum Universale

Visual acuity should be measured in photopic conditions (160 cd·m?2) with high-contrast levels (dark stimuli on a white background). The standard optotype is the Landolt C. The thickness of the ring and the width of the gap are 1/5 of the outer diameter (Figure 5).

    VA refers to the smallest optotype identifiable by the patient at a certain distance. It can be expressed as the reciprocal of the angular size of the gap (or of the thickness) of the smallest resolvable optotype at a given distance: this is, in fact, the inverse of the MAR. If the gap of the smallest resolvable Landolt’s C subtends 1 min arc (considered as the minimum angle of resolution in normal subjects), VA will be: 1/MAR =1/1’=1.0. If the smallest resolvable C has a gap of 5 min arc, the corresponding VA is lower: 1/5=0.2. Instead, if the gap of the smallest identifiable C subtends 10 min arc, VA will be 1/10=0.1.

    A different notation is linear, and expresses VA as the Snellen fraction: d/D, where d is the distance in meters (or feet) at which the test is administered and D corresponds to the distance in meters (or feet) at which the gap of the ring subtends 1 min arc. If the chart is placed at 6 meters and the gap of the smallest resolvable symbol subtends 1 min arc at that distance, VA is 6/6=1.0. On the other hand, if the gap of the smallest resolvable symbol subtends 1 min arc at a distance of 60 meters, this means that the smallest resolvable C has an interruption wider than the previous one, and, in fact, VA is lower: 6/60=0.1. The Snellen fraction can be reported in decimal notation, for example: 6/6=1.0=10/10; 6/60=0.1=1/10.

    This fraction corresponds to the ratio between the maximum distance at which the symbol is recognized by the observer and the (predetermined) distance of the symbol from the observer. So, if the predetermined distance of the symbol is 6 meters and the maximum distance at which the observer can recognize the symbol is 6 meters, VA will be 6/6=1.0. If the maximum distance at which the observer can identify the same symbol is reduced to 3 meters, VA is 3/6=0.5. Likewise, if the maximum distance at which the observer can correctly name the letter of the same size is only 1 meter, VA is 1/6=0.16.

    To summarize: