Radiometry is the measurement of radiant power or energy within that part of the electromagnetic spectrum that is optical, meaning it is refracted by glass or can be focused by a lens. This includes microwave, infrared, visible and ultraviolet wavelengths approximately in the range of 1 millimeter to 100 nanometers (10^-3 to 10^-7 meters, or frequencies of 3 x 10¹¹ to 3 x 10¹⁶ Hz). Radiometry excludes radio waves, xrays and gamma rays.

The standard radiometric device is a vacuum glass bulb with a wheel of paddles inside, each paddle painted black on one side and white on the other; the wheel rotates when exposed to heat, light or an ultraviolet lamp.  

In radiometry, electromagnetic power is measured in watts (joules per second), which can in turn be converted into other units of energy or power. Actinometry is electromagnetic power measured in number of photons per second. Radiometry provides the fundamental link between our visual sense and the physics of matter and energy.

(A comment on terminology. Energy is the potential to cause a change in matter, for example a change in its structure, temperature, location, speed or direction of movement; it is measured in joules, roughly the amount of energy required to raise an apple 1 meter off the ground. Power is energy emitted within a fixed time interval, equivalent to the potential speed or rate at which a change in matter occurs; it is measured in watts (joules per second). Intensity is the quantity of power radiating into a fixed solid angle or projected area of space.)
 

The photopic sensitivity curve is scaled so that 1 watt of radiant flux at a wavelength of 555 nm ("green" light) equals a luminous flux of 683 lumens (diagram, right). (This odd number was chosen to provide continuity with the inherited, historical measures of light — as emitted from a single burning candle or lamp, or through an aperture the width of a pencil lead placed over white hot platinum.) The photopic curve then determines the proportional weights used to convert energy at other wavelengths into light.
 

The Nine Photometric Elements. The nine elements necessary to define the spatial geometry of light are:

1. An imaginary point source to stand for the spatial location of the light source and to represent the radial property of the emitted light.

2. An imaginary measurement sphere that is centered on and completely encloses the point light source (the geometry of luminous flux).

3. An imaginary aperture or opening in the measurement sphere, made by cutting away a specific surface area (A) from the sphere, that reduces the total emitted light from the point source to the light that radiates through the aperture (the geometry of luminous intensity).

4. A straight line that defines the average direction of light emitted from the point source through the aperture.

5. The physical distance (D) measured along the direction of light from the point source to the surface of the measurement sphere, or to a surface that is equal in area to the aperture in the sphere (a key element in illuminance).

(The measurement aperture, direction of light and distance are combined as a single unit of measurement, the solid angle, which is the ratio between the measurement aperture emitting light, or the surface area receiving light, and the squared distance of the aperture or surface from the point source: A/D². See solid angle geometry and inverse square law.)

6. The angle of incidence (θ_i) between the direction of light and a line perpendicular to the plane of a physical surface receiving the light (see cosine correction for surfaces).

7. The image area (S) of the physical surface of the light source, as viewed from a point on the surface that receives the light (the key element in luminance).

8. The angle of emittance (θ_e) between the direction of light and a line perpendicular to the surface of an extended light source (see cosine correction for light sources).

9. The pupil area admitting light to the eye (the key element in retinal illuminance).

In combination, these nine elements restrict the light to five different measurement geometries: each geometry creates a different unit of light measurement. The diagram below provides a summary and visual mnemonic for these five measurement units — luminous flux, luminous intensity, illuminance, luminance and retinal illuminance — the geometry that defines them, and the units of measurement based on them.
 

the relationships among the five photometric units

1/683 watt emitted = 1 lumen

As explained above, this fractional unit of power (the watt) was adopted to remain consistent with historical units of light measurement: one lumen is roughly equal to the total light emitted by a single wax candle.

Luminous flux is the generic term for the visible power a light source emits per second. Total luminous flux is specifically the light emitted by the light source in all directions. The spatial geometry of this concept is equivalent to enclosing the light source within a measurement sphere that captures on its inner surface all the light emitted (diagram, right).

The radius of the sphere does not change the total amount of light falling on its interior surface. So we can imagine that the sphere is so enormous that the light source is in proportion just a single point source at its center. Because all light radiates from the center of the sphere, all the light is perpendicular to the interior surface of the sphere. This describes the radial geometry of light.

In practical situations, luminous flux is calculated by measuring the output from a light source from many different angles at equal distances, then integrating these over a spherical area; or by measuring the reflected light at one point inside a diffusing sphere, and extrapolating that quantity to the total spherical surface. In fact, the quantity is usually obtained by measuring the light from a single direction and distance (illuminance or luminance), and then calculating backwards from that.

The solid angle is essentially a "window" or aperture cut into the measurement sphere. To maintain the radial geometry of light, the average direction of light we want to measure must be centered within the area of this aperture, and the light must be both a point source and located at the center of the sphere when the aperture and direction are defined.

This aperture removes a spherical surface area from the measurement sphere, and this is the area of the solid angle. This aperture area can be any size or shape, but the standard or unit solid angle is the steradian, equivalent to a square aperture 57° on a side or a circular aperture 65° in diameter. The steradian defines luminous intensity as lumens per steradian or candelas (cd):

1/683 watt emitted into 1 steradian
= 1 lumen/steradian
= 1 candela

The steradian encloses an area equal to the square of the radius of the measurement sphere, or a square radian. It is the unit solid angle because it is defined on a unit sphere (of radius 1), which makes its area equal to 1. As a result, the steradian aperture area is equal to 1/4π (roughly 1/12th) of the total surface area of a sphere. Thus, assuming a point source that radiates equally in all directions:

Luminous intensity captures the notion of a light source as having a brightness or power in a specific direction: street lights illuminating the pavement underneath, ceiling lights illuminating an office work area, a spotlight turned toward a cabaret singer, the sun shining toward the earth.

However, luminous intensity is still a "source centric" or abstract measure of light, because we have not specified the distance to a viewer or illuminated surface, nor the size of a physical surface that receives the light. Luminous intensity is an abstract measure of the quantity of radiant power or flux density within a standard solid angle.  

The solid angle geometry is analogous to a cone or pyramid, in that it always has three attributes (diagram, right): (1) an apex or point, corresponding to the point light source, (2) a straight line central axis, corresponding to the average direction of light we want to measure, and (3) an interior angle measured at the apex between opposite sides of the cone, if the cone is cut vertically along its central axis, corresponding to the angular subtense (θ) of the solid angle. If we continue the analogy and assume the solid angle has a circular cross section (like a cone), then the solid angle ratio is:

This formula just illustrates that we can define a conical solid angle without knowing the area of the base of the cone or the distance of the base from the apex.

To transform the solid angle ratio into an actual measured area, we need to specify two numbers to fit into the ratio: the area of the surface receiving the light, and the distance of the surface from the point source. The only complication here is that the projected area is defined on a measurement sphere centered on the point source, and not on the physical surface area that actually receives the light.

As a simple example, we can cut across the light cone with a flat surface, creating a "base" for the cone; or we can place the cone around the surface of a sphere such as the moon (diagram, right). The curved surface will have a greater surface area than the flat surface, but curved or flat, each surface receives the same quantity of light enclosed by the light cone, and we define this quantity on the measurement sphere.  

To calculate the projected area in relation to a specific physical surface, we perform three steps. First, we measure the apparent dimensions of the physical surface, in degrees of an arc, as viewed from the location of the point light source. Thus, the moon is a sphere, but from any viewpoint on earth it appears to be a flat disk in the sky: we simply measure the width, in degrees, of this disk. This is the angular subtense of the surface area in degrees (AS_degrees).

Second, we convert the angular subtense into radians, which is the appropriate unit of distance on the surface of a sphere. The conversion formula is:

Third, we calculate the spherical solid angle area, using radians as the dimensional units in the usual formulas for the area of plane figures. This results in the solid angle area expressed in steradians. Thus, a circular measurement area defines a solid angle as:

and a square measurement area defines a solid angle as:

solid angle (steradians) = AS_radians².

For example, from the earth, the moon appears as a circular disk with an average diameter (angular subtense) of 0.52° or 0.0091 radians. So the solid angle defined by the moon's spherical surface as viewed from the earth is π*(0.0087/2)² or 0.000065 steradians.

A Simplified Solid Angle. All of that is accurate, but also arcane and a bit cumbersome. In practical viewing situations it is usually sufficient and much easier to measure (or estimate) the distance of the light source than its angular subtense; and in almost all situations we can assume, with negligible loss in accuracy, that the receiving surface is flat rather than three dimensional — especially when the width of the surface is very small compared to its distance from the light source. Provided those conditions are true, then the area of a solid angle simplifies to:

Inverse Square Law. Once we have established the steradian in terms of surface area and distance, we invite the comparison of the same surface area at different distances from the light source, for example the increase in light on an open book if we move closer to a reading lamp. These adjustments are all governed by a simple relationship, the inverse square law.

Because the sides of a solid angle and the beams of light it contains both radiate from the point source, the total luminous intensity radiating into a solid angle remains constant regardless of the distance from the point source where we may measure the light. However the cross sectional area becomes larger, so the energy is more spread out (diagram, below).
 

solid angle and standard surface area
the solid angle is used to measure luminous intensity, which does not change with distance; the surface area is used to measure illuminance, which does change with distance

 
But if we are interested to find the change or difference in the quantity of light falling on a constant surface area (the same physical surface) that is placed at two different distances from the light source, then the surface may equal the cross section of the solid angle at one distance (D_a), but cover only a small part of it farther away (D_b). So the quantity of light falling on the surface must decrease — but by how much?

The inverse square law defines the relationship between the two distances and their corresponding luminous intensities (I_a and I_b) as:

I_b = I_a * D_a²/D_b²

Which means: if a quantity of light I_a is incident on a surface area at the old distance D_a from the light source, and the surface is then moved to a new distance D_b from the light source, then the new quantity of light incident on the surface I_b is the old quantity of light increased or decreased by the ratio of the two squared distances.

Because foreshortening always makes the surface area as seen from the point source appear smaller, the surface always defines a smaller solid angle and therefore receives less light. The amount of this reduction in the incident light is determined with the cosine correction for foreshortening. We insert, into the formula for the solid angle, the cosine of the angle (θ_i) between the direction of incident light and a line perpendicular to the surface (diagram right):

If a surface is perpendicular to the direction of light then the angle of incidence θ_i = 0, the cosine = 1.0, and there is no change in the amount of light falling on the surface. If the angle of incidence θ_i = 45°, then the cosine = 0.707 and the quantity of light incident on the surface is reduced to 71%.

With the simplified formula for the solid angle, an understanding of the inverse square law, and the cosine correction for foreshortening, we can develop the two most common — and "viewer centric" — measures of light: illuminance and luminance.  

Illuminance is the quantity of light incident on a surface from a light source or sources. No information about the distance or size of the light source(s) is necessary. However, the standard measurement unit of illuminance is defined as the light energy incident on a surface area of one square meter at a distance of one meter from a point light source, in one second. This yields lumens per square meter (lm/m²) or lux (lx):

1/683 watt incident on 1 square meter
= 1 lumen per square meter
= 1 lux

There are several obsolete or nonmetric definitions of illuminance, including the foot candle (1 lumen incident on 1 square foot). As there are roughly 10 square feet in one square meter, the foot candle defines a larger unit of light:

1 foot candle = 10.76 lux.

The illuminance on a surface area 1 meter square has a simple relationship to the two previously defined measures of output from a light source, luminous flux and luminous intensity:

In formula [1], we derive luminous intensity by dividing the luminous flux, the total output of light in all directions, by the number of steradians in the surface area of a sphere (4π or about 12.57); then in both formulas we use the inverse square law to adjust luminous intensity for the distance of the light source from the light receiving surface. Thus, a nonreflecting 60W incandescent bulb emits about 600 to 840 lumens; lumens denotes luminous flux, so by formula [1] its illuminance at 3 meters is about 5 to 7.5 lux.

Recessed or reflector bulbs emit their luminous flux in one direction, as a diffuse beam or light cone. The solid angle of this light cone varies across different types of lighting, but a handy rule of thumb is that the cone fits within a solid angle of one steradian. Then the rated luminous intensity of the light in lumens is identical to its illuminance in lux (1 lumen equals 1 lux at 1 meter distance), and only the inverse square correction for different distances is necessary. For a 60W reflector or recessed light, rated at 650 lumens, the illuminance at 3 meters is about 72 lux.

If the light is obliquely rather than perpendicularly incident on the surface, then the illuminance is less, and the luminous intensity must be multiplied by the cosine correction for foreshortening:

Finally, there is a third definition of illuminance, based on the luminance and image size (solid angle) of the light source:

[3] illuminance = luminance * solid angle S

where luminance is measured in candelas per square meter, and illuminance is expressed in lux.

Because illuminance is commonly used in architectural or task specifications, it is natural to think of it in terms of how lighting appears in a given situation. But this can be misleading, and three key points will help clarify what illuminance actually measures.

The first and most important point is that illuminance is never directly visible as a quantity of light. We only see its reflected image as the luminance of physical surfaces. If the light source were behind us, and there were no surfaces in view (or the surfaces completely absorbed light), we would look into total darkness. If we turned to look directly at the source of illumination, we would perceive the luminance of the optical image of the light source on the surface of the retina, which depends not only on the quantity of illuminance but also on the visual size of the source that emits it. Our judgment that areas are brightly or dimly lit is actually based on our perception of the light reflected from familiar surfaces, such as white walls or the pages of a book, and covertly on our perception of visual adaptation cues, such as contrast and color.

Second, by itself illuminance does not specify light sources — it describes the total light power incident on a specific surface at a specific location in space. It has nothing to do with the power, size or number of light sources. It is equivalent to the "blind" skin sensation of heat on the skin that can be induced equally by the distant sun or by a nearby bare light bulb. The sensation, by itself, tells you nothing about the source.

Third, in real world situations illuminance is very closely related to the geometry of obscuring objects in relation to light sources: it does not depend on light sources alone. For example, on a clear sunny day, a book held in the shadow of a stop sign on a country road is illuminated by the entire visible sky; but if we stand in the shadow of a large building that blocks out half the sky, the illuminance is reduced by almost half. Similarly, if a single north facing window illuminates a room, drawing the blinds from a second window doubles the illuminance. In forests, canyons, alleys or overcast days, the reduction in illumination from the daylight maximum is equal to the amount of light obstruction.

Fourth, distance has a huge impact on illuminance through the effect of the inverse square law. At one meter, a single candle (1 candela) yields an illuminance of 1 lux. At 10 meters, the candle produces an illuminance of 0.01 lux. To get back the illuminance of 1 lux from a distance of 10 meters, we would have to use 100 candles.

This dependence on distance makes illuminance the key architectural measure of lighting, and the physical dimensions of an interior space, along with the activities that must be performed in the space, determine architectural lighting requirements. The same level of illuminance can be provided by many different lighting systems (number, power and physical size of light sources); illuminance only specifies the total light output that the system must deliver.

There are many illuminance standards for the amount of light desirable in different architectural settings or for different tasks. Office lighting standards require illuminances in the range of 300 to 500 lux at work surfaces; home lighting levels are typically lower. (See this section for a broad comparison of illuminance levels.) Many eyestrain problems are created by the effects of glare (reflected light) or excessive light contrast, and not by inadequate illuminance levels.

Lighting engineers and interior designers measure illuminance with an illuminance meter or light meter, which determines the amount of light through the electric current produced by light energy falling on a photosensitive surface. Lighting engineers often use a light meter that collects light arriving from a wide area, so that the measured illuminance corresponds to light sources of any size and shape in any direction; these all sum to the total light incident on the measurement device. Photographers similarly use a light meter to estimate the average quantity of light available to alter photographic film; however, they typically measure only the light reflected from an average gray card, or the average light reflected from only the surfaces included in the image.  

Luminance is the illuminance incident on a surface area, divided by the angular width of the source as viewed from the surface. It is simply the ratio of light received to the visual size of the light source.

Luminance is the most context specific definition of light, because it is based on two solid angles, or a solid angle and a surface area, which gives candelas per square meter (cd/m²):

1/683 watt incident on
1 square meter at 1 meter distance
= 1 candela/meter²

the basic geometry of luminance
luminance is the solid angle of point luminous intensity, times the surface area of the light

 
This geometry yields five equivalent definitions of luminance based on different measurement units:

Formula [1] shows that luminance is defined by three geometrical quantities: the two surface areas (or a surface and an aperture area) and the common distance between them. Thus, the surface area of the light source (circular in the diagram) is defined by π*X². (The area of a square light source would be (2X)².) The circular area of the light receiving surface (or aperture) is π*Y². This surface (or aperture) is at distance D from the source.

It is usually convenient to divide the distance squared into one or the other surface area to create a solid angle, as in formulas [2] and [3]. The light receiving aperture area defines the solid angle A, equal to π*Y²/D² steradians; the light emitting source area defines the solid angle S equal to π*X²/D² steradians.

As always, it is convenient to express X and Y as a visual angle in radians, as this excludes the surface geometry of both the source and the aperture area. If Y is the angular subtense (in radians) of the aperture as measured from a point s on the source and X is the angular subtense of the source measured from point d at the aperture, then their solid angles are π*Y² steradians and π*X² steradians.

The solid angle A defines the proportion of luminous flux passing through the aperture from any single "point source" s on the surface of the light source. The total number of points illuminating the aperture is equal to the physical area of the source (π*X²). So the solid angle A based on the spherical area of the aperture is multiplied by the area of the source, and this is divided into the luminous flux.

Note that distance is only included once in a luminance calculation, so two solid angles are never required. In formula [4], luminous intensity already contains one solid angle (in the "aperture" area of the steradian solid angle), so this is scaled by the source area only. In formula [5], illuminance only contains a surface area (the area of the surface/aperture that receives the light), so this must be scaled by the solid angle to the source (its visual size at the viewing distance).
 

Formula [6], based on illuminance (easily measured with a photometer) and including the cosine corrections necessary for specific lighting problems, is probably the most commonly used calculation of luminance.

Now that I've introduced all the standard formulas, let's consider the perceptual implications of luminance.

First, luminance describes physically extended light sources, not imaginary point light sources. Image area is really the key concept here, since all physical lights or reflecting surfaces must have a measurable surface area, or a visual width and height (angular subtense), no matter how small or far away the source may be. (Note that the bottom limit on visual width is determined by the "pixel" structure of our retina or CMOS chip, or the "granular" structure of a film emulsion; these impose an angular subtense on the imaging of very tiny sources, such as stars.)

Second, luminance is the "area intensity" of the light source. It does not just depend on the quantity of light reaching the eye, but the degree to which the illuminance originates in a visually compact source.

As a simple example: you step into a business office and notice that the floor is brightly illuminated. Simply by looking at the floor and the material it is made of, you can infer the approximate level of illuminance, for example as compared to the illumination produced on the floor by direct sunlight. But you cannot see the luminance of the light source itself: to do that, you must look up. If the illuminance originates in several "spotlight" ceiling fixtures, each of which would appear very small from your vantage, they would probably be uncomfortably bright to look at — they would have high luminance. Or if the illuminance originates in an entire ceiling covered by diffusing light panels, which spread the light over a large area, the panels may appear comfortably dim — they would have low luminance. Lighting engineers apply this fact in the design of light fixtures that deliver the same illuminance while spreading the source luminance over a larger visual area.  

Third, the "area intensity" of luminance is constant across distance: the luminance of a light is the same, whether it is 1 meter or 100 meters away! Since the illuminance of a light source is proportional to the inverse square of its distance, the incident light decreases as a light moves farther away. But the visual size of the light also gets smaller from our viewpoint, and by the same inverse square proportion. As a result, the ratio between incident light and visual size remains constant. As lights recede from us, they become dimmer but also proportionally more concentrated in visual area, so we perceive the source as having a constant environmental brightness. The same is true for relative luminance (illuminance times surface reflectance): material surfaces appear to have the same lightness regardless of distance.

Fourth, luminance is the photometric unit that most closely approximates the perceived brightness of a light or the lightness of a surface as viewed by a camera or human eye. If illuminance is the measure of incident light, luminance is the most appropriate measure of perceptible light. Luminance is always visible, and is therefore complementary to illuminance, which is always invisible.

However, despite the fact that luminance is constant across changes in distance from the light source, luminance and the perception of brightness/lightness are only loosely related. Perceived brightness depends on the apparent distance of the light (lights appear fainter as they move farther away), and the lightness of surfaces depends on the local contrast with other surfaces; and both judgments are affected by the level of luminance adaptation.  

Luminance & Optics. The luminance of a light source as imaged in an optical system, such as a camera or the eye, introduces some specific issues.
 

the pinhole geometry of luminance
the solid angle of the image equals the solid angle of the light source surface; both are constant across distance

 
In the simplest case, a pinhole camera contains no lens, only a hole that is extremely small in relation to the distance D to the light source and the focal length F to the image plane. The pinhole causes each point s on the source to project a single point image of itself on the image plane. As the solid angles S and I are therefore equal, the total light passing through the pinhole is equal to the illuminance (not the luminance) of the source at the pinhole.
 

luminance [image] = illuminance/F²

What happens if we increase the size of the pinhole aperture, to let in more light? This increases the area admitting light:

luminance [image] = aperture area * (illuminance/F²)

which produces a much brighter but optically blurred image. The blurring effect of increased aperture is overcome by a lens (or parabolic mirror).
 

the optical geometry of luminance
the solid angle of the image is proportional to the ratio of the distances D/F

 
The solid angles I and A on opposite sides of the lens are no longer equal — the lens refracts the light "rays" into a wider solid angle. However, the two solid angles have in common the surface area of the aperture or lens opening, therefore the ratio of the two solid angles equal to the ratio of the two distances, D²/F². This is the power or light concentrating capability of the optical system.

Assuming D is very large relative to F (as for binoculars or a telescope), then the solid angle I becomes larger, and the power of the optical system increases, as the focal length F becomes shorter. When the lens produces a focused image, then the physical image area is proportional to the physical source area in the ratio F²/D² — that is, the higher the optical power, the smaller the focused image.

Each ratio is a reciprocal of the other, so they neatly cancel each other out when the solid angle I is multiplied by the image area (or when solid angle A is multiplied by the source area). As a result, for a completely transparent lens:

luminance [image] = luminance [source].

This essentially restates the third luminance property of light sources — luminance remains constant across changes in distance — because the image of an object in any optical system grows larger or smaller in the same inverse square relation to the object distance.

The quirk here is that the image illuminance increases with optical power (shorter focal length), just as it does with increased aperture. This is because a shorter focal length images a larger solid angle (field of view), and this "wide angle" image captures more light at the same time that it reduces the image size of every image element, increasing its luminance. Both effects increase the incident light (illuminance) at any point within the image area and the luminance (brightness) of the image. For this reason, a wide angle lens exposes a photographic film more quickly than a long focal length telescopic lens of equal aperture, and a magnifying glass ignites a piece of paper in sunlight, because the sun's heat is condensed from the aperture area of the lens (its shadow area on the paper) while the short focal length concentrates this light into a tiny spot.

Luminance & Surfaces. The luminance of reflecting surfaces is potentially complex and depends on (1) the illuminance onto the surface, (2) the surface reflectance or albedo of the surface (the proportion of light falling on the surface that is reflected from it), (3) the angle of incidence of the light, (4) the angle of view to the surface, and (5) how much the surface diffuses or randomly scatters the light. These issues are explored on a later page, but a few points should be mentioned here.

There are several alternative luminance measures that attempt to equate the luminance of surface with the illuminance incident on it. The most common are the metric apostilb and millilambert (preferable to the inconveniently small lambert) and the USA foot lambert:

1 candela/meter² (1 nit) = 3.1416 apostilbs
= 0.3142 millilambert
= 0.2919 foot lambert

These are all related to the general formula for the luminance of a surface that is viewed perpendicular to the surface and illuminated from all sides by a perfectly diffuse light:

Thus a surface with a reflectance of 0.4 and incident illuminance of 500 lux will have a nondirectional luminance of about 200 apostilbs or 64 cd/m².

Note that reflectance is not the lightness (L), which is a visual sensation, but the proportion of the total incident light reflected by the surface. Rescaling this quantity on pi is necessary because light arriving from a single direction onto a physical surface is scattered or diffusely reflected in all directions, dimming it from any single point of view.

However, this straightforward definition of reflectance (as the proportion of incident light that is reflected) applies only to a surface viewed under a completely encompassing and evenly diffusing light source. It is more convenient in graphic arts to view a surface perpendicularly and to illuminate it from one side (at a 45° angle of incidence) to minimize gloss or reflections from the surface of photographic prints, oil or acrylic paintings, mosaics, and the like. This requires use of the reflectance factor, which is reflectance calculated as the amount of incident light reflected from a surface as a proportion of the incident light that would be reflected, under identical lighting and viewing conditions, by a perfectly reflecting diffuser (or Lambert surface, a pure "white" surface that reflects all incident light equally in all directions). Reflectance is a constant for any surface, but the reflectance factor of a surface can change substantially, depending on the angle of incident light and the angle of view.

In most situations surfaces reflect to the eye only a small portion of the light incident on them: a "middle gray" (with L = 50) reflects less than 20% of the incident light. In addition, surfaces are typically much larger in visual size than the light source, and the luminance of equally illuminating light sources varies with their size. This paradoxically causes smaller light sources to appear brighter at the same time that it causes the surfaces under them to appear dimmer in comparison.

As an explicit example, assume an evenly diffusing (matte) white surface 1 meter square, placed 5 meters below a perfectly diffusing light panel 1 meter square that provides 100 lux of illumination to the surface. Then the light source, viewed from 5 meters, will appear to have a luminance of 2500 cd/m² while the surface will have a luminance of about 30 cd/m²: the light source appears more than 80 times brighter than the surface it illuminates. Replacing the diffusing panel with a single 6" spot lamp that again provides 100 lux of illuminance will increase the luminance of the source to over 137,000 cd/m², while the surface luminance is unchanged; now the light appears almost 4600 times brighter than the surface. Thus, diffusing panels allow the same luminance output to reach across larger architectural spaces with the same surface luminance effect; for the same reason, the ground appears brighter than the sky on diffusely illuminated overcast days. In general, natural light sources (the sun and concentrated incandescent lights such as a bare tungsten filament) always look far "too bright" or glaring compared to the illuminance they provide on surfaces. Alternately, sources that are comfortable to look at directly (such as the full moon, a candle or a small flashlight) typically produce illuminance that is useful only in near darkness.

Paradoxically, glossy or shiny surfaces will usually appear darker than matte surfaces having the same reflectance, unless the light source is reflected directly to our eyes. This is why the highly reflective waters of a sea or lake appear dark in comparison to highly reflective beach sand: the sand diffuses light equally in all directions, whereas the water reflects light primarily in the direction where the sun's image is visible on its surface.  

Retinal Illuminance is a measure of the amount of light that actually enters the eye. It is measured in trolands and is derived as the luminance of the light source multiplied by the area of the observer's pupil in square millimeters. Because pupil sizes vary from one individual to the next and within the same individual across different light intensities, the troland has only an empirical or observed relationship to source luminance; but at typical indoor levels of illumination (around 300 lux):

1 candela/meter² = ~10 trolands

Thus, the moon's retinal illuminance in trolands is greater at night than it is during the day, because at night our dark adapted pupils are larger.

The troland is primarily used in vision research, and to standardize measurement across different individuals, an artificial pupil is placed in front of the viewers' eyes. This is a small hole in an opaque screen that is of a fixed diameter smaller than the minimum pupil diameter found in normal eyes; the area of the artificial pupil is used to calculate the troland.

The troland does not take into account most of the perceptual changes involved in luminance adaptation between day and night, so (like luminance) the troland does not describe very accurately the subjective sensation of brightness. It simply allows more precision in the estimate of light actually incident on the retina.

James Calvert has posted a useful Illumination tutorial that explains the geometrical definitions and standard formulas for photometric units.

the colormaking attributes

differences in lightness
hue and chroma held constant

Brightness and lightness are correlated with the luminance of a surface or light source. This means that brightness and lightness usually go up or down as the color luminance goes up or down, but whether and by how much depends on the viewing context. Let's first review the relationships among context factors and then summarize how they affect brightness/lightness perception. (An expanded version appears in color in the world.)
 

context factors creating the perception of brightness/lightness

 
We perceive physical environments because of the light incident on material surfaces, which depends on the source intensity or illuminance of the light source. The illuminance is the luminous intensity of the light source reduced by its distance from the illuminated surfaces. Averaged across all directly lit surfaces, this is the scene illuminance.

Illuminance is separate from the source image or luminance of the light source, which depends on the source intensity, its distance, and its visual size from our viewpoint. An extended, diffuse light source, such as an overcast noon sky or ceiling light panel, can provide substantial illuminance but, as a source image, appear very dim. This is because, at equal illuminance, luminance increases as the visual size of the source image gets smaller: an incandescent filament has much greater luminance than a light panel.

Environmental surfaces reflect more or less light depending on their surface reflectances. The combination of scene illuminance, shadows and surface reflectances defines the surface luminance range — the variations in the brightness of the physical enviroment. This is the anchor of luminance adaptation for two reasons: the luminance of surfaces is constant across distance (as with lights); and, for diffusely reflecting surfaces, luminance is not significantly affected by the angle of view or the angle of incident light.

Exactly how luminance adaptation occurs is not clear, but it apparently requires three simultaneous adjustments in the visual response: (1) a receptor gain adaptation to the average scene luminance (the adaptation gray, L_g,equivalent to a reflectance of about 13%), (2) a cognitive lightness anchoring that links the highest surface reflectance (no more than 5 times the adaptation gray) to the perception of "pure white" (the adaptation white, L_w), and (3) a perceptual expansion or contraction of the lightness range so that a surface presenting a luminance of about 1/5 of the adaptation gray is perceived as black (the adaptation black). As a result, color areas with luminances within the range L_w to 1/20L_w are perceived as objects varying in lightness, or color with some gray content.

The lightness range orphans many specific color areas that have greater luminance than the adaptation white, including areas of spot illumination (sunlight falling on the floor through a window), gloss or specular reflections, and secondary or primary light sources. These appear as lights varying in brightness, or color with some brilliance content. A stimulus darker than the adaptation black is invisible unless silhouetted against a lighter background, where it appears as a void. Brightness and lightness are both necessary to perceive the total range of luminances, from voids to source image, that can appear in physical environments.

Brightness perceptions are powerfully affected by the level of luminance adaptation and the luminance of the surrounding area. Lightness perceptions are remarkably consistent and stable, provided all surfaces are under the same illuminance; but lightness differences can be powerfully affected by the perceived spatial geometry of material surfaces and light sources, especially when these define the perception of average luminance, spot light and shadow.  

Within this general context description, the brightness/lightness of a color area depends on:

• Luminance adaptation. The visual adaptation to light intensity sets the perceptual boundary between surface lightness and light emitting brightness, and sensitivity to luminance differences within each range.

In most environments, the anchor for light adaptation is the average quantity of light reflected from surfaces — the scene luminance range — which is actually the reflected image of the scene illuminance. For both surfaces or lights of constant luminance, brightness decreases as light adaptation increases, and conversely brightness increases as dark adaptation increases. The lights and colors of a film appear dim as we enter the theater but brighten after our eyes become adapted to the dark, though there is no objective change in the luminance levels of the film. Even "gray" sidewalks or building walls have a high brightness, and automobile colors appear more vivid, as we exit the movie matinee, but these effects are muted as our eyes adjust to the light. In the same way, emitting lights appear to grow dimmer, whites appear brighter, surface colors appear more chromatic, and the contrast among whites, grays and blacks is greater, as scene illuminance increases, although these effects partially disappear as we adapt to the new illuminance level.

Under mesopic or scotopic vision (dark adaptation) we also experience a sensory change in the appearance of lightness: lightness contrast declines and "white" surface appears perceptually to be gray, as compared to the memory color of a white surface under photopic illumination. Under scoptopic vision only light emitting sources (such as the moon) appear perceptually as a "pure white".  

• Luminance Contrast. Lightness and brightness are local contrast judgments, not direct perceptions of light acting on the retina. So the relationship between a color's luminance and its perceived brightness or lightness is strongly affected by the visual context.

The key factor is relative luminance contrast. The light emitting or "brilliance" quality of brightness is perceived in color areas with 2 or more times the luminance of a white surround, or roughly 40 times the luminance of a dark gray or black surround. Lights appear brighter in relative darkness because of the substantially reduced surround luminance and lower luminance adaptation. And brightness contrast is increased if the brighter color area is made visually smaller, even when the contrasting color areas are surfaces of constant reflectance.

At night a flashlight appears "bright", and "brighter" than a candle, because the contrast is with a dark surround a dark adaptation; under a noon sun, both the candle and flashlight are invisible, because they produce a negligible luminance increase in relation to the average surface luminance and the eye's light adaptation. "Bright" also describes specular reflections that are visually much smaller than the source image, and surfaces whose luminance exceeds the current adaptation white due to spot illumination.  

A restricted range of luminance contrasts usually creates the lightness scale of grays. It is sometimes claimed that we cannot see the color "gray" in lights, but this is belied by the grays in the diagram at right, which are generated by the pixel sized lights in your computer monitor. We cannot see gray in lights viewed in isolation or as recognizable sources of illumination; the lights appear veiled or dim instead.

Lightness and brightness are complementary regions on the luminance dimension: normally lightness masks direct perception of brightness, and vice versa. The "blacks" in a television picture have the same absolute luminance (brightness) as the "gray" monitor screen when the television set is turned off: they produce a black color in the video image through contrast with higher luminance pixels around them.

Lightness is substantially affected by the contrast between a color area and its surround. The lightness of a color area can change, sometimes radically, depending on the lightness of surfaces that are visually next to or behind it (simultaneous contrast). In particular, a dark background or surrounding color will make a color area appear lighter; a light valued surround will make the color appear darker. A classic example is the full moon in either the day or night sky, which appears white although it is actually very dark (its albedo, equivalent to its reflectance, is 7%).  

• the scene illuminance changes by a large amount (the sun comes from behind a cloud; we exit a movie theater)

• the local illuminance changes (we move an object from shadow to light, we turn on a desk lamp)

• there is a brighter or darker spot illumination on a surface (cast shadows, volume shadows, a beam of light on the floor)

• we see a surface reflection (the moon on water, the sun in an automobile windshield)

• we see a luminous color area against a dark surround, or a void within light reflecting surfaces.

In all these cases we see light as a distinct attribute that is more or less separate from surface.

In general, we see illuminated darks as darker than their actual reflectance, and shadowed lights as lighter than their actual reflectance. In the example (at right, top), the central "dark gray" tile on the illuminated side of the cube is the same monitor luminance and measured lightness as the "white" tiles on the shadowed side, which is evident when all other tiles in the form are removed (diagram, right, bottom).

The spatial interpretation of illumination differences between light and shadow, and the apparent match in of surface patterns on all sides of the object, obliterate a direct comparison of the brightnesses: again, lightness masks the perception of brightness. The tiles on the shadowed side are perceived to have a higher lightness because the visual system compensates for the effects of the virtual shadow.  

• Spot Illuminance. Finally, the brightness or lightness of surfaces depends on the continuity of the scene illuminance.

In most cases of spot illuminance (a local area of increased illuminance), the visual system registers the absolute increase in luminance as an increase in surface brightness. (Refer to this discussion for the distinction between relative and absolute luminance changes.) Similarly, a local are of reduced illuminance is perceived as a shadow. We don't see patches of sunlight through leaves as white spots on the ground; we see them as brighter versions of the same lightness visible in the surrounding shadows.

A surface can appear mysteriously darker or lighter than it normally appears if we cannot perceive the relative illuminance difference in a visual comparison. If we are sitting in a dark room, and see the sunlit asphalt pavement outside through a narrow opening in a curtain, the pavement can appear white or light gray rather than black. In the same way, a black paper hung in complete darkness and illuminated by a narrow beam of intense light will appear quite white, so long as nothing else in the room is visible. Its appearance snaps to black if a gray or white surface is also placed in the beam of light.

It is even possible to contrive situations in which the spot illuminance cannot be perceptually separated from the spatial definition of surface patterns or surface edges, or attributed to visible light sources or cast shadows in the scene. For example, a beam of light or shadowing form can be arranged so that the edges of the light or shadow corresponded exactly to the edges of a single white or black tile in a checkerboard floor. In that situation the discrepant surface luminance will appear as an isolated gray tile, or — if the spot illuminance change is large enough — as a square light source embedded in the floor, or as a square hole.

Terminology. Artists usually talk about a painting without concern for the lighting of the situation where it is viewed, and they interpret landscape values into paint values that will appear under different kinds of illumination. For those reasons, the related color judgment of lightness (or the artists' term, tonal value) is the concept to use when discussing works of art; brightness should be used to describe the landscape, studio or gallery lighting.  

Hue. This is the most familiar color attribute, the one that answers the familiar question, what color is it?

Hue is identified by a categorical basic color name such as red, yellow, green or blue; or a compound of two basic names such as yellow green; or a secondary color name such as orange. The example below shows several different hues of equal lightness and hue purity.
 

However, monochromatic lights change hue slightly if their brightness or chroma changes (as discussed here), and can shift substantially in contrast to other hues around them (as discussed here). So the match between a specific hue and a spectral wavelength is relative to the viewing context. That is, any color stimulus can be described by a dominant wavelength, but the dominant wavelength does not define the appearance of the color stimulus in all situations.

If many wavelengths are involved, the hue is determined as the average or geometric mean of all the wavelengths on a chromaticity diagram, not on the linear spectrum. That is, the hue created by a mixture of "red" and "violet" light (at the ends of the spectrum) is not green (in the middle of the spectrum) but purple (outside the spectrum, but between red and blue on a hue circle). For these extraspectral hues, the dominant wavelength is the complementary color ("green" monochromatic light) that exactly neutralizes the color mixture, denoted by a "c" placed before the wavelength number. The dominant wavelength of magenta is c530 nm.  

As this matching procedure implies, hues are limited to spectral (prism) colors and the extraspectral mixtures of spectral "red" and "violet" light. In English these hue names are magenta, red, orange, yellow, green, cyan, blue and violet or purple; compounds of these names such as blue violet or yellow green; and specific names for saturated colors such as scarlet, orange, chartreuse or turquoise.

Hue specifies the location of the color around the circumference of a hue circle, not any color location toward the center. Names of dull or muted colors such as white, gray, black, brown, maroon, pink, tan, gold, russet, olive and so forth do not describe spectral colors, and this rules them out as hue names, even though they may be appropriate answers to the question, what color is it?

Even so, artists should learn the correct hue designations for dull colors. "Brown" for example is technically a near neutral, dark valued orange with a dominant wavelength around 610 nm; "olive" is an dull, mid valued yellow with a dominant wavelength around 570 nm. You will never be comfortable describing your coffee as dark orange and your martini olive as dark yellow, but that is what they are; and accurately recognizing the hue of any surface color will help you to mix that color using a color wheel and to understand how the color is likely to change appearance under different types of lighting or from light to shade.

Hue is an attribute of both unrelated and related colors. We can easily identify the hue of traffic lights at any time of the day or night, and we can judge the hue of any surface provided that we know whether the illumination is bright or dim, and "white" or tinted.  

In general, apparent hue remains constant across wide changes in daylight illumination. In particular, changes in the sun's light from morning to afternoon, or in cloud cover, don't significantly affect hue perception. However, the contrast between similar hues, and their saturation, does appear to increase as illumination increases, and under dark adaptation (at night) hue perception in surfaces disappears, though we can see hue in lights (such as the planet Mars and distant traffic lights).

Reliable hue recognition can go awry in several unusual or extreme situations: (1) the surface is viewed without surrounding colors and without an accurate idea of the intensity and color of the light source; (2) the viewer has been adapted to one colored light source, and the illumination changes to another color or to white; (3) the hue is viewed in contrast to adjacent color areas of strongly different color and brightness; (4) the illumination has an intense (pure) color; (5) the spectral power distribution consists of a broken spectrum that emits only a few wavelengths or many wavelengths at very different intensities; (6) colors are viewed at extremely high luminance levels that saturate or overwhelm the photoreceptor cells; or (7) colors are viewed through a positive or negative afterimage.

Most of us are familiar with the grossly distorted automobile colors that appeared under yellow sodium vapor lights, or the dulling effect of fluorescent lights on reds and yellows. Abrupt changes in lighting color, for example when we step from daylight into the red light of a photographic darkroom or bar, produce especially inaccurate hue judgments. Color distortion is obvious in surface colors around sunrise or sunset, but this effect is familiar enough, and sufficiently minimized by discounting the illuminant, that it has a trivial effect on hue recognition.

Terminology. Artists use both unrelated and related color judgments to determine the paint mixtures needed to match colors in the environment. Related color judgments refer to the "true" or local color as it would appear on a normally lit surface (which is how artwork is typically displayed), even when we see the surface under unusual lighting conditions. Monet's advice, that the artist simply match the hue of a retinal color patch, means a painter should ignore local color and instead match the hue as it appears under the influence of any contextual factors.  

Hue Purity. The third and last colormaking attribute is the clarity or intensity of hue, again where hue has the limited meaning of monochromatic spectral colors and extraspectral mixtures.

Hue purity ranges from intense or highly chromatic for pure hue sensations to neutral or achromatic for completely colorless (white, gray or black) sensations. However it is common to find a very chromatic color (such as a "blue" monochromatic light) described as saturated, pure, bright, brilliant, rich, vivid, luminous or glowing, and an achromatic or near neutral color as unsaturated, impure, dirty, dull, dead, veiled, dark, pale, whitish or subdued.

The substantial overlap in the adjectives that describe chroma and brightness (and between both of them and the adjectives that describe vitality and intelligence) signifies the sensory and "moral" similarity between the two. However it is a parallelism rather than a polarity: chroma has its null state (gray) and brightness its null state (darkness). They are otherwise polar opposites: luminance is a broadband quality while chroma is targeted to spectral subunits; the most saturated possible hues, spectral lights, appear black if viewed as surface colors; and a strong luminance contrast by itself can produce both a high chroma and a luminance color perception.

The example below shows variations in the chroma of scarlet at constant hue and lightness.
 

The problem with the "wavelength purity" definition of hue purity is that the eye does not respond equally to the "breadth" of wavelength mixtures. The span of "red" wavelengths labeled "medium" in the diagram would appear just as saturated as a single wavelength of "orange" light, because the "red" wavelengths do not lose hue purity when mixed. In contrast, if the "medium" span of wavelengths were centered on the "yellow green" wavelengths, the resulting mixture would appear desaturated almost to "white". Emittance and reflectance curves do not represent these quirks of color perception, so hue purity cannot be inferred from the "wavelength purity" of the color stimulus.  

A second solution: define hue purity in terms of a standard light mixture. Hermann von Helmholtz, the 19th century surveyor of colormaking yardsticks, proposed that Sättigung was the proportional mixture of "white" and pure monochromatic light. This confines all hue purity measures to a single wavelength standard, creates a constant hue purity "ruler" (a mixing line between 0% at "white" and 100% at the pure spectral hue), and defines a practical method to manipulate hue purity in a color vision experiment (mix "white" and single wavelength lights of equal brightness). Best of all, this method matches an observer's experience of colored lights, which always appear to whiten as they are desaturated.

So we have a third solution: hue purity is the chromaticity distance from the white point to the color in a chromaticity diagram, in which by definition all colors have equal luminance. This is an alternative definition of chroma in unrelated colors:

4. Chroma refers to the attribute of a visual sensation which permits a judgment of the amount of pure chromatic color present, regardless of the amount of achromatic color (CIE, 1982).

This is a sensible definition in terms of an established color model or color space, as it creates concentric circles of equal chroma centered on the white point (diagram, right); as brightness or lightness go up or down, the circles become concentric cylinders centered on the achromatic gray scale. But it is a peculiar peceptual definition: if color is salt and achromatic content is water, it amounts to saying that you can taste that there is one tablespoon of salt in a glass of water, a bucket of water, or a swimming pool. Indeed, as more "white" light is added to a stimulus of constant chroma, the saturation appears steadily diluted.  

In any case, we now see that Helmholtz's Sättigung points to yet another definition of hue purity that requires a color model for its measurement:

5. Excitation purity refers to the chroma of a color area judged as a proportion of the chroma of the monochromatic hue of the same brightness and dominant wavelength.

So the chromaticity plane or chromaticity diagram in a color model is used to define the chroma of the spectrum locus at a given luminance or lightness, and the chroma of the stimulus is divided by this hue specific quantity. As a result, excitation purity and chroma will yield very different estimates of hue purity, depending on the hue of the color (see diagram caption, above right).  

Context & Hue Purity. Unfortunately, a chromaticity diagram is a map of color sensations, a map that depends on the way we define receptor responses to light and on our assumptions about how cone outputs are combined and weighted in color perception. In short, we are still not talking directly about color perception, the psychological side of psychophysics.

However, across the normal range of photopic (daylight) vision (excepting extremely bright surfaces or lights), chroma and saturation are constant across changes in illuminance provided that all color areas in the scene are equally illuminated. This is embedded in the definition of chroma as the colorfulness of an area judged as a proportion of the brightness of a similarly illuminated white. As illuminance increases, the colorfulness increases, but so too does the brightness of a white area; the ratio between them remains constant. In the diagram (right), the surrounding black or white areas, which appear to be similarly dark and only slightly contrasted under low illuminance, become more highly contrasted under high illuminance (the Stevens effect); but the red also increases in colorfulness in relation to the brightness of white, and color contrasts increase to match the increased contrast across lightness gradations. The result is constancy of the chroma.

Next, consider the relative luminance contrast between the color area and its surround: increasing color luminance or decreasing surround luminance increases colorfulness, chroma and saturation. (This was studied by Ralph Evans under the rubric of brilliance.) The diagram below illustrates the basic effect.
 

luminance contrast effects on hue purity

 
In this example, which only hints at the actual impact of environmental luminance contrasts, the violet and magenta color areas each have a characteristic luminance (CIELAB lightness L*) at maximum chroma, as shown in relation to the adaptation white or black at left. Each color is shown (center) within a surround of matching lightness, and then (right) within a surround that is of a higher lightness (for the magenta) or of lower lightness (for the violet). Although the effect is small, you should see the violet in the darker surround as more saturated, and the magenta in the lighter surround as less saturated.

What's more, the contrast between a surface color and a brighter surround induces a quality of "blackness" in the color that does not appear in colors perceived as isolated lights or as surfaces brightly lit within a dark surround. Gray and some unsaturated "warm" colors (such as olive, brown or maroon) only live within the limited luminance contrasts that produce the appearance of blackened surface colors. Lights viewed against a dark background do not appear gray or brown but instead as dim white or orange lights.

In general, provided the color luminance remains constant, minimizing surround luminance increases colorfulness. Or, if the illumination on a color area is selectively increased while surrounding color areas are kept at reduced illumination, the colorfulness of the color area increases. This is a gimmick widely exploited in "tourist trap" art galleries: by illuminating individual paintings with tightly vignetted spotlights, in a gallery space that is otherwise dimly lit, the colorfulness and lightness contrast in the paintings is artificially increased.  

Now, it is possible to eliminate most or all of the distorting contrast effects in surface color displays by viewing them against an achromatic background of the same lightness. This yields a definition of saturation that is specific to surface colors:

3b. Saturation is the colorfulness of an area judged in relation to an achromatic (gray) area of equal lightness.

The robustness of this saturation as a perceptual judgment is that any perceptible difference between the color area and the surround is, by definition, entirely due to the colorfulness or chromatic intensity. Shown below is the original red chroma series against an achromatic background of matching lightness: