Figure 2.18(a) shows that, when the distance of a real object to a collecting lens is larger than the focal length, a real image will result. Placing the object between the focal point and the lens will result in a virtual image.

Angular magnification

A distant object projects a small image on the retina; by bringing it closer, the image is enlarged. While the limit for how close an object can be brought and still be imaged sharply on the retina (the near point) depends on a person’s age (see later), normal reading distance is taken to be 25 cm. By using a magnifying glass, the object (e.g. a text) can be brought closer than the near point and still be sharply focused. When the object is placed at the focal point of the magnifying glass, the object will appear to have been moved to infinity, spanning a visual angle u0. If, when seen at a distance of 25 cm without the magnifying glass, the object is found to span an angle u, then the angular magnification, M, of the magnifying glass is defined as:

M ¼ u0=u

M, then, is the ratio of the angular size of an object in the focal point of a magnifying glass, divided by its angular size at 25 cm from the eye – without any magnifying glass. For small angles we can replace the angles with the ratio between the object hight and the object distance (i.e. u ¼ tan u ¼ y=0:25 m), where y is the hight of the object in meters:

M ¼ ðy=f Þ=ðy=0:25Þ ¼ 0:25=f ¼ P=4

Angular magnification is thus one-quarter of the power of the magnifying glass. This magnification applies for an object in the focal point of the magnifier and for an image seen at infinity, e.g. without accommodation. A magnifying glass with focal length of 5 cm has a power of P ¼ 20 diopters, and it will magnify an object 5 . Magnification greater than 3 – 4 for a simple bi-convex lens is not practical because lens aberrations will be conspicuous. If the image is seen with accommodation, it will be slightly more magnified than when seen without accommodation.

Imaging in the eye

The eye consists of many different parts that contribute to imaging. Most of the refraction occurs at the cornea, with its refraction index n close to 1.38. The lens, with n ¼ 1:44, would have a greater collecting power in air, but since it is embedded in fluids which also have a high index of refraction, its refracting power is reduced

(aqueous humor and vitreous humor both have n ¼ 1:34). However, the lens is able to increase its curvature and thus increase the refracting power. This allows focus on objects close to the eye (to accommodate). Accommodation means changing the lens’ focus from infinity to a shorter distance. This adjustment can be expressed in diopters in the same way as the power P of the lenses in a pair of ordinary glasses. The ability of the lens to change its curvature, and hence its accommodation power, deteriorates with age. For persons older than 25 years the accommodative power of the eye is typically less than 10 diopters.

When swimming underwater, the refractive power of the eye is dramatically reduced. The lens is unable to accommodate sufficiently to fully compensate for the reduced refraction at the cornea. The refractive power of the cornea drops from 60 m 1 in air to about 1.8 diopters under water. With an additional 10 diopters available by accommodation, the total refractive power underwater is 11.8 diopters. This renders the eye farsighted or hyperope. Goggles trapping a layer of air between the glass and the eye will restore normal refraction.

In a model eye, where we need not consider small differences in refractive indices, we can consider the eye to be a homogeneous optic medium with only one surface of refraction and one refractive index, n0. The following general formula can be derived from Snell’s law of refraction:

n=a þ ðn0 nÞ=r ¼ n0=b0

Here a is the object distance, and b0 is the image distance within the medium, while n and n0 are the refraction indices of the medium containing the object and of the medium containing the image, respectively. If the eye is surrounded by air, we have:

1=a þ ðn0 1Þ=r ¼ n0=b0

By definition, the power of the refracting surface relative to air is

P ¼ n0=f 0 ¼ ðn0 1Þ=r

Using the concept of vergence, we can rewrite Snell’s law of refraction for one refracting surface:

V þ P ¼ V0

If the surrounding medium is not air with an index of refraction equal to 1.00, but another medium with refractive index n, we have:

P ¼ n0=f 0 ¼ ðn0 nÞ=r

Let us see what implications this has for seeing underwater. We write n ¼ 1:33 for water and n0 ¼ 1:34 for the average eye. The radius of curvature of the cornea is r ¼ 0:0057 m. The equation above for P gives us a power of 1.75 diopters. Underwater this gives a focal distance of about 76 cm inside the eye, whereas the focal distance would be 2.25 cm in air.

This reduced spherical eye simplifies the calculations by assuming only one refracting surface and only one curvature (Meyer-Arendt, 1995). In reality there are many transitions of refracting indices (cornea/aqueous humor, aqueous humor/ lens, lens/vitreous humor). In addition, the lens accommodates by increasing its curvature. If we insert the following values for the reduced model eye, n ¼ 1:00, n0 ¼ 1:34, r ¼ 0:0057 m, we obtain for the focal length inside the model eye,

f 0 ¼ 22:5 mm

Going in the opposite direction, from eye to air, by setting b0 ¼ 1, we get

f ¼ 16:8 mm

The two focal lengths are related to each other by the expression nf 0 ¼ n0f , but the reduced eye has only one magnitude of the power:

P ¼ ðn0 nÞ=r ¼ ð1:34 1Þ=0:0057 ¼ 60 diopters:

Models of the eye

Instead of the values for a reduced spherical eye used above, one might have used a model with somewhat different values (Schober, 1957): only one refracting surface with the same refraction index as water (n ¼ 4=3), a diameter of the eye equal to the focal length on the image side (i.e. inside the eye) of 2.4 cm, and a radius of curvature of 0.006 m. These values give a power of 56 diopters, not far from the diopters of real eyes.

The schematic eye of the Swedish ophthlmologist A. Gullstrand (1862–1930) is yet another simplified model eye with the following values:

Since the cornea has a refraction index of about 1.38 in air and a radius of curvature of 7.80 mm at the first refracting surface, it alone accounts for 49 of the 56 diopters.

Farsightedness and nearsightedness

In the normal eye, objects are imaged on the retina, about 2.4 cm from the cornea. In nearsighted eyes (myopia), the eyeball is usually longer than this or the refraction is too strong, so that the image of distant objects falls on a plane in front of the retina [see Figure 2.19]. This nearsightedness can be corrected by concave (negative) lenses which move the image plane farther away from the lens.

Figure 2.19 Refraction of light in a normal eye, a nearsighted (myopic) eye, and a farsighted (hyperopic) eye. In a myopic eye, the image is in front of the retina and can be brought into place by a diverging lens. In a hyperopic eye, the image is behind the retina, and this refraction error is corrected by a converging lens.

An eyeball that is too short causes farsightedness (hyperopia) by bringing the image plane behind the retina. Such conditions are corrected by convex lenses that collect the light and shorten the distance to the image plane, so that it coincides with the retina.

Mild forms of myopia, hyperopia and a lens aberration called astigmatism (see below) can also be corrected by changing the curvature of the cornea by using a laser (LASIK). In cases of myopia, the cornea is flattened; by increasing the radius of curvature the focal length is increased, reducing the eye’s refracting power. In hyperopia one does the opposite, increasing the refractive power. Every pulse of the laser removes an extremely thin sheet of corneal tissue, only about 0.25 mm.

Figure 2.20 The curvature of the cornea can be altered by laser treatment. A thin sheet of the cornea is evaporated by means of a computer-guided laser. About 10 mm is removed per diopter. A myopic person who needs a correction of 5 diopters must have about 50 mm of tissue removed from the central part of the cornea, corresponding to a reduction of the thickness of the cornea by about 10%. The figure shows correction of myopia (a), hyperopia (b) and astigmatism (c). The upper parts of the figures show the correction as seen from above.

Ideally, this operation provides the patient with optimal refraction without glasses or contact lenses (see Figure 2.20).

Children can focus and see sharp images at distances shorter than 10 cm, but since the ability of the lens to accommodate declines with age, it is increasingly difficult to achieve the lens curvature needed in order to focus on close objects. The near point moves farther away from the eye, and after a certain age books must be held at arm’s length. This condition is called presbyopia and can be corrected with convex lenses.

As the ability to accommodate deteriorates with age, individual differences in accommodative power increases. Typical accommodation range as a function of age is shown in Figure 2.21(b). In early years, the accommodation range can be 12–16 diopters, whereas after 50 years of age it is typically less than 2 diopters.

Using the lens formula and the concept of vergence

Example 1

A nearsighted woman has her far point at a distance a ¼ 0:2 m. She has a 4 diopter accommodation range. What would be the power of glasses that would allow her to see distant objects?

If we assume that the image distance of her eye is 2.4 cm, then from the simple lens formula, 1=f ¼ 1=a þ 1=b, we get

P ¼ 1=f ¼ 1=0:2 þ 1=0:024 ¼ 46:7 diopters

Because this simple formula is valid only for thin lenses, the calculated power is wrong, but we can live with that since, in this example, we only care about changes of powers, the absolute power of the eye being irrelevant. To move the woman’s far point, a, to

Figure 2.21 (a) The distance to the near point (the shortest distance of focus) increases with age.

(b) The width of accommodation in diopters between the far point and the near point decreases with age. The hatched areas illustrate the natural variation within the population.

infinity, P ¼ 1=0:024 ¼ 41:7 diopters. This means she needs glasses with 5 diopters (¼ 41:7 46:7).

With an accommodation range of 4 diopters, maximum refractive power without glasses will be 46:7 þ 4 ¼ 50:7 diopters. Using the lens formula again and substituting the near point x for a, we get x ¼ 0:11 m, her near point without glasses. When using eyeglasses of 5 diopters, and maximum accommodation of 4 diopters, then maximum total power is 41:7 þ 4 ¼ 45:7 diopters, and the near point x will now be

45:7 ¼ 1=x þ 1=0:024; x ¼ 0:25 m

Example 2

A farsighted person has his near point at a distance of 1 m. What power must his eyeglasses have to move the near point to 0.25 cm?

When focusing at 1 m (a ¼ 1 m), the power of the eye is P ¼ 1 þ 41:7 ¼ 42:7 diopters. From example 1, we know that focusing at 0.25 cm requires a power of 45.7

distance of 0.2 m. For an object at infinity, V ¼ 0, and P ¼ 55:8 diopters. Thus, in

order to move the far point to infinity, she would require 5 diopter eye glasses, just like we found before.

The result obtained for the required corrections is the same with the two methods, although the numerical values for the power of the eye are different. In prescribing glasses, only differences in refracting power are of interest. The numerical value of the power of the eye itself is a constant that is cancelled out by subtraction.

Where is the image plane in the retina?

The image plane of the fovea is normally close to the receptor pedicles (Williams et al., 1994). The cones are about 80 mm long (0.08 mm), corresponding to 0.2 diopters. It is believed that imaging can take place along the whole length of the receptors without compromising the sharpness of the image, so determining refraction at a resolution finer than 0.2 diopters is therefore not necessary.

Lens aberrations

It may come as a surprise to learn that even the normal eye suffers from a series of optical defects. In fact, the defects are so severe that an optician trying to sell lenses of a similar poor quality would ruin his reputation (a comment usually referred back to Hermann von Helmholtz). We shall take a closer look at some of these defects, or aberrations. Aberrations seen for white light, as a result of the refractive index changing with the wavelength of the light (dispersion, see Figure 2.8), are called chromatic aberrations. Other deviations that occur even when the light is monochromatic are called monochromatic aberrations.

Chromatic aberrations

As a result of dispersion, shortand long-wavelength light is not focused together in the same plane on the retina. Normally, wavelengths around 580 nm are in focus, rendering the eye nearsighted for short-wavelength light (blue and violet objects) and farsighted for long-wavelength, red light. In other words, in normal white illumination, the short-wavelength components are imaged somewhat in front of the retina, whereas the long-wavelength components are imaged a little behind the retina. This difference in focus is called longitudinal chromatic aberration [Figure 2.22(a)]. The images formed by wavelengths other than 580 nm are therefore not rendered sharply on the retina. The images formed by short and long wavelengths are also of different sizes, since the blue image closer to the cornea is seen as larger than the red image. This is due to transversal (or lateral) chromatic aberration.

The dependence of refraction on wavelength leaves the eye with about 2 diopters higher refracting power for short wavelengths around 400 nm than for long wavelengths around 700 nm. Figure 2.22(b) shows the difference in diopters for all

(a)

f

(b)

Relative defocus (diopters)

Wavelength (nm)

Figure 2.22 (a) Longitudinal chromatic aberration is caused by different wavelengths having different focal lengths. (b) Lens powers necessary to bring monochromatic lights between 370 and 750 nm in focus on the retina, relative to an initial focus at 575 nm.

wavelengths in the visible spectrum when 575 nm is in focus on the retina. From this diagram one can see that 700 nm needs about 0.4 diopter more than 575 nm to be in focus, and 400 nm needs a 1:7 diopter correction. This corresponds to a difference in refractive index of about 0.02.

When an object is illuminated by white light, the image on the retina will be sharp for one wavelength range and out of focus for another. The image will have colored borders – similar to that of a cheap lens – but we are usually not aware of chromatic aberration in our own eyes. One reason may be that the central foveola (where we have the sharpest vision) is free from the blue-sensitive S-cones, making this part of the retina insensitive to short-wavelength light. Another possibility is that the nervous system, in some unknown way, corrects for chromatic aberrations.

It is not possible to remove these color defects from a lens, but it is possible to combine two or more lenses in such a way that the aberrations of one lens are to some

P 1 (crown)

P 2 (flint)

P =P 1+P 2

Figure 2.23 An example of an achromatic lens composed of crown glass and flint glass. The lens power of the combination is the same for all wavelengths.

extent cancelled by the aberrations of the other(s). Figure 2.23 shows an example of a relative simple achromatic lens. Here, the color aberrations are neutralized by combining a convex lens, made of flint glass, with a concave lens made of crown glass. It can be seen from the table in Figure 2.23 how the index of refraction and the lens power varies for each lens, and that in the combination the effective power, P ¼ P 1 þ P 2, is constant for all wavelengths. This lens combination is therefore free from chromatic aberrations.

Correcting for the difference in refraction for short and long wavelengths does not lead to a significant improvement of visual acuity. However, it has been claimed that yellow sunglasses (that absorb short wavelength light and transmit long wavelength light) lead to higher acuity. One likely explanation is that, with removal of the shortwavelength, blue light, there are fewer chromatic aberrations and therefore sharper border contours in the image.

Spherical aberration

The relatively simple lens formulae used in this chapter are, strictly speaking, only valid when the light rays form a small angle with the optical axis. In reality, light at a distance from the optical axis will also contribute to image formation, and all light rays from one object point are generally not going to meet at one point in the image, even if the light is monochromatic. This affects the sharpness of the image. Monochromatic light passing near the periphery of a lens is refracted more than that passing close to its center. This can be viewed as a prism effect and is called spherical aberration (see Figure 2.24). The same thing happens in the eye, and its focal length decreases with distance from the optical axis. Consequently, this aberration is more disturbing the larger the pupil diameter. A small pupil size yields a greater depth of focus and requires less accommodation.

∆f

Figure 2.24 Longitudinal spherical aberration can be characterized by the difference in focal length, f , between central and peripheral light beams.

Astigmatism

In the eye, astigmatism is caused by irregularities in the cornea that lead to a variable curvature of its surface, i.e. the radius of curvature depends on orientation. Sometimes, the difference resembles that of a cylinder lens, with one orientation having a greater curvature than the other. In the presence of astigmatism, the sharpness of contours depend on their angle relative to the horizontal; horizontal and vertical stripes will not be imaged in the same plane and their contrast on the retina will be different. This defect can be compensated for by a cylinder-lens correction in the eyeglasses.

When optical and other distortions of vision occur in children, they should be corrected for as early as possible in order to give the neural system a chance to develop normally. When severe distortions prevail after pre-school age, it might be difficult to achieve the desired improvements. Such conditions are known as amblyopia.

Adaptive optics

The correction of the eye’s main optical aberrations, such as defocus and astigmatism, by spectacles and contact lenses has become routine. Spectacles have been used for more than 700 years, and astigmatism has been corrected since shortly after 1801, when Thomas Young discovered this defect in the human eye. A pupil diameter of 3 mm is considered to be optimal for optical performance for a wide range of visual tasks. While increasing pupil size would remove the deleterious effect of diffraction, dilating the pupil would increase the blur due to several kinds of optical aberrations. If all aberrations could be compensated for, one would not only achieve improved spatial resolution in vision, but one would also increase the resolution of images taken of the retina. This goal appears closer after the recent developments in adaptive optics. The development of new and rapid methods for measuring the lower-order as

well as the higher-order aberrations of the eye has provided the necessary background to develop technologies that correct for them. Such adaptive corrections have been achieved by deformable mirrors, and a retinal camera has been constructed that can take images of the retina, with a resolution that allows discrimination of single receptors (Miller, 2000). The first image of the distribution of cone receptors in the retina was taken with a camera using adaptive optics (Roorda and Williams, 1999; see also Figure 3.15). Such images could not be obtained earlier because the imperfections of the eye’s optics caused too much blur. It is conceivable that this technology for imaging the finer details of the retina, down to the size of receptor cross-sections, will allow for the monitoring of physiological and pathological changes during retinal diseases.

Adaptive optics, which also compensate for higher-order aberrations such as coma and spherical deviations, bears promise of improving vision as well. The largest benefits can be expected for large pupils where diffraction plays no role. The effect of adaptive optics would be to render scenes and objects in finer detail; they would appear crisper and with higher contrast. In an eye provided with adaptive optics, the spacing of the retinal receptors would set the final, neural limit of visual resolution. For large improvements of the eye’s optics, the cone mosaic would be too coarse in comparison to the optical resolution, and the finer details of the optical image would pass unnoticed by the neural system. This supernormal vision may heighten contrast sensitivity and visual resolution, but it is not clear how these improvements, achievable in the laboratory, would transfer to everyday use. How could these adaptive corrections be applied? Traditional spectacles are not the best means because the eyes rotate relative to a fixed spectacle lens. The most likely choice is to design special contact lenses. Another option might be refractive surgery (see also Figure 2.20).

Spectral power distributions

Every light source has its characteristic spectral energy or effect distribution. Figure 2.25 shows a few examples of radiant energy distributions of light as a function of wavelength. The distribution, P , of power (expressed as Joules per second or Watts) in the spectrum characterizes energy per second and nanometer:

P ¼ P= ðW=nmÞ

This quotient describes the concentration of power within the small spectral range centered on , with , P and being magnitudes that can be measured. P can change smoothly throughout the spectrum, as for light from an incandescent lamp [Figure 2.25(a)], or in a discontinuous manner, as for a fluorescent tube with emission lines due to gas discharges [Figure 2.25(b)].

Transmission through a filter

Figure 2.26 (a) Reflection of light from a surface. (b) Transmission of light through a filter. P denotes the incident spectral power distribution from the light source, and that of the transmitted or reflected light. represents the spectral reflection factor and the spectral transmission factor. See also Color Plate Section.

has shone on them, which color sinks deepest into the snow and which one is the highest?

Pressed magnesium oxide (MaO4) and snow reflect light diffusely and have spectral reflection factors that are close to 1.0 for all wavelengths (in practice this means about 0.95). Black velvet is blacker than most materials; here nearly all light is absorbed.

Transparent materials, such as colored glass and color filters, have uneven spectral transmission curves telling us how much light passes through the filter for each wavelength [Figure 2.26(b)]. The spectral transmission factor is defined as follows:

¼ =P

The curves in Figure 2.27 are examples of spectral reflection or transmission curves. The optical density, D, of a filter is defined as D ¼ logð1=Þ (a filter with a transmission factor of 10 percent has an optical density D ¼ 1:0).

Some surfaces and filters are strongly spectrally selective, meaning that they reflect or transmit only narrow spectral bands. Interference filters, for instance, have such narrow transmission bands, down to a few nanometers bandwidth at 50 percent transmission.

The narrower the spectral band a surface reflects, the less light it will return, and the blacker it will appear. The way in which the perceived color of a surface is determined by its reflection spectrum is not a trivial matter, and it will be dealt with later in Chapter 5.

Visual acuity

The minimum distance between two borders that can be resolved is usually given as an angle, expressed in minutes of arc (arcmin; Figure 2.28). Visual acuity is a

Figure 2.28 Visual angle is the angular size of a distant object at the eye.

measure of how fine details a person can resolve at maximum contrast. Acuity can, for instance, be determined using a Snellen chart to read small letters [Figure 2.29(a)], or by a Landolt ring where the task is to indicate the orientation of the gap [Figure 2.29(b)]. In recent years, several other tests have been made available, among others a grating test for the measurement of contrast sensitivity and grating acuity. The task is to detect a pattern of thin black and white bars – a grating – from a homogeneous area. This can, for instance, be done by asking the subject to report the orientation of the grating, whether it is vertical or horizontal. Figure 2.29(b) also gives an example of Hyva¨rinen’s (1992) optotypes (figures) that can be used to determine acuity in children.

If , measured in arcmin, is the smallest angle that can be resolved between two adjacent black bars of the grating, then the numerical value for the visual acuity (VA) is given by:

VA ¼ 1=ðarcmin 1Þ

For 200–300 lx illuminance, a value of 1.0 is considered normal.

Converting a width measured in cm to an angle measured in degrees is relatively easy if one remembers that 1 cm at a distance of 57.3 cm from the eye equals 1 . An acuity

(a)

(b)

Figure 2.29 (a) A letter chart of 100% contrast used in testing visual acuity (VA). (b) Different optotypes and a grating also used to test visual resolution.

( )

Figure 2.31 The dependence of visual resolution on eccentricity in the retina. Visual acuity is given relative to its maximum value in the fovea.

Visual acuity is best in younger years, except for infancy, and is said to decrease steadily after 40 years of age (see Figure 2.32). As a rule, upon reaching an age of 80 years, acuity is halved, but the spread of the given data indicates great variety between individuals.

Visual acuity (1/α)

Age (years)

Figure 2.32 Visual acuity decreases with age, particularly after the age of 40. The gray area indicates the spread based on published data.

contrast sensitivity is also a useful tool to evaluate the visual ability of visually impaired people. In order to appreciate the information inherent in a contrast sensitivity function, we shall sketch some of the mathematical background that makes this test so valuable.

A square wave grating, with its sharp edges of abrupt transitions in contrast, can be described mathematically as a sum of particular sinusoidal gratings of fixed contrasts (amplitudes) and spatial frequencies (see Figure 2.34 for an explanation). The same applies to other contrast waveforms (triangular waves, saw-tooth, etc.). Sinusoidal spatial variations of contrast therefore have a particular relevance; they are more fundamental than square wave modulation or other wave variations of contrast. Thus, if we could determine an optical system’s contrast rendering for all sinusoidal functions, we would be able to calculate the contrast rendering of all other distributions of contrast.

In electrical engineering, for instance in analyzing electrical signals in electronics and sound in acoustics, an analysis based on fundamental sinusoidal waves is common. For instance, the rendering of sound of a loudspeaker is characterized by looking at distortions in particular frequency bands, similar to the contrast rendering of an optical system. Most sounds, such as spoken words, are a sum of many simple, sinusoidal sound waves of different frequencies each, with their own amplitudes and phase. Sound is, of course, described in the time domain, whereas images are described in the spatial domain, but the principle of the mathematical treatment of these modulation transfer functions is the same.

At the bottom of Figure 2.34 the luminance distribution of a square-wave grating is shown, and above it how this waveform is obtained by adding several odd harmonic frequencies to the fundamental frequency !:

L ¼ Lo þ Lmid½sin !x þ ð1=3Þ sin 3!x þ ð1=5Þ sin 5!x þ ð1=nÞ sin n !x&

Here, ! is the angular frequency, x is the distance in degrees along the x-axis, and n are the odd numbers 1, 3, 5, 7 etc. If the length of a period is , the spatial frequency, f, in cycles per degree is given by

f ¼ 1= ðc=degÞ

The angular frequency is defined as ! ¼ 360= ¼ 360 f .

Figure 2.34 shows that even a sum of only the three first parts of the expression above the fundamental (the third and the fifth harmonics) approaches a square wavemodulated stimulus (more so when the test is far from the viewer). With the first eight harmonics the result is even better, and with all of them a perfect square wave is generated. The amplitude of the final square wave is /4 of the fundamental at the top of the figure. This method of analyzing a particular waveform can be used also in two dimensions, for example to describe a pattern in a picture. The pattern can theoretically be described by a set of spatial sine functions that runs over the image in different directions.

The modulation transfer function

We have just seen how the contrasts in a picture can be broken down to a sum of sinusoidal frequencies with different amplitudes and phases. This means that the contrast rendering properties of an optical system can be characterized by measuring its reproduction of the contrasts of many sinusoidal gratings with different spatial frequencies. Figure 2.35 shows how this is represented by the modulation transfer

pupil, compared with a perfect optical system with the same aperture. The dashed curve shows the image contrast of different sinusoidal gratings of 100% contrast. When spatial frequency increases, the contrast of the reproduction decreases. For perfect optics and 2 mm aperture size, diffraction leads to the loss of all contrast at 62 cycles/deg. The eye loses all contrast at a somewhat smaller spatial frequency, around 55 cycles/deg.

function (MTF). In this diagram the contrast rendering factor for an original with 100% contrast is plotted for several spatial frequencies. The examples are for perfect (only diffraction-limited) optics and for the eye.

The solid curve in Figure 2.35 applies to the human eye with a 2 mm pupil diameter. The dashed curve is for perfect optics with the same pupil size. Contrast is reduced most for the higher spatial frequencies (as the bars of the grating get narrower), and finally approaches zero at high frequencies. One can also see how even the best optical system reduces the contrast of small details, this limitation being due to diffraction in the entrance aperture. In such an optical system, 100 percent contrast in the original will be reduced to 0 percent at 62 c/deg.

The example shows that, in the eye, the contrast rendering for the same pupil size reaches zero a little earlier, around 55 c/deg, corresponding to a bar width of 0.55 arcmin (which corresponds to a grating acuity of 1.8).

MTF curves are measured empirically, and they give a good description of the quality (resolution, sharpness and contrast rendering) of optical instruments and imaging systems. Examples of such systems are combinations of camera, electronic transmission and a monitor, such as for closed circuit TV (CCTV) or TV broadcasting. Similar curves are used in elaborate tests in vision research, and particularly in low-vision research. Later we shall take a closer look at this application.

Is the image contrast always lower than the object contrast? Not necessarily. In photographs and in copies it is possible, through image manipulation, to give the reproduction a higher contrast than the original.