Carl Bergmann (1814-1865) and the discovery of the anatomical site in the retina where vision is initiated.

A preeminent quest of nineteenth-century visual neuroscience was to identify the anatomical elements of the retina that respond to light. A major breakthrough came in 1854, when Carl Bergmann discovered through direct observation that the human fovea contains only rods and cones. On this basis, he argued that these must necessarily be the light-sensitive elements (i.e., photoreceptors) that initiate vision. Bergmann also argued that Henle's fibers form part of the necessary anatomical link between these distal elements and the proximal ganglion cells, which transmit visual signals to the brain via the optic nerve. However, despite his achievement, Heinrich Müller, not Bergmann, is remembered as the discoverer of human photoreceptors in the literature. This article seeks to correct the record. It situates Bergmann's work alongside that of his contemporaries, sets out his arguments and the critique he received using archival documents, and makes this history more accessible for current readers by comparing what was said to what we know now. We argue that Bergmann's arguments are at least as compelling as those of Müller, and that he should be recognized as a codiscoverer of the anatomic site in the retina where vision is initiated.

Catastrophe optics theory unveils the localised wave aberration features that generate ghost images.

Monocular polyplopia (ghost or multiple images) is a serious visual impediment for some people who report seeing two (diplopia), three (triplopia) or even more images. Polyplopia is expected to appear if the point spread function (PSF) has multiple intensity cores (a dense concentration of a large portion of the radiant flux contained in the PSF) relatively separated from each other, each of which contributes to a distinct image. We present a theory that assigns these multiple PSF cores to specific features of aberrated wavefronts, thereby accounting optically for the perceptual phenomenon of monocular polyplopia. The theory provides two major conclusions. First, the most likely event giving rise to multiple PSF cores is the presence of hyperbolic, or less probably elliptical, umbilic caustics (using the terminology of catastrophe optics). Second, those umbilic caustics formed on the retinal surface are associated with certain points of the wave aberration function, called cusps of Gauss, where the gradient of a curvature function vanishes. However, not all cusps of Gauss generate those umbilic caustics. We also provide necessary conditions for those cusps of Gauss to be fertile. To show the potential of this theoretical framework for understanding the nature and origin of polyplopia, we provide specific examples of ocular wave aberration functions that induce diplopia and triplopia. The polyplopia effects in these examples are illustrated by depicting the multi-core PSFs and the convolved retinal images for clinical letter charts, both through computer simulations and through experimental recording using an adaptive optics set-up. The number and location of cores in the PSF is thus a potentially useful metric for the existence and severity of polyplopia in spatial vision. These examples also help explain why physiological pupil constriction might reduce the incidence of ghosting and multiple images of daily objects that affect vision with dilated pupils. This mechanistic explanation suggests a possible role for optical phase-masking as a clinical treatment for polyplopia and ghosting.

Astigmatic Stokes lens revisited.

Stokes lenses are variable power astigmatic lenses comprising of, in its standard version, two pure cylindrical lenses of equal but contrary power that rotate in opposite directions. Here, we present an optical device with variable and continuous astigmatic power which is based on a modified Stokes lens where two sphero-cylindrical lenses (in the form of pure astigmatic lenses) are combined in the classical way but merged with another fixed pure astigmatic lens for improving the capabilities of the resulting optical device concerning the expansion of the astigmatic range without worsening the dioptric power step resolution. The performance of this device is theoretically analyzed in virtue of the power vectors formalism including a three-dimensional (3-D) representation of the generated dioptric power as a function of both the meridian and the rotation angle between the cylinder's axes. In addition, we have assembled a lab-made prototype of the proposed modified Stokes lens and validated its theoretical behavior by dioptric power measurements with an automatic focimeter. As conventional Stokes lenses, the applications of this new optical device range from astigmatism compensation in optical instruments to measurement of refractive error in subjective routines with the previously commented improved capabilities.

The blur horopter: Retinal conjugate surface in binocular viewing.

From measurements of wavefront aberrations in 16 emmetropic eyes, we calculated where objects in the world create best-focused images across the central 27\(^\circ\) (diameter) of the retina. This is the retinal conjugate surface. We calculated how the surface changes as the eye accommodates from near to far and found that it mostly maintains its shape. The conjugate surface is pitched top-back, meaning that the upper visual field is relatively hyperopic compared to the lower field. We extended the measurements of best image quality into the binocular domain by considering how the retinal conjugate surfaces for the two eyes overlap in binocular viewing. We call this binocular extension the blur horopter. We show that in combining the two images with possibly different sharpness, the visual system creates a larger depth of field of apparently sharp images than occurs with monocular viewing. We examined similarities between the blur horopter and its analog in binocular vision: the binocular horopter. We compared these horopters to the statistics of the natural visual environment. The binocular horopter and scene statistics are strikingly similar. The blur horopter and natural statistics are qualitatively, but not quantitatively, similar. Finally, we used the measurements to refine what is commonly referred to as the zone of clear single binocular vision.

Resolution acuity across the visual field for mesopic and scotopic illumination.

We investigated the classical question of why visual acuity decreases with decreasing retinal illuminance by holding retinal eccentricity fixed while illumination varied. Our results indicate that acuity is largely independent of illuminance at any given retinal location, which suggests that under classical free-viewing conditions acuity improves as illumination increases from rod threshold to rod saturation because the retinal location of the stimulus is permitted to migrate from a peripheral location of maximum sensitivity but poor acuity to the foveal location of maximum acuity but poor sensitivity. Comparison with anatomical sampling density of retinal neurons suggests that mesopic acuity at all eccentricities and scotopic acuity for eccentricities beyond about 20° is limited by the spacing of midget ganglion cells. In central retina, however, scotopic acuity is further limited by spatial filtering due to spatial summation within the large, overlapping receptive fields of the A-II class of amacrine cells interposed in the rod pathway between rod bipolars and midget ganglion cells. Our results offer a mechanistic interpretation of the clinical metrics for low-luminance visual dysfunction used to monitor progression of retinal disease.

Combining optical and neural components in physiological visual image quality metrics as functions of luminance and age.

Visual image quality metrics combine comprehensive descriptions of ocular optics (from wavefront error) with a measure of the neural processing of the visual system (neural contrast sensitivity). To improve the ability of these metrics to track real-world changes in visual performance and to investigate the roles and interactions of those optical and neural components in foveal visual image quality as functions of age and target luminance, models of neural contrast sensitivity were constructed from the literature as functions of (1) retinal illuminance (Trolands, td), and (2) retinal illuminance and age. These models were then incorporated into calculation of the visual Strehl ratio (VSX). Best-corrected VSX values were determined at physiological pupil sizes over target luminances of 104 to 10-3 cd/m2 for 146 eyes spanning six decades of age. Optical and neural components of the metrics interact and contribute to visual image quality in three ways. At target luminances resulting in >900 td at physiological pupil size, neural processing is constant, and only aberrations (that change as pupil size changes with luminance) affect the metric. At low mesopic luminances below where pupil size asymptotes to maximum, optics are constant (maximum pupil), and only the neural component changes with luminance. Between these two levels, both optical and neural components of the metrics are affected by changes in target luminance. The model that accounted for both retinal illuminance and age allowed VSX, termed VSX(td,a), to best track visual acuity trends (measured at 160 and 200 cd/m2) as a function of age (20s through 70s) from the literature. Best-corrected VSX(td,a) decreased by 2.24 log units between maximum and minimum target luminances in the youngest eyes and by 2.58 log units in the oldest. The decrease due to age was more gradual at high target luminances (0.70 log units) and more pronounced as target luminance decreased (1.04 log units).

Retinal Image Formation and Sampling in a Three-Dimensional World.

In this review, I develop an empirically based model of optical image formation by the human eye, followed by neural sampling by retinal ganglion cells, to demonstrate the perceptual effects of blur, aliasing, and distortion of visual space in the brain. The optical model takes account of ocular aberrations and their variation across the visual field, in addition to variations of defocus due to variation of target vergence in three-dimensional scenes. Neural sampling by retinal ganglion cells with receptive field size and spacing that increases with eccentricity is used to visualize the neural image carried by the optic nerve to the brain. Anatomical parameters are derived from psychophysical studies of sampling-limited visual resolution of sinusoidal interference fringes. Retinotopic projection of the neural image onto brainstem nuclei reveals features of the neural image in a perceptually uniform brain space where location and size of visual objects may be measured by counting neurons.

Customized models of ocular aberrations across the visual field during accommodation.

We aimed to create individual eye models that accurately reproduce the empirical measurements of wave-front aberrations across the visual field at different accommodative states, thus providing a mechanistic explanation for the changes in the eye's aberration structure due to accommodation. Structural parameters of a generic eye model were optimized using optical design software to account for published measurements of wave-front aberrations measured for 19 individuals at 37 test locations over the central 30°-diameter visual field at eight levels of accommodative demand. Biometric data for individual eyes were used as starting values and normative data were used to constrain optimizations to anatomically reasonable values. Customizations of the accommodating eye model accurately accounted for ocular aberrations over the central 30° of visual field with an averaged root mean square fitting error typically below 0.2 µm at any given field location. Optimized structural parameters of the eye models were anatomically reasonable and changed in the expected way when accommodating. Accuracy for representing spherical aberration was significantly improved by relaxing anatomical constraints on the anterior surface of the lens to compensate for not including gradient-index media. Use of the model to compute pan-retinal image quality revealed large penalties of accommodative lag for activating photoreceptor responses to the retinal image.

Calculation of the geometrical point-spread function from wavefront aberrations.

PURPOSE: This report uses the principles of geometrical optics to compute the optical point-spread function (PSF) from the wavefront error function. METHOD: Step 1 uses Prentice's rule to determine the spatial form of the PSF established by tracing a field of rays from the eye's exit pupil to the retina. Ray vergence is related to the slope of the wavefront error function, which enables the mapping of light rays to produce a retinal 'spot diagram'. Step 2 completes the PSF by assigning an irradiance value to each ray in the spot diagram. RESULTS AND CONCLUSIONS: Spot irradiance is inversely proportional to the Gaussian curvature (i.e. the product of principal curvatures) of each local region of wavefront error surface centered on the corresponding ray. The Gaussian curvature, in turn, may be computed as the determinant of the vergence error matrix associated with each point on the wavefront error surface. Elements of the vergence error matrix consist of sums and differences of the local power vector components M, J0 and J45 . This method is shown to be equivalent to published derivations of the geometric PSF using the Jacobian of the ray mapping function and equivalent also to the Hessian of the wavefront error function. Examples are presented for the familiar cases of spherical and astigmatic blur as well as for higher order aberrations and the formation of caustics in the retinal image.

Ricco's law and absolute threshold for foveal detection of black holes.

Visual detection of small black objects surrounded by a light background depends on background luminance, pupil size, optical blur, and object size. Holding pupil and optics fixed, we measured the minimum background luminance needed for foveal detection of small black targets as a function of target size. For all three observers, absolute threshold varied inversely with target area when disk diameter subtended less than 10' of visual angle. For target diameter ≥10', threshold remained constant at about 0.3 Td, which was also the absolute threshold for detecting light spots 10' or larger in diameter on a black background. These results are consistent with Ricco's law of spatial summation: a "black hole" is just detectable when the background luminance is sufficiently high for its absence inside the Ricco area to reduce 555 nm photon flux by 7500 photons/s, which is the same change needed to detect light spots on a black surround. These results can be accounted for by a differential pair of Ricco detectors, each about the size of the receptive field center of magocellular retinal ganglion cells when projected into object space through the eye's weakly aberrated optical system. Statistical analysis of the model suggests the quantum fluctuations due to internal, biological noise (i.e., "scotons") are a greater handicap than the photon fluctuations inherent in the light stimulus at absolute foveal threshold.

Psychophysical study of the optical origin of starbursts.

Starbursts seen around small bright lights at night have been attributed to optical scatter, diffraction, or aberrations. We manipulated pupil aperture and aberrations to investigate the entopic appearance of perceived starbursts. The impact of circular, annular, and wedge-shaped pupil apertures, and spherical aberration sign and magnitude were used to identify pupil sub-apertures responsible for each radial perceived starburst line. Local intensity distributions within the starbursts mapped onto unique sub-regions of the pupil of both phakic and pseudophakic eyes, consistent with the hypothesis that ocular aberrations are the cause of starbursts. In paraxially focused eyes, the size of starbursts is predicted by the amount of spherical aberration, and starburst orientation is either the same or 180 deg rotated from the pupil region that creates each starburst line. No starbursts are seen when the pupil diameter is smaller than 3 mm. Replacing the eye's natural lens with a radially symmetric and optically homogeneous intraocular lens reduced the observed number of starbursts by 50%. Geometrical optics modeling including the measured aberrations of an individual eye can reveal point spread function structure that captures some of the key elements of the entopic perceptions.

Small Text on Product Labels Poses a Special Challenge for Emerging Presbyopes.

SIGNIFICANCE: Approximately 10% of the lowercase text on nonprescription drug labels is smaller than the 1 mm required by the Food and Drug Administration. The small size, combined with the progressive decline in accommodative amplitude and gain, poses a reading challenge for middle-aged emerging presbyopes. PURPOSE: The purpose of this study was to evaluate the impact of progressing presbyopia and near adds on the ability of middle-aged patients to read small text routinely encountered on product labels. METHODS: Geometrical optics was used to determine the impact of changing viewing distance, accommodation, and pupil size on retinal blur size. We photographed 261 consumer product labels in grocery, personal care, and nonprescription (over-the-counter) drug categories and used character recognition software to identify and size 255,298 printed letters. We computed the impact of viewing distance on the ratio of blur to letter detail and used published blur ratios of ≤4 and ≤2 to identify the conditions that allowed for letter recognition and proficient reading, respectively. RESULTS: Median/mode lowercase letter heights (in millimeters) were 1.39/1.16, 1.29/1.15, and 1.18/1.01, respectively, for groceries, personal care, and over-the-counter drug categories. Despite the increased angular subtense of approaching letters, blur ratios generally increased with reduced viewing distance because of increased defocus. Increased viewing distance decreased blur, but small (e.g., 1 mm) letters became too small to read proficiently (angular size <10 arcminutes) for distances beyond 37 cm. With larger pupils, blur ratios were too large to support proficient reading when accommodative amplitude dropped to ≤3 diopters. An add power sufficient to bring the far point closer than 37 cm was required to proficiently read small text. CONCLUSIONS: Product labels, especially nonprescription drug packages, typically use fonts that are too small to be read proficiently by unaided emmetropes with emerging presbyopia. This problem can be ameliorated by correction of presbyopia at an earlier age and with higher add powers.

The effect of spherical aberration on visual performance and refractive state for stimuli and tasks typical of night viewing / Efecto de la aberración esférica sobre el rendimiento visual y el estado refractivo para los estímulos y tareas típicos de la visión nocturna

Purpose: The aim of this work was to examine the impact of Seidel spherical aberration (SA) on optimum refractive state for detecting and discriminating small bright lights on a dark background. Methods: An adaptive-optics system was used to correct ocular aberrations of cyclopleged eyes and then systematically introduce five levels of Seidel SA for a 7-mm diameter pupil: 0, ± 0.18, and ± 0.36 diopters (D )mm-2. For each level of SA, subjects were required to detect one or resolve two points of light (0.54 arc min diameter) on a dark background. Refractive error was measured by adjusting stimulus vergence to minimize detection and resolution thresholds. Two other novel focusing tasks for single points of light required maximizing the perceived intensity of a bright point's core and minimizing its overall perceived size (i.e. minimize starburst artifacts). Except for the detection task, luminance of the point of light was 1000 cd m-2 on a black background lower than 0.5 cdm-2. Results: Positive SA introduced myopic shifts relative to the best subjective focus for dark letters on a bright background when there was no SA, whereas negative SA introduced hyperopic shifts in optimal focus. The changes in optimal focus were -1.7, -2.4, -2.0, and -9.2 D of focus per D mm-2 of SA for the detection task, resolution task, and maximization of core's intensity and minimization of size, respectively. Conclusion: Ocular SA can be a significant contributor to changes in refractive state when viewing high-contrast point sources typically encountered in nighttime environments

Objetivo: El objetivo de este estudio fue examinar el impacto de la aberración esférica de Seidel (AS) sobre el estado refractivo óptimo para detectar y discriminar las luces brillantes de pequeño tamaño sobre un fondo oscuro. Métodos: Se utilizó un sistema de óptica adaptativa para corregir las aberraciones oculares de ojos bajo cicloplegía, e introducir sistemáticamente cinco valores de AS para una pupila de 7 mm de diámetro: 0, ± 0,18, y ± 0,36 dioptrías (D) mm-2. Para cada valor de AS se solicitó a los sujetos que detectaran un punto de luz, o resolvieran dos puntos (cada punto subtendía 0,54 minutos de arco de diámetro) sobre un fondo oscuro. El error refractivo se midió ajustando la vergencia del estímulo, para minimizar los umbrales de detección y resolución. Los sujetos realizaron además otras dos tareas observando un sólo punto luminoso y en las que tenían que maximizar la intensidad percibida del núcleo del punto luminoso o minimizar el tamaño de la imagen percibida (es decir, minimizar el "starburst"). Excepto para la tarea de detección, la luminancia del punto de luz fue de 1000 cd m-2 sobre un fondo negro con un valor inferior a 0,5 cd m-2. Resultados: La AS positiva introdujo cambios miópicos respecto al mejor enfoque subjetivo para letras oscuras sobre un fondo luminoso sin AS, mientras que la AS negativa introdujo cambios hipermetrópicos respecto al enfoque óptimo. Estos cambios fueron -1,7,-2,4,-2,0, y -9,2 D de enfoque por D mm-2 de AS para la tarea de detección, la tarea de resolución, la maximización de la intensidad del núcleo y la minimización de su tamaño, respectivamente. Conclusión: La AS ocular puede ser un factor que influye significativamente en los cambios en el estado refractivo, al visualizar las fuentes puntuales de alto contraste típicas de los entornos nocturnos

Normative best-corrected values of the visual image quality metric VSX as a function of age and pupil size.

The visual image quality metric the visual Strehl ratio (VSX) combines a comprehensive description of the optics of an eye (wavefront error) with an estimate of the photopic neural processing of the visual system, and has been shown to be predictive of subjective best focus and well correlated with change in visual performance. Best-corrected visual image quality was determined for 146 eyes, and the quantitative relation of VSX, age, and pupil size is presented, including 95% confidence interval norms for age groups between 20 and 80 years and pupil diameters from 3 to 7 mm. These norms were validated using an independently collected population of wavefront error measurements. The best visual image quality was found in young eyes at smaller pupil sizes. Increasing pupil size caused a more rapid decrease in VSX than increasing age. These objectively determined benchmarks represent the best theoretical levels of visual image quality achievable with a sphere, cylinder, and axis correction in normal eyes and can be used to evaluate both traditional and wavefront-guided optical corrections provided by refractive surgery, contact lenses, and spectacles.

What is a troland?

Trolands are a widely used measure of retinal illuminance in vision science and visual optics, but disagreements exist for the definition and interpretation of this photometric unit. The purpose of this communication is to resolve the confusion by providing a sound conceptual basis for interpreting trolands as a measure of angular flux density incident upon the retina. Using a simplified optical analysis, we show that the troland value of an extended source is the intensity in micro-candelas of an equivalent point source located at the eye's posterior nodal point that produces the same illuminance in the retinal image as does the extended source. This optical interpretation of trolands reveals that total light flux in the image of an extended object is the product of the troland value of the source and the solid angle subtended by the source at the first nodal point, independent of eye size.

Two-dimensional simulation of eccentric photorefraction images for ametropes: factors influencing the measurement.

PURPOSE: Eccentric photorefraction and Purkinje image tracking are used to estimate refractive state and eye position simultaneously. Beyond vision screening, they provide insight into typical and atypical visual development. Systematic analysis of the effect of refractive error and spectacles on photorefraction data is needed to gauge the accuracy and precision of the technique. METHODS: Simulation of two-dimensional, double-pass eccentric photorefraction was performed (Zemax). The inward pass included appropriate light sources, lenses and a single surface pupil plane eye model to create an extended retinal image that served as the source for the outward pass. Refractive state, as computed from the luminance gradient in the image of the pupil captured by the model's camera, was evaluated for a range of refractive errors (-15D to +15D), pupil sizes (3 mm to 7 mm) and two sets of higher-order monochromatic aberrations. Instrument calibration was simulated using -8D to +8D trial lenses at the spectacle plane for: (1) vertex distances from 3 mm to 23 mm, (2) uncorrected and corrected hyperopic refractive errors of +4D and +7D, and (3) uncorrected and corrected astigmatism of 4D at four different axes. Empirical calibration of a commercial photorefractor was also compared with a wavefront aberrometer for human eyes. RESULTS: The pupil luminance gradient varied linearly with refractive state for defocus less than approximately 4D (5 mm pupil). For larger errors, the gradient magnitude saturated and then reduced, leading to under-estimation of refractive state. Additional inaccuracy (up to 1D for 8D of defocus) resulted from spectacle magnification in the pupil image, which would reduce precision in situations where vertex distance is variable. The empirical calibration revealed a constant offset between the two clinical instruments. CONCLUSIONS: Computational modelling demonstrates the principles and limitations of photorefraction to help users avoid potential measurement errors. Factors that could cause clinically significant errors in photorefraction estimates include high refractive error, vertex distance and magnification effects of a spectacle lens, increased higher-order monochromatic aberrations, and changes in primary spherical aberration with accommodation. The impact of these errors increases with increasing defocus.

The effect of spherical aberration on visual performance and refractive state for stimuli and tasks typical of night viewing.

PURPOSE: The aim of this work was to examine the impact of Seidel spherical aberration (SA) on optimum refractive state for detecting and discriminating small bright lights on a dark background. METHODS: An adaptive-optics system was used to correct ocular aberrations of cyclopleged eyes and then systematically introduce five levels of Seidel SA for a 7-mm diameter pupil: 0,±0.18, and±0.36diopters (D)mm-2. For each level of SA, subjects were required to detect one or resolve two points of light (0.54 arc min diameter) on a dark background. Refractive error was measured by adjusting stimulus vergence to minimize detection and resolution thresholds. Two other novel focusing tasks for single points of light required maximizing the perceived intensity of a bright point's core and minimizing its overall perceived size (i.e. minimize starburst artifacts). Except for the detection task, luminance of the point of light was 1000cdm-2 on a black background lower than 0.5cdm-2. RESULTS: Positive SA introduced myopic shifts relative to the best subjective focus for dark letters on a bright background when there was no SA, whereas negative SA introduced hyperopic shifts in optimal focus. The changes in optimal focus were -1.7, -2.4, -2.0, and -9.2D of focus per Dmm-2 of SA for the detection task, resolution task, and maximization of core's intensity and minimization of size, respectively. CONCLUSION: Ocular SA can be a significant contributor to changes in refractive state when viewing high-contrast point sources typically encountered in nighttime environments.

Reducing starbursts in highly aberrated eyes with pupil miosis.

PURPOSE: To test the hypothesis that marginal ray deviations determine perceived starburst sizes, and to explore different strategies for decreasing starburst size in highly aberrated eyes. METHODS: Perceived size of starburst images and visual acuities were measured psychophysically for eyes with varying levels of spherical aberration, pupil sizes, and defocus. Computationally, we use a polychromatic eye model including the typical levels of higher order aberrations (HOAs) for keratoconic and post-LASIK eyes to quantify the image quality (the visually weighted Strehl ratio derived from the optical transfer function, VSOTF) with different pupil sizes at both photopic and mesopic light levels. RESULTS: For distance corrected post-LASIK and keratoconic eyes with a night-time pupil (e.g., 7 mm), the starburst diameter is about 1.5 degrees (1 degree for normal presbyopic eyes), which can be reduced to ≤0.25 degrees with pupil sizes ≤3 mm. Starburst size is predicted from the magnitude of the longitudinal spherical aberration. Refracting the eye to focus the pupil margin also removed starbursts, but, unlike small pupils, significantly degraded visual acuity. Reducing pupil diameter to 3 mm improved image quality for these highly aberrated eyes by about 2.7 ×  to 1.7 ×  relative to the natural pupils when light levels were varied from 0.1 to 1000 cd m-2 , respectively. CONCLUSION: Subjects with highly aberrated eyes observed larger starbursts around bright lights at night predictable by the deviated marginal rays. These were effectively attenuated by reducing pupil diameters to ≤3 mm, which did not cause a drop in visual acuity or modelled image quality even at mesopic light levels.

Interaction of aberrations, diffraction, and quantal fluctuations determine the impact of pupil size on visual quality.

Our purpose is to develop a computational approach that jointly assesses the impact of stimulus luminance and pupil size on visual quality. We compared traditional optical measures of image quality and those that incorporate the impact of retinal illuminance dependent neural contrast sensitivity. Visually weighted image quality was calculated for a presbyopic model eye with representative levels of chromatic and monochromatic aberrations as pupil diameter was varied from 7 to 1 mm, stimulus luminance varied from 2000 to 0.1 cd/m2, and defocus varied from 0 to -2 diopters. The model included the effects of quantal fluctuations on neural contrast sensitivity. We tested the model's predictions for five cycles per degree gratings by measuring contrast sensitivity at 5 cyc/deg. Unlike the traditional Strehl ratio and the visually weighted area under the modulation transfer function, the visual Strehl ratio derived from the optical transfer function was able to capture the combined impact of optics and quantal noise on visual quality. In a well-focused eye, provided retinal illuminance is held constant as pupil size varies, visual image quality scales approximately as the square root of illuminance because of quantum fluctuations, but optimum pupil size is essentially independent of retinal illuminance and quantum fluctuations. Conversely, when stimulus luminance is held constant (and therefore illuminance varies with pupil size), optimum pupil size increases as luminance decreases, thereby compensating partially for increased quantum fluctuations. However, in the presence of -1 and -2 diopters of defocus and at high photopic levels where Weber's law operates, optical aberrations and diffraction dominate image quality and pupil optimization. Similar behavior was observed in human observers viewing sinusoidal gratings. Optimum pupil size increases as stimulus luminance drops for the well-focused eye, and the benefits of small pupils for improving defocused image quality remain throughout the photopic and mesopic ranges. However, restricting pupils to <2 mm will cause significant reductions in the best focus vision at low photopic and mesopic luminances.

Variation of axial and oblique astigmatism with accommodation across the visual field.

In this study we investigated the impact of accommodation on axial and oblique astigmatism along 12 meridians of the central 30° of visual field and explored the compensation of corneal first-surface astigmatism by the remainder of the eye's optical system. Our experimental evidence revealed no systematic effect of accommodation on either axial or oblique astigmatism for two adult populations (myopic and emmetropic eyes). Although a few subjects exhibited systematic changes in axial astigmatism during accommodation, the dioptric value of these changes was much smaller than the amount of accommodation. For most subjects, axial and oblique astigmatism of the whole eye are both less than for the cornea alone, which indicates a compensatory role for internal optics at all accommodative states in both central and peripheral vision. A new method for determining the eye's optical axis based on visual field maps of oblique astigmatism revealed that, on average, the optical axis is 4.8° temporal and 0.39° superior to the foveal line-of-sight in object space, which agrees with previous results obtained by different methodologies and implies that foveal astigmatism includes a small amount of oblique astigmatism (0.06 D on average). Customized optical models of each eye revealed that oblique astigmatism of the corneal first surface is negligible along the pupillary axis for emmetropic and myopic eyes. Individual variation in the eye's optical axis is due in part to misalignment of the corneal and internal components that is consistent with tilting of the crystalline lens relative to the pupillary axis.