Notes on Color Vision Theory

7/31/2019 Notes on Color Vision Theory

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J. M. Williams Color Vision Notes 1

Notes on Color Vision Theory

by John Michael [email protected]

2012-06-11

Mathematical and physical assumptions, with graduate-level commentary, onthe content of MacAdams' Sources of Color Science textbook.

Keywords: color vision, color, color science, vision, wavelength, brightness,

luminance, illuminance, line element, trichromatic, Schroedinger, Helmholtz,

differential equation, calculus, derivative, integral, tensor, Grassman's laws, just

noticeable difference, spectral locus, pigment

Copyright (c) 1978, 2012 by John Michael Williams. All rights reserved.

[email protected]


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Preface

This is a reprint of notes and exercises slightly revised from a series of lectures given in

the summer of 1978 at Southern Illinois University, Carbondale, as part of a graduate-level

advanced seminar on vision, hosted by Professor Alfred Lit.These Notes include only the topics on color vision by the present author.

The coverage below is very much up-to-date in 2012, even after more than 30 years,

because the understanding of quantitative aspects of human color vision has changed little

since the middle of the twentieth century. A few minor differences, such as the order of

arguments in integrals, will be noticed.

The reading references for the Notes are for the book, Sources of Color Science, by D.

L. MacAdam -- Cambridge, Mass.: TheMIT Press, 1970. This book of writings by classical

and modern experts was required of the attendees of the lecture series. This book is out of

print at present, but it is available in libraries and from Google and other online suppliers.

A few of the explanations below are meaningful only in the context of the MacAdam book.

However, considerable additional explanatory material is included in these Notes, which were

written to be understandable to anyone with some knowledge of the problems and a semester

or so of calculus. All the theory and practice of color vision is heavily mathematical, so a great

deal of the content of these Notes is devoted to a review of certain specific mathematical topics

possibly overlooked or forgotten by the attendees.

Readers seeking further insight into human color vision should supplement the

MacAdam text with the authoritative coverage in Color Science (2nd ed.), by G. Wyszecki

and W. S. Stiles -- John Wiley and Sons: New York, 1982. Other relevant material may be

found in Vision and Visual Perception, by C. H. Graham -- John Wiley and Sons: NewYork, 1985; inHuman Color Vision, by R. M. Boynton -- Holt, Rinehart & Winston: New

York, 1979; in Visual Perception, by T. N. Cornsweet -- Academic Press: New York, 1970;

and, in theHandbook of Chemistry and Physics.

Original

Day Topic

June 12: I. Early Formulations of Color

June 13: II. The Nineteenth-Century Chromaticity Diagram

June 14: III. The Nineteenth-Century Trichromatic Retina

June 15: IV. Color PhotographyJune 16: V. Basic Color Operations

June 19: VI. Introduction to the Line Element

June 20: VII. The Line Element

June 21: VIII. Colorimatry

June 22: IX. Colorimatry and Psychophysics

June 23: X. Review (not included here)


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Basic Terms and Definitions for the Readings and these Notes

Note: These terms and definitions may be somewhat informal and are tailored for the current

course. They provide supplementary explanations for all the special terms used in the

MacAdam textbook and other similar publications. Readers new to the numerical analysis of

vision especially should understand the difference between radiometric and photometricquantities.

1. A light. A stimulus defined by its physical properties and capable of evoking a response of

color.

2. color. A response to light energy which makes lights discriminable independent of their

spatial characteristics such as size, shape, distance, or association with an object. For present

purposes, this word will be used to refer to any difference, including a luminance difference,

which is reduced to near-zero during color matching.

3. reflection (of light). A change in the direction of flow (= flux) of light energy which leaves

the wavelength unaltered and which, at a smooth, abrupt interface, obeys the laws of rayoptics governing reflection.

4. refraction (of light). A change in the direction of flow of light energy which depends on a

change in wavelength and which, at a smooth, abrupt interface, obeys the laws of wave

optics governing refraction.

5. refractive index (of a medium). The ratio of the speed of light in a vacuum to the speed of

light in that medium. For a speed of light c = in a narrow frequency band d, the change in

speed at an interface is given exactly by the change of wavelength . Refractive index n in

general depends upon (a) frequency and (b) the direction of the E-vector (polarization) of

light in the medium.

6. absorption (of light). Diminishment of energy of light because of conversion of some or all of

the energy to heat or to the motion of atoms or charged particles in the absorbing medium.

7. pigment. Something which absorbs light.

8. visual pigment (of the eye). A pigment which supplies energy for physiological processes

governing vision.

9. absorbance spectrum (of a substance). A graph of light energy which is being absorbed as a

function of the wavelength or the frequency of incident light. The incident light is

assumed, or its units of measurement are normalized, to have equal energy flux in each small

interval d or d.

10. reflectance spectrum (of a substance). A graph of light energy which is being reflected as a

function of wavelength or frequency . Again, an equal-energy incident light is assumed or

is normalized.

11. action spectrum (of a response). A graph of some measure of a response as a function of

the wavelength or the frequency of a light-stimulus which is or was being presented. The

response may be an operant [in the sense of operant conditioning]; or, it may be some neural or

biochemical reaction occurring in the visual system.


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12. radiance (of a light source). This is a radiometric term. Radiance is measured in units of

power (or flux)Pradiated toward some receiving surface. This is power per unit solid angle

per square meter of source surface. Radiance is defined in small regions dA near any point on

a source in terms of radiant emittance P/A at that point; the energy involved is that which

leaves the region in a narrow pencil (cone) including a given angular direction toward the

receiving surface. The orientation of the element of area dA with respect to the radiatingangle involves a cos factor for which = 0 if dA is perpendicular to the direction . All

this leads to a differential definition of radiance N at a given wavelength as

N = ____d2P _____ (1)

(cosdA d)

13. irradiance (of a receiving surface, caused by a source). This is a radiometric term in units

of powerPincidental (but not necessarily absorbed) per square meter of receiver surface.

Irradiance is defined in small regions dA near any point of the receiving surface in terms of

the radiant flux per unit area P/A at that point. Radiant flux is energy flow per unit time

per unit area and varies inversely as the square of the distance between source and receiver.

This flux also varies as cos , with the direction of orientation of the receiver element dA

with respect to the source direction.

A differential definition of irradiance H at a given wavelength therefore is

H = dP / dA (2)

14. luminance (of a light source). A photometric term corresponding to source radiance

adjusted for the sensitivity of the human eye. To obtain luminous flux F from radiant flux at

photopic (color-visible) levels of illuminance, the relative luminous efficiency curve V is used:

F=k visible

PVd (3)

Then, a differential definition of the luminance L of a source becomes,

L = d2 F_ _ (4)

cos dA d

as in the definition of radiance above. At scotopic levels, the Purkinje-shifted scotopic relative

luminous efficiency curve V' is used in (3).

15. illuminance (of a receiving surface caused by a source). A photometric term

corresponding to irradiance, but adjusted for the sensitivity of the human eye. Once the

source luminous flux is obtained as in (3) above, a differential definition of illuminance E is

given by,

E=dF

dA(5)

This term is extremely important in experimental and diagnostic computations. A

corrected retinal illuminance E' may be obtained from the transmittance T of the ocular

media. For such a correction, it would be possible first to compute a corrected luminous flux

by

corrected F '= k visible

PTVd (6)


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Next, the definition (4) above may be used:

corrected L'=d

2F '

cosdAd=

1

cos

d

d(dF '

dA)

Using (5) above, and for retinal and distant source surfaces perpendicular to the optic

axis, we then find

L' d=dF '

dA= kdE ' ; (7)

and the final transmittance-corrected troland value E' then would be

E'= :pupil area

L' d= L' S (8)

in which by convention S = total pupil area in multiples of a 1 mm2 artificial pupil (to

standardize the area of visual input just before the eye).

The troland value E commonly in use is uncorrected and is based on a luminance Ldirectly taken from equation (4) above:

E= L S (9)

To control for the Stiles-Crawford effects, a different measure of retinal illuminance may be

defined by making L a function of pupillary entry , which is measured by displacement from

the visual axis. This just means that the illumination of the eye is displaced from the imaging

center of the pupil and/or is no longer exactly perpendicular to the pupil. In this context, the

Stiles-Crawford effects mean that

L

0 (10)

and so,

troland valueE ' '= pupil area

[ pupil radius

L

d] d

= S pupil radius

L

d (11)

16. hue (of a color). A change in hue depends mainly upon a change in wavelength of a light,

but it also may depend upon changes in purity and/or luminance of that light. In terms of

generalized differential quantities, if

d (hue) =(color )

d+

(color)(purity)

d(purity) +(color )

(L)dL ;

then, for a change in hue,

(color )

> >(color)

(purity)or

(color )(L)

.


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17. saturation (of a color). A change in saturation depends mainly on a change in spectral

purity of a light, but it also may depend upon changes in wavelength and/or luminance. To

restate this in generalized differential terms, if

d (saturation) =(color)

d +

(color)

(purity)d (purity) +

(color)

(L)dL ;

then, for a change in saturation,

(color )(purity )

> >(color)

or

(color)(L)

.

18. brightness (of a color). A change in brightness depends mainly upon a change in

luminance of a light, but it also may depend upon changes in wavelength and/or spectral

purity. To restate this, if

d (brightness) =(color)

d +

(color)(purity)

d (purity) +(color)

(L)dL ;

then, for a change in brightness,

(color )(L)

> >(color)

(purity)or

(color)()

.

19. primary (arbitrary choice of a standard light). This refers to a light which may or may not

be spectrally pure (= "homogeneous") and which is intended to be mixed with other primaries

to match an unknown light in a colorimeter. A primary color is the color of a primary light,

which latter, for measurment purposes, may be reflected by a pigmented surface, transmitted

by a pigmented filter, dispersed by a prism or grating, emitted by a filter or plasma, etc.

20. fundamental. This technical term has two major meanings:

(a) A mathematical function of (or ) which is used as a basis for explaining color-

matching behavior in a colorimeter; or,

(b) an action spectrum of a visual pigment as inferred (usually) from color-matching

behavior.

21. Trichromatic theory refers to the hypothetical three, and only three, fundamentals which

are necessary to explain normal human color vision.


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Short Table of Derivativesd f(x)d( x)

1.da

dx= 0 , for any constant a.

2. ddx

(a x+ b) = d(ax)dx

+ d(b)dx

= adxdx

+ 0 = a , for constants a & b, and any variable x.

3.d

dx(a x2 + bx+c) =

d (ax2)dx

+d(bx)

dx+

dc

dx= 2 a x+ b + 0 = 2 a x + b .

4.d

dx(a x3 + bx2 + cx+ d) =

d(ax2)

dx+

d(bx)

dx+

dc

dx= 3a x2 + 2b x + c + 0.

5.d

dx(x) = d

dx(x

1

2) =1

2(x

1

2) =1

2 x

1

2

=1

2x=

1

2

xx

.

6.d

dx(ln x) =

d

dx(log

ex) =

1

x.

7. d

(1

x)

dx=

d

dx(x1) = ( 1)x2 = x2 =

1

x2.

Some Basics of Partial Derivatives

Partial derivatives F x

are for functions F of more than one independent variable;

they are called "partial" in part because solving for one variable is only part of the solution and

usually affects the attempted solutions of all the others. Examples of typical partial-

derivative expressions are: f : f(x , y) , f(x) , f(x1, x

2, x

3) , etc. General approaches to

solving partial derivatives can be extremely complex and are too much for this presentation, so

we only touch on a few simple aspects here.

Partial derivatives algebraically can be simpler than other derivatives in that all

variables remain constant except one; they usually are more complicated in that the

expressions f x

, fy

, etc. generally must be treated as single numbers and cannot be

solved in quotient form to obtain differentials such as dx, dy, or df. The symbols "x", "y", or

"f" are meaningless for differential equations.

Example of an analysis of a simple partial differential equation:

Let the equation be f(x , y , z) = 3 x2y + yz2 + x+ 5 .

This example is extremely simple, as shall be shown.


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Three partial derivatives are possible, in this simple example allowing terms in square

brackets [ ] to be held constant:

(a) F x

=([3y ]x2 [ + yz2] + x[ + 5])

x= 3y(2x) + 0 + 1 + 0 = 6 x y + 1;

(b) Fy

= ( [3 x2]y + y [z2] [ + x+ 5])

y= 3 x2 + z2 + 0 = 3 x2 + z2;

(c) Fz

=( [3 x2y ] + [y]z2 [ + x+5])

z= 3 x2 + z2 + 0 = 3 x2 + z2 .

Here are two examples of how to solve for a differential in a derivative expression:

(a) Ifdf

dx= x , solve for df:

dfdx

= x => df = x dx= x1 /2 dx.

(b) If x2 = 2y1/2 , solve for dx:

1. Find y =f(x) by algebra: x4 = 4y => y =1

4x

4.

2. Then, differentiate -- using, if necessary, the table above:

dy

dx=

d f(x)dx

=1

4(4) x3 = x3 =>

dy

dx= x3 .

3. Finally, solve for dx, using algebra:

dy

dx= x3 => dy = x3 dx => dx =

dy

x3

-- which is the answer, if x 0 .


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Early Formulations

Lecture I: Reading assignment is Text pp. 1 - 61, for Day 1 of the lectures.

1. Isaac Newton used a prism to disperse the sun's rays into a spectrum. He then picked out a

small region of the spectrum and found that this region could not be dispersed further. The

insights he developed were that elongated, multicolored images of the sun meant "compound

light"; elliptical images meant light which was less "compounded"; and, circular images meant

pure light ("regular, [un]dilated" rays).

2. Newton found that his homogeneous rays (= spectral lights) could not be changed further in

color but could be mixed again to make white light.

3. Newton's color circle was based on a center-of-gravity approach: By analogy, lights made

brighten in various regions of the circle weighed down the visual system in those regions; the

resulting tilt determined the final color seen in a mixture.

4. Grassman's Laws. These, like "Maxwell's" equations of electromagnetism, actually are a

collection of principles discovered by several contemporaries and predecessors (see Graham,

1965, pp. 371 - 372). The concept of dominant wavelength is most important in

understanding these Laws: Not only is every color compoundable from the lights of the

spectrum (Newton); but, also, every color and/or its complementary color is compoundable from

one "dominant", unique spectral light of wavelength , plus a certain amount of white light.

Grassman's fourth assumption corresponds to Abney's law, which states that

L = VNd , (1)which, ifL is determined by flicker photometry in a 20 field, may be used to define luminance.


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For example, if L1(white) = V(N1)d (2)

and L2(green) = V(N2)d , (3)

then the term, L1 + L2 = L(white1 + green2), will be defined as,

V(N1 + N2)d . (4)And, in particular, ifL1 = L2,

L1

+ L2

= 2 L1

= L(2N). (5)

Given this, we then have,

VNd +VNd = 2VNd = V(2 N)d . (6)Hence, "luminance" makes possible a color metric.


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Day 1 Exercises

1. A derivative dy/dxmay be viewed as the slope of a line which rises a certain distance dy for

each horizontal run of distance dx.

a. What is the slope dy/dxthat the straight line,y =f(x) = 5x+ 6, has at the following

values ofx?

x= 0

x= 3

x= -1.

b. A parabola with formula,y =f(x) = x2, is drawn below. Using a ruler to estimate the

slope dy/dx, letting dx= 1.0 cm, at the following values ofx:

x= 0

x= 1

x= -1

x= 2x= 3.

c. For the parabolay =f(x) = x2 above, use the "Short Table of Derivatives" above to find

the general derivative dy/dx(which is exact for any value ofx):

dy/dx =


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d. Complete Table 1 below, using your answers from parts b and c above:

x f(x) dy/dxby ruler(1-cm runs ofdx)

dy/dxby formula from

the "Short Table"

01

-1

2

3

Table 1

e. For the parabolay =f(x) = x2 graphed above, use a ruler and let dx= 2 cm. Estimate

dy/dxat x= 0, x= 1, x= 2, x= 3, and x= -1 as before. Compare these new estimates ofdy/dxwith the values you have tabulated in Table 1. Are your answers closer for large or small

runs ofdx? Is your error in applying the ruler important here?

2. Consider the unknown functiony =f(x) graphed here:


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a. Use a ruler and 1.0-cm runs ofdxto estimate df(x)/dxat x= 0, x= 1, x= 2, . . ., x= 10.

Enter these estimates into the top row of Table 2 below.

b. Use the "Short Table of Derivatives" to find general derivatives of the following two

functions:

g(x) = x

1

2 and h( x) = ln(x).

For values ofxgreater than or equal to 0, these functions provide upper (g) and lower (h)

bounds on the unknown functionf.

c. Compute the specific values that dg(x)/dxand dh(x)/dxassume at the values ofx= 1 to

x= 10; enter them into the second two rows of Table 2 below.

This exercise is intended to show how the two alternative "theories" g(x) and h(x) might

explain the "data" off(x). Notice that Table 2 compares only the derivatives of the theoretical

functions and data, not the functions or the data points themselves. One result of this is that

constant displacements ofg(x) or h(x) from the dataf(x) are eliminated (recall the first formula

of the Short Table) and need not be taken into account. Only the shapes of the derivative

graphs, governed by the varying tangent slopes of the graphs, remain.

d. Compute the three Pearson product-moment correlation coefficients r of the three rows

of data you entered into Table 2 in the previous steps. Square each r to obtain a variance and

see how the derivatives of the data and the derivatives of the theories match. Which ofgor h

explainsfbetter?

x 1 2 3 4 5 6 7 8 9 10

df(x)/dx

(1-cm rows)

dg(x)/dx

(formula)

dh(x)/dx

(formula)

Table 2


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The Nineteenth-Century Chromaticity Diagram

Lecture II: Reading assignment is Text pp. 62 - 96, for Day 2 of the lectures.

1. Maxwell defines a "color triangle" with primary colors at its corners. Only three primaries

are required because normal color vision is trichromatic. A plane surface containing

Maxwell's triangle will substitute in appearance for a normally-visible three-dimensional

space if the sum of luminances (= total luminance) of the three primaries is held constant.

Two primaries would define only a line rather than a plane.

2. Maxwell's colorimeter involved the matching of lights in a bipartite field.

Two different bipartite fields are shown here: ====================>

3. Helmholtz gives an interesting and merciful account of Goethe's little-known and totally

wrong theory of color perception. Also, to understand physiological optics, it is important to

note the difference between color mixing for paint pigments versus for lights.

4. Helmholtz defines on p. 95 a "color pyramid" or "color cone" which is very important to

understand. Interested readers may find an expansion of this discussion in Cornsweet's

chapter 10.

5. Helmholtz has changed Newton's color circle into a more accurate flattened shape

(Helmholtz's p. 96, Figure 4). In this representation, colors of the spectral lights are on the

perimeter, while colors of mixtures of those lights lie within.

Back to Derivatives: Concept and Usage of Differentials

6. Consider the derivative,dy

dx= x2 , (1)

which happens to be the derivative of a function such as y =f(x) = (1/3) x3 + any constant.

Under certain conditions, dy and dxmay be considered numbers and, thus, they may be

manipulated algebraically. For example, from (1),

dy = x2dx (2)

or, dx=dy

x2

= x2 dy . (3)

Such expressions as

dx+ dy = 3 (4)

or, (dy)2 + (dx)2 = 0 (5)

then may have algebraic or geometric meaning independent of any need to solve for a function

y =f(x) in which dy/dxmight have a derivative. In such use, for example, dy or dxalone might

stand for a small error in estimating the value of some quantity; likewise, it might stand for a

"jnd" ("just-noticable difference" by human perceptions), for a small change in a stimulus


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value, etc. -- again, independent of any need to form the ratio dy/dxor to solve for a function of

which the ratio is a derivative.

In standard usage, the ratio dy/dxis called a derivative; dy or dxalone is called a

differential. Differentiatinga function may mean either finding a derivative or finding

one or more of the associated differentials.

7. Consider a three-dimensional space defined by orthogonal x,y, and z axes. Assume that the

nature of the space is unknown, but that small changes in x,y, and/or z produce small and

possibly measurable changes in some dependent variablef=f(x, y, z). In terms of

differentials, this is the same as saying that a small change dfoccurs as a result of small

differences in dx, dy, and/or dz.

If we now introduce small changes in dx, dy, or dz one-at-a-time, we can measure the

dependence ofdfupon the appropriate small difference in the corresponding differential. This

then suggests the meaning of measuring thepartial derivatives off. The results of such

measurements would be written in general as follows:

f x

= g1(x , y , z )

fy

= g2(x , y , z )

f z

= g3(x , y , z ) ,

(6)

and these could be estimated or calculated at various points (x,y, z). In general,g1,g2, andg3

would be different functions and would vary differently at different locations (x,y, z) in the

three-dimensional space.

8. Consider the Pythagorean Theorem as applied to differentials:

In two dimensions, the theorem would state that

(ds)2 = (dx)2 + (dy)2 ; (7)

in three dimensions,

(ds)2 = (dx)2 + (dy)2 + (dz)2 . (8)

So, when a small change ds takes place in any arbitrary direction, the distance of the

change can be expressed in terms of the differentials dx, dy, and dzalong the coordinate

axes. This distance ds, then, may be defined, as above, by

(ds)2 = f(dx,dy,dz) = (dx)2 + (dy )2 + (dz )2 . (9)

Of course, the value offhere also would vary with the actual location (x,y, z) at which the

change occurred -- and so, therefore, would ds.


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Day 2Exercises

1. Suppose we have f(dx ,dy ,dz) = aln(x) + bln(y) + cln(z) . (10)

Use the "Short Table of Derivatives" to find

f x , fy , and f z .

These partial derivatives each show howfchanges when only one ofx,y, or z is changed

slightly while the other two are held constant.

2. Consider a small change or horizontal run ds in the location of the point (x,y, z) such that

(ds)2 = (dx)2 + (dy)2 + (dz)2 . (11)

This actually tells us that for any functionf(x, y, z) which depends on location in this

space, the independent variables x,y, z, and s are related mutually so that

dx/ds gives the slope or rate of change of location in the xdirection,dy/ds gives the slope or rate of change of location in they direction, and

dz/ds gives the slope or rate of change of location in the z direction.

Furthermore, for any small change ds in location, the dependent variablefis related to x,y,

and z such that

f/xgives the rate of change inffor a change of location in the xdirection only,

f/y gives the rate of change inffor a change of location in they direction only, and

f/z gives the rate of change inffor a change of location in the z direction only.

Finally, we may wright df/ds to represent the total rate of change inffor a small change

ds in any arbitrary direction.It seems reasonable, then, to apply the Pythagorean Theorem here to obtain

(dfds )2

= (f xdxds )2

+ (fy dyds )2

+(f z dzds)2

. (12)

In terms of differentials, then, (12) becomes

(df)2 =(f xdx)2

+(fy dy)2

+(f z dz)2

= (f

x)2

(dx)

2

+ (f

y)2

(dy)

2

+(f

z )2

(dz)

2

.

(13)

Assuming that this result is correct, use it to find an expression for (df)2 for the specific

functionf(x, y, z) given in (10) above.

3. Suppose dfnow to represent the greatest difference in two lights R, with R1 =f(x, y, z) and

R2 =f(x + dx, y + dy, z + dz) such that R1 and R2 cannot quite be discriminated under a given

set of experimental conditions. Thus, dfis a jnd.

a. How are equation (10) above and the answer to eq. (13) above related in this contex?

Take the answer for #2 above to be (df)2 = a2 (dx/ x)2 + b2 (dy /y)2 + c2 (dz/ z )2 . How does this


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relationship compare with that between Fechner's and Weber's laws?

b. Suppose that in general a small change ds is given by (8) above. Ifds is required to be

exactly 0 for all allowable changes dx, dy, and dz, what (real) values might dx, dy, and dz

assume?

c. Suppose ds again is given by (8) above. Ifds is allowed only to equal some constant kfor all possible (real-valued) changes dx, dy and dz, how are dx, dy, and dz constrained

geometrically?

d. Suppose ds again is given by (8) above. If (ds)2 is constrained to equal some constant k

and dx, dy, and dz are transformed to some new coordinate space so that dx' = (dx)2, dy' = (dy)2,

and dz' = (dz)2, then how are dx', dy', and dz' constrained geometrically in this new space?


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The Nineteenth-Century Trichromatic Retina

Lecture III: Reading assignment is Text pp. 97 - 126, for Day 3 of the lectures.

1. On pp. 97 - 98, Helmholtz defines the fundamentals in terms of responses of classes of visual(neural) fibers at some level in the visual system. These classes correspond to the color

channels of more modern theories.

2. Von Kries suggests that for visual-pigment classes A, B, C, D, . . . with action spectraAS,BS,

CS,DS, . . . and spectral retinal illuminance ES,

elementary visual responseA = aAEd = a Aelementary visual responseB = bBEd = bBelementary visual responseC = cCEd = cCelementary visual responseD = d

DEd = d D

. . . etc .

(1)

The numbers a, b, c, d, . . . in (1) are functions of time and represent the sensitivity of the

retina with regard to the respective elementary visual response A,B, C,D, . . ..

Now, because normal color vision is known to be trichromatic, there must exist exactly

three orthogonal matching functions , , and involved when two lights are matched in color.

If the colors are matched in a bipartite field , then, at match, the lights must not be

discriminable. This means that, at match,

1

= 2

1 = 2and

1=

2

(2)

regardless of whether or not (E1)S = (E2)S for any S.

But, , , and are functions ofA,B, C,D, . . .. Thus, at match,

1(A

1, B

1,C

1, D

1, ...) =

2(A

2, B

2,C

2, D

2, ...)

1(A

1, B

1,C

1, D

1, ...) =

2(A

2, B

2,C

2,D

2, ...)

and 1(A

1, B

1, C

1,D

1, ...) =

2(A

2, B

2,C

2,D

2, ...) ,

(3)

which may be rewritten as

1(a A

1,b B

1, ...) =

2(a A

2, b B

2, ...)

1(a A

1,b B

1, ...) =

2(a A

2,b B

2, ...)

and 1(a A

1,b B

1, ...) =

2(a A

2, b B

2, ...) .

(4)

Assuming that there are only three distince visual pigments and (at some level) only

three corresponding elementary responsesA,B, and C, it then follows that, at match,

A1

= A2, B

1= B

2, and C

1= C

2. (5)


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Von Kries' point is that if there were, say,fourelementary responses, an assumption

which may be contrasted with the idea of "central connections" of Guild on pp. 204 - 207, the

three matching functions could remain equal for numerous arbitrary values of A1,A2,B1,B2,

and C1, C2. For example,A1 = A2 might be possible at match. So, a match would be a

fortuitous occurrence depending strongly, in particular, upon the precise values ofa, b, and c,

as in eq. (4), which prevailed. However, matches empirically are found to persist despitechanges in retinal adaptation (= sensitivity-): Therefore, three and only three elementary

visual responses are likely to exist.

In this connection, Von Kries goes on to discuss the effects of adapting different

subregions of the retina to different lights.

3. Von Kries' "invariable points" on the color chart also may be called copunctal points. The

coordinates (x,y) of such points are invariant under projective transformation.

Day 3Exercises

1. A completely randomized two-way ANOVA (ANalysis Of VAriance) assumes sources of

variance A, B, AB, and S/AB (S = sum of squares within cells = "subjects within AB"). For a

random-effects model with very large sample size n, the ANOVA table takes the form

Source df Effect SS

A p - 1 j [A] - [x]

B q - 1 k [B] - [x]

AB (p - 1)(q - 1) jk [AB] - [A] - [B] - [x] (6)

S/AB (n - 1)pq (ijk) [SAB] - [AB]

Total npq - 1 T [SAB] - [x]

The math model for this design may be written

SABi jk

= j

+ k

+jk

+ (ijk) , (7)

and the corresponding sums of squares will add as

SST = SSA+ SSB + SSAB + SSe. (8)

These "sums of squares" refer to values obtained off(A, B, S), the dependent variables in

the ANOVA. Assuming no interaction AB, which is to say that SSAB = 0, and assuming acontinuous variablef, suppose that the SSequation (8) were rewritten in differential form:

df =fA

dA +fB

dB +f e

de . (9)

(a) Iff(A, B, S) is found experimentally not to depend on A for small, nonzero changes dA,

rewrite equation (9) to reflect this finding.

(b) Suppose that neither A nor B could be shown experimentally to depend on the error

term e under any circumstances. Divide both sides of (9) by de to obtain an expression df/de


20/50


for the dependance of small changes infupon small errors.

Notice that if neither A nor B depend on e, as supposed, then dA/de and dB/de both must

equal 0.

(c) Suppose that B in equation (9) above cannot be shown to depend on e (viz. dB/de = 0).

Then (9) may be reduced to

df

de=

fA

dA

de+

f e

de

de; (10)

or, df=fA

dA +fe

de . (11)

The differential form (11) of this "ANOVA" thus suggests some such expression as

(variance in dependent variablef) = (Aeffect) * var(A) + (error effect) * var(e) . (12)

2. Linear Independence: By definition, if A, B, and C are linearly independent variables,

then for any choice of coefficients c1, c2, and c3,

c1A + c2B + c3C = 0

if and only ifc1 = c2 = c3 = 0. Otherwise, the three of A, B, and C are linearly dependent.

(a) Suppose A, B, and C are linearly independent. Also suppose that

A + 2B + 3C + kD = 0 (13)

holds among these variables. What is the only value that kD can assume which will make A,

B, and C become linearly dependent?

(b) Solve equation (13) for D.

-- Answer: D = A

k

2B

k

3C

k. (14)

(c) Which three coefficients r,g, and b respectively of A, B, and C in equation (14) make D

equal to ( r g b )A

B

C

?

-- Answer: r = 1

k; g =

2

k; b =

3

k.

(d) In equation (14) above, if D happened "momentarily" to assume the value 0, then A, B,

and C would become linearly dependent under whatever conditions made D = 0. However,

what value ofk is absolutely forbidden mathematically in equation (14)?

(e) Suppose k approached 0 in an orderly fashion in equation (13) above. What would

happen to the dependence/independence of A, B, and C?

(f) Suppose k approached 0 in an orderly fashion in equation (14). How would D have to

be changed in order to keep the equation true?


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Color Photography

Lecture IV: Reading assignment is Text pp. 127 - 133, for Day 4 of the lectures.

1. F. E. Ives refers to Maxwell's diagram (as on Text p. 68). The corner (= primary) colors of

the triangle in that diagram locate the dominant wavelengths of the respective light-mixturecurves (= color-mixture curves) of Figure 6 on his p. 128.

2. Ives' photographic films respond to lights to become his color records.

3. Ives' color records, correctly chosen, will reproduce the color of white light if exposed to

white light and then properly projected.

4. The Ives color records are made with bandpass color-curve filters. The lights used to

project the resulting colored image must be spectrally pure ("homogeneous") so that adequate

saturation is available in the mixture.

5. Pigments used in printing colored photographs are chosen by shadowing the corresponding

projecting lights and matching the light reflected from the pigments to the light of theshadowed areas.

For example, a spectrally pure green light at 527 nm (527 nanometers in wavelength)

would be regorded in the green record but not in the red or blue records. If projected, the

green record would be transparent, while the red and blue records would be opaque; therefore,

the projected image would be seen as green. If printed, the red-shadowed pigment would be

laid down, the green-shadowed pigment would not be laid down, and the blue-shadowed

pigment would be laid down; therefore, in white light the printed image would match a

mixture of spectral (red + green) added to spectral (green + blue), causing a result of one part

red, two parts green, and one part blue, which would be seen as a desaturated green.


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Day 4Exercises

1. Euclidean insights. A three-dimensional coordinate system divides Euclidean R3 into 8

octants. We shall be concerned only with the Ioctant in which the coordinates all are

positive.

(a) The plane x+y + z = 1 intersects the coordinate axes at three points. Mark thelocations of these three points on the axes drawn here:

(b) If two planes intersect, and they are

distinct, they must do so along straight lines.

Therefore, connect the three points you

marked in (a) with straight lines. These

lines will show the intersections of the plane

x+y + z = 1 with the three coordinate planes

x= 0,y = 0, and z = 0 and will form a triangle

in a new plane.

(c) Any three points in R3 can define a

triangle. Draw a dotted line L which passes

through the origin O: (x,y, z) = (0, 0, 0) and

also passes through the centerPof the

triangle which you drew in (b) above; this

line L is perpendicular the the plane of the

triangle.

(d) What are the R3 coordinates of the

pointP?

(e) What is the distance from O (centerof the coordinate axes) toP?

(f) Now, inscribe or sketch a circle in the triangle of (c) above. Under proper conditions,

an arc of this circle could correspond to the spectral locus of all lights of luminance

proportional to the distance |OP| and exciting the eye in the various amounts || = x, || =

y, and || = z. Here, , , and would be the orthogonal matching functions derived for a

given set of three primaries; your circle thus would define the eye's response to the lights of

the spectrum as matched by mixture of the three chosen primaries. All points of the circle

would be equidistant from O and therefore would represent equal brightness responses

operationally best defined as equal luminances.

If Helmholtz's "color pyramid" (or Schroedinger's "spectral bag") were inscribed in the

triangle, and if the axes were not scaled in equal luminance units, Figure 8 of the Text would

result. The "cross section" shown in Figure 8 also could correspond to a planar equal-scaling

of spectral-light luminances.


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2. Color centroids. For each of the following figures, sketch a vertical line passing through the

centroid (center-of-gravity) of the function drawn:

3. The dominant wavelength of a light is given by

=PVd

P

V

d (1)

and corresponds to the centroid of its luminous flux density as a function of wavelength.

Draw a vertical line through the approximate dominant wavelength of the following

distributions of luminous flux, and make a guess as to what would be the color displayed by

the corresponding light.


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4. Draw a color circle (such as Helmholtz's on p. 96) and mark two points representing the

effect of the two different lights on the visual system as follows: Put one point somewhere on

the spectral locus (arc of the circle) to indicate a saturated color; put the second point

somewhere else. Construct a solution for the color of the mixture of your two lights. How

would the computational operations leading to the solution be expressed in terms of dominant

wavelengths?

5. Vectors may be transformed to other vectors by multiplying them by matrices of numbers.

For example, if y = (y1,

y2,

y3) , and x= ( x

1,x

2,x

3), then a 3 x 3 matrixA , written as

A =a

11a

12a

13

a21

a22

a23

a31

a32

a33

(2)

may be used to compute y , given x , by the

transformed vector y = A x. (3)

Here,A is called the matrix of the transformation.

Given specific values for the components x1, x2, and x3 ofx , and for the nine entries inA,y is computed as follows:

y1

= a11

x1

+ a12

x2

+ a13

x3

y2

= a21

x1

+ a22

x2

+ a23

x3

y2

= a31

x1

+ a32

x2

+ a33

x3

.

(4)

Questions for topic (5):

(a) If we have A =1 0 00 1 0

0 0 1

, what is y = A x when x= (1,1,1) ?

(b) WithA as in question (a), find y when x= (1,2,3) .

(c) WithA as in question (a), find y when x= (2,2,2) .

(d) If A =1 1 1

0 0 0

0 0 0

, find y when x= (1,1,1) .

(e) WithA as in question (d), find y when x= (1,2,3) .

(f) If we have A =0 0 0

1 1 1

0 0 0

, find y when x= (1,1,1) .

(g) WithA as in question (f), find y when x= (1,2,3) .


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6. Cramer's rule may be used to solve systems of simultaneous linear equations. Although the

procedure may seem overcomplicated, each step is simple, thus greatly reduce typing or

computing errors. To use Cramer's rule to solve three simultaneous equations in three

unknowns x1, x2, and x3, follow this procedure:

(a) Rewrite the equations in the form of (4) above. y = (y1,

y2,

y3) will be replaced by the

constant terms (if any) in the equations; the coefficients of the unknowns (x's) will become the

respective entries in the matrixA of (2) above.

(b) Compute the determinant |A| of the matrixA as follows:

A = a11

(a22

a33

a23

a32

) a12

(a21

a33

a23

a31

) + a13

(a21

a32

a22

a31

) (5)

The determinant of a matrix is a number, not a matrix, and it will equal 0 if the three

equations (4) of our matrix are not independent.

(c) Now use the matrixA to form three new matricesA1,A2, andA3. This may be done

as follows, by replacing the columns ofA one-at-a-time with (y1,y

2,y

3):

A1

=y1 a

12a

13

y2 a22

a23

y3 a32

a33

, A =a

11y1 a

13

a21

y2 a23

a31

y3 a33

, A =a

11a

12y1

a21

a22

y2

a31

a32

y3

. (6), (7), and (8)

(d) Now find the determinants |A1|, |A2|, and |A3| by using the procedure of Step (b)

above. For example, to form the determinant ofA1, wherever an a*1 appears in the matrix

(2), replace it with ay* of that value:

A1 = y1(a22a33 a23a32) a12(y2 a33 a23y3) + a13(y2 a32 a22y3) , (9)

and correspondingly for |A2|, and |A3|.

(e) The solutions to the three simultaneous equations now have been found, because the

four determinants yield,

x1

=A1A

, x2

=A2A

, x3

=A3A

. (10)

Schroedinger uses Cramer's rule on pp. 147 ff.

Questions for topic (6):

(f) Solve for x using Cramer's rule if y = A x , y = (0,0,0), and A =1 0 0

0 1 0

0 0 1

.

(g) If possible, solve for x if y = A x , y = (0,0,0) , and A =1 1 1

1 2 1

0 0 0

.


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(h) If possible, solve for x if y = A x , y = (0,0,0) , and A =1 1 1

2 2 2

0 1 1

.

(i) Solve for x if y = A x , y = (1,2,3) , and A =1 0 0

0 1 20 2 1

.

7. Affine geometry. If the original matrixA in (2) above contains no x's (values ofx), the

transformation which is defined byA will be linear. An affine transformation is a linear

transformation used in place of some nonlinear transformation in order to simplify

computations. Affine geometry is a study of invariances under various (or all possible) affine

transformations in Rn.


27/50


Basic Color Operations

Lecture V: Reading assignment is Text pp. 134 - 154, for Day 5 of the lectures.

1. Schroedinger's gore (first used as "gores" on p. 140) evidently is a plane triangular sector

with sharpness corner at the origin.

2. Discussion of p. 145. Schroedinger refers to areas such that for the given choice of

primaries,

area = (spectrum)

x1()d =

(spectrum)

x2()d =

(spectrum )

x3()d ; (1)

thus, at this step in the exegesis,

1 unit of x1 + 1 unit of x2 + 1 unit of x3 = 3 units of white. (2)The match of a mixture of three lights on the left with one light on the right of (2)

requires a color match by a trichromatic observer.

With this in mind, consider small wavelength intervals d in the spectrum. For anysuch choice of , we have x

1() , x

2() , and x

3() as in (1) above; we also have, for a

standard white light with irradiant flux () , a ratio of the flux f() of any visible light to

the flux of white in any small interval d . This ratiof()()

specifies the spectral radiant

flux in intervals d just as well as f() alone does. The given light with flux f()therefore may be entered into the color-mixing equations by the components

f()

()

x1(),

f()

()

x2() ,

f()

()

x3() ; (3)

or, in vector notation, by

f()()

x() . (4)

The color coordinates of the light f() thus may be expressed by this vector in R3:

{spectrum}

f()()

x()d , (5)

which represents the three numbers

f()()

x1()d , f()

()x

2()d, f()

()x

3()d . (6)

The source radiance of each of the three components of x() mentioned in (2) abovethen may be adjusted so that the three numbers in (6) above become equal (numerically).

Then, the units in (2) each may be renamed as "1", so that they are equal when the three

components in (6) are equal.


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So, the final color coordinates of the light with flux x() will be in ratios

renamed unitsof x()d original (white) unitsof x()d

, (7)

which, in effect, were obtained by a scaling change of the results of (2) to conform with (5).

3. Concerning pp. 147 - 148, Schroedinger's equation (8) may be written in vector notation as

the scalar product ("dot product")

F = xF , (8)

which is exactly the same as

F = x1 F1 + x2 F2 + x3 F3 .

Schroedinger's equation (9) may be rewritten as

(A

B

C) =a

1a

2a

3

b1

b2

b3

c1

c2

c3

F1

F2

F3

= (what we shall call matrixK)F

1

F2

F3

. (9)

So, Schroedinger's equation (10) becomes

F = y (A , B , C) , (10)

which yields y (A , B , C) = x F from (8) above;

or, y (KF) = x F from (9) above.

Therefore, it follows that y K = x , which, transposing theKvector, is the same as

KT

y = x , (11)

in which Cramer's rule may be used to solve fory and thus obtain Schroedinger's equation

(15).

4. Discussion of p. 149. The development here may become clearer if seen as follows:

Suppose

(AB

C)=

a1

a2

a3

b1b

2b

3

c1c

2c

3

F1

F2

F3

= (what we shall call matrix K')F

1F

2

F3

. (12)

Now, recalling that Schroedinger's Wrepresents the color of the undispersed light at some

luminosity, the assumption that

A + B + C = F1

+ F2

+ F3

= W (13)

requires that, in terms of 1 x 3 matrices,


29/50


(1 1 1)(ABC) =(1 1 1)F

1

F2

F3

. (14)

Premultiplying (12) by (1 1 1) and substituting (14) yields

(1 1 1) F

1

F2

F3

=(1 1 1) K' F

1

F2

F3

, (15)

or,

(1 1 1) = (1 1 1) K' . (16)

Rewriting (16) in terms of simultaneous equations, we obtain the final result,

a1

+ b1

+ c1

=1

a2

+ b2

+ c2

=1

a3

+ b3

+ c3

=1 ,

(17)

which is Schroedinger's (18).


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Introduction to the Line Element

Lecture VI: Reading assignment is Text pp. 155 - 167, for Day 6 of the lectures.

1. An affine transformation of the color-mixture coordinates cannot be used as a measure of

relative brightness, saturation, etc. because of the Fechner (Weber) relation, which is notlinear. As Schroedinger points out, for a difference A between two lights, it is possiblethat an equality such as

|A (1 + )A | = |10A (10 + )A | (1)

might hold arithmetically; however, the difference on the left may be discriminable while that

on the right may be not. Therefore, the Euclidean color-space difference A does notmeasure discriminability, and this is exactly why a line element is needed.

2. To expand further on the point just made above (and on p. 156 of the text), call s a

discriminability function (dissimilarity function) which is at a relative minimum whenever two

lights x and y are at their least discriminable. We can write,

s = s (xy) = s (x1,

x2,

x3

; y1,

y2,

y3) . (2)

Ifx and y are not much different, then we have

y = x+ dx (3)

in vector notation; or,

y1

= x1

+ d x1

y2

= x2

+ d x2

y3 = x3 + d x3

(4)

in simultaneous-equation notation.

Ifs is a continuous, differentiable function, there are any number of polynomial infinite

series which might be used to express s (and likewise its differential ds). Schroedinger follows

Helmholtz here in choosing to take only the quadratic and lower terms of such a series when it

is used to express (ds)2 . Therefore, (ds)2 will be considered to be equal to a quadraticpolynomial such that

(ds)2 = ai k

dxidx

k, a

i k= a

ki(5)

in tensor notation (see below) -- which is the same as

(ds)2 = i = 1

3

k =1

3

aik

dxidx

k, a

i k= a

ki(6)

in more explicit summation notation.


31/50


Here, (6) may be expanded as

(ds)2 = a11

d x1

2 + a21

d x2d x

1+ a

31d x

3d x

1

+ a12

d x1d x

2+ a

22d x

2

2 + a32

d x3d x

2

+ a13

d x1d x

3+ a

23d x

2d x

3+ a

33d x

3

2;

(7)

or,

(ds)2 =(1 1 1)a

11dx

1

2 + a21

d x2d x

1+ a

31d x

3d x

1

a12

dx1d x

2+ a

22d x

2

2 + a32

d x3d x

2

a13

d x1d x

3+ a

23d x

2d x

3+ a

33d x

3

2

. (8)

Notice that the column vector (or 1 x 3 matrix) on the right side of (8) can not be obtained

from any possible matrix of constant coefficients a 'i k . In other words,

(ds)2 (1 1 1)a '

11a '

12a '

13

a '21

a '22

a '23

a '31

a '32

a '33

dx1

dx2

dx3

(9)

for any matrix of constant a 'i k

.

Thus, the transformation ofdxdefined by (8) can not be linear and must be written as

a11

d x1

a12

d x1

a13

d x1

a21

d x2

a22

d x2

a23

d x2

a31

d x3

a32

d x3

a33

d x3

dx1

dx2

dx3

. (10)

The line element developed below will be based on the assumption that all ds are equal

for just-discriminable lights; so, therefore, the color mixture coordinates based on any chosen

set of primaries may be scaled to express ds in equal measure in all small regions of color-

matching space. This equal measure is the metric referred to by Schroedinger.

3. On p. 157, the dxi

are invariant, but the ai k

will change under transformation.

4. Concerning p. 158, recall that ds is a differential ofs. To measure the dissimilarity, which

is the same as to obtain s for any two lights x and y , we may write

s = min{ x ,y}x

y

ds . (11)

In particular, ifx and y are just barely discriminable,

s = x

y

ds = 1 , by definition. (12)


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5. Concerning p. 160, Schroedinger's equation (2) means that

i = 1

3

k =1

3

ai k

dxidx

k= constant , (13)

consistent with (6) above.

The summation convention of tensor calculus is that all summation signs should be

omitted whenever subscripts are repeated. Two examples of the summation convention are as

follows:

(a) "ai k

xi" means

i

ai k

xi

, which would be a vector inRk . (14)

(b)a

i kx

i

xl=

ai l

xi

xk, k , l = 1,2,3; kl , (15)

means

i =1

3 ai k

xi

xl=

i =1

3 ai l

xi

xk; or , (16)

a1k

x1

xl

+a

2kx

2

xl

+a

3kx

3

xl

=a

1lx

1

xk

+a

2lx

2

xk

+a

3 lx

3

xk

for k = 1,2,3; l = 1,2,3; but kl ,

(17)

which last describes six simultaneous equations.

6. On page 161, Schroedinger's equation (4) represents a line integral which will be evaluated

along a path consisting of straight lines between the three points, (k, l, m) = (1, 2, 3), (k, l, m) =

(2, 3, 1) and (k, l, m) = (3, 1, 2), all of which lie in the plane k + l + m = 6. If this line integral

in fact equals 0, then (a) the equal-brightness function will be analytic, (b) Schroedinger'sequation (3) will be integrable, and (c) "brightness" will be meaningful as a psychological

response allowing of a color metric.

7. Concerning p. 162, Schroedinger's brightness convention is that not only do color matches

remain matches when luminances are varied proportionally, but also that the brightnesses of

the (two) matched colors also will remain equal. Only when matched-brightness and

matched-color are operationally defined differently is this convention meaningful. This point

is discussed later by Guild.

8. To make Schroedinger's p. 163 math more explicit, here is what he is doing:

If h = ai k

xi

xk

, (18)

then ln(h) = ln(ai k

xix

k) . (19)

So,

ln(h) xl

= xl

[ln(ai k

xix

k)] . (20)


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But (18) above holds for (x) = chosen as a constant multiplied. Therefore, (20)becomes,

ln(h) xl

= xl

[ln() + ln(ai k

xix

k)] (21)

= ln() x

l

+ x

l

[ln(ai k

xix

k)] (22)

= 0 + x

l

[ln(ai k

xix

k)] (23)

Applying a generalized form of formula #6 of the "Short Table of Derivatives" to (23),

ln(h)

xl=

1

aik xi xk

xl

(ai k

xix

k) . (24)

Recalling the summation convention and expanding the remaining partial derivatives on

the right in (24),

ln(h) x

l

= ( 1aik

xix

k) x

l

a11

x1

2 + a21

x1

x2

+ a31

x1

x3

+ a12

x1

x2

+ a22

x2

2 + a32

x2

x3

+ a13

x1

x3

+ a23

x2

x3

+ a33

x3

2

. (25)

For l = 1, the partial derivative on the right side of (25) becomes

( )

x1

= 2 a11

x1

+ a21

x2

+ a31

x3

+ a12

x2

+ a13

x3

; (26)

for l = 2, it becomes

( )

x1

= a21

x1

+ a12

x1

+ 2a22

x2

+ a32

x3

+ a23

x3

; (27)

and for l = 3, it becomes

( )

x1

= a31

x1

+ a32

x2

+ a13

x1

+ a23

x2

+ 2 a33

x3

. (28)

Inspecting (26), (27), and (28) and recalling that ai k

= aki

, it becomes evident that any

of (26), (27), or (28) can be written in the form

( ) x

1

= i=1

3

2ai l

xi

. (29)

Given this, we see that (25) above may be written

ln(h)

xl

=1

ai k

xix

k

i=1

3

2 ail xi , for l = 1, 2, 3 . (30)


34/50


According to the summation convention, (30) is the same as

ln(h)

xl

=

2i = 1

3

ai l

xi

k = 1

3

i = 1

3

ai k

xi

xk

; (31)

and, dividing both sides of (31) while recalling formula #2 of the "Short Table" yields

ln(h1 /2) x

l

=

i

ai l

xi

i

k

ai k

xix

k

, for l = 1 , 2, 3 . (32)

This is the same as Schroedinger's equation (6) except that it appears in terms of h1 /2

instead ofh alone. As Schroedinger explains on p. 165, this is because ifh = (some constant) is

to "split up the color[-matching] space in the manner of onion shells" (p. 162), the xl = (l2) must be squared to produce spherical surfaces. Recall that the actual calculations were doneon the right side of (32): So, the measure actually used must be the square of the h appearing

on the left of (32). Therefore, in terms of the units in Schroedinger's line element, equation

(32) should be rewritten as

ln(h) x

l

=

i

ai l

xi

i

k

ai k

xix

k

, l = 1 , 2, 3 , (33)

which is identical to Schroedinger's equation (6).

9. Concerning p. 163 - 164, we recall that Helmholtz's line element comes from (5) above, witha

i k= 0 , ik , and

ai i

=1

3x

ix

i; (34)

or,

(ds)2 = ai k

dxidx

k= a

i idx

idx

i

=1

3 [ dx1x1

2+

dx2

x2

2+

dx3

x3

2 ]= Schroedinger's equation (8), corrected.

(35)

10. On p. 164, still referring to Helmholtz's line element, Schroedinger notes that for a

brightness function h, using (33) above,

ln(h) x

l

=1

3

xl

xl

2=

1

3xl, l = 1, 2, 3 . (36)


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36/50


ai

also will change.

Because a light is specified in terms (a) of a plane of constant brightness and (b) of a

vector x perpendicular to that plane in small regions around x , small but discriminablechanges in brightness will be in the direction ofx . Thus, the following is required so thatadditivity of brightness can be guaranteed:

d x1

: dx2

: dx3

= x1

: x2

: x3

. (47)

To extend this affinte additivity to the entire color-matching space, Schroedinger

suggests therefore that a transformation

1

= (a1

x1)1/2

2

= (a2

x2)1/2

3

= (a3

x3)1 /2

(48)

be performed. Then, for constant brightness h1 /2 = a

1x

1,

h = constant = 1

2 + 2

2 + 3

2. (49)

So, the total differential d will be of the form

d =

x1

dx1

+

x2

dx2

+ x

3

dx3

; (50)

or,

d =

l

xldx

l, l = 1, 2, 3 . (51)

Using (48) above,

dl

=1

2(a

lx

l)1 /2 a

ldx

l; (52)

so, recalling the summation convention,

(d)l

2 =1

4(a

lx

l)1 /2 a

l

2 dxl

2 =1

4

al(dx

l)2

xl

. (53)

Thus, Schroedinger's Euclidean line element will be in terms of

(ds)2 = 4 [(d 1)2 + (d2)2 + (d 3)2]

= a1

(d x1)2

x1

+ a2

(d x2)2

x2

+ a3

(d x3)2

x3

, (54)

and this satisfies the additivity-of-brightness requirement because it was derived to do so.

12. Going on to p. 166, we see now how Fechner's law will be satisfied: The luminance-based

Weber fraction would not be constant, given (54) above. Schroedinger corrects this by

transforming the family of spheres of (49) above into unit spheres, for any given light x


37/50


(corresponding to the point Fof Figure 12 of the text). This transformation in space is thesame as dividing (ds)2 by ax = a

1x

1+ a

2x

2+ a

3x

3in xspace: This transformation

makes the line element of (54) become, finally,

(ds)2 =1

a1 x1 + a2 x2 + a3 x3 [a

1

(d x1)2

x1+ a

2

(d x2)2

x2+ a

3

(d x3)2

x3] (55)

as given in equation (12) of the text.

Applying the summation convention, (55) becomes

(ds)2 =1

ai

xi[ ak (dxk)

2

xk

] , i , k = 1, 2, 3 . (56)


38/50


The Line Element

Lecture VII: Reading assignment is Text pp. 167 - 182, for Day 7 of the lectures.

1. On p. 167 and following, a match will be made for two lights, x and y = x + dx , for

which x , y , and

dx all lie on the spectral locus of Schroedinger's figure 8. An error inbrightness x

imay exist (recall p. 156), so, at match, one light may have coordinates

(1 +) xi, while the other light may have coordinates x

i+ dx

i. Here, dx

i(or dx) reflects

a response of the observer; xi

(= x) reflects an error in brightness matching. In an actual

matching experiment, dxi

may happen to equal xi, but only by chance.

2. Consider equation (13) on p. 167: Recall the first mathematical expression on p. 166; in that

case, if is such that h + h yields a barely-discriminable change, this would be the sameas saying that dh = h is a barely discriminable change in h at the prevailing level ofh.Therefore, if

dh = h , (1)

then =dh

h=

1

hdh = d [ln(h)] . (2)

Now, returning to the problem, we may treat (ds)2 as a sum of squares:

Let (ds)2 = SSdifferences

(3)

= i

k

ai k

( xi dx

i)( x

k dx

k) . (4)

For fixed i and k, this is an

error in brightness minus an

error in setting;

and this is the corresponding

error in setting minus the

error in brightness.

So, using (2) above,

(ds)2 = i

k

ai k

(xi

d (ln h) dxi)( x

kd(ln h) dx

k) (5)

= i

k

ai k

(xix

k)

xi

xi

d(lnh) dx

i

xi

xk

xk

d(lnh) dx

k

xk

(6)

Using (2) above, = i k ai k xi xk ( d ln(h) d ln ( xi) )( d ln(h) d ln (xk) )

= i

k

ai k

xix

k[ d ln( hxi) ] [ d ln( hx

k) ] ;

or, (ds)2 = ai k

xix

kd[ln( xih ] [ d ln( xkh ] , (7)

Which is Schroedinger's equation (13).


39/50


3. The result in equation (7) above then is used by letting 1

= (a1x

1)1/2 , etc., as in (48) of

the preceding lecture above, and this allows Schroedinger's equation (14) to be obtained for the

standard wavelength discrimination experiment.

4. Concerning pp. 167 - 170, Schroedinger's equation (15) may be written

ds = [a

1x

1

h (dx1d 1x1

dh

d

1

h )2

+a

2x

2

h (dx2d 1x2

dh

d

1

h )2

+a

3x

3

h (dx3d 1x3

dh

d

1

h )2 ]

1

2

d ; (8)

or,

ds = [(a1 x1h )(dx1x1

dh

h ) + (a2 x2h )(dx2x2

dh

h )+ (a3 x3h )(dx3x3

dh

h )]1

2

. (9)

Becausedx

1

dand

dh

d are functions of in (8) above, Schroedinger's equation (15)

expresses

ds = f() d , (10)

which makes ds a function of for the given primaries (x1,

x2,

x3) and the transformation

matrix of ai k

. Equation (10) above therefore stands for Schroedinger's equation (15) and

may be used to convert experimentally observed errors d in wavelength discrimination intocolor-matching errors (dx/ x)i and vice-versa, a result made possible by the line element.

5. Comment on Schroedinger's equation (24) on p. 171: To scale the unit sphere, we must have

(ds)2 = 4[(dr)2 + r2 (d)2]

r2= 4 [(drr )

2

+ (d)2] ; (11)or,

(ds)2 = 4 [(d ln(r))2 + (d)2] . (12)

But (12) is in the form of the Pythagorean theorem; so, the space must be Euclidean and the

geodesics must be straight lines.


40/50


6. On p. 172, to evaluate

s (y , z ) = Y

Z

ds , (13)

the domain of the line integral (= the geodesic line) must be found first. Then, the measure of

dissimilarity s (see p. 156) may be computed by integrating (13) along the geodesic.

7. Concerning pp. 173 - 174, geodesics in space lie on planes through the origin; theseplanes become cones in xspace. Planes of constant brightness h are of the form

xl = const; (14)

or, in more explicit notation,

x1

+ x2

+ x3

= const. (15)

The coordinate axes l trace the chromaticity diagram on the planes of (14) or (15), and

this is a triangular diagram in the transformed space described by Schroedinger.

The quadratic form (1) of page 157 makes the value = 1 trace an ellipse around eachx in the plane of the chromaticity diagram. All geodesics therefore are segments of ellipses;and, so, the integral of (13) above will be evaluated along a segment of an ellipse inscribed in

the chromaticity diagram and intersecting the points y = Y and z = Z .

8. Continuing on p. 175: But, there are two such ellipses in general; and, the shorter one of the

two will not be tangent to the chromaticity diagram.

Therefore, Y

Z

ds will be evaluated along this nontangent path.

In particular, this means that = ( 1 , 2 , 3) will not have a zero

component in the domain of integration; and, so, neither will

x = (x1

+ x2

+ x3) -- nor will the algebraic-product functions

g() = 1

2

3or r (x) = (x

1 x

2 x

3) . (16)

Note that these last two may be used to test for the correct geodesic.

9. Concerning p. 176, on a geodesic line as in (12) above, we have

d ln(r) = kd , (17)

in which k, a constant of proportionality, is specific to the geodesic between

Y = y and Z = z .

10. Also on p. 176, only under one or both of the following conditions can the elliptical

geodesics degenerate into straight lines in x-color-matching space:

(a) Y ( = y ) and Z ( = z ) differ only in luminance; or,

(b) Yand Zrepresent complementary colors.


41/50


11. On p. 179, Schroedinger suggests a use for the line element in defining a "constant hue" for

colors obtained by varying the purity of a spectral light.

12. Concerning p. 180, and relevant to p. 179, it should be noted that Schroedinger's line

element does not predict the Bezold-Brucke effect in which a large change in luminance can

cause a change in hue.

Day 7Exercises

The first one of these two exercises is based on some preliminary calculation in part

assuming that the reader can access one of the figures in the textbook by Grahan (1965)

referenced in thePreface above, or can access some other reference containing "MacAdam

ellipses". The second exercise provides everything necessary from the textbook. Both

exercised may be read simply to strengthen understanding of the mathematics of these Notes.

1. For dh = 0 (i. e., for constant h), equation (8) above reduces to

ds = [ a1 x1h (dx1d 1x1)

2

+a

2x

2

h (dx2d 1x2)

2

+a

3x

3

h (dx3d 1x3)

2

]1

2

d . (18)

For just-discriminable lights, ds by assumption is the same throughout all of color-

matching space. Therefore, we may set ds = 1 and allow the constant h to be expressed in

terms of the coefficients ai. Solving (18) for d then yields

d =

[

a1

x1

(

dx1

d

)

2

+a

2

x2

(

dx2

d

)

2

+a

3

x3

(

dx3

d

)

2

]

1

2

. (19)

To obtain a wavelength-discrimination function using equation (19), we first note that on

a plane of constant luminance, x1

, x2

, and x3

are not independent but must be related by

x1

+ x2

+ x3

= const = c . (20)

Therefore, from (20),

x3

= c x1

x2

; (21)

and, differentiating (21),

d x3d

= d cd

d x1d

d x2d

= d x

1

d

d x2

d . (22)


42/50


Substituting (21) and (22) into (19) then yields

d =

[(a1x

1

+a

3

(c x1

x2) )(dx1d )

2

+

(a2

x2

+ a3(c x

1 x

2) )(

d x2

d )2

( 2 a3(c x1

x2) )(d x1d ) (d x2d ) ]

1

2

. (23)

We now have formulated d in terms of four constants a1, a2, a3, and c, as well as interms of the (x1, x2) coordinates of a chromaticity diagram and its spectral locus.

Along this spectral locus, the slopes of tangent lines will be expressible in terms ofdx

2

dx1

,

while small changes in wavelength d will be related to small changes dx1

in x1 and small

changes dx2

in x2. The four constants will depend on the primaries chosen and on other

characteristics of the transformation yielding the particular chromaticity diagram being used.

In the present example, we shall use the graphical plot of theXYZchromaticity diagram

of Graham (1965), figure 13.15, p. 391. The same graph may be found in Wyszecki and Stiles

(1982, Figure 2 (5.4.1), p. 308) and in other reference works. The figure is labelled

"MacAdam Ellipses of 1942", but we shall not be concerned with the construction by MacAdam

for these exercises. The table at the end of this chapter contains a few sample data points

drawn from this figure.

The abscissa in the MacAdam figure corresponds to x1 in (23) above; the ordinate (y axis)corresponds to x2.

To simplify computations at the expense of accuracy, let us begin by fixing

a1

= a2

= a3

= c = 1 ;

equation (23) above then becomes

d =

[( 1x

1

+1

(1 x1

x2) )(dx1d )

2

+ (1

x2

+1

(1 x1

x2) ) (

d x2

d )2

( 2(1 x1

x2) )(d x1d ) (d x2d ) ]

1

2

. (24)


43/50


It is now convenient to find d , the error in wavelength discrimination, as a function of :

(a) Choose some particular on the spectral locus.

(b) Read off the coordinates x1 and x2 (= xandy) of that point on the spectral locus,

using the abscissa and ordinate of Graham's Figure 13.15.

(c) Estimate d x1

/ d (= d x / d ) by using a small interval near along the

spectral locus. On the abscissa, read off how much xchanges (= x ) in this interval. Forexample, choose = 475 nm; let = 10 nm, which means 5 nm in each direction along thespectral locus, starting from 480 nm. The spectral locus of course is not parallel to the

abscissa.

Doing this, I obtained a change in xequal to 0.04 (= .10 at = 475 nm minus .06 at = 485 nm.). I wrote this change as "-0.04" because xis decreasing whenever is increasingin this region of the spectral locus. Given this, I calculated

d x1

d = approx. x1

= 0.04

10= 0.004 . (25)

(d) Now estimate d x2

/ d (= d y / d ) by using the same small interval , but

measuring on the ordinate to obtain x2

(= y ) for the given change in along the

spectral locus. Doing this, I obtained x2

= 0.12 at = 480 nm. Therefore, I calculated

dx

2

d = approx.

x2

=

0.12

10= 0.012 . (26)

(e) Use your copy of the XYZ chromaticity diagram to repeat steps (a) - (d) for as many

values of as desired. After that, compute d using (24) above and graph the result.

2. Using the procedure of the previous exercise 1, I have prepared three sets of estimates of the

human wavelength-discrimination function (2o field) in the table below. In that table, in the

four columns just to the right of the leftmost column, I have entered values read fromGraham's 1965 "MacAdam ellipse" (Figure 13.15). I obtained these values by the procedure

just described in Exercise 1 above.

My estimates from the procedure in Exercise 1 are subject to errors which particularly

affect the squared terms in (24) above, so I also am supplying more exact estimates from

Wyszecki and Stiles (1967, Table 3.2, p. 240, the x and y rows only. In the 1982 edition,

the relevant data are in Table II (3.3.1) on pp. 736 - 737). These data are indicated by the

asterisked-labelled entries in the middle of the table below.

In the rightmost three columns of the table below, I have given (a) the rough

approximation (a1

= a2

= a3

= c = 1) computed directly from equation (24) above; (b) a

"better" approximation using eq. (23) with c = 1, and new ai computed from known data as

described below; and, finally, (c) a "best" approximation using equation (23), c = 1, the ai as in

(b), and tabulated chromaticity data from Wyszecki and Stiles (1965, Table 3.2).


44/50


45/50


Table of Exercise 2 Results

Coordinates

onXYZ

Spectral Locus

( * = .001 )

Change in Coords

with Change in along theXYZ

Spectral Locus

( * = .001 )

d = Average Error in WavelengthDiscrimination

(arbitrary units)

x1 x2d x

1

d

d x2

d

a1 = a2 = a3

= c = 1[ using graph + eq.

(24) ]

{ai} forXYZ

system, c = 1

[ using graph + eq.

(23) ]

{ai} forXYZ*,

c = 1

[ using tabulated

data & eq. (23) ]

460.13

.144*

.03

.030*

-.0015

-.0016*

.002

.0017*77.80 26.16 30.77

470.12

.124*

.06

.058*

-.003

-.00026*

.005

.0047*41.90 14.80 15.48

480.08

.913*

.13

.1327*

-.004

-.0041*

.012

.0114*24.76 9.08 9.65

490.04

.0454*

.295

.295*

-.004

-.0045*

.022

.0212*18.09 7.46 7.74

500.01

.0082*

.54

.538*

-.002

-.0020*

.024

.0242*18.35 9.25 9.15

510.02

.0139*

.75

.750*

.004

.0035*

.015

.0157*24.80 17.44 16.66

520 .07.0743* .83.834* .007.0075* .002.0014* 32.36 137.69 197.37

540.23

.230*

.75

.754*

.007

.0073*

-.0055

-.0057*11.13 47.55 46.02

560.37

.373*

.625

.625*

.007

.0071*

-.007

-.0069*5.04 34.10 34.57

570.44

.444*

.55

.555*

.0075

.0070*

-.007

-.0069*6.86 31.99 32.54

580.50

.514*

.49

.487*

.007

.0066*

-.007

-.0066*

7.11 30.20 31.99

600.62

.628*

.37

.373*

.007

.0045*

-.007

-.0045*7.10 26.24 41.05

620.68

.692*

.30

.309*

.007

.0021*

-.007

-.0020*7.10 23.63 84.05

640.71

.719*

.28

.291*

.007

.0009*

-.007

-.0009*7.10 22.83 181.06


46/50


Schroedinger Summary, Colorimetry, & Psychophysics

Lectures VIII & IX: Read Text pp. 183 - 245, for Days 8 & 9.

1. On p. 184, Schroedinger recommends using subjects as their own controls; he describes a

procedure based upon a psychophysical method of limits as well as one based upon a method ofaverage error.

2. On p. 189, Schroedinger's equation (4), reminiscant of equation (28) on p. 173, gives a

specific evaluation of the integral; this evaluation becomes simpler if the lights being matched

either have equal luminances or differ only in luminance.

3. On p. 190, the "Figure 17" mentioned here by Schroedinger has been omitted from the

MacAdam text, but probably it was similar to Figure 13.

4. On p. 192, we find that Schroedinger's expression for saturation is derived by substituting

x ' = (1,1,1) into his equation (5) on p. 189. Thus,

s = ds = 2 Arccos [ (x1 x '1)1

2 + (x2

x '2)

1

2 + (x3

x '3)

1

2

(h h')1

2 ] (1)= 2 Arccos [( x1)

1

2 + ( x2)

1

2 + (x3)

1

2

[( + + ) h]1

2 ] . (2)This ends commentary on the Schroedinger line element; we pick up here with J.

Guild's more recent elaboration.

5. Concerning p. 194, around 1930, Guild and W. D. Wright gathered data on color matching

which has been accepted ever since, with very minor changes, as the standard for a 2 o

artificial-pupil matching fields.

6. On p. 198, it should be mentioned that action spectra could be measured at any of the four

levels of the "reception system" described.

7. On pp. 201 and following, we find that an operational definition of trichromatic vision would

be in terms of the three control knobs on the apparatus, which three always would suffice to

make a color match. Guild is assembling an associationistic theory of color perception.

8. On pp. 218 and following, Guild is using the word "color" to refer more specifically to the hue

(and possibly saturation) of a light. This is ordinary usage, especially when the user is

examining the semantics of color names such as red, green, blue, etc.


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9. On p. 223, Guild suggests that A

may be affected by

simultaneous color contrast in relation to the adjacent half-field:

Therefore, if

EA A = EB B , (3)

the standard radiance EA must establish a match governed also by a contrast-affected ratio

describable as B

/ A

: This means,

EA

=

B

A

EB

, (4)

in which EA represents the standard radiance being matched, EB represents the matching

radiance being manipulated, and B

/ A

represents a "brightness factor". This expression

may be rewritten as,

EA

= NB

EB

, (5)

for Nsome number.

10. On pp. 227 - 228, Guild carefully distinguishes the operational definition which yields the

relation versus the abstracted definition which, for example, might be written for a dictionary.

The values ofNB, NC, . . ., etc. all must be constant in order to enable luminance to be

additive; and, the luminance, to be a useful quantity, must be proportional to the radiance of a

light.

The constancy ofNoccurs if and only if V (the modern variable representing a"luminosity curve") is of invariant shape. Therefore, color matches cannot be predicted in

mesopic ranges using additive luminances; for this reason, V may not be used validly to

describe a photometric stimulus being presented at such levels.

[Luminous efficiency at mesopic and scotopic levels is referenced in Wyszechi and Stiles,

1982, section 5.7.2 (part viii), p. 406 and section 4.3.2, p. 256, respectively; and elsewhere.]

11. On p. 230, Guild describes the color-matching response by the integral, A=aAEd ,(recall Von Kries on chromatic adaptation, pp. 109 ff.), but not by a differential such as

illuminance E= (dF/dA) . Thus, a matching field must not be imaged at only one point on

the retina of the eye.

12. On p. 233, Guild's analogy, color : brightness :: shape : size, should be ignored.

Three radiances suffice if subtractive mixing (that is, adding light of one of the primaries

to the standard half-field as in the mixing field illustrated with (9) above) is feasible in the

given apparatus.


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If, at match, for sourcesA,B, C, andD,

left half-field = right half-field

such that EA

NA

+ EB

NB

+ EC

NC

= ED

ND

, (6)

then the field radiances Emust be related at match to the numbered indicators n in the

colorimeter by the field-luminance factors N'as follows:

nA

N 'A

+ nB

N 'B

+ nC

N 'C

= nD

N 'D

, (7)

which, of course, comes from matching-equation (6) just above. In (7), the total adjustment nDin the illuminance of the standard is given by

nD

=1

N 'D( n

AN '

A+ n

BN '

B+ n

CN '

C) , (8)

in which N'D is the luminance factor of the standard light.

13. To explain p. 234 in further detail: Because equation (7) above holds at a colorimetric

match, it also expresses a fixed relation among the lightsA,B, C, andD. So, lightD now is

replaced with a standard light S(which also happens to fix the distance of the color-matching

plane from the origin in color-matching space). Then, each term on the left in equation (7)

becomes at match exactly one unit Uof luminance, for use in all subsequent operations.

Therefore, for the standard-light match, equation (7) is replaced by

UA

+ UB

+ UC

= 3 US

. (9)

Because three units of luminance on the left of (9) by definition now match three units on

the right, (9) now may be rewritten asm

AN '

A+ m

BN '

B+ m

CN '

C= m

SN '

S, (10)

which corresponds to equation (7) above. But, both (7) and (10) express matches, so the

relation of the lightD to the standard Smay be expressed in terms of the colorimeter settings

for these matches, as follows:

nA

mA

UA

+n

B

mB

UB

+n

C

mC

UC

= nD

N 'D

. (11)

Furthermore, the instrumental factor nD for lightD may be eliminated; this, because, if

equation (9) holds, then we must have

nA

mA

+n

B

mB

+n

C

mC

=n

D

1= n

D. (12)

Using this relation, equation (11) may be rewritten as

nA

mA

nD

UA

+n

B

mB

nD

UB

+n

C

mC

nD

UC

= UD

; (13)


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or, + + = UD

,

which corresponds to Guild's equation (9) on p. 235 and is a typical datum ("trichromatic unit")

obtained during colorimetry, using a given instrument or other specific piece of equipment.

14. Continuing on p. 235, the , , and of Guild's equation (9) specify a line in color-matching space passing through a point with coordinates (, , ) and also through theorigin. All points with coordinates in ratios x

1: x

2: x

3= : : lie on this

Notes on Color Vision Theory

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