Ординатура / Офтальмология / Английские материалы / Seeing_De Valois_2000

Using this pupil function as a multiplication factor in the integrand allows us to define the integral over the whole plane of the aperture, in which case Equation 20 becomes

−∞ −∞

If this last integral doesn’t look familiar, think of x , y as spatial frequency variables in the plane of the pupil, and think of xˆ, yˆ as spatial variables in the image plane. Now, except for the scaling constant C in front, the Fraunhofer diffraction pattern is recognized as a two-dimensional inverse Fourier transform of the incident wavefront as truncated by the pupil function. This seemingly miraculous result yields yet another astonishing observation when the incident wavefront is a plane wave. In this case the field amplitude U is constant over the aperture, and thus the di raction pattern and the aperture are related by the Fourier transform. A compact notation for this Fourier transform operation uses an arrow to indicate the direction of the forward Fourier transform

In words, Equation 23 says that the amplitude U of the light distribution in a distant plane due to diffraction of a monochromatic plane wave by an aperture is proportional to the inverse Fourier transform of the aperture’s pupil function P.

Equation 22 has had a major impact on optics, including visual optics, in the latter half of the 20th century because it brings to bear the powerful theory of linear systems and its chief computational tool, the Fourier transform (Bracewell, 1969; Gaskill, 1978; Goodman, 1968; Williams & Becklund, 1989). Although cast in the language of di raction patterns, Equation 22 is readily applied to imaging systems by generalizing the concept of a pupil function to include the focusing properties of lenses. By thinking of the pupil function as a two-dimensional filter which attenuates amplitude and introduces phase shifts at each point of the emerging wavefront, a complex-valued pupil function P(x ,y ) may be constructed as the product of two factors

where D(x , y ) is an attenuating factor, and W(x , y ) is a phase factor called the wave aberration function, which is directly attributable to aberrations of the system. This maneuver of generalizing the pupil function captures the e ect of the optical system without violating the arguments which led to the development of Equation 23. Thus, the complex amplitude spread function A(x,y) in the image plane of an aberrated optical system, including di raction e ects, for a distant point

16 Larry N. Thibos

source of light equals the inverse Fourier transform of the pupil function of the system,

A graphical depiction of this important relationship is shown in Figure 9a,c. Ordinary detectors of light, such as retinal photoreceptors, are not able to

respond fast enough to follow the rapid temporal oscillations of light amplitude. Instead, physical detectors respond to the intensity of the light, which is a realvalued quantity defined as the time average of the squared modulus of the complex amplitude. Consequently, the intensity PSF is given by

where A* denotes the complex conjugate of A. A graphical depiction of this important relationship is shown in Figure 9c,d.

Taken together, Equations 25 and 26 say that the intensity PSF, which is a fundamental description of the imaging capabilities of the eye’s optical system, is the squared modulus of the inverse Fourier transform of the eye’s pupil function. The next section shows that the pupil function may also be used to derive another fundamental descriptor of the eye’s imaging system, the optical transfer function. As

FIGURE 9 Fourier relationships between fundamental quantities associated with an optical imaging system.

will be shown, both of these descriptors can be used to compute the retinal image of an arbitrary object in a straight forward manner.

C. Linear Systems Description of Image Formation

One of the major paradigm shifts in optics this century has been the treatment of imaging systems, including the eye, as a linear system characterized in the spatial domain by the PSF. (For an historical account, see Williams & Becklund, 1989). It doesn’t matter whether the image is well focused or blurred, di raction-limited or aberrated. The key assumption is simply that the PSF is invariant to lateral (i.e., orthogonal to the optical axis) translations of the point source. In the theory of linear systems, this property is called space-invariance, but in optics it is called isoplanatism. The special significance of the linear systems approach to the eye is that it allows us to easily compute the actual retinal image (which is normally inaccessible to an outside observer) from knowledge of the PSF and the spatial distribution of intensities in the object.

Although the eye’s PSF varies significantly across the visual field, it is not unreasonable to assume spatial invariance over small patches of the retinal image. Within such a patch the image is conceived as the superposition of a myriad PSFs, one for each point in the object and scaled in intensity according to the intensity of the corresponding point in the object. For ordinary objects there is no fixed relationship between the phases of light waves emitted from di erent points on the object. Such light sources are called spatially incoherent, and for such sources the intensities of elementary PSFs in the retinal image are real-valued quantities which add linearly.Thus the retinal image may be represented by a superposition integral that is equivalent, under the assumption of spatial invariance, to a convolution integral. Using * to denote the convolution operation, we can summarize the imaging process by a simple mathematical relationship

An example of the application of Equation 27 to compute the retinal image expected for an eye with a 4-mm pupil su ering from 1 diopter of defocus is shown in the upper row of Figure 10. For computational purposes, the upper-case letter in this example was assumed to subtend degree of visual angle, which would be the case for a 3.3-mm letter viewed from 57 cm, for ordinary newprint viewed from 40 cm, or for letters on the 20/80 line of an optometrist’s eye chart. Additional examples of computed retinal images of this sized text viewed by an eye with a 3- mm pupil and various amounts and combinations of optical aberration are shown in the lower row of Figure 10. To make these calculations, Van Meeteren’s power series expansion of the wave aberration function in dioptric terms was used (van Meeteren, 1974).These results demonstrate that the e ect of optical aberrations can be to blur, smear, or double the retinal image depending on the types of aberration present and their magnitudes.

18 Larry N. Thibos

Top: An example of the computation of the retinal image (including di raction e ects) as the convolution of an object and the point-spread function of the eye. To perform the computation, the square field was assumed to subtend 2 2 visual angle, the eye defocused by 1 diopter, and pupil diameter 4 mm. The spread function is enlarged threefold to show detail. Bottom: Additional examples of the blurring of an individual letter of the same angular size as above. Letter heightdegree of visual angle, pupil diameter 3 mm. D diopters of defocus, DC diopters of cylinder (astigmatism), DSA diopters of longitudinal spherical aberration.

In the general theory of Fourier analysis of linear systems, any input function (e.g., an optical object), output function (e.g., an optical image), or performance function (e.g., an optical PSF) has a counterpart in the frequency domain. In optics, these correspond respectively to the frequency spectrum of the object, the frequency spectrum of the image, and the optical transfer function (OTF). By definition the OTF is a complex-valued function of spatial frequency, the magnitude of which is equal to the ratio of image contrast to object contrast, and the phase of which is equal to the spatial phase di erence between image and object. These two

components of the OTF are called the modulation transfer function (MTF) and phase transfer function (PTF), respectively.

The link between corresponding pairs of spatial and frequency functions is forged by the Fourier transform. For example, the intensity PSF and the OTF are a Fourier transform pair

A graphical depiction of this important relationship is shown in Figure 9b,d. The physical basis of Equation 28 derives from the fact that in the frequency domain the elemental object is not a point of light but a sinusoidal grating pattern. In this way of thinking, a visual target is defined not by the arrangement of many points of light but by the superposition of many gratings, each of a di erent spatial frequency, contrast, and orientation. Given that a single point of light has a flat Fourier spectrum of infinite extent, forming the image of a point object is equivalent to simultaneously forming the image of an infinite number of gratings, each of a di erent frequency and orientation but the same contrast and phase. Forming the ratio of image spectrum to object spectrum is trivial in this case, since the object spectrum is constant. Therefore, the variation in image contrast and spatial phase of each component grating, expressed as a function of spatial frequency, would be a valid description of the system OTF. Thus the PSF, which expresses how the optical system spreads light about in the image plane, contains latent information about how the system attenuates the contrast and shifts the phase of component gratings. According to Equation 28, this latent information may be recovered by application of the Fourier transform.

A frequency interpretation of the input–output relationship of Equation 27 requires an important result of Fourier theory known as the convolution theorem. This theorem states that the convolution of two functions in one domain is equivalent to multiplication of the corresponding functions in the other domain (Bracewell, 1969). Applying this theorem to Equation 27 summarizes the imaging process in the frequency domain as a multiplication of the complex-valued object

Given the above result, two important conclusion may be drawn. The first is a Fourier transform relationship between the PSF and the pupil function. As a preliminary step, evaluate the squared modulus of the amplitude spread function in both domains by using the convolution theorem and the complex conjugate theorem (Bracewell, 1969) in conjunction with Equation 25:

F

A(x,y) A* (x,y) → P(x′,y′) * P(−x′, −y′).

20 Larry N. Thibos

It is customary to translate this convolution relationship into an auto-correlation relationship, denoted by the pentagram ( ) symbol (Bracewell, 1969, pp. 112, 122), using the rule

In words, Equation 32 says that the intensity PSF is the inverse Fourier transform of the auto-correlation of the pupil function.

The second conclusion we may draw from the preceding development completes the matrix of relationships diagrammed in Figure 9. Because the OTF (Equation 28) and the autocorrelation of the pupil function (Equation 32) are both Fourier transforms of the PSF, they must be equal to each other

T( fx , fy ) = P(x′,y′) P(x′,y′).

A graphical depiction of this important relationship is shown in Fig. 9a,b. This last result puts the pupil function at the very heart of the frequency analysis of imaging systems, just as for the spatial analysis of imaging systems. It also lends itself to an extremely important geometrical interpretation, since the autocorrelation of the pupil function is equivalent to the area of overlap of the pupil function with a displaced copy of itself.

On a practical note, to use the preceding results requires careful attention to the scale of the (x ,y ) coordinate reference frame in the pupil plane (see Goodman, 1968, p. 117). The simplest way to deal with this issue is to normalize the pupil coordinates by the pupil radius when formulating the analytical expression for the pupil function. Then, after all computations are completed, the frequency scale may be converted into physical units by appealing to the fact that the cuto spatial fre-

where d is pupil diameter and is wavelength. By convention, the magnitude of the OTF is always unity at zero spatial frequency, which is achieved by normalizing the magnitude of the pupil function by pupil area. For example, in an aberra- tion-free system, the pupil function has value 1 inside the pupil and 0 outside. For a system with a circular pupil, such as the eye, the OTF by Equation 33 is simply the area of overlap of two circles as a function of their overlap, normalized by the

area of the circle. By symmetry, the result varies only with the radial spatial fre-

In summary, the pupil function (Figure 9a), the PSF (Figure 9d), and the OTF (Figure 9b) are interrelated characterizations of the incoherent imaging characteristics of an optical system such as the eye. Of these, the pupil function is the most fundamental, since it may be used to derive the other two. However, the reverse is not true in general because the lack of reversibility of the autocorrelation operation and the squared-modulus operation indicated in Figure 9 prevents the calculation of a unique pupil function from either the PSF or the OTF. It should also be kept in mind that the theory reviewed above does not take into account the e ects of scattered light, and therefore is necessarily incomplete.

D. Empirical Evaluation of the Eye as an Imaging System

1. Point Spread Function

All three of the optical performance metrics described above have been used to evaluate the imaging performance of the human eye experimentally. Historically, the first measures were of the line-spread function (LSF), the one-dimensional analog of the PSF (Flamant, 1955; Krauskopf, 1962; Westheimer & Campbell, 1962). The technique was to use the eye’s optical system to image a white line source onto the retina, and the light that is reflected out of the eye is imaged by the eye a second time onto a photodetector. Because the light that forms the aerial image captured by the detector has been imaged twice by the eye, the technique is known as the double-pass method. In order to make inferences about the single-pass behavior of the system from double-pass measurements, it is necessary to make assumptions about the nature of the fundus as a reflector. Early evidence suggested the fundus acts as a perfect di user that does not contribute a further spread of the image (Campbell & Gubisch, 1966), but this point continues to be debated (Burns, Wu, Delori, & Elsner, 1995; Gorrand & Bacom, 1989; Santamaria, Artal, & Bescós, 1987).

These early experiments indicated that the white-light LSF of the eye for foveal vision has the narrowest profile for a pupil diameter of approximately 2.5 mm, which is about one-third the maximum physiological pupil diameter attainable under dim illumination. For smaller pupils, the LSF closely matched the wider profile of a di raction-limited system, whereas for a larger pupil the LSF was much broader. Taken together, these results indicated that di raction dominates the LSF for small pupils, and aberrations dominate for large pupils, with the optimum trade- o occurring for a medium-sized pupil. Similar measurements in the peripheral visual field revealed a loss of image quality attributed to oblique astigmatism and the transverse e ects of chromatic aberration.

22 Larry N. Thibos

The main technical di culty of the early experiments was in recording the extremely faint aerial image of the fundus reflection from a source that was kept dim for safety reasons, so as not to risk damaging the retina. Subsequently, the invention of lasers and microchannel light intensifiers allowed the development of instrumentation sensitive enough to record the monochromatic, two-dimensional PSF by the double-pass method (Santamaria et al., 1987). These results indicated a lack of circular symmetry in the PSF, as would be expected of an eye with some degree of astigmatism. With the new technology it became possible to reliably survey the eye’s PSF across the visual field. When the eye is left in its natural state, the PSF varies dramatically from central to peripheral vision due to the presence of oblique astigmatism plus additional defocus, which varies depending on the particular location of the fundus relative to the midpoint of the astigmatic interval (Navarro, Artal, & Williams, 1993). However, when these focusing errors are corrected with spectacle lenses, then the quality of the peripheral optics are much improved (Williams, Artal, Navarro, McMahon, & Brainard, 1996).

Scattered light from the ocular media adds a broad tail to the PSF, which is di - cult to measure experimentally. Nevertheless, despite uncertainty about the height and extent of this tail of the double-pass image measurements, Liang and Westheimer have demonstrated that reliable information on PSF shape in the central 7- arcmin radius can be obtained (Liang & Westheimer, 1995). When these data are coupled with psychophysical techniques for estimating the background retinal illumination due to widely scattered light, it was possible to synthesize the complete PSF, which takes into account all three factors of di raction, aberrations, and scatter.

2. Optical Transfer Function

Given the Fourier transform relation between the PSF and the OTF (Equation 28), it might be thought that the OTF could be derived computationally from empirical measurements of the PSF. Unfortunately, the double-pass through the eye’s optics forces the light distribution in the aerial image to have even symmetry regardless of any asymmetry in the single-pass PSF. As a result, the simple doublepass technique is not capable of recording odd aberrations such as coma, or transverse chromatic aberration. This implies that although the modulation component (i.e., MTF) of the OTF can be inferred from PSF measurements, the phase component (i.e., PTF) cannot (Artal, Marcos, Navarro, & Williams, 1995). Given this limitation, Figure 11a shows MTFs calculated by Williams, Brainard, McMahon, and Navarro (1994) from PSFs recorded by the double-pass method in the central field using a 3-mm artificial pupil. These results indicated that for foveal vision in an eye with a medium-sized pupil, optical imperfections reduce retinal contrast for most spatial frequencies by a factor of from 3 to 5 compared to retinal contrast when di raction is the only limiting factor. Similar experiments performed in the peripheral field (Williams et al., 1996) yielded the MTFs shown in Figure 11b. These results show that image contrast is reduce by an additional factor of 2 or

Empirical modulation transfer functions (MTF) for the human eye with 3-mm pupil (0.633 m light). (a) Comparison of the di raction-limited MTF with the interferometric (subjective) and double-pass (objective) MTFs averaged for three observers. (Redrawn from Williams et al., 1994).

(b) MTFs of eyes for various eccentric locations in the visual field when astigmatic and defocus refractive errors are corrected. (Redrawn from D. Williams et al., 1996.)

3 as objects move farther into the midperipheral field. Modifications of the dou- ble-pass method to make it more sensitive to asymmetric aberrations indicate the existence of significant amounts of coma and other odd-symmetric aberrations in some individuals (Artal, Iglesias, Lopez-Gil, & Green, 1995; Navarro & Losada, 1995).

One of the awkward features of the double-pass method is that light used to make the measurements is reflected from the fundus, which means the measurement light is not the light absorbed by photoreceptors for vision.Therefore, it is useful to have another technique available for measuring the MTF, which has a closer connection to vision. A technique based on the principle of Young’s interference fringes consists of imaging a pair of mutually coherent points of light in the pupil plane of the eye (Le Grand, 1935; Westheimer, 1960). Once inside the eye, the coherent sources produce high-contrast, sinusoidal interference fringes directly on the retina. Because the optical system of the eye is not required to form a retinal image in the conventional sense, the interferometric method is often said to “bypass the optics of the eye.” Although this may be true for monochromatic fringes, the eye’s optics can have a major a ect on fringe contrast for polychromatic interference fringes (Thibos, 1990).

Campbell and Green were the first to use the interferometric method to measure the MTF of the eye psychophysically (Campbell & Green, 1965). In this classic technique, the human observer is asked to adjust the contrast of the fringes until they are just visible. Next, the observer repeats this judgment for sinusoidal gratings generated on the face of an oscilloscope or computer monitor. Since the visual stimulus is imaged on the retina by the optical components of the eye in the second experiment, but not in the first, the ratio of threshold contrasts in the two experiments is the modulation transfer ratio (i.e., the loss of contrast due to imperfect imaging). Replication of this experiment at di erent spatial frequencies thus pro-