Constancy and contrast IIIB

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~~~t~~y, University of Amsterdam, Nether fan& 0

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CTIVF STIMWLW%(CMIClUSiQn) ity in

8.5.

calmsalr&do

. . . . - . . . . . . . , .p.

IVE MEASUREMENT OF CONSTANCY

9.

e measures ctive

scale

of constancy

AND CONTRAST.

and contmt

.....

i0.i.

The objective law.

.........

.....................

REFERENCES

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e

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l

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,

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. . . . . . . . . . . . . . . . . a . . . . . . , . . . .

Ambiguity

297

300 300 325

PRQCESIi OF PEhCEPT’ION ....................

10.

8.5.

ships.

334 334 352

in Causal

In the foregoing section the standard of ‘balanced’ perception e unbiased point of view has beea descAbed, which case is defined ective and the subjective. with respect to the two ‘sides’ of reality, the Th.e corresponding mirror model of perce (fig. 40) fhows conlplete s:-mmetry with respect to thi: mirror plane, the personal constant equalling its universal average &3ticl 1 : 1); it would then be possible to interpret the subjective dekption of reality, as to its forrrt, as the symmetrical counterpart--the mirror image -of the objective description, vice versa. This does not :irrrply a causal relationship: by a mirror (as well as by a lens) pairs of conjugate points are determined. in space; this geometrisally determined relIationship only implies that two sets of possibilities are ken, a one-to-one correspondence occurring between the e”,elments of e one set and the elements of the other set. geometrical correspondence nothing, can be said about the realization of the given possibilities, nor (ow far nor in which direction the reahzation would go (each point of a pair of conjugate points could be the image of the other). P_

* I3W.s 25, 222-292.

I,

II,

Wa:

Acta Psych., 1964, 22, 272-320; 1965, 24, 91-166; 297

6966,

II the objective escription of the results of the ex casesL of ambiguity are nd (sections 8.1 and symmetry of possibilitie follows that a subjec to be anticiprite The origin of the cbjective ambiguity can easily be seen: the bri ratio depends on twlo factors, the illumination tio e’ and the c” ratio x; these factors are not yet ratio is given. However, the ambiguity occurs if both factors are given the brightness ratio is This can be represented as follows: 13&S]

e’lx.=

SB . . . .

l

biguj ty,

.

e’/x-=+ SB . . . . . . no

biggity,

where the brl he same is found if the contrast r i a (cross) ratio of brig x4==

/x---+ =

Sc

.

.

.

.

.

.

ambi

SC . . . . . . no am‘bi

91 Ib ma : this more clear, a comparison may be made with the image that is tially forned by a magic lantern; it would only be realized if a screen wet-Is olffivef, to what e the ‘true’ image would be realized would depencf e of the sc th respect toi the lens of the luntesx

P.

0

z--

H

0

ga *

sf

QD

s

F-i

3

L.

3

0 ?

C.

s

Gf

3

CD

6

0

it-

L.

0

P+

F=+

c4

=r 0

..

2P

F -.

5:

L.

3

2

Id.

LL.

z

6 L.

0

id

Y

. W.

3

ire&ion away from the mirror, an towards the miI~OF.9” survey of the different cases of a combined, takes the following form:

biguity

i

ectio ein

132aJ [132b] ‘two su ective variables being t to be distinguished in view of their functional dl n in the subjective whole (just as ective stimulus in the objective whole). barrier, separates the objective and subjective variables. From [132] it follows that a causal relationship introduced the &jective and subjective variables would show ambi direction, i.e., if the inner movements are thought to be caus uter movements, or if the outer movements are thought to arise from the inner movements: in the first case (direction from the left to the right in [132]) thert! would be subjective ambiguity, in the secon right to left} objective ambiguity. conclusion can then be drawr.. that any direct correspondence ueed between the objective and the subjective reality (in the sense lelism”) would c1priori be ambiguous ; however, the indiir al comparison of the two realities, both of them being referred eal minor plane (as described in section 8. , would never give rise to any ambiguity! erg y

and z

are

OBJECTIVE MEASUREMENT

9. .

E

bjecf ive

OF CONSI ANCY

AND

CONTRAS

awes of Constancy and Contrmt

e literature, different formulae are given fair the measurement of try ti the mirror model refers only to the real and the in section 8.4, in the virtual the form of the actual has given. Since the objective and the subjective system:; in the model are frmct&xxzZ wholes, their ‘form’ is to be inteqxeted in a functional sense, thi,: direction of the movements and the amb@ity that might occur in 4 fr3 be real (and theeobjective svstem virtual) iguities will be found as indicated in [1301and [ 13I].

ig. 41. Ikterminatiori of the Brunswik ference value YD. point of r Jbjective equality, 5 ection factors of the Dbjects corn the line of constant object ulus point, I identity tio. cts actually used, s sti CIline of object identity,

arently, neither the re on factors, nor the i in themselves, necessary ut only the ratios x

3

I

=-.-x

x-e’ L--e’

I’@

line s restricts the stimulus li.ne, the co e non-correc than the non-corre formula used fcr its measureme runswik gales; thu b?th segzxnts. $‘P 3 and SI, decrease, by the same amoun the denominator in S ly the values of the 0nstancy used so far:

termination of the de the variable, ;he illumination ratio would s~xve as well, thou

s equivalence has been y point, in the y of illuminations. ly the same as t

pointof subjective stant ilhninati

on

by the ratio of the same lime segments

the inversion in the determination of the value of now be writte;? PS/PI.

[

in fig. 42 Ilie illumination ratio has be:en change value of ED’, the value of EL being kept co,nstant. The c - EL’ being varied and ED’ kept const;mt---would produce a secxxrd point S and a second point I g. 43), where stkm us equivalence and identity of illuminations would be obtained as well. In the figure the symmetrical exper’ment has been shown in the same diagram; here x is varied by ta’king rD as the variable, I”Lbeing kept constant. In the diagritm in fig. 143generalized co-ordinates (v) have been u:,ed, just as in fig. 14; in the lower half of the diagram they are to be interpreted as E’-values, in the upper half as r-values. The value of‘[BAR]’following from fig. 43 differs from the value fo so far; since in the present case BP/OH = l/x and BS/OB = 1/e’,

-m-q_

Fig. 43. Second determination of [BR]‘, reference values I-L and ED’. VDco-ordime m or ED’ in the darker pati of he field, VI,co-ordinate r~ or EL”in the fully Iigbd semifield, o I i idmtity line, 12 identity point, SZ stimulus point.

e’ in 113

by the inverse values. The

in gives the same value. ince x C 1, it follows fr0 [136] that this value will be greater than ‘I, except for the extreme values of constancy, 0 and 1, s well as its rtlatio ira a sec”lld paper (26); sult of the same expericting either the distant object onto the ear object onto the distant plane. Thus, her object can be used as thP variable, the other object then being constant; in the present diagrams the analogue is shown in fig. 41 (line AI) and fig. 43 (line BI) where the reflection factor of one object is kept constant, that of the other is used as the variable. he inverse determinations shown in fig. nd in the lower half 3f escribed by BrunsvLl; : fig. 43 could easily be applied to the experim instead of one of the objects being projected, it would, in thought, hdve to be displacSzd parallelly in the direction of the other object until jhe two objects are in perspective; the displacement would directly Fhow to what extent the one object had ‘approached’ the other in perception. Then, the distance ratio of the objects would be used as the variable in the calculation instead of the size ratio, just as, in the inverse determination of brightness constancy, the illumination ratio is taken tcl be the variable instead of the object ratio. AS Brunswik points out, each of the two forms of the ratio might be used in the corn arison of experiments, but, since the values are not the same, a choice must be made. The choice would also depend on practical considerations: if the size of the near object is va-ied in the experiment, it would be simpler to calculate the first value of the ratio; if the distant object is varied, the second calculation would be simpler. A general principle would seem to be imp ,cit here: in tl-e evaluation of tl1.cresults of an experiment, the variable actually used in the performance of the experiment should be adhered to. n the second evaluation

anerr, so in three eases the

the aumt of the experiments, the card in t9re darker semi-

be represented by a number of points lying on the line of constant value r~ determined e tied card. Al these points twould form part of a series, 9 series being S and 1. Thus,, in the description, the: method the present experiments appears to fit in with the atio, since all the pairs of cards conside?.*edbe1 n athe same way, if the converse method of m;atching were used, the card in the ighted semi-field being the tied one, it would fit in Ah the second determination of the Brunswik Ratio, represented in fig. -23.All the pairs of cards considered would then be represented in the diagram by points of the series SZIZ,on a Cne of constant

a&y been mentioned, the description given of the presen

e given the form of sn. e essmtially the same ic; both cme al value in [EM], Can

atio, since the variables rposc,

formu

tan oc( - tan y’ > 0, ------_ tan Cki’ this inequ-zlity gives a thi rived frorr_the ratio

nts v&h identical bmkgrounds, wil! not be negative-as every formula used so far.

X]‘, reference values point, x identity point corres

and EIR’. ing to P.

rAll

be written

line’ for measuring two segments marked off c-. that line by the rays s, he symmetrical line (in respect of the line o = i) wo emit. etrical err then give the reference line in

308 the condition Several kinds of reference line wo all the PSEs on the same line 0 PSEs should give the same value two he segments. One possibility, including the usu the Brunswik Ratio, would arise from a linear relatio rD and r~ (or, in the Symmetrical ex eriments, between EL’ and ED’). In the diagram, the reference line is then a straight line, its slope determining the ‘weights’ of the two varia les rD and r~ (or in the constant reference value. For instance, in the usual of [BK]’ @g. 41), the reference value is entirely determined by Wl9’a_ (fig. 43), the reference value is in fig. 42). In the determination oic rgls_ entirely determined by r~ ;:)I’ &‘nr), and in the case of [~R]s’ (fig. the weights of the two variables v are balanced, since here any point of the reference line shows the constant reference value I’D + VL. Generally, the weights of the two variables in the refe e value are in inverse proportilon to tlbz segments OA, and OB ma off by the reference line on the axes c-*fco-ordinates. These weights are relatively the same for any line of a set of parallels; moreover, the parallels will show the same ratio SP/SI if the points P and S lie on the same lines OP and s, respectively. From this it follows that, using a straight reference line, and with the relative weights of the variables in the reference va kzp? coastant, all the PSEs on the same line OP are described as equivalent, giving the same value [BR]’ of constancy. The straight reference lines of any pair of symmetrical experiments are entirely determined by the choice of the common identity point on the line ~1=I i, relative to the PSE. Thus in figs. 41 and 42 the reference lines are determined by the identity point I = 11, in fig. 43 by 12, in fig. 44 by x = 13. Each choice of identity point implies a special weighting of the variables r and E” in the calculation ctf [M?]‘, and thus in all these cases different values of [RR]’ will be fouti3. Iln fig. 45a, b the 1alues of [RR] are represented for two diKerent values of tan hi’, taking tan y’ to be the ndent variable. The curves are segments of hyperbolas, all of through the same terminal points, where Izonstancy would &c values 0 and 1, respectively. Only three graphs have been she y refer to the above three cases, i.e., of identilty points 11, x, and Iz. ff only positive weights of the variables are used, curves 11 and I2 form the limiting curves of the whole set.

_,,__-_-___-\

/Iii

tan ai =I

._--_,

1 I

0c

Fig. 45. t :ompariso,n of values [BR]’ of generalized Brunswik Ratios, (a) tan oii’ T=0.5 ; b) tan osi’ = 1; (c) comparison of (a) and (b). 11 usv+J determination of Brunswik Ratio, reference values m, EL’; 12 second Jeterrnination of [RI<]‘, reference values r~. En”: x I= Id third determination of [BR]‘, reference values Ym, Em’.

he same value is ith the helg a%[31] and [46] the limit o

point on the same line

compared. The figure shows that the graphs of value tan bci’ = 1 are set of hyperbolas in the reduced diagram for any limiting curves of t entity point used. nly a small past of the upper area in the di can +qe rea’tied directly-the limit (not indicated in the figure) formed by a curve that does not differ considerably from the gr of value tan ai’ = &. om the diagrams in figs. 41 4 it is evident that the usual me is the only method where each of termining the value of [ can be c;xAuated dir&y. This, however, does not imply that the second termination, by means of the identity point 2, would be less usable. he zero poles so far considered (of constancy as we1 as of contrast, les s and c) patis through the point 0 indicating zero reflection; a ioned, however, another interpretation of tion 7.9, and here the be possible (note 56, sting zero absorptio ass through the point i at case, wou d be the only value of [B

in

le aforegoing, it follows that in the o cctive cfescription iments different reference systems can be chosen ; all these cases rent values of ‘constancy’ would be obtained. same reference system but the choice would imply that arbitrariness herefore, the results of the in the description of perception.

ht it may seem that only the addition of one fixed ray is necessary, since 1’ the fixed line o = i already occurs. in the determination of by accepting in the system of description one of the limits choice would be introdu er, it will be clear that the identnty point, her. of reality and disr ive reality (fig. 29), must have a function ain which is at the ten that differs widely from that of the outer limits of that damair:.

then either

or

hese two possible values of (=:

n be given in one fo

where the cross-ratio he calculation of G el to an axis of eoreduced to a single ratio. considered to be nearly pa will then be very large of their ratio will be arallehsm is obtaine en tuaily their relative ratio-in the limit cross-ratio is then d A = x, and [Ml]

zf-,log 2/x.

f course, by using the reference line parallel to the other axis of co-ordinates (fig. 43), the calculation of C1 will be sknplified as weI\. oirnt A then lies at infinity, and the redu.ced c ass-ratio is determi e line segment ; since t?xxx segments are and to the values l/x a I/e’, hit: vaiues il\f C (the: signs in [ 11 remain the same since no are substitute for x and e’ but also re The reference line in fig. makes the calculation of both single rat& d ;.s [I 391necessary; by means of [31 61their values are to be x and &, respectively, and again the same values of :he caste of a reference line of arbitrary slope, the values of the d e’, but the cross-rati he latter can easily itrary) reference line, identity line o. ” follows from the similarity of triangles

Iu4 Geometri~~Zy this follows by drawing a line P hen the relationships PA/PI3 = y drawing SS’ parallel to QB (point Sf 0

%..

ig. 46. nation of the cross-ratio of constancy ( ty point; 0 line of t ratio determined b

referencz line; orrected stimulus iQ, PI,reflection factors.

ince IA” =

’

“‘)the followi

in the

ingone

the values x an e’ by :ie orther, I&HZ value is eliminated.; it then follows that &/x OF

of

the choice of reference line.

314

K. W. MOED

From [142] ancI [ 1431 it is evid.ent that the value in [1 the sign, has been determined unique(:v in the objective sense (namely, reover, the functional by means of the litits of difference of the !limitin

be clear. Points P, S, a relations in respect of points A’ and El’, i.e., by the ratios thus the limiting points measured. The identity for instance, point P is pared, but its value with the value IA”/1 determined by the illumination c f tile objects, but its val only equals the illumination ratio e’ if compared with the v of point I. Here:, the value lA’/IB’ is ihe reference value, de the identity point II In the cross-ratio this reference value has been eliiminated, thus the value of CR is objectively invariant. In section 8.1 the value e’/X (= e!’grD/IL) was fol’!?d to where RD and BL are the corrected values of the objective brightnesses of the objects compared. Thus the value of log CR in [ 1411can be written :

Pw

log CR = Z!Zlog BD/BL.

The use of the brightness ratio is known in ex erimental practice

(in non-corrected form, so as the ratio of physical br htnesses). Helson, in his paper “Some Factors and Implications of Color Constancy’, describes experiments on constancy where the ratio of luminaxrces of the disks co:mparecl served as ‘measures of compensation’. This method of measuremefrt ‘has been described : e red3ction screen technique of Katz wa nnecessary since we co measure the luminances of the disks with a beth illuminometer. Instead of -using any of the formulae for consta ouched in terms of degrees white of disks in free and reduced observation woehave used the ratio of luminances in apparent footcandles of the two disks as measures of compensation.. The larger this ratio is the greater the nsation (sincethe eye has had to compensate for greater differences in illumination.’ (27; pp. 251-2X9.95 inves&ation quoted dates from 1943; the original paper has bwn reprinted in Perception’., edited by Beardslee -and Wertheimer. T!k experimental

htne- ratio, +-

owever, the compensation is object ratio, line segment r. hus the same term is -+ site ere a clear example contradiction, caused by tsing the same term objective sense (corn re the controversies ntioned in section 8.2). e objective c_Dmpenree magnitudes, all ured on the same objective cornpar ison, part of the whole can l---its realization (in s reality) does not ive reality. In the sar re relationship t with again from the sub.jecti\fe point 3f view; then pzrt B of e

the whole is realized (or: “compensated’) and part Y remains virtual. 3 the directed lisle segment he value of log CR in [l #2rences of the variables A3in fig. 33. In this figure the logarithm +have been represented; however, in t e objective system of ciescription developed in part II of this pape the deviations have be*zn used throughout, The value of a deviatio , on an equally stepped scale, is half the value of the corresponding ifference; since in the symmetric model of description (section 5.7) the reflection factors as well as the values of the illuminations are measured on the same equally stepped rithmic values, the brightnesses are also equally stepped on that scale (this follows from log E’ -I- log r = log B + constant). Thus, in order to make the objective measure of constancy fit in with J,hc objective system of description used so far, the factor 4 must be added arrarrlgementusea t>y H&on shows the possibility of variation in three directions, the importance of which is stated as follows: ‘mile the mai.n variable of i!lterest to USIwas reflectance of background, since this factor has either been overlooked Or too casually dismissed, reflectance of the shadowed disk, and depth of shadow were also investigated, The interrelations of these three variables seemed to USto be of paramount importance in the explanation of phenomena of visual comPensation’(27; p. 252).

Constam@ywill then e

The value of the logarithmic deviation fro “bcnearly the same as the value of the non-l when the deviations arc small. (AB < 1). may differ considerably: the upper limit of A B i crea~:s until, in the cas? of d B = 1, its value would be infinite. section 5.10, sho\v,0 the general rela%onship between the two types

0 0

can be wiritten

ere the positive sign in [ 45] had to be chosen in order

h negative, SIP being measured in. the negative &#&, itself being negatixe negative dire fcpllows from the positive G3rmeasuring the angles y’ and ai’ : an increase of th and s run along the reference line in :.he pasSIre direstion, so ef ‘line sqment e reference line were stance of poin :gative sign in of [PS]h positive. ss-ratios in [ 142] and [ 1 in [146]: hyperbolic measure, just as

1h h ==:

d.

rry refemace line, then, the relations’hi

common factor log

r

e’ = log

eing omitted, cxxrespon

e especially suitabl on a r written

Y US nce line of constant

r comparing case: of slight an es log I’Lof the rD the value in [I 511 may be

constant. 1523 is CL Tb oules:, atio, cmrecte he value [I&i’. in El respect to the line s equivalence I Thus, the * to be the ‘absolute’ form of the that form which does not depend on any arbitrarily dmen refkencc: value.. runswik agrees with houless tlhat the substitution of logarithms for

y

when the “span” bet

shovvs that with

means of a diagra

f3aforegoing ir. has

c / /

5 experiments on contrast have rence values, the res en the same form. ’

320

H.

IL W.

OED

the identity line (iin bsth c:xpeCtientsj, but the zero poles are functionally different. Thus, clonclusion 111for experiments on constancy is also valid for experiments on contrast, the zero pole of constancy being replaced by the zero pole of contrast ‘The same method may serve to obtain the objective meastire {contrast, the objective measure of constancy being defined. T SEs me referred to the poles, as in the determination of the .Ratio, only the zero pole of constancy has to be replacL3 bv the zero pole of contrast in order to obtain the correspondiag Ratio for measuri contrast.96 In formula 11341 of the Brunswik Ratio the illumination ratio (e’) must then be replaced by the background ratio (G). The first value of the Ratio thus obtained is (x -- G)l(l - G), correspondin [i?JZ]‘lin 11341;the values corresponding to [MS]‘2and [BR]‘:Bfollow i the same way from [136] and [ 1381. The above measures of contrast will have the same value in two conjugate experiments if the background ratio and the object ratio calculated in both casts in the same way, e.g., by dividing the lowtr value of the reAection factor by the higher, hence [153]

G=

R1IRz-c 1,

and 11541

x = r1/r2 c 1.

Of course, t.he second condition is already implied in the first, since the lower r-value will occur in the semi-field of the lower -value, (conclusion IIB) It will be remembered that in the experiments on constancy tne object ratio has been de-fined as r&D (< l), thus, by virtue of the above defirlitions, the values of x and G, just as that of e’, will always be 1~s ihan unity in each type of experiment. In the diagram, by using straight reference lines, th,e ratio of line se‘gmentswill be the same on any line of a set of parallels, rays OP and 0C (or OC*) being kept conscant. The reference line may go throuc$ the zero pole, and especially the line passing through the zero pole and its conjugate, C and C*, may be practicable. nes c (= OC) and c* (= OC*) indicate contrast equivalence, just as line 5 stimulus equivalence. Observing that contrast equivalence may be obtained by either a (second:) PSE lying on line c, or a (second) ---96 in fact lines OC and OC* may be used to represent the poles C and C*, just a.8 in conci

usion 115,section 7.3.

entsare ~1, diagram indicates bserving that. in the have interchanged their positions same measure of

of contrast if the

ferencc line used Just as in the experiments on constance , in a\1 the contrast exlPt:riments the card in the semi-field faathc$t away from the wkdow was made the fixed one. The different cards used in the other semi-field, in order to obtain a matching pair, would gike a e value ~2in the L~-,~of experiseries of points in the di,jgram lying on a line of co ments (zero pole Cg, and of constant value ~1 i case of eqkriments B* (zero pole C*). Thus, in the description, the method of n~atch~ng used in the B-experiments appears to fit in with thai measurement of contriist where ~2, R:I are the reference values (analogous to the determination of [BRJ’I in fig. 41): the matcS_ringused in :he B+-experiments would fit in with that measurement where 1’1~Rr are tKrereference values (analogous to the dc_cerbGation of [BR]2’ in fig. 43).

In order to define a measure of contrast that does not depend on any choice of reference value here again the invariance of cross-ratios, measured on arbitrary refei*encc lines A’B’, may bc used. y the rays tL>C P two variable points are marked off’ on the refi:rence line; in relation to the fixed limiting points A’ and B’ they determine the single ratios G and X, respectively, th*: unit of mezjurement being given by the value IA’/IB’ of the identity point I (compare formulae [143] and [lJ2]). The value of the cross-ratio is then determined by

[1551

log CIW = & log G/x,

where G/x = Rl/Rz : r& equals the contrast ratlo, Cl/C2 = 2, already defined (formulae [I 161to [I 19]). Here the Lding in section 8.2, namely - .. w Reference lines of ‘symmetrical’ experiments are identical here, since in the case of contrast there is no inversion; thus C and P interchange on the same line.

that the rela?ion of contrast of Zhe object an mined by R/r = C, is affirmed; in the camp ratio C’$Ts is a measure that does not depe vake. By adckg the factor Q in [155], in order to convert the logarit diEmma.ts in deviatiLons (from contrast equivalence), the obj ned by the hyperbo measure of “contrast’ is found to be dete measup*e(1distatnce ofthe variable points C ( ‘fine u; ,ec1: [I 5621-j

and F: ?.56b]

[C

h-

--

8

log Cl/C%

Observing that the value of log C’#?2 is negative (since G c: x), tht.: signs of the distances CP and C* are so chosen as to th:: positive direction rJn the referen line. However, in 81sicription of the exjperiments given in section 5.16, conjugate experiments on c,Dntrast are considered to be equivalent; thus, the (opposite) signs in [156a] and [156b] will not have any real meaning in respect of the measure of contrast itself. The signs only serve to correlate the measures of constancy and of contrast: in section 8.1 it has been found that the A-experiments on constancy can be directly compared with the IV-experiments on contrast, and, as the formulae [146] and [l:idb] show, the signs in these cases are op_posite (ratios < 1 in both cases). This would still be the case if the positive direction on the reference line *werer*eversed,or if the reversed distances, PS and PC , were measured; thus, just as the sign in [156] with respect to contrast, the sign in the measure of constancy (in 11461)has no real meaning in the definition of cotlstancy itself. The signs in the measures of constancy and contrast are like the indications ‘north’ a~1 ‘south’ used for the poles of a magnet: determining only the opposition of the poles but no intrinsic properties of tb ‘: poles themselves. FL.>* 48 shows the above opposition; the results of the Aexperiments are measured here on the same reference line the two measure,ments will be easily comparable?8 It will be seen that 88 RMs P in fig. 48 may be the equivalent reprwentatwes of the actual PSEs ljic;; on the same lhw OP.

Fig. 48. Objective measures of constancy and contrast. A3 reference line; A, B limiting points, I identity point; s line of stimulus equivalence, c* line of contrast equivalence; lines of constant objlect ratio determined by the PSE; [SP]h objective measure of constancy, [VP ]h objective measure Of contrast.

yperholically measured distances X)hequal the Values -I- log x a.nd - &log x respectively (see formula [ 1471). oreover, just as [SI]lI eq I), the same relationship will apply to ination ratio e’ bein replaced by the background ratio G, and the sign reversed: n the 4idimnce liil

Thus the relationship [?58]

shown on the reference line

11591

log

6 =

, c’orres c + log

log

st

e rcl

Jlt+,

in fig. 34 (the common factor -

34 no inversion occurs and in x have the same form, vljvs, where VI < 19.

fig.

correlating in log x+ might in gene the difference between [159] a for constancy. nstead of the value of [C* *]h itself, a ratio its 0 and 1 may be used, analogous to the ratio [MO]

cSch would be especially suitable for cornpairing experiments of di values of background deviation. 0f course, by varying rr, and r2 being constant, the value in [MO] may be written in co-ordi analogous to [ 11521; by using the co-ordinates log r1 of the poi and I on a reftxence line of constant value Q (= Rz) the ratio of contrast will then be:

[162)

log kc

--

lcsg r1.p

i0g

I-

log r1.T;’

r2 =

rl,c

l?:: >

RI.

ere Q ar il r2 are the co-ordinates of the given , or, in case r2 +of an (objectively) equivalent point of the same object ratio rl/rs. n general, if the values of e’ and G are determined by the evaluations, ulae [I511 and [MI] for the ratios of constancy and of contrast be more practical than [ 1521 and l’! ” 6 l]. n the ratios of constancy and or’contrast the line segments compared in the same direction, thus the (algebraic) signs occurrin in the easure!, of ronstancy and of contrast (in [146] and [ 156]) “rravenow y be added to the values of the ratios again as for instance by assigning the ratio of constancy , and the ratio of contrast values from 0 to 1. nts on constancy are then distinguished from arby

objective vaiue of \I [lIO, [119]]

stimulus function = cross-ratio of

2

constancy, contrast

[nzq, [131]

relative deviation of relative value o

[I;

logarithmic di

fig,i. 32, 34

31, [KY%] [I.: 41, [123]

arithmic deviation of objective measure of idem, hyperbolically

constancy9 contrast measured

ratio of constancy, contrast

@.s. 32, 34 [146],1[I%]

El4, I15451 W],

WQ]

of cards. The difference between this actual limit (35, i case) .a& the ideal limit (=

:itself in the performance of the experiment.

livould be valid in every case.

cornbinatioras would be 122,but in the present descr ption value of tan y’ are considered to be equivalent.

resent case the

for example, could be

Id always be possible (objectively!).

ease as their distances from 0 increase. Since the

*-experiments would then

the cas9 cannot ents in fig. 33, but it can be

scale measures one card step.

of

course,

rence r contains a who

th.e o

hercethe objective brightne enms i~mpliesthe equality consequently it has been it kll always be possible to match pairs of car experiments would give exactly the same brightness Fig. 49 shows an example where the logarithmic n&ion has been dividec! corresponding ratios th.e relative deviations are calculated.1~ 1

--+

A,

Fig. 49. The objective scale. _

P&stepped segment - - - - - - 1I-stepped segment, Ai,

of hywrbola,

gure shows only the one quarter of the domain in figs. 33 an tan y’ and dB or .dc are positive; in order to apply it to the &e 33 a ck~kwi~ tmmof !S” would be necessary, c ad in the case of the 35 a turn of 180’.

steps, the symmetry

of

101 A card difference measuring the same number of steps as the difference c in t fig. 3Q v.:guld give in the A-experiments the same contrast diflerence since here the contrasts are in (inverse) proportion to the reflection factors, the backgrobaJs being identical.

330 ~;he objective scale is not linear, but hypxbolic; i has hen shown for the steps formed by the backgr tions, and, of course, the same arguments apply to the inter S~AQS giving an equally stepped series on the same (logan as the baickground steps. Fig. 49, in a sense, demonstrates the relation tween the objective scale and a set of hyperbolas; any number of steps deter of a hyperbola in the figure, as shown for 11 and ivided into that number of steps. re shows that each point marked off on the line segme:nt number of its steps into two parts, which correspond to the two co-ordinates of the given point, measured by step-numbers on the two axes of co-ordinates. Conversely, the sum of these co-ordinates, measured on the objective scale, will be the same for any point on the line segment; this constant. sum is the parameter of the hyperbola (i.e., the objective deviation tan W’ or tan /?‘, measured OII the same objective scale). It then follows that in the case of a standardized arrangement formulae [ 1151 and [ 1271 for the objeetive matching of experiments A and B* can be written:

where /Ink and dnB* are the object-diRerences in the experiments A and B*, measured by the number of card steps, and LIII the number of ste*pc,on the line segment concerned (determined by 1log e’ IA or log G [B* respectively). In the present experiments a non-standardized :*et of cards is used; owing EOthis, the number dn is not constant in a series of matchings: blc 25 it varies from 94 to 13Qin tl matchings Al, B to 31 in AZ, B*, dnd in table 26 from 1 P to 15 in matchi in all these cases the matching itself is fairly accumte, as can be checked in columns 1 log x 1 of the tables. In section 5.7 it was found that the standardized described by means of two basic constants: the by the; extreme r-values of the series of cards, and t:le The constants k and $1also suffice for calculating by the logarithmic value-step

arrangement could ratio k determined number of cards, n. the objective scale : .

rwmber of cards only.

zs

e !atter vaiue it follows and their relative deviation fferent values of the logarithmic ient number of terms; in most ut 50 terms would

r of 40 t0 50 steps used so far in experiments on constancy contrast (section 1.1) appears to be a standard set by natural developent, human vision bei adapted SO as to meet practical exigences in the comparison of co1 S. It is not measurement of difference limeni, under quite artificial conditions, that is im ed in these experiments, ut just ordinary comparison of colours curring in everyday life under quite natural conditions imptying a fish on the whole situation; ere a gamut of the above number of fixed steps apparently suffices. Following this line of thought, the adaptation of vision must have been determined by the limits set by objective reality-here the extreme white and black occurring in everyday life. In the present experiment the ratio of their values was 1 : 46; a ratio of 1 : 60 was found by and other investigators (section 3.5); Munsell has obtained the extreme value of 1 : $5, which will certainly exceed the range of everyday cases. These figures are in surprising concord with the above number of 40-50 shades (96 being definitely too high, as found by Burzlafff). 1 gamut in subjective reality shows the same number of fixed steps as the measurement of the range ay1 a linear scale shows fixed units of measurement-the unit being determined by the termi black of the series ! 102 It is, however, only the two wholes that can compared in this way, not the single steps isolated from the whole, That the series of neutral colours is to bL considered as a whole will be clear from the objective description by means of the objective scale. lo2 In earlier investigations, e.g., of Lehmann (13; p. 510), degrees of black instead of degrees of white were indeed used as units of measurement.

H.

332

K. W.

MOED

ra.nge c f the given 14 steps follows from log k = - 1.400; the riitio k = 1 : 25 would actually be determined by the physical prop&es of the two dye-stuffs used. If the number of steps were to be chanl;;ed tcj 15, this w,ould not be possible by the simple addition of one more step: the additional step would belong to another whole, its range; following feom k = - 1.500 and the ratio k = 1 : 32 being 1n fig.

49 the

by oth\er dyes. ]LnI he sa_meway, a change to a series of 13 steps would not 5e obtained by the onG;sioc of one terminal step: either another whole determined by o*ther dyes would be obtained, or-in the event of the same dyes king; used-the same whole would still be given, but n a truncated, incomplete %vay. In both examples all the terms of the series should be calculated again, observing the changed number of steps (and the same range); of course, the same would be necessary if the range itself were changed Thus it has been found that any change of the series would affect each of its terms;; the terms zire not independent of each other, but they are given as parts of a whole and any change of that whole would affect all its parts. Actually, in experiments on constancy or contrast a series of neutral colours has often been ‘completed’ by the addition of one or more shades of extreme black or white; in the present exiperiments it was on;y certain difficultier:, in the evaluation that eventually led to the conclusion that the addition would not be legitimate. It might be compared with the ‘completion of the keyboard of an organ by adding at both its terminals a few keys of a piano! Tk:te question then arises : what are the ideas that have prompted several investigators to accept this, in fact, very strange procedure? Firstly, the idea of completeness seems to have obtruded itself (which implies the recognition of the series as a whole). MoreqDver, the aforesaid concord ofnumbers might point to a certain idea of baldmce. For instance, in the present e>;perime:nts the original series of 36 steps gave a ratio 1 : k = 22.1; after the addition of white and black cards there were 40 steps, and th.e value 1 : k became 45.4 (see table 1, observing the correction [lo]). At the time that the cards were made there was no explicitly given idea at all, but in the reconstruction of what has really happened, the idea of balance may be recognized. From th.e k-values it appears that the ‘balancing of the series’ by the addition of’ cards has gone a little too ,far, and indeed the added black has &en described in

CONSTANCY

AND CONTRAST

section 1.1 as a ‘very dee

333

here a small deviation might be see idea of which appears implicit in ed whole, a special group of objective scales e ‘bala,nced” scales, dei;tsLsed by the equaiity IV===f/k,

is the number of steps. eneral, since N must be a whole number, condition [HA] carrot be completely satisfied; with a series of 50 steps, the maximum deviation of N from the calculated value would be I ‘$6.

ram [ 1631 (where N = rzraced scale is found to be

[J653

1) and

[

1643 rthe value-step b’ of the

6’ = k log k.

; moreover, the entire This scale is determined by one constant ized by means of this experimental arrangement can row be st letely determined by the one constant, i.e., the arran dary line of the semi‘break’ occurring in objective reality at t fields. In fig. 49 line segment of 14 steps does not represent a balanced number of steps is too small (l/k = 25); also the segment series, since of 11 steps is not yet perfectly balanced (1 /k = 12.6). The line segment of 10 steps would show the one balanced series in this figures (log k = - 1.OOO,thus 1/k = 10). In the given arrangem.ent of cards, where l/k = 45.4, the number of steps (40) is a little too small, apart from the other deviations from a standardized series.

The objective scale is not linear, but in figs. 29 and 49 it is constructed by means of a linear scale; a linearly graduated ruler has been used to mark off the calculated values. Thus there are in fact two scales: a linear scale which forms the basic metric system on the paper? and the hyperbol’rc scale which in a way is ‘projected’ on to that basic system. This can be represented by means of Ihc mirror model. It is well known that a mirror ca!n give a quite distorted image, i.e., if its surface is curved. In the present case, however, the ,mirror is perfectly plane, Snt the objective space itself is ‘curved’ (as is the su’bjective space, see

334

section 8.4). Here the same efikct of distortion hours when the objet spa= is projected on to the mirror plane (atnd, OR the other s plan?, the subjective space); there are, then, in fact three scales, one (basic) linear scale and two) distorted scales?@ In the distortion of the whbAe,its structuring into parts is preserved: if one looks into a distorting Error, the details of a face, ears, eyes, etc., can still be identified-though with some difficulty. This constancy of structure may be the base of the concord of number found with respect to the present series, hich all refer 40 t same phenomenally given reality. 10.

‘%E

PROCESS

OF PERCEPTION

10.1. The Objective Law

In part XIof this paper a uniform description of the objective conditions has been obtained by observing the deviations from the standard case of continuity at the common boundary-line of the two semi-fields. Physically, the break in continuity can be measured as a jump in the luminance at the bound line; the luminance ratio found here equ the ratio of the objective ightnesses in the two semi-fields (see sections 5.3, 5.4, 5.5, and 8.3)Jw The ratio X of the objective brightnesses in the semi-fields can be written : r

whe:e ED’ and EL’ are the objective values of the illuminations in the two semi-fields, &) and RL the reflection factors of the backgroundq; e’ and G are the ratios &‘/EL’ and RD/& reSpdV&L With reference: to the objects in the semi-fields, the ratio X can be split up into two factors:

WI

X = e,‘G = (d/x-) (G/x+),

where the ratio of the reflection factors of the objects is written x- = rL/rD and x+ = PD/rL, thus x- = l/x+.

lm The xale on the linear ruler and the equally stepped logarithmic Scaxeare related to each other by a constau;t scale factor, thus there is in f(Wtonly one linear =le (moreover,the value-step6 or 5 could be made unity by adapting the base of the logar.%hmsin [I631 or iUS]. loQ With Astandardized arrangement th(:&minimum value of this ratio in the spcctal ti equals the constant k de&mining the balanced objective scale ([1651).

or, in shorter W

tation,

*

z

-I- e; 4‘-

quation one or two of the direction; in may the general experiments), depending on the algebraic of the logarithmic differences. means of the stimulus functions, &3 and &, the ratio can be

[I 701 thus, (III) in any experimem t two sti’mulus functions, cifconstancy and sf contrast he constant being objectively given as the ratio of the objective brightnesses of the two semi-fields. In logarithmic form this can be written: [171]

log

x =- log

%B+ ‘lag SC.

‘701,as well as the equivalent forms [I673 to [ %7 11, are id; the objective relationship represented by these equations may be called the objeche baw: it does not depend on.;; objective comparison but is valid for any pair of objects, and any combination of conditions, If in the logarithmic forms of the law (formulae WY W91TIWI\p the terms are divided by a factor 2, a relation be ween logarithmic deviations is given; the two terms on the right-hand side of these equations are then the absolute measures of constancy and contrast defined in section 9.9,; the term on the left-hand S’de is the ‘field’ determined d in the same abs4ute way. by the objective conditions, Since this field forms a whole, the objective law can 1be written:

inany

t comtancy and con d together as the two parrs into which the objectively determined whole is dividedby the objects. (IIIa)

expeximen

he two objective conditions refer to different aspects of objective reality, which are not dire&ly comparable but in perception are tog&her in the form of a (complex w ale (sections 5.10 to 5.12). stimulus function this complexity has been preserved: the two SB and SC are not directly comparable, since each of them refers to one of the objective conditions only (thus to one aspect of o reality), but they are bound together in formul [170] in the same way e va’luese’ and G of tlle conditions in [M]. Physically, the of luminance serves as the mediate to unite the two aspects of reality (section 5.10); objective&, the concept of objective brightness (section 8.3). According to the mirror model, the stimulus function projected on to the mirror plane serves as the ideal model for the action reflected back into objective reality. This ideal model is given in the form of a complex whole, and, as observed in section 5.12, the two aspects of such a whole will not merge into one another and lose their individuality, but each of them, in that whole, will preserve the possibility of being identified. However, when the ideal model is being realized in the object, the two aspects, identified separately, cannot both serve as a model at the same time. One of them is identified and serves as the model, observed by the outer eye; the other remains then virtual-it could be seen as a model and realized from another point of view or in another experiment. This may be compared with a painter who is painting a portrait of a model: it would be impossible to paint the front and the back of t!le ode1 in one and the same portrait-to paint the other side the pai;lter vJould have to walk around the model and paint another portrait, or the model itself would have to make a half-turn. In the same way, in the present case one would have to ‘walk around the mirror’ in the way dexribed in section 8.4; in the description the other aspect would then be the model, seen by th.e inner eye and realized in the subject. Or, by the other method, another experiment would have to be srt up with the stimulus functions themselves interchanged; the other aspect would then be the model, seen by the outer e an-3 realized in the object. From the above it follows that two ki of r:xperiments will arise, thos on constancy and those on contrast; in the first the n&ion SB

minatio: is deviate of experiment will arise nditioris do not

ealt with in th s part of the pajq either Ilc~ws,from [ 1Si W-ICI obsllrving the that in the A-experiments on c,ll:TL?ncy ,k’-- SI3’s_ plies, and in the B.

WI hese two equations D 541

@XP*4

*-experiments on contrast

x=

x+~sIc].

may be combined in the form x=

x+*x-

the other fact or would where either X- or x+ is realized by the o be the value of x in another experiment o the same relative deviation ill or AC
In this for&m of the objective law, hJ measures the objectively given field, determined by the difference log e’ or log 6; of the illuminations or backgrounds; JZ+and n- are the two ,a~~rts---representing constancy and contrast respectively-into which this field is divided by the objects In any experiment one part is realized, the other remains virtual; the

(since here e’ -

1).

he three numbers in [I 751are integers if t andardized; in the ease of a non-standa ay aArise,but they woul not really imply a gain in accuracy. n order to visualize the objective law, the two aspects which ca bjectively be distinguished in experiments on constancy and contrast are indicated in a diagram (fig. 50). In fig. 50 the reference line for measuring constant ly used for measuring constancy, but a o would serve the purpose. The two a measured wwithreference to the limiting po 61 and [M])~ or th,e objective scale sho n the latter case both aspec s, and the entire field (between lines o, s or o, c) as well. scale, marked off on an arbitrary reference line between the limitin is divided into two half-scales by the identity point on line o, one .afe for the upper half of the diagram, the other for the lower half. On a reference r to line cb the two half-scales are identical, just as they are on a r of the domain of objective reality (Part III, fi . 29); in that case they have the as the scales in 3g. 49. By means of rays pa ejected on to any other re

ment, however, ii woul ce then the ratios ca. ple, in an experiment on eps, the

value of n- arises, owing to irregularit&s

GO

t on constancy

Fig. 50. Interrelatiorl of constancy and contrast ), “Contrast” ==logarithmic contrast of contrast; equivalence, 0 line of pair of objects, ~1,~2 r&:ction facto63 of objects.

and subjective equality obtaine with nos. 21 an ards, ‘contrast’ would be 5113 d ‘constancy’ S/l is in th.k simple way that of constancy, the values of the geometrical series. ig. 50 shows the complete synthesis of ‘constancy” an the two aspects zre inseparably bound tog section 4.5 the operational synthesis has reover, it is shown that in the comparison of ex with experiments on contrast the two aspects In the fcbregr;;ng there would seem to be a contra icition with respect *. In the h-experiments the stimulus experiments A and ideal model of the action n of constancy, SB, should be *-experiments the s ( i.e., of the choice of c function of contrast, SC. owever, in formulae [I 721 and [ 1731 of these experiments, compared with the generai formula [ 1701,tlhe other stimulus function has been replaced by the object ratio x and thus the ‘wrong’ function seems t;) be ‘realized’. he seeming contradiction is eliminated by the recognition one ideal model of the action: ‘unity’, appearing in the form of ‘i This mcdel is not directly accessible, but the deviation from identity is given, pointiq indirectly to the identity itself. Any ‘realization’ can 0 an approach towards the ideal model, and will thus imply a &crease in the given deviation from identity. deal standard case would be the ideal situation, both the objects eir envircbnments being perfectly identical. ThereFore, t ode1 for all other case if it were reali is in rtself the ideal changes would gilIrea deviation from identity.107 _.- __ ‘consfancy’ has been maintained in order to elucidatt:: the ns of well-known terms, but the term ‘logarithmic brightness ore direct and ob.jective description---in the experiments on would increase as the obje o rlstancy’ cviation increases2 Other objections ncy’ have been given b lson (27 ; p. 25 1) ; the term ‘compenIson as a substitute would, in the present objective description, 2 experiments on contrast. caneidentity could be substituted for another; this, hwsever, would and the corresponding points in the diagrams (on the same d bo be equivalent.

(ex

(SC

=

x.._ =

1; exp.

ent ; as shown in ce from each other in the differences f course, the latter

functions

SB and Sc will then alse decrease.

ight’ stimulus function indeed acts ence, in each experime as the model. ince both stimuhts functions are given, bound together as a comple>; whole, a problem then arises with respect to the meaning he other stimulus function in the performance of the experiment. 50 it will be cl ar that by the other firnctio,il a deviation icated which iurcreases as the deviation of t!le objects again it is found that this function definitely does not ensionless relatio In the contrary, in slo far as the v stimulus function is re ected in the choice of the objects, it will counte -balance the function of the ideal model! cessively greater objective dilTerences fro , the deviation from the identity shown ideal model will decrease, but the secok deviation from identity by the other s ulus function) will increase. At a certain (determine point a ‘balance’ will be obt d ; this b4ance is subjectively determine

thus the results o owever, Ser. hed if t1h.eideal

letely rea

[ I731 GUI thcrli ,-. ( B II. ;e

x = x+.

( c zzzz ; ex

)

)

n formulae [17T] to [18

two limits are de experiment), where either oile or the other of t e general formula [ 1701is unity; these limits af values given in [lCN[Iand [lOa]. f course, in tk two limiting cases the one factor S(===1) c omitted in [170], and the one term log S (= 0) in the logarithmic form

afthat formula as well ; thus, if the value of a stimulus function reaches unity, the fu.nction disa ears out of the formulae. This may be compared with the actual formin: f unities described in section 8.4; at the very moment that unification occurs the objects disappear out of sight. nity, appeikri:>g 21s identity, is considered to

e the only one ideal! of the action, ij: follows that the field of a will lie between deal 1imil.s given in [NO] and [ 1031 (which rmulae arc now in [170], undei- the condition that either $3 = 1 or SC = 1).

n section 7.3 the results of the A and -experiments are found twezn the limits set by the two objectively determined ideal ues 108;the theoretical argument can now be advanced that ir of ebjerzts cali tained only in the field the two stiml nctions counteract here. of the objects exceed the limits of the field, bo*thsti an increasing; deviation from the idea!. valu the same ckectian and both counteract this (choice of objects r rat then be obtained.109 1-e~l.xrimentsthe lower limit has been passed;, this may be caused imperfkxxion ~9’tile actual standard case, which shows the same irregularities. Ia practke ancjther two limits will arise from the absolute values of the brightintensit ESare too low or too high, the function of the causal trains will the Edsduations (section 3.3) the process of unification itself was them

L.

3

ka

L.

3 0

v1

L.

r?

WJ

C.

C.

i2 r-+

L.

L.

L

-?*

n

344

oice l:$’ the objects. In formula determined objectively as X(= SBO or y a non-caustllly determined choice of objects, however, causally entails a de SthdUS

fUIlCtiOn: SC =

x-

Or

SB = x-j-.

or ScO remains in the formulae as other experiment, the two aspe same experiimient envisaged from th example of the painter who is pai In the ser:ond case, i.e., in the expkment, SIBor SC would be realized in the su description this realization can only be ‘virtual’. n divilded into two parts, the one part be other part 0nLy virtually. his partition of the field, a of the two parts in the object and subject respectively, can be visualize the mire model, as shown in fig. 51. In fig. 5liI\, just as in fig. 37, the object A’ deviates frvlri tLt: o ich wouI!(;lgive a ‘true’ image of the ich. could T>eidentified with I). ed into subjective reality, as $ this line of (deviating) actio ’ represe ots the difference between obje er (sec:l’ilBn 8.4) ; similarly, the distance eciive ‘self’ differs from the his ‘true’ self (i.e., the seJf that could be identified with the Sm

cl. However, the virtual parts

se points are considere without breaking them the front of the

art of log .Cvirtually realized in the subject. Other nctstions ,ee filgs.37 am.I 39.

rst sheet, the virtual part on the then be obtai e upper part about the harimntd sily check for as a whole: around the axis unti he mirror mod plane; the meaning of this rema

n fig.

axis in

5 lb the gap I’[

as been ‘redzmced”to its essentials wit g_ 51~ it has then been ‘normaliz are shij:‘ted(without their distance:; fr

and .EA is drawn arbi

would not represent a.ny decrease at all-there

ay, in the virtual part of

_I_ mkE2 fn fig. 51 the notations I = A alld H

have been maimaiijed in order to georrretrical transforma tiorls in that figure. owever, with a view to

ey were exper

ents on

cont9a5+t,

ents at the same time; however, rception these notations could be misleading, sk;e neither I[, I rceived in the unities formed. Therefore, in using the no nd S (subject point). points might be denoted as

348

“ELK.

W.

MOED

that of the spposite experiment will be virtually known (in the inverse form-as ‘not realizeri in the experiment actually performed in objective reality’). ?C’hesame a.I;tpliesto the sub_iectivedescription; the interchange of obj,ectiwe and sul:!iective reality then entails the interchan actual and the virtu!al experiment 11:)what is virtually kno one clescription appl:ars to be actually realized in the other. Both experiments, Ihe one that is actually realized in objective reality and the other, which is a(:tually realized in subjective reality, refer to one and the same experiment performed ; the duplicity arises here from the obser?rer’s envisaging the one ‘act’ pej*formed from two opposite points of vie-w in s~ccessic~~~. For this very reason it is possible to speak of ‘one a& the same’ experiment, recognizing the identity of the experiment itself. In the objective and the subjective description of the same experiment thy number N of potentialities will be the same. To elucid,,G this, a comparison may be: drawn with the keyboard of a musical instrument where each key presents the possibility of a note to be realized. If the instrument is looked at from two different points, i s form may seem to be diff’erent, and the geometrical dimensions measured relatively to each other in two hoto,graphical pictures will then differ. However, the number of keys1 ill lbe the same; since the number is not perceived directly, it does rot have any form which might change, nor does it show any relation!Aip which might vary-ilumbers are given in an ideal and absolute way, not dependent on the varying perceptions. The whole formed by the potentialities is determined by the discontinuity at the bonnda.qGne of the semi-fields, which can be measured objectively. In figs. 32 and 34 this whole is represented by the upper line constancy ;and log G in that OTB segments, log e’ in the experiment contrast; in 11711its value is genera indicated as log X. By evaluatirlg log X on the objective slcale (which in the logarithmic diagrams is equally s*epped) the nu&er lV of potentialities is obtained. As fig. 32 and 34’ shlovd, the objectively determined whole of the potentialities is divided lby the objects into two parts, one part being realized in the objecr difference log X, the other part virtual.114 The 118

section 8.4, nl.~te74. lp4 fn fig. 32 a11 -hrec lint: segments will go in the same direction if the signs are for correlatingonly. thus if the object difference is mer,sured in ihat figure as log x-. In fig. 34 the oki& difference log x+ has been rep xsented.

CONSTANCY AND C

objectively realized in an

3 her experiment

of the

e J g;IIyteexlJerjrr,ent

poslle point of view; 51 the realization:; in bject) are represented ‘he figure shows the following proportion

:

which is not affecte by the two geometrical transformations shown. hus, the ratio of the two reGzations, m the actual and the virtual object, is determined by the ratio of line segments EA. and in the normalized model (fig. 51c) are measured on the horizontal axis. In [1X] the number of potentiatlitles realized in the two cases is denoted by n+ and n-; one of the two numbers (depending on the type of experiment) is actually realized, the other virtually. The relationship between the whole formed by the potentialities (the entire ‘potential’) and its two parts can then be written [182]

log x =

which is evident from [174]F In [175] and [ 182] corresponding terms are in direct proportion; they measure the same on two different scales, both equally stepped. Thus, in fig. 51 the ratio of the line segments AE and E*I* will also represent the ratio of the two parts of the whole representeci in [I 823, and, with the pro er scale factor, these line segments themselves will represent the two parts themselves. The ectire distance AI* then represents the value log X; the part of this whole which is actually realized (in objective reality) is generally indicattd in fig. 51c as log X, which in experiments on constancy will be log X- a;ld in experiments on contrast log X-1_. Of course, line segments AA’ and I*1 ’ (the ‘realizations’) must represent the two parts of the whole on a scak that is ako equally

115 In [ 1821, as in [174], x+ and x- refer to differ, nt experiments (here: the actual and the virtual experiment in the objective description), hence X+X- # 1; since bcth experiments themselves refer to one and the same experiment performed, the vakle of log X in the virtual experiment will be equal to that in the ~tual one.

350

13. K.

W. MOED

stepped (like the scale: of EA and E*I*), otherwise would be invalidated; however, here another scale

e proportion or may be us

since chonstancy and contrast are found to be bomd together insep:arably, it will be c ear that in the absolute sense the distinction between experiments on constancy and those on CO trast disappears ; thus the results of a;tl thesle experiments can be cctjmbined in order to determine one and Ithe sa.me personal constant, as will bc discussed further in Part 111~. Only from the relative point of view does the distinction hold, so the objective categories described in Part II (section 5X are still valid; however, what from the objective point of view is an experiment on constancy is from the subjective point of view an exrcriment on contrasr, and vice versa. As a further consequence of the synthesis described in this section, cortil-ast must be recognized as a concept of exactly the same generality as constancy; thu.s in all the cases ih which constancy is measured (of brightness, size, form, etc.); contrast should also be considered. Generally, then, with respect to any objectively measurable aspect of the objects, both experiments on constancy and their counterparts on co!gtr?astshould be &signed (observing the a perational synthesis described in section 4.5); in every case the domain of objective reality can then be explored systematically (see fig. 27, section 5.16) and the values of the personal constant determined. The possibility of realizing this programme will be discussed-from. the theoretical point of view-in a subsequent part of this paper. ‘TJCdl:y, not only the ratio,’ of the values of constancy and contrast (which is related to the personal constant) but also the product of these values must be considered. In the product the two values are bound together in ai.rably,since, if one factor is omitted, the wh.ola=disappealrs (this result not obtained when the values are merely ad:ded together). Thus, by mlAtiplying the values of ‘constancy’ lend ‘contrast’ obtained in one an the sa.me experiment, their synthesis can be expressed ; conversely, these values can be recognized as the two factors contained ilo a quantity of universal character with respect to any aspect of the sbjlects. In phiysics such a universal quantity is formed by the ‘wcrk 116 By using two non-commensurable scale factors the realizations can be distinished from that which is potential (compare formula [83]).

CONST

35

and measurable as c?n ‘extenrealized in subjective reality and onsequently, the product of the values of experiment can be ught in the process of implications of this ill be I.Jiscusseld in

iie ‘constancy’ and 4contrast’ are foun to be compiementary, anti nerality, a remarkable difference between the two concepts cd. When it is endeavoured to measure and describe them objectively, two part of a broken whore are found; in the one part the idea of contrast is st 1 preserved and can be recognized, but in the other part--the ‘brightness deviation’, which shoul a measure of constancy ea of constancy has disappeared. 0 the term is still used for the sake of convenience, so as to lin with commonly used concepts; but in the objective description of the result of the measurement the term has in fact ost its sense; nor could it be used in the subjective description, since the only difference woulld be that the ‘real and the ‘virtual’ had interchanged. .A first conclusio with respect to this difference found between constancy and contrast can be drawn here (the further discussion on the terminology must be postponed to a later stage) In the foregoing,, contrast has been efined as a relationship betbeen two objects (th.e object proper and its surroundings, see section 8.2, formula [l I$]); constancy, however, can be considered to be implicit in the idea of ‘unity’ and, in that sense, it can only be complete-so it would not rnakt; sense to speak of, say, 70 ‘A constancy as the outcome of an. experiment.. ‘Unity’ cannot be directly perceived-when an attempt is made, at the critical moment the percept is lost (section 8.4, note ‘72; section KI.4, note 112) ; similarly, ‘constancy’ cannot be c:xplicitly conceived-when an, attempt is made (e.g. by measurement), in the result obtained the concept is lost. The conclusion

ofChapter

10 will be priblished as Part IIlc

352

l-23 24.

25. 26.

27.

28.

29. 30.

H. K. W. MOED

see Part I, Acta Psych., 1964, 22, 3 19-320. BIERG~~N, El.,E'tv~crlution Matrice. Paris, 86f.h ed., 1959. -I h pe&e et Ze muuvunt. Pa&, 3fst ed., 1955. BRUNSWIK~ E., Bit: Zug&@iehkeit van Gegenstinden fiia die und deren quantiitative Bestirxunung. (No. 1 of: Unt Wa$rnehmun~pgggensttide, (::d.by E. Brunswik), A 1933, 88’, 373-4X& HEWN, H., Some Factors and Impliczctions of Color Constancy. In: Recodings in Perce~~tion,ed. by D. C. Beardslee and M. Wertheimer, PrincetLjwn, 1960, 24.3-266. The original paper in 9. Opt. Sot. Amer., 1943, -73, 555-567.. HUGENIWLTZ,, P. TH., Tgd7 02 creativitGt. Amsterdam, 19!Ky. KLAGES, L., IX(TCbxndlag~~~der Charcikterkunde. Leipzig, t;sh ed., 1928. P~~MAI~, L,, TOLMAN, EC., Brunshvik?; Probabilistic Functionalism. In : Ps~9d&g~9, A Study of a Sfence, Study I, vol. i, ed. 4y S. Koch, New York, 1959, 502-5613.