A Virtual Retinal Display For Augmenting Ambient Visual Environments

by Michael Tidwell

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Chapter 2: Imaging Characteristics of the Human Eye

2.1 Introduction

The eye is a subsystem of any visual display system. A fundamental understanding of the eye's visual resolving abilities and limitations is essential in understanding any optical system designed for human viewing [2]. Section 2.1 describes the field of view of the human visual system and also the eye's resolving ability over that field. Section 2.2 discusses limitations to resolution including the diffraction limit and optical aberrations. Section 2.3 presents information on estimated retinal illuminance and the wavelength transmission characteristics of the eye. Display quality issues such as contrast, contrast ratio, and contrast modulation are related to estimated retinal illuminance in Section 2.4.

2.2 Field of View and Resolution of the Eye

2.2.1 Overview of Resolution and Field of View of the Human Eye

A single human eye sees roughly a 140 [deg.] field horizontally [8]. The eye's resolution is highest in the foveal region of the retina which constitutes 2-4 [deg.] about the optical axis of the eye. In the foveal region, the eye can detect an angular separation of about one arc minute. The eye can also rotate 45 [deg.] about its vertical axis. Therefore the highest resolution portion of the eye can be directed at any time to view any part of a 90-95 [deg.] section of the horizontal monocular field of vision. Accordingly, any display system designed for human viewing should attempt to present equally high resolution over the entire display field [8].

2.2.2 Resolution and the Eye

To understand the eye as a subsystem decoupled from a previous optical system, we assume ideal conditions entering the eye. We assume ideal conditions to be an undistorted scan of perfectly collimated ray bundles as shown in Figure 2.1.


Figure 2.1: Schematic representation of retinal scan.

The lens and cornea of the eye focus the collimated ray bundles to a spot on the retina. The size of the spot formed on the retina is inversely proportional to the resolution of the system. A smaller focused spot allows the possibility of more resolvable spots spread over the same unit of area on the retina. The number of resolvable lines in the display, N, is given by:

N = q / Df

where Df = the angular extent of each spot and q = the instantaneous field of view. For any display, a higher number of resolvable points corresponds to higher information content in the display.

2.2.3 Optical Limitations to Resolution

Different phenomena contribute to the size of the spot on the retina. Some are as follows:

Diffraction Limit

The diffraction limit of light is an optical phenomena which, for a circular aperture, sets a lower bound on the diameter of the focused circular spot. The minimum spot diameter D, of a beam passing through an optical system is,

D = 1.22 l f/a

where l = wavelength of the light, f = the focal length of the optical system, and a = the radius of the of the system's limiting aperture. This minimum spot diameter is called the Airy disk and represents the diameter of the first dark fringe in the concentric fringe pattern, a result of interference, of monochromatic light passing through a circular aperture. For the eye, with a pupil diameter of 2 [mm] in bright green light [3], the Airy disk diameter is

D = (1.22)( 555x10-9)( .02)/.002 = 6.8 [mm].

The angular extent of a spot on the retina is

Df = tan-1[D/feye]

where feye = the focal length of the eye. For the above bright green light adaptation example, the angular extent of the focused spot is

Df = tan-1[6.8 x 10-6/.02]

= 1.17 [arcmin].

Optical Aberrations

A perfect point spot can only be formed by convergent spherical rays. Imperfect lenses and mirrors and even the eye introduce wavefront aberrations to the light waves in any optical system. The most common types of aberration in an achromatic optical system are [2,24]:

1) Spherical aberration - caused by a change in focal length as a function of lens radius.

2) Coma - caused by the difference in lens power between the far (top) and near (bottom) reaches of a lens for an off axis point (centered below optic axis). Coma causes a comet shaped point to be formed instead of a circular one.

3) Astigmatism - caused by an optical system with a given focal length in one plane, say parallel to the earth (sagittal plane), and a different focal length in the orthogonal plane (tangential plane).

4) Field curvature - caused when the focal plane of the system is curved and not flat.

Chromatic aberrations occur in polychromatic systems and arise from lens performance which is dependent on the wavelength of light passing through the system. The two most common chromatic aberrations are [3]:

1) Longitudinal chromatic aberration - Occurs when the focal length of a lens is wavelength dependent. The result of longitudinal chromatic aberration is a different focal plane location longitudinally for each wavelength passing through the system. The difference in focal length for different colors is due to the refractive index of the material being wavelength dependent.

2) Lateral color - caused by the off-axis imaging characteristics of the lens being

wavelength dependent. The result of lateral color is an off axis polychromatic point being imaged to a rainbow of different image height locations for different wavelengths.

2.3 Estimated Retinal Illuminance

2.3.1 Theory and Calculation of Estimated Retinal Illuminance

The relationship between estimated retinal illuminance and scene luminance is important in understanding the display described in this thesis. As the display in this thesis contains no screen or real object, it is impossible to discuss the brightness of the display in terms of luminance. In terms of brightness, estimated retinal illuminance is a common denominator, so to speak, of screen based display systems and retinal scanning displays systems. The estimated retinal illuminance is [36]:

I (trolands) = R x pupil area (mm2) x scene luminance (cd/m2)

where I = retinal illuminance, "pupil area" refers to the area of the pupil of the eye, and

R = the effectivity ratio. The effectivity ratio, R, allows for the Stiles-Crawford effect and is,

R = 1 - 0.0106d2 + 0.0000416d4.

where d = the eye's pupil diameter in millimeters. As shown by dimensional analysis on the equation for I , trolands reduce effectively to the units of optical power per unit steradian.

The Stiles-Crawford effect describes the contribution to brightness sensation of light entering different points of the pupil (i.e. light entering the center of the pupil contributes more to the sensation of brightness than does light entering farther from the pupil center). Some standard scene luminance values, L, and their corresponding Stiles-Crawford corrected estimated retinal illuminance values, I, are given in Table II.1 [36,37].

Table II.1. Standard scene luminance values and corresponding estimated retinal illuminance values.

Type of Scene
Approximate Luminance

[cd/m2]
Estimated Retinal Illuminance [trolands]
Clear day
104
3.0 x 104
Overcast day
103
4.5 x 103
Heavily overcast day
102
9.5 x 102
Sunset, overcast day
10
1.5 x 102
1/4 hour after sunset, clear
1
20
1/2 hour after sunset, clear
10-1
2.0
Fairly bright moonlight
10-2
0.23
Moonless, clear night sky
10-3
2.7 x 10-2
Moonless, overcast night sky
10-4
3.0 x 10-3

2.3.2 Transmission Characteristics of the Ocular Media

Transmission losses in the eye result from scattering and absorption in the cornea, lens, aqueous humor, and vitreous humor. The transmittance of the ocular media is a function of the wavelength of the light traveling through the media. Figure 2.2 shows a plot of the total transmittance of the ocular media as a function of wavelength [38].

Figure 2.2: Transmittance of the ocular media vs. wavelength.

2.4. Image Quality as Related to the Eye

2.4.1. Introduction

Measurements of display image quality depend heavily on two display characteristics, resolution and "contrast" (see subsequent sections). It is virtually fruitless to discuss image quality in terms of either resolution or "contrast" without including the other. Definitions for display resolution, contrast, contrast ratio, and modulation contrast are given in Sections 2.4.1-2.4.4. Whenever possible, the meanings of the terms are related to the effect or result at the retina.

2.4.2 Display Resolution and the Eye

The resolution of a display can be defined as the angle subtended by each display resolution element. For a screen (CRT or LCD) based display, the angular extent of each pixel element determines the resolution. For the VRD, the angular extent of each spot on the retina dictates the system resolution. A spot of extent h on the retina allows for an angular resolution of,

q tan-1[h/feye]

where feye is the focal length of the eye. Display resolution is often measured in cycles per degree for periodic gratings such as bar patterns or sinusoidal gratings.

2.4.3 Display Contrast and the Eye

The contrast, C, of a display is the ratio of the difference between the maximum display intensity and the minimum display intensity divided by the maximum. In other terms [40],

C = (LDmax - LDmin) / LDmax

where LDmax = the maximum display luminance and LDmin = the minimum display luminance. Extending the definition of contrast in terms of estimated retinal illuminance gives

C = (IDmax - IDmin) / IDmax.

where IDmax = the maximum estimated retinal illuminance due to the display and IDmin = the minimum estimated retinal illuminance due to the display. In other words, the values of IDmax and IDmin correspond to the estimated retinal illuminance values of displays with luminance values of LDmax and LDmin respectively. In the case of a retinal scanning display, as in this thesis, estimated retinal illuminance is a preferable measure of display brightness as there is no screen in the system.

2.4.4 Display Contrast Ratio and the Eye

The contrast ratio, CR, of a display is the ratio of the maximum display intensity to the minimum display intensity. In other terms [40],

CR = (LDmax/LDmin)

where LDmax = the maximum display luminance and LDmin = the minimum display luminance. Extending the definition of contrast in terms of estimated retinal illuminance gives

CR = (IDmax/IDmin)

where IDmax = the maximum estimated retinal illuminance due to the display and IDmin = the minimum estimated retinal illuminance due to the display. The values of IDmax and IDmin correspond to the estimated retinal illuminance values for displays with luminance values of LDmax and LDmin respectively.

2.4.5 Display Modulation Contrast and the Eye

The modulation contrast, CM, of a display is the ratio of the difference between the maximum display intensity and the minimum display intensity divided by the sum of the minimum and maximum intensities. In other terms [40],

CM = (LDmax - LDmin) / (LDmax + LDmin)

where LDmax = the maximum display luminance and LDmin = the minimum display luminance. Extending the definition of contrast in terms of estimated retinal illuminance gives

CM = (IDmax - IDmin) / (IDmax + IDmin)

where IDmax = the maximum estimated retinal illuminance due to the display and IDmin = the minimum estimated retinal illuminance due to the display. In other words, the values of IDmax and IDmin correspond to the estimated retinal illuminance values of displays with luminance values of LDmax and LDmin respectively.

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