Retinal image formation

Though its optical components are made of living tissue instead of glass, the eye forms a remarkably good retinal image. The quality of this image depends on diffraction at the pupil, aberrations in the cornea and lens, light scatter in the optical media, and the optical properties of the retina. Fourier optics succinctly describes the effects of diffraction and aberrations. For an excellent treatment of Fourier optics in imaging systems generally, see Goodman (1996). See Thibos (2000) or Packer and Williams (2003) for treatments pertaining to the eye.

The Wave Aberration

Figure 50.1A shows a perfect and an aberrated eye forming a retinal image of a distant object such as a star. The parallel light rays arriving at the eye from the star can be described alternatively as a series of planar wave fronts. The wave front is always perpendicular to each ray at the point of intersection. The optics of a perfect eye transform this planar wave front into a spherical wave front which collapses to form a compact light distribution on the retina. The perfect eye delays light traveling through different parts of the pupil so that all the light takes exactly the same time to reach the retinal location where the image of the star is formed.

Figure 50.1..  

A, Retinal image formation for an aberration-free eye. Parallel rays from a distant point source converge inside the eye to a single point on the retina. If light is described as a wave, image formation consists of a planar wave front incident on the cornea that is transformed into a spherical wave front, which collapses to a single point on the retina. B, In the case of all real eyes, aberrations deviate the rays so that they do not converge to a point. Alternatively, the wave front is distorted from the ideal spherical shape, and the retinal image is blurred.

In the aberrated eye shown in Figure 50.1B, the wave front is not delayed by the proper amounts and the wave front inside the eye departs from its ideal spherical shape. It fails to collapse to a compact point at the retina, and blurring is inevitable. Aberrations arise from several sources, such as a misshapen cornea or lens, and these errors add. The contributions of the cornea and lens to these errors are often compensatory, especially in young eyes, so that the overall optical quality of the eye is superior to that of either element alone (Artal et al., 2001). We can sum all the errors experienced by a photon passing through the cornea and lens and assign the sum to the point in the entrance pupil of the eye through which the photon passed. It is convenient to express the total error in a unit of distance, such as micrometers (µm), indicating how far the distorted wave front departs at each point in the entrance pupil from the ideal wave front. A map of the eye's entrance pupil that plots the error at each entry point is the eye's wave aberration. The wave aberration captures in one function all the aberrations for a particular wavelength of light that ultimately influence image quality at a particular location on the retina. Figures 50.2A–D are wave aberrations for four typical human subjects. Like a fingerprint, the wave aberration is different in different people, though there is often some degree of mirror symmetry between eyes of the same person (Liang and Williams, 1997; Porter et al., 2001). Figure 50.2E shows the wave aberration for an ideal eye, which is zero at every point in the pupil.

Figure 50.2..  

A–E, Wave aberrations of real eyes are shown in A–D; the flat wave aberration of an ideal, aberration-free eye is shown in E. The pupil diameter was 5.7 mm. F–J, The point spread functions computed from the wave aberrations shown in A–E. K–O, The result of convolving the point spread function with the letter E, which subtended 0.5 minute of arc. Note the increased blur in the real eyes compared to the diffraction-limited eye. The calculations were performed assuming white light, axial chromatic aberration in the eye, and that a wavelength of 555 nm was in best focus on the retina.

Decomposing the Wave Aberration into Individual Aberrations

To understand more clearly what is wrong with the optics of an individual eye, one can decompose the wave aberration into a number of component aberrations. A Zernike decomposition is used in much the same way that Fourier analysis is used to decompose an image into spatial frequency components. Zernike polynomials are defined on the unit circle, which approximates the natural shape of the eye's pupil. Figure 50.3 shows the pyramid of Zernike modes, akin to the periodic table of elements. Three modes, piston, tip, and tilt, that cap the pyramid have been omitted because these modes do not influence image quality. Modes are characterized by their radial order, which refers to the exponent that describes how the function behaves in the radial direction from the center of the pupil.

Figure 50.3..  

The pyramid showing each of the Zernike modes in radial orders 2 through 5, along with their names and their designation (OSA Standard). Tip, tilt, and piston, which would normally cap the pyramid, have been excluded because they do not influence image quality.

Figure 50.4 shows the average magnitude of these modes in a normal population (Porter et al., 2001, see also Castejon-Mochon et al., 2002). The higher-order monochromatic aberrations were measured with a Shack-Hartmann wave front sensor in a population of 109 normal subjects for a 5.7 mm pupil size. The magnitude corresponds to the absolute value of the Zernike coefficient associated with each Zernike mode. Each Zernike coefficient is the root mean square deviation of the wave front from the ideal wave front for each particular Zernike mode expressed in microns. The largest monochromatic aberration of the eye is typically defocus, followed by astigmatism. These are the only aberrations that conventional spectacles and contact lenses correct. However, normal eyes have many monochromatic aberrations besides defocus and astigmatism (Berny and Slansky, 1969; Howland and Howland, 1977; Liang and Williams, 1997; Liang et al., 1994; Sergienko, 1963; Smirnov, 1961; van den Brink, 1962; Walsh and Charman, 1985; Walsh et al., 1984; Webb et al., 1992). Generally speaking, Figure 50.4 shows that aberrations corresponding to more rapidly varying errors across the pupil (i.e., higher radial order) have smaller amplitudes and make a smaller contribution to the total wave aberration for normal eyes.

Figure 50.4..  

Mean absolute root mean square (RMS) values for 18 Zernike modes as measured from a population of 109 normal human subjects for a 5.7 mm pupil. The percentages above some modes indicate the percentage of the total wave aberration variance accounted for by each mode. The inset excludes the first Zernike mode corresponding to defocus and has an expanded ordinate to illustrate the magnitudes of higher-order aberrations.

The Pupil Function

The retinal image is made up of light that arrives from many different locations in the pupil. In addition to depending on the eye's aberrations, the quality of the retinal image depends on how much light entering each location in the pupil actually gets caught by the retinal receptors. The pupil function succinctly captures both the wave aberration and the variation of the effectiveness of light at different points in the pupil. The pupil function is

where (η,ς) are two-dimensional spatial coordinates in the entrance pupil and λ is the wavelength of light used to measure the eye's wave aberration, W(η,ς). P₀(η,ς) is the amplitude transmittance across the eye's optics, the truncating effect of the iris being the most important. The variation in the absorption of the cornea and lens across the pupil is small enough in the normal eye that it can usually be ignored. However, because of the antenna properties of cones, the quantum efficiency of the retina depends on the entry point of light in the pupil. Though this effect, known as the Stiles-Crawford effect, is caused by the optics of the retina, it is equivalent to reducing the amplitude transmittance toward the margin of the entrance pupil (see Chapter 53). Therefore, the generalized pupil function for the eye includes the directional sensitivity of the retina in P₀(η,ς).

The Point Spread Function

In evaluating the quality of an imaging system such as the eye, it is often useful to characterize the system's point spread function (PSF). The PSF describes the distribution of light in the retinal image when the eye is viewing a distant point source, such as a star, and provides a complete description of image quality at that retinal location for a given wavelength of light. Figures 50.2F–J show the PSFs corresponding to the wave aberrations of Figures 50.2A–E, respectively. Because of aberrations in real eyes, the PSFs are broader and more irregular than the aberration-free PSF for the same 6 mm pupil size, shown in Figure 50.2J.

The PSF is the squared modulus of the Fourier transform of the pupil function. That is,

where the Fourier transform is given by

This is of fundamental importance, because it allows us to compute the retinal image of a particular object, a single point of light, based on knowledge of the pupil function. Knowledge of the retinal image of a single point of light allows us to compute the retinal image of any object, as we describe later. This is a very powerful result, because it provides a firm quantitative link between aberrations in the eye and their impact on the retinal image.

Diffraction

An analysis based solely on geometrical optics might lead one to suppose that the PSF in the aberration-free eye of Figure 50.2J would be a single point. But even in this case, the light is spread out somewhat across the retina due to diffraction at the pupil, an inevitable consequence of the wave nature of light. The PSF shown in Figure 50.2J has a bright central core, called the Airy disk, surrounded by dimmer rings. The PSF can be described quantitatively by

where I(r) is normalized intensity as a function of distance r from the peak and J₁ is a Bessel function of the first kind. The radius of the PSF, r₀, expressed in radians and measured from the peak to the first point at which the intensity is zero, is given by

where λ is the wavelength of light and α is the diameter of the circular pupil. Because the width of the diffraction-limited PSF is proportional to wavelength and inversely proportional to pupil diameter, the retinal image quality of this ideal eye is optimum at large pupil sizes and short wavelengths. Figure 50.5 illustrates the inverse relationship between PSF diameter and pupil size for an aberration-free eye.

Figure 50.5..  

PSFs for an ideal, aberration-free eye limited only by diffraction for various pupil diameters. Note that the diameter of the PSF is inversely proportion to pupil diameter. (Courtesy of Austin Roorda.)

Light Scatter

In addition to blurring by diffraction and aberrations, the eye's PSF is blurred by light scatter in the anterior optics and retina (Vos, 1963). The sources of intraocular scattered light are (1) forward scatter from the cornea, (2) forward scatter from the lens, (3) forward scatter from the retina, and (4) back scatter from the fundus. Roughly a quarter of the scatter comes from the cornea, another quarter from the retina, and the remaining half from the lens. Since most of the forward scattering is by relatively large particles, the scattered light in the eye does not show a strong wavelength dependence (Wooten and Geri, 1987).

Scatter tends to contribute predominantly to broaden and increase the skirt of the PSF. This has the effect of adding a veiling illumination to the retinal image, reducing image contrast at relatively low as well as high spatial frequencies. In the young eye, scatter is not a major source of image blur. It becomes important primarily in the aging eye (Guirao et al., 1999; Westheimer and Liang, 1995), or in the young eye when the observer is required to detect a relatively dim object in the vicinity of a much brighter object. The PSF due to scatter can be summed with the PSF due to diffraction and aberrations, at which point the retinal image can be computed for any object, as described below.

Computing Retinal Images

Once we have the PSF, the retinal image for any arbitrary object can be computed either in the spatial domain or in the frequency domain. In the spatial domain, the intensity distribution of the image, I(x, y), is the convolution of the PSF with the intensity distribution of the object, O(x, y). That is,

where M is the magnification between the object and image planes. The convolution of two functions f(x, y) and g(x, y) is

In practice, the computation of the retinal image is more efficient in the spatial frequency domain. In that case, the intensity distribution of the object, O(x, y), is Fourier transformed to provide the object Fourier spectrum, σ(f_x, f_y). That is,

The object Fourier spectrum is then multiplied by the optical transfer function (OTF) of the eye to give the image Fourier spectrum:

By taking the inverse Fourier transform of the image spectrum, one obtains the retinal image

The OTF is the autocorrelation of the pupil function. Alternatively, the OTF can be computed by taking the Fourier transform of the PSF. The OTF is complex, consisting of two parts, a modulation transfer function (MTF) and a phase transfer function (PTF). The MTF indicates how faithfully the contrast of each spatial frequency component of the object is transferred to the image. The PTF indicates how individual spatial frequency components of the object have been translated in the retinal image.

Estimates of the Point Spread Function and Modulation Transfer Function

Figure 50.6 illustrates how the PSF changes with pupil diameter for a typical human eye when both aberrations and diffraction are taken into account. At small pupil sizes, aberrations are insignificant and diffraction dominates. The PSF takes on the characteristic shape of the Airy pattern, with a wide core and little light in the skirt around it. At larger pupil sizes, aberrations dominate. The PSF then has a small core but reveals an irregular skirt that corresponds to light distributed over a relatively large retinal area. A pupil size of roughly 3 mm in diameter represents a good compromise between blur due to diffraction and aberrations (Campbell and Gubisch, 1966). In this case, the full width at half height of the PSF is approximately 0.8 minute of arc, corresponding to nearly twice the width of a cone at the foveal center.

Figure 50.6..  

PSFs for a typical real eye for various pupil diameters. Defocus and astigmatism have been zeroed. Note that diffraction controls the PSF shape for small pupils, whereas aberrations dominate for large pupils. (Courtesy of Austin Roorda.)

Retinal image quality is often represented by the MTF instead of the PSF, mainly because only relatively recently has it been possible, with the advent of wave front sensing, to recover the PSF. Only the MTF was accessible with earlier methods of measuring the optical quality of the eye, such as laser interferometry and the double pass technique (Artal et al., 1995). The MTF by itself is not as complete a description of the eye's optics as the PSF, because it does not include the PTF. Image quality in the human eye depends on the PTF when the pupil is large (Charman and Walsh, 1985). Furthermore, accurate phase information is important for the perception of complex scenes (Piotrowski and Campbell, 1982). Nonetheless, the MTF has its uses, especially in situations where the contrast sensitivity of the eye is involved. In that case, knowledge of the MTF of the optics allows the segregation of optical and neural factors in visual performance.

The solid curves in Figure 50.7 show MTFs for a diffraction-limited eye with pupil diameters ranging from 2 to 7 mm calculated for a wavelength of 555 nm.

Figure 50.7..  

MTFs calculated for eyes with pupils ranging from 2 to 7 mm. In each panel, the solid line shows the MTF for an eye whose optics suffer only from diffraction, λ = 555 nm. The dashed lines show the mean monochromatic MTFs of 14 normal human eyes. MTFs were computed from wave aberration measurements obtained with a Hartmann-Shack wave front sensor and a 7.3 mm pupil (Liang and Williams, 1997). The dotted lines show the MTFs expected in white light, taking into account the axial but not the transverse chromatic aberration of the eye. The eye was assumed to be accommodated to 555 nm, and its spectral sensitivity was assumed to correspond to the photopic luminosity function. The MTFs were calculated without defocus and astigmatism by setting the appropriate Zernike terms in the wave aberration to zero. This is not quite the same as finding the values of defocus and astigmatism that optimize image quality, as one does in a conventional clinical refraction. Had such an optimization been performed, the white light and monochromatic MTFs would have been more similar.

Figure 50.7 also shows the mean monochromatic MTFs (dashed lines) computed from the wave aberration measurements for 14 eyes made with a Shack-Hartmann wave front sensor (Liang and Williams, 1997). For small pupil sizes, the MTF is high at low spatial frequencies owing to the absence of light in the skirt of the PSF. For large pupil sizes, however, the MTF is reduced at low spatial frequencies due to the large skirt in the PSF that aberrations produce. However, at high frequencies, the MTF is higher than that for small pupils due to the narrower core of the PSF. The implication of this is that although a 2 to 3 mm pupil is commonly said to represent the best trade-off between diffraction and aberrations, there is no single optimum pupil size. The optimum size depends on the task. Smaller pupils minimize optical aberrations for visual tasks involving low spatial frequencies. Larger pupils transmit high spatial frequencies for tasks that involve fine spatial detail even though they suffer more aberrations. If the goal is to resolve very fine features in images of the retina, then larger pupils transfer more contrast. With increasing pupil diameter, the difference between the diffraction-limited and real MTFs grows due to the decrease in the contribution of diffraction to retinal blur and the increase in the role of aberrations.

Retinal Images in Polychromatic Light

In everyday viewing situations, the eye is confronted with broadband light instead of monochromatic light. The assessment of retinal image quality then demands that the effects of chromatic aberration be taken into account. Chromatic aberration arises because the refractive index of the ocular medium increases with decreasing wavelength. Consequently, any light rays incident on the cornea will generally be bent more if the wavelength is short than if it is long. See Thibos (1987) for a review of chromatic aberration in the eye. There are two kinds of chromatic aberration, axial and transverse. The main effect of axial chromatic aberration is to cause the Zernike polynomial corresponding to defocus to vary with wavelength. All the other aberrations retain roughly the same amplitudes when expressed in microns (Marcos et al., 1999). Across the visible spectrum from 400 to 700 nm, the difference of focus caused by axial chromatic aberration is about 2.25 diopters. Transverse chromatic aberration can manifest itself either as a lateral displacement of a single point in a scene or as a magnification difference of an extended object. In the typical human eye, transverse chromatic aberration does not reduce image quality very much in the fovea.

The blur produced by chromatic aberration does not have especially important effects on spatial vision. The largest aberrations in human eyes are defocus and astigmatism, followed by the aggregate effect of all the remaining, higher-order monochromatic aberrations. Chromatic aberration is not as large as the combined effect of the higher-order aberrations, and its influence on vision is less pronounced. Campbell and Gubisch (1967) found that contrast sensitivity for monochromatic yellow light was only slightly greater than contrast sensitivity for white light. Visual performance on spatial tasks usually depends very little on the S cones, the cone class that would generally experience the greatest retinal image blur due to axial chromatic aberration. Moreover, the proximity of the L and M absorption spectra means that the deleterious effects of axial chromatic aberration will be similar for both cone types. The reduced spectral sensitivity of the eye at short and long wavelengths also reduces the deleterious effects of chromatic aberration. In fact, the 2 diopters of defocus produced by chromatic aberration will have an effect on image quality similar to that of 0.15 diopter of monochromatic defocus, an amount that is close to the limit of what can be detected (Bradley et al., 1991). A less widely recognized reason that chromatic aberration is not more deleterious is that it is overwhelmed by the numerous monochromatic aberrations. These aberrations, most of which spectacles fail to correct, dilute the impact of axial chromatic aberration (Marcos et al., 1999; Yoon and Williams, 2002).

Figure 50.7 compares the MTFs in monochromatic (dashed curves) and broadband (dotted curves) light. These curves were obtained by computing the separate MTFs at each wavelength and then computing from these the mean MTF, weighting the influence of each separate MTF by the photopic luminosity function.