Ординатура / Офтальмология / Английские материалы / Optics of the Human Eye_Atchison, Smith_2000

206 Aberrations and retinal intag): qllality

.........

Spatial frequency (c/deg)

Figure 18.12.Comparison of in-focus white-light MTF

with defocused monochromatic MTFs for the Chromatic eye, which is free of monochromatic aberrations (pupil

diameter 2.5 mm, white light provided by a P4 phosphor, reference wavelength 589 nm). Also shown is

the neural contrast threshold of one subject from Campbell and Green (1965). Intersection of the MTF

curves with the neural contrast threshold predicts the cut-off spatial frequency. Based on Figure 4 of Thibos et al. (1991), with data kindly provided by Larry Thibos

and with permission from The American Academy of Optometry.

sinusoidal target, leading to loss of image

contrast. This has its greatest effects for target orientation at right angles to the decentration

direction, with up to three times loss of resolution for 3 mm displacements of small artificial pupils (Figure 18.13)(Green, 1967;

Thibos etal., 1991).

The Stiles-Crawford effect

The Stiles-Crawford effect has often been implicated by vision researchers to explain the

difference between expected and actual findings - for example, the failure of depth-of-

field to decrease with increase in pupil size as quickly as expected (Campbell, 1957; Tucker and Charman, 1975;Charman and Whitefoot, 1977; Legge et al., 1987). However, the results of theoretical investigations using the apodization model suggest that the Stiles-Crawford

effect is not important for spatial resolution for in-focus imagery, even in the presence of

considerable aberrations (Metcalf, 1965;

Krakau, 1974; van Meeteren, 1974; Carroll,

1980). Even in the presence of defocus, the influence of the Stiles-Crawford function

should be small (e.g. van Meeteren, 1974; Atchison et al., 1998a) (Figure 18.14). The Stiles-Crawford effect is expected to influence optimal refraction in the presence of aberrations, but the magnitude is likely to be smallfor example, approximately 0.20 at 10 c/deg in Figure 18.14.

Pupil decentration

Decentration of the eye's pupil induces

additional optical aberrations, such as transverse chromatic aberration and coma, which

decrease spatial visual performance (Green, 1967; van Meeteren and Dunnewold, 1983; Artal et al., 1996). The Stiles-Crawford effect may be of assistance to spatial vision by reducing the influence of the aberrations of the parts of the pupil furthest from the peak of the Stiles-Crawford effect; Bradley and

-0.5

...

~

'"c

~ -I

c

.2 co

:;

-g -1.5

E

'0

jOJ) -2

Spatial frequency (c/deg)

Figure 18.13.White-light MTFs for the Chromatic eye, which is free of monochromatic aberrations, for various

displacements of a 2.5 mm diameter pupil perpendicular to the grating orientation (white light provided by a P4

phosphor, reference wavelength 589 nm). Also shown is the neural contrast threshold of one subject from

Campbell and Green (1965). Intersection of the MTF curves with the neural contrast threshold predicts the

cut-off spatial frequency. Based on Figure 9 of Thibos et

al. (1991), with data kindly provided by Larry Thibos and with permission from The American Academy of

Optometry.

0.8

<2:!.. 0.6

'"c

~c 0.4

0

.~

:;'8 0.2

::E

0

fJ= 0 mm- 2

Defocus (D)

Figure 18.14.Modulation transfer as a function of defocus at various object spatial frequencies (cycles/ degree) when there is +1.0 0 of primary spherical aberration at the edge of a 6 mm diameter

pupil in 605 nm wavelength light. Results are shown with and without Stiles-Crawford apodization of f3 =

0.17, which is near the 97.5 per cent upper limit (Applegate and Lakshminarayanan, 1993).The Stiles-Crawford effect has only a small influence on

image quality, which is greater when the defocus and spherical aberration are in the same direction than when

they are opposed. Data from Figure 6 of Atchison et al. (1998a),with permission from The Optical Society of America.

Thibos (1995) refer to this as an anchoring role. A number of studies have shown that subjective transverse chromatic aberration declines as pupil size increases; for example, Ye et al. (1992). This effect reduces as luminance is reduced, thus indicating that it is a retinal effect and due to the Stiles-Crawford effect.

Peripheral vision

The optics associated with the peripheral retina are poor, mainly because of focusing errors in the form of oblique astigmatism and field curvature (see Chapter 15). Retinal

image quality declines steadily with object angle (Jennings and Charman 1978, 1981; Navarro et al., 1993 and 1998). However, as shown in Figure 18.15,when the periphery is carefully refracted, the image quality

208 Aberrations and retinal illlage quality

fitting central modulation transfer functions of the eye by the function

where a is the spatial frequency, ac is a spatial frequency scaling constant at each pupil size, and n is a shape factor constant. Their fits are

accurate only for lower spatial frequencies. At higher frequencies, errors arise because actual

modulation transfer functions have a finite

resolution limit and the above function does not. Later, Jennings and Charman (1997)

found that this approximation was useful for the peripheral field, with n remaining relatively constant at about 0.9 out to 40° eccentricity and ac declining steeply over this range. Navarro et al. (1993) and, later, Williams et al. (1996) used the fit

MTF(a) =(1 - C)exp(-Aa) + Cexp(-Ba)

(18.22b)

where a is again the spatial frequency in cycles/degrees and A, Band C are constants for a particular object angle for a range of object angles. The results of these are shown in Figure 18.15.

Summary of main symbols

PSF point spread function LSF line spread function

CSF contrast sensitivity function MTF modulation transfer function

OTF optical transfer function

PTF phase transfer function G(a) optical transfer function

a spatial frequency (c/radian)

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210 Aberrations and retinal image quality

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Introduction

In any optical system, the ultimate precision in focusing is set by the ability to detect errors in focus. The range of distances over which

the system's detector cannot detect any change in focus is called the depth-of-field,

and this range may be specified by a movement of the object plane or by the corresponding movement of the image plane. Because these two distances are usually different, some textbooks differentiate between depth-of-field, a movement of the object plane, and depth-of-focus, a movement of the image plane. In vision science, depth- of-field is usually expressed as a change in vergence, which has the same value in both object and image space. Distinctions may still have to be made between object and image

space quantities in some circumstances, such as when using visual optical instruments. However, in this chapter there is no need to distinguish between object and image situations.

The definition adopted here for depth-of- field is the vergence range of focusing error t1L, which does not result in objectionable deterioration in retinal image quality. This is sometimes referred to as the total depth-of-

field. This can be determined according to a number of criteria (Experimental results, this chapter). Some studies mentioned in this chapter used half the total depth-of-field and expressed values as ±.:1L (e.g. Campbell, 1957). When referring to such studies in this chapter,

their numbers have been doubled. This is satisfactory where depth-of-field is measured

in just one direction from a focus position, or where simple theory is used in which the depth-of-field is symmetrical about the position of focus.

The depth-of-field sets the precision to which refractive state, including the ampli-

tude of accommodation, can be measured by subjective methods. It also determines the distance range for which a target can be seen clearly when using visual optical instruments, such as simple magnifiers, microscopes and telescopes. For example, the depth-of-field of

a simple magnifier or microscope of magnification M can be given as a total distance &m in object space. This distance is related to the depth-of-field of the eye t1L by the approximate equation (Smith and Atchison, 1997)

As another example, the depth-of-field of a two-lens afocal telescope of magnification M

can be given as a total vergence in object space t1Lt' and this is related to the depth-of-field of the eye & by

Increasing the depth-of-field is advantageous in some circumstances. For example, the agerelated reduction in amplitude of accommo-

dation can be ameliorated by increasing depth-of-field. One way to do this is to intro-

duce additional aberrations. This approach has been used with contact lenses for

214 Miscd/all('(1I1S

presbyopes but, unfortunately, this reduces peak visual performance (Bradley et al., 1993; Plakitsi and Charman, 1995).

Depth-of-field depends upon several factors, including the following:

1. Optical properties of the eye

pupil diameter (interacts with the other optical properties)

accommodation level monochromatic and chromatic aberrations

diffraction.

2.Retinal and visual processing properties photoreceptor size and ganglion cell density

visual acuity and contrast thresholds disease in ocular pathway.

3.Target properties

luminance spatial detail contrast

spectral profile, e.g. colour.

Depth-of-field in the eye can be explained at a simple level using the defocus blur disc model of defocused systems (Chapter 9) and the size of the detector elements in the image plane. In an aberration and diffraction-free system, the image of a defocused point is a

defocus blur disc. If this disc is smaller than a detector element, the system will not be able to detect defocus. Defocus will be detectable

only once the defocus blur disc overlaps at

least two detectors. Because this model neglects aberrations, diffraction and how the

visual system processes retinal images (e.g. interactions between adjacent receptors), it is

a crude model and cannot be expected to predict accurately the depth-of-field of the

eye.

In the following sections we look at experimentally determined values, and consider models, such as the above defocus blur disc model, that can be used to predict depth-of-field.

Experimental results

There are several criteria for measuring depth-of-field according to a focusing error

range that do not cause an 'objectionable deterioration in retinal image quality'. Six of these

are considered here. Because these criteria

may depend upon different optical, neural and psychological factors, it is not always

meaningful to try to compare depth-of-field results based on different criteria. The common feature of experimental results according to most criteria is that depth-of-field decreases as pupil size increases, at least out to 5-6 mm pupil diameters.

Criterion 1: the range offocusing errors for which no perceptible blur of a target is noticeable

This criterion is relevant to subjective refraction and determining the amplitude of accommodation. The vergence of the target is varied, and the extremes at which the target first appears to be blurred are measured. The

:1.5

c....0.... Atchison<,Iat. (1997)

~~,O

~

J: r.s

15.

c" I.ll

o.s

Pupil diameter (mm)

Figure 19.1. Depth-of-field as a function of pupil size. Details of studies are as follows:

Campbell (1957): threshold blur, retinal illuminance

constant (corresponding to 318 cd/rn? at 1 mm pupil diameter), one subject.

Ogle and Schwartz (1959): 50 per cent probability limits

of correctly resolving checkerboard with equivalent letter size 6/7.5, subject jTS.

Charman and Whitcfoot (1977): limits of depth-of-field give 95 per cent correct identification of the direction

of movement of laser speckle, mean of six subjects, error bars indicate ±1 standard deviation of subjects.

Atchison 1.'1 al. (1997): threshold blur, 6/7.5 letter E, mean of five subjects, error bars indicate ±1 standard deviation of subjects.

details of presentation can vary. The target can be moved backwards and forwards to locate

the range within which it appears to be in focus (Campbell, 1957; Atchison et al., 1997).

Two targets may be presented, either in succession or simultaneously side-by-side,

one in focus and the other with various levels of defocus, and the subject is asked to decide which is not in focus (Jacobs et al., 1989).

Studies using this criterion show that the depth-of-field decreases with increase in pupil diameter (Campbell, 1957; Atchison et al., 1997) (Figure 19.1), increasing target luminance and correction of longitudinal chromatic aberration of the eye (Campbell,

1957). As an example of the dependence on pupil size, in Campbell's study the depth-of-

field decreases from 1.7 D to 0.3 D between the pupil diameters of 1 mm and 7 mm (Figure 19.1). Depth-of-field is smallest for target sizes near the visual acuity limit, and increases slowly with increase in target size (Jacobs et al., 1989; Atchison et al., 1997) (Figure 19.2). Jacobs and colleagues (1989) measured the threshold of just detectable change in defocus of an already defocused target and found that the threshold was slightly less than the depth-of-field.

................. Jacobs rt al. (19119)

4.2 mm pupildiameter

1.0 ---..- Atchison et at, (19971

4 mm pupildiameter

e~ O.H

'tl

..~:. 0.6 o

J:.

Q..

~ 0.4

0.2

Target detail size (log min arc)

Figure 19.2. Depth-of-field as a function of target size. Jacobs et al. (1989)measured the depth-of-field in only

one direction from the optimum focus, and their results

have been doubled so that they are comparable with those of Atchison et al. (1997).

Del'fll·(lf-field 215

Figure 19.3. Theoretical modulation transfer as a function of defocus at 5 and 10 c/deg when there is OD or +1.0 D of primary spherical aberration at the edge of a 6 mm diameter pupil in 605 nm wavelength light.

Results are shown without Stiles-Crawford apodization. Data of Figures 5 and 6 of Atchison et al. (1998), with

permission from The Optical Society of America.

Criterion 2: the range of focusing errors for which the visual acuity or contrast sensitivity does not decrease below a particular level or by more than a certain amount

This criterion has two aspects, which can be

explained with reference to Figure 19.3. This figure shows theoretical 'through-focus' modulation transfer as a function of two

levels of spherical aberration for spatial frequencies of 5 and 10 c/deg. Considering just the 10 c/deg plots and adopting an absolute level of depth-of-field as that for which modulation transfer values are greater than 0.5, the no-aberration condition has a depth-of-field of 0.4 D. However, the aberration condition has a depth-of-focus of 0 D

because its modulation transfer is always less than 0.5. If the modulation transfer require-

ment is lowered to 0.3, the aberration condition has greater depth-of-field (0.7 D) than does the no-aberration condition (0.5 D).

If we adopt either a relative loss in modulation transfer or a particular proportional loss in modulation transfer, relative to the

216 Miscellalleous

peak value, the no-aberration condition always has smaller depth-of-field than the

aberration condition (considering only the defocus region within which the modulation transfers first fall to zero). This is because the rate of loss with modulation transfer away from the peak value is always slower for the former than for the latter condition. This aspect of criterion 2 has been used in some studies comparing the depths-of-field for different bifocal contact lenses, e.g. Plakitsi and Charman (1995).

A couple of studies are now mentioned which use the first aspect of criterion 2. Ogle and Schwartz (1959) determined the defocus providing 50 per cent and 99 per cent probability levels for correct recognition of checkerboard patterns of various sizes. As expected,

depth-of-field was larger for the 50 per cent level. The depth-of-field decreased with

increase in pupil size (Figure 19.1) and with

decrease in target size. Tucker and Charman (1975) found similar results with letter

targets.

Tucker and Charman (1986) estimated depth-or-field using the visibility of sinusoidal modulated (80 per cent) luminance targets as the criterion. As spatial frequency increased, the depth-of-field decreased. For

example, at a 10 cd/m2 luminance level, the depth-of-field was zero for 30 c/deg, 1 0 at 20

c/deg, and 5 0 for 10 c/deg. Increasing target luminance from 0.001 to 10 cd/m2 increased depth-of-Held. Tucker and Charman found

little difference in depth-of-field between 3 mm and 7 mm pupil diameters.

Criterion 3: the range offocusing errorsfor which changes in contrast are not detected for a target in longitudinal sinusoidal motion

The subject views a target through a Badal optical system. The target is usually periodic - i.e. the luminance profile is generally that of a sine wave or square wave. As measured as an

image vergence at the eye, the target is moved forwards and backwards in sinusoidal

motion. The peak-to-peak amplitude of the movement is varied until the subject can

detect apparent variation in the target's contrast (for non-periodic targets some other

criterion can be used, such as the appearance of blur or changes in target shape).

Similar to the results of Jacobs et al. (1989) with criterion 1/ the depth-of-field is a minimum when the centre of the range is slightly off-set to one side of the optimal focus (Campbell and Westheimer, 1958; Walsh and Charman, 1988). Typically, the depth-of-field at optimal focus is 0.6 0/ while the minimum depth-of-field is approximately 0.20 (Walsh and Charman, 1988). The effect of increasing pupil size is to decrease the difference between the optimum focus and the centre of the range at which the minimum depth-of- field occurs. Walsh and Charman investigated a range of variables, including target colour, luminance and temporal frequency.

Criterion 4: the range offocusing errors for which a laser speckle pattern appears to be stationary

The laser speckle method for measuring refractive errors was described in Chapter 8. Briefly, a subject views a speckle pattern

produced by a laser beam reflected diffusely from a moving surface such as a rotating

drum. The speckle pattern appears to move at increasing velocity as the refractive error relative to the surface increases. This tech-

nique can be adapted to depth-of-field

measurement by determining the focus range over which the subject cannot reliably

distinguish the pattern's direction of move-

ment.

Results with this technique show that pupil size has a strong effect on depth-of-field (Ronchi and Fontana, 1975; Charman and Whitefoot, 1977). Charman and Whitefoot (1977) found that depth-of-field decreased with increase in pupil size up to about 5 mm diameter; their values are slightly smaller than Campbell (1957) using criterion 1 for nearly all pupil sizes (Figure 19.1).

Criterion 5: the range offocusing errors for which the accommodation response does not change

Accommodation response can occur for target movements as small as 0.10 (Ludlum et al.,

1968), even if this produces no perceptible blur (Kotulak and Schor, 1986).

Criterion 6: the range offocusing errors which degrades retinal image quality below a particular level or by more than a certain amount

The retinal image quality referred to here may be measured by the point spread function, line spread function or modulation transfer

function, or by their derivations, such as halfwidth of the point spread function (see Chapter 18).The modulation transfer function

is closely related to the contrast sensitivity function mentioned with criterion 2, and we used the modulation transfer function to explain two aspects of criterion 2.

Using the double-pass point spread function technique, Artal et al. (1995) determined through-focus modulation transfer at

2.6 cl deg in patients with monofocal and multifocal intraocular lenses. Marcos et al. (1999) used the criterion of a quantity, similar to the peak of the double-pass intensity point spread function, decreasing to 80 per cent of its value at optimum focus. They found that rather than decreasing, the average depth-of- field for three subjects increased slightly (although non-significantly) for increase in pupil diameter from 4 mm to 6 mm.

Modelling depth-of-field

Various models will be discussed here that can be used in understanding the factors that affect depth-of-field, by examining the optics of defocused images using two image quality criteria at various levels of complexity.

Criterion 1: The range offocusing errorsfor which no perceptible blur of a target is noticeable

The threshold of blur can be modelled by examining the image of a defocused point source of light and making assumptions about

the threshold of defocus. We will begin by considering the simplest of models, assuming

Depth-of-fiet« 217

that the system is aberrationand diffraction-

free and, therefore, a point is imaged as a point. When this point is defocused, its image is a uniformly illuminated blur disc as already referred to in the previous section.

In Chapter 9, equation (9.17), the angular diameter tPof the defocus blur disc was given

as

where 0 is pupil diameter and .1L is refractive

error. Using (total) depth-of-field to replace the refractive error, this equation becomes

We will now assume that the depth-of-field is

set by the range over which this defocus blur disc is smaller than a certain threshold dia-

meter. If this diameter is tPth' the depth-of- field is

This equation predicts that the depth-of-field is inversely proportional to the pupil dia-

meter.

The smallest meaningful estimate of tPth is obtained using the diameter of a foveal cone, which is approximately 0.003mm (Polyak, 1941). At the back nodal point, a distance of 0.003mm subtends an angular diameter of

-= 0.003/17 -= 0.000176 rad (-= 0.61 min. arc)

where 17 mm is the approximate distance from the back nodal point to the retina. Substituting this value for tP\ll in equation (19.3) gives the threshold of defocus as

field of 0.1160, which is much less than the experimental values given in the preceding section. Therefore, there are serious flaws in this model.

Effect of diffraction and aberrations

Because of diffraction and aberrations, the focused image of a point is not a point but a

patch of light - the point spread function as discussed in Chapter 18. The size of this patch depends upon the pupil diameter, since both diffraction and aberrations depend upon the size of the pupil.