JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

Discrimination StepELLEN HISDAL

Institute of Physics, University of Oslo, Blindern, Oslo 3, Norway

(Received 15 September 1967)

A theoretical expression is derived for the least distinguishable step in the number of pulses from anarea whose image is focused on and measured by a single photoemissive detector. For large numbers ofpulses this discrimination step is found to be equal to 6.8 times the rms fluctuation of the number of pulsesat the given level. For the threshold step, the 6.8 factor must be replaced by 4.1. The maximum noise-to-signal ratio is computed ior the cases of sample and path noise. A formula is found for the total number ofdiscrimination steps in a macrocell whose photon number may vary between two given limits. Satisfactoryagreement is found between computed and experimental values found by other researchers for the dis-crimination step of temperature, the discrimination step for the eye, the number of discrimination stepsin a TV picture element and the number of equal lightness steps in a grey sample as given by the formulaof the original Munsell system. A theory which assumes that in observing areas of even radiance, the eyeadds up the signals from neighboring retinal detectors can explain some experimental data for the eye.INDEX HEADINGS: Detection; Information theory; Vision.

IN this paper, the problem of the just-detectablechange of the number of photons coming from a

given area or sample is treated. The electromagneticradiation from the sample is assumed to be focused onand measured by a single photoemissive detector. Wecall this just-detectable change the "discriminationstep" in accordance with the nomenclature used byWright' for the eye. If we assume that the cones of theretina act like photoemissive detectors with a givenquantum efficiency, the problem treated here is never-theless more restricted than the problem of the dis-crimination step mentioned by Wright, where usually afield of view is assumed which comprises many detec-tors or cones. Our results can be applied in two cases.The straightforward case is the discrimination stepobservable by a single detector. The other case is thediscrimination step observable by a detector array, if theinformation obtained from the array is processed insuch a way that the numbers of pulses from the dif-ferent detectors are added up. In this case we couldobtain the same result with a single detector, having thesame size as the whole array.

By a "just-detectable change," we mean a changewhich can be interpreted by the observer as being dueto new physical conditions and not to statistical fluctua-tions of the photon number. Thus, if the opaque sampleemits thermal radiation, the change must be due to achange of the emittance of the sample surface or to achange of the surface temperature of the sample. Ifthe sample reflects a given incident beam, we must havea change of the reflectance of the sample surface. Thefinite size of the discrimination step is due, in the firstplace, to the quantized nature of the electromagneticradiation and in the second place to the above men-tioned statistical fluctuations of the number of photonswhich are present in every photon beam.

I. THRESHOLD SIGNAL

Assuming that we start with intensity 0, we havepreviously shown in a short note,2 that the smallest

I W. D. Wright, Researches on Normal and Defective ColourVision (Henry Kimpton, London, 1946).

number of pulses from the source beam which willgive a just-noticeable or threshold signal is approxi-mately 17.

We wish to define more clearly what we mean by athreshold signal. Suppose that the source beam isincident upon a reflecting sample and that we measurethe reflected beam. The simplest definition would be tosay that the threshold signal is the weakest incidentbeam which can tell us whether a sample is or is notpresent to reflect it, or, what amounts to the same,whether the sample on which the beam falls has areflectance 0 or 1. A definition equivalent to this is torequire that a measurement with the threshold beanmmust give us one bit of information about the reflectanceof the sample. We saw, however,2 that it is reasonable todefine a somewhat lower limit for the average informa-tion in a threshold measurement, namely n2m= 4 nat=0.72 bits. This choice comes about in the followingway. When a sample is observed with a given beam andwith a detector array composed of many photoelectricdetectors, the total information obtained will increaseas the resolution of the array increases. It reaches amaximum when each detector in the tightly stackedarray is so small, or when the observation period is soshort that the average information from such a singledetector is 2 nat. For a still higher resolution, the totalinformation decreases rapidly. For beams in the Wienregion, the average information per macrocell is3

i2 = 4 ln ((ni2 )/2r). (1.1)

By a macrocell we understand that part of the beamwhich impinges upon one detector of the array duringone measuring interval t. (ni 2 ) is the average number ofpulses per macrocell when a sample with reflectance 1is used. From Eq. (1.1) we obtain

(fli 2 )m = 27we = 17.1 (1.2)

pulses for i2 == 2 nat.Using the reasoning of Ref. 3, we can now answer the

question as to what the average information of 4 nat

2 E. Hisdal, J. Opt. Soc. Am. 57, Il1 (1967).

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VOLUME 58, NUMBER 7 JULY 1968

ELLEN HISDAL

per measurement, obtained with the threshold signal,refers to. We have a priori 18 equally probable measur-able reflectance values, equally spaced from 0 to 1.These correspond to 0, 1, 2, - 17 pulses from thereflected beam. The measurement tells us which of thesereflectance values the sample actually has, and shouldtherefore give in 18= 2.9 nats of information. However,because of the statistical fluctuations of the number ofpulses, there is an uncertainty or error probabilityconnected with every measurement which reduces theaverage information to - nat. If this uncertainty werenot present, then one photon would have been suffi-cient to give us one bit of information, assuming adetector with quantum efficiency 1.

We thus determine the threshold signal, and as weshall see below, also the discrimination step, from therequirement that a measurement must give us, on theaverage, - nat of information. Of course there willalways be a certain latitude in the way a threshold signalis defined, just as Rayleigh's criterion for the minimumresolvable angular distance cannot be taken literally tothe last decimal place but is still a useful quantity.

The radiation from the sample is assumed to bethermal and, to simplify the treatment, quasimono-chromatic. The methods of Ref. 3, Sec. 5 allow us, how-ever, to extend the treatment also to the polychromaticcase. The number of pulses per macrocell is assumed tobe large compared to 1.

The fluctuation formulas of which we make use arevalid in general for polarized beams. In the Wien limitof small degeneracies they are, however, valid forbeams with arbitrary degree of polarization, and alsofor unpolarized beams.

The statistical distribution of the number of pulsesfrom a single-mode-laser beam is the same as for athermal beam in the Wien approximation, 3 namely aPoisson distribution which can be approximated by a

gaussian distribution for large pulse numbers. TheWien approximation formulas for the threshold step andfor the discrimination step of the number of photonscan therefore also be applied to single-mode-laser beams.

II. THE DISCRIMINATION STEP

Suppose that we start with a number of photonsnl, different from 0, coming from the sample. We wishto find the smallest distinguishable increment Anl ofthe photon number. The corresponding numbers ofpulses are 112 and A&z2. We call this smallest distinguish-able increment the "discrimination step." We assumethat

Ani/ni= An2/•2<<1. (2.1)

The result is then applicable to all cases in which thederived formula for An2 satisfies the inequality (2.1).The equality sign in (2.1) follows from the relation

f1 2 =anll where a is the quantum efficiency of thedetector.

E ., Hisdal, J. Opt. Soc. Am. 57, 35 (1967).

The problem of the discrimination step can be solvedif we identify the pulse level 112 from which we start witha noise beam. In a recent paper 4 we distinguished be-tween two types of noise beams, sample noise and pathnoise. Both of these can be applied to our problem.

We start with sample noise. The incident signal beamgives (;Z2) pulses when it is reflected from a sample ofreflectance 1. The sample has a temperature which issuch, that if it were black, it would give rise to (A2)

noise pulses owing to its thermal emission. Alternatively,if the temperature of the sample is higher than that ofthe incident beam, we consider the thermally emittedradiation from the sample as the signal, (ni2) being thepulse number due to a black sample. The incidentradiation is now the noise beam, giving rise to (4l)

pulses when it is reflected from a sample of reflectance 1.We make use of the latter case for our problem, andillustrate it by an example.

The sample is the surface of a small object in an ovenwhich is originally in thermal equilibrium with thesurrounding walls. The sample can then not be seenor detected. We now raise the temperature of the sampleslightly, and wish to find the smallest temperaturedifference between the sample and the oven radiationso that the sample can be detected. The number ofpulses per macrocell from any area in the oven exceptthe sample is (,l2). We identify (A2) with the lower levelof the step which we wish to find,

U2= (Il2)- (2.2)

The number of pulses from the sample is (ni2) if it isblack, while if the reflectance of the sample is equal to1, the number of pulses from it will be (,M2). We ask aboutthe smallest value of the quantity

An2 = (1i2)- (M12) (2.3)

which can still give us some information about thereflectance of the sample.

As regards the detector, we require that it willrecord a genuine increase (i.e., an increase which is notdue to fluctuations) in the number of photons from thesample, or a difference between the number of photonsfrom the sample and its surroundings. The discrimina-tion step will be the same whether the informationwhich we seek refers to a change of the reflectance of thesample or whether it refers to a change of surface tem-perature of a sample of given emissivity.

Reasoning in the same way as in the case of thethreshold signal, we determine (nil2) so that for a given(Ml) we obtain on the average 4 nat of informationabout the reflectance of the sample in a measurement.As we saw in Ref. 4, Sec. 3, this choice assures that thesize of the sample corresponds to the resolution neces-sary for maximum information with the given noisebeam. Because of (2.1), it follows from Eqs. (2.2) and

I E. Hisdal, J. Opt. Soc. Am. 58, 977 (1968).

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9)ISCRIMINATION STEP

(2.3) that

An2ln2= ((ni)-(42))/&2) = (di2 -82)/82<<1, (2.4)

i.e., we have large sample noise. di2 = (n12)/z and82=(42)/z are the degeneracies corresponding to (ni 2 )

and (42).

i3, the average information in a measurement of thesample, is given by Eq. (2.23) of Ref. 4. The last termin the curled bracket of this equation may, because of(2.4), be written as the derivative of 1(82),

dE[l(2)]/d8 2= ln(1+8 2 )+1n8 2+2. (2.5)

The right-hand side of Eq. (2.5) was obtained bydifferentiating Eq. (2.18) of Ref. 4. Substituting fromEq. (2.5) into the expression for i3, we obtain for i3

in the limit of large sample noise

i3= ln[(di 2 -8 2 )/8 2 ]+42 ln(z/27re)-ln[(1+8 2 )/62j. (2.6)

Here we have replaced ln((ni 2 )-(Q2 )+1) by ln((n)2)-(g2))=ln(di 2 -8,2)+1nz. Putting i3 of Eq. (2.6) equalto 2 in order to find the discrimination step, we obtain

(ii2-32)/62= cE(1+b2)/62Z1', (2.7)

wherec ( 2 7r)we= 6.814. (2.8)

The left-hand side of Eq. (2.7) may be written as((ni2)-(12))/(Q2). On the right-hand side we replace62z by (42). This gives

(ni2)-(412) = c[(U2) (1±+2)]. (2.9)Now the right-hand side of Eq. (2.9) is c times a(,2)

where a(,Q2 ) is the nms fluctuation of the number ofpulses in a beam of (42) pulses. 3 Using Eqs. (2.2) and(2.3), we finally obtain from Eq. (2.9) for the dis-crimination step

n 2 with the noise (n2) and the extra step An2 with thesignal (ni 2),

l 2 = (q2)

An 2 = (n12),

(2.12)

(2.13)

and we ask about the smallest value of (n12) which canstill give us information about the reflectance of thesample.

Again we assume that An2 is small compared withn 2 , which means a small signal-to-noise ratio in thepresent case according to Eqs. (2.12), (2.13),

,An2 /n 2= (ni2)/(q2)<<1. (2.14)

i3 is found from Eqs. (4.10) and (4.11) of Ref. 4.Using a procedure similar to the one we used for samplenoise, we obtain again Eqs. (2.10) for the discriminationstep.

We have thus determined the discrimination step bythe requirement that we must obtain on the average4 nat of information in a measurement. According toRef. 4 this information refers to whether the reflectanceof the sample is 0, 1/An 2 , 2/An 2 , * - A n2/An2 , theuncertainty due to fluctuations of each measurementreducing the information to 4 nat on the average.However, as we saw earlier in this section, An2 maybe identified with the discrimination step, irrespectiveof what our measurement referred to. It is thereforevery gratifying that both types of noise beams give thesame answer for An2 /n 2 . As we assumed differenta priori knowledge for the two noise-beam cases, wecould not be quite certain that this would be the case.

The discrimination step AT of the temperature of asample which emits thermal radiation can be found, ifwe use the former reasoning in which the discriminationstep was compared with the difference between a signaland sample noise (Eq. 2.3).

In the Wien limit, we have for the degeneracy,dij=n 1 /z, of the beam emitted from sample

An2 = c~n2(1+d 2)]'= co-(n2), (2.10)where

d2 = n2/z. (2.11)

The discrimination step An2 is thus equal to 6.8 timesthe rms fluctuation of the number of pulses.

In the Wien and Rayleigh-Jeans limits, the equationsfor An2 become

,Am2=cn2)

An 2= cn2/z1 .

(2.10 W.)

(2.10 R.-J.)

A similar computation can be carried out for thecase of path noise. Here a noise beam of 2)2 photons permacrocell is superposed upon the signal beam from thesample somewhere in the path between the sample andthe detector. The noise beam may be due to thermalemission from the atmosphere or from mirrors or lensesin the path. In this case, we identify the lower level

di,= ee-hPkT, (2.15 W.)

where e is the emittance of the sample. Differentiationof Eq. (2.15 W.) with respect to T gives for; Adildil = Aml/nl = An2/n2

An 2/n 2 = (hv/kT2)AT, (2.16 W.)

and using Eq. (2.10 W.), we obtain for AT/T

AT/T= ckT/hPvn21 , (2.17 W.)

AT being the temperature-discrimination step and Tthe absolute temperature of the sample.

The corresponding equations in the Rayleigh-Jeansregion are

di,= EkT/hV,

An2 /n2= AT/T,

(2.15 R.-J.)

(2.16 R.-J.)

AT/T= c/z2. (2.17 R.-J.)

jully 1968 987

ELLEN HISDAL

It is easily proved that the discrimination step Au2,as given by Eqs. (2.10), is well within the range ofvalidity of our theory as expressed by the inequalities(2.9) and (4.15) of Ref. 4.

III. MAXIMUM ALLOWABLE NOISE-TO-SIGNAL RATIO

We found the discrimination step of Eqs. (2.10)by the device of comparing the lower flux level of thestep with a noise beam. We may, however, be interestedin the actual case in which either sample or path noiseis present. From Eq. (2.4) for sample noise, we obtainfor 52/di2 = (M2)/(n1i2) the value 1- (An12/ns2). Combiningthis with Eqs. (2.10), we obtain

(62 /dj2)m= 1-Ccr(}t 2 )/)1 2 (3.1)

(62 /d6i2)= 1-C/u192

(32/di2).l= I1-c/zi

(3.1 W.)

(3.1 R.J.)

for the maximum sample noise-to-signal ratio whichstill gives us information about the reflectance of thesample. The same ratio for the case of path noise,(22 /di2).= (n72)/(ni 2 )m, is equal to (n72)/A112 according toEq. (2.13). Using Eqs. (2.10) and (2.12) we obtain

(02 /di2)m= (2j2)/[Cu_(1?2)]= 0.147(1n2)/0(n12)

(6 2 /di2)m= (ni2)/C2

= 2.154X 10-2(n i2)

(021di2).= z1/c= 0.147zi. (3.2 (R.-J.)

The values of 62/di 2 and of O2/di 2 in the last rows ofTables III and IV of Ref. 4 may be obtained fromEqs. (3.1 W.) and (3.2 W.).

IV. NUMBER OF STEPS

The problem of the number of distinguishable steps ofa photon signal with a given maximum level has beentreated by Gabor' in the Wien limit. Gabor used therms fluctuation of the number of photons as a measurefor the discrimination step, while we shall use theexpressions given by Eqs. (2.10). The formulas whichwe obtain are valid for arbitrary degeneracies of thebeam. On the other hand Gabor also included the case ofa thermal-noise beam superimposed upon the signal.This case will not be treated here.

We wish to compute the total number of distinguish-able steps or discrimination steps in a macrocell whosepulse number n2 may vary from 2t2min to 1 t

2mnx- For agiven n2, let us choose an interval 5

nm2 which is so largethat it contains one or more discrimination steps An2 .The total number of steps Si in the interval 6fl2F is then

Sk-= 8lk2 /A112 (4.1)

if An2 satisfies the requirement that it does not varyappreciably in the interval 8nm2k. We denote An2 as a

3 D. Gabor, Phil. Mag. 41, 1161 (1950).

function of 112 by A1 2 (n12), and define a quantity a bythe equation

,A112 (112 + A 2)/An 2(n2) = l+a.

Our requirement can then be expressed by

a<<1.

(4.2)

(4.3)

Developing A,, 2 (n 2 +An12 ) into a Taylor series, andretaining only the first two terms because of (2.1), weobtain from (4.3) and Eqs. (4.2) and (2.10) therequirement

(c/2)(1+2112/z)/[12(1+n2/z)] <<1. (4.4)

The total number of steps is S==ISk. If (4.3) issatisfied, we may replace this sum by an integral,3)222 in Eq. (4.1) being replaced by dn 2,

S= / dn2/An2.J 2.d

(4.5)

Substituting for An2 from Eq. (2.10) and solving theintegral, we obtain for S

S= (z'/c)[ln{1+2x/z+ 2 (x/z+x2/z2)? }] Xfl29ax (4.6)

(3.2)In the Wien approximation we have that l2max/Z<<1.

(3.2 W.) The requirement (4.4) now becomes

fl2min>> (C/2)2 =11.6, (4.4 WV.)

and for the number of steps we obtain from Eq. (4.6)in this approximation

S = (2/C) (n 21n,- It in )nv

where, according to Eq. (2.8)

2/c= (2/1re2).= 0.293.

(4.6 W.)

(4.7)

Because of the requirements (2.1) and (4.4 W.) wemay not assume nt2min==0. However, if 112m. is verylarge, e.g. 103 or larger, the error due to the contribu-tion to the integral (4.5) from the small values of n2will be small. An approximate expression for the totalnumber of steps when the pulse number in the macrocellmay vary from 0 to n2nX is then obtained from Eq.(4.6 W.) by putting 2n2min equal to 0,

S= (2/c)n2mni. (4.8 W.)

If both limits for n2 lie in the Rayleigh-Jeans region,i.e., if 1Z2 min/Z>>1, we obtain from Eq. (4.6)

S= (z1/c) ln(1n2may/nZ2min). (4.6 R.-J.)

The requirement (4.4) can be shown to be the same as(2.1) in this case.

If sz2 1,,D/z>>1 and n2min=0, the error introduced intothe expression for S from the contribution of the small122 values in the integral (4.5) will be small. Using the

Vol. 58

989DISCRIMINATION STEPJuly 1968

limits 0 to il2rn, in Eq. (4.6), we obtain for the totalnumber of steps in this case

S= (zl/c) lnk>xs. (4.8 R.-J.)

The information content per macrocell in the Wienlimit is, according to Ref. 3 Table I, given by

i2 ln (n,_,/27r). (4.9 W.)

Substituting for n2ma, from Eq. (4.8 W.), we obtain the

following relation between i2 and the number of steps S

z2 =ln[(e/2)S]= In(1 36S). (4.10 W.)

This equation is valid for large n2max and for beams inthe Wien limit.

In connection with experiments on pulse-code-modu-lated TV transmission6 it may also be of interest tocompute the information content in a TV picture whenthe radiance values of each picture are quantized, theintervals between neighboring steps being given by thediscrimination step according to Eq. (2.10 W.). Ifall discrimination steps have the same a priori prob-ability, then the information content per pictureelement is equal to the logarithm of the number S ofdiscrimination steps. But if equally spaced reflectancevalues of the sample are equally probable, as we have

assumed, then we have different a priori probabilitiesfor the different discrimination steps according toEq. (2.10 W.), and the information i2 per picture elementshould therefore be smaller than InS.

Let pA be the probability that the photon number ofthe element lies inside the jth discrimination stepAn2j

pj A112 1/fl 2 onax. (4.11)

The information content is then

S

*I26 -x pi hnpi. (4.12)j=1

When we approximate this sum by an integral and useEq. (2.10 WV,) for An.:, we obtain in the Wien limit

for large 1 t2max

Al2n(n 2 x/2 (4.13 lW.)

This relation is valid instead of Eq. (4.9 W.) when we

have a sample with quantized radiance levels. We haveneglected fluctuations in the derivation of Eq. (4.13 W.),

i.e., we have assumed that the quantized radiance levelsent out by the picture element is that level whichcorresponds to the nearest reflectance value of thesample. As neighboring levels are separated by 6.8times the nns fluctuation, the correction to Eq.(4.13 W.) when fluctuations are taken into accountshould be small.

Substituting for f 2ma,, from Eq. (4.8 W.) into Eq.

6T. S. Huang, IEEE Spectrum 2, 57 (1965).

(4.13 W.), we obtain

i2q ln[(e /2)5j= ln(0.82 4S). (4.14 W.)

We see that, according to Eq. (4.14X W.), the informa-tion content from an element with quantized radiancelevels is smaller than InS because of the differenta. priori probabilities of the steps. X'hen there is no such

quantization, the information content is increasedaccording to Eq. (4.10 W.). This is reasonable, becauseif we divide a discrimination step into two halves, anda picture element sends out a radiance corresponding tothe upper half, then, despite the fluctuations, there is alarger probability that the reflectance of the samplecorresponds to a radiance lying in the upper half of thestep than to a radiance lying in the lower half andtherefore we obtain more information in a picture withcontinuous radiance levels. The relation i2= InS whichis usually used in connection with TV pictures, liesbetween the relation (4.10 W.), valid for a continuouspicture and (4.14 W.) valid for a picture with quantizedradiance levels.

The quantization noise,'- i.e., the information lossdue to quantization of the radiance levels, is given bythe difference between (4.10 W.) and (4.14 W.),

2q =2 -,2q = Ine'= 2 nat. (4.15 W.)

We see that j2q is independent of S. It is thereforerelatively large for a picture element whose maximumradiance is small.

We have not taken noise in the transmission channelinto account in the present treatment.

V. EXAMPLES

VWe use the numerical data of the previous paper, 4 re-

ferring to the experiment of Ginsburg, Fredrickson, andPaulson9 on target distinguishability, to illustrate thediscrimination step of temperature. The temperature ofthe sample or target is assumed to be 'rd= 200C and thewavelength at which we measure is assumed to beX= 4.75 uim, from which we obtain hi/kTd= 10.33.

Thenu-mber of pulses perrn-acrocellis' (ii 2)= 3.885X 1Og.We assume that thermal background radiation isincident upon the target in the pertinent wavelengthregion. We wish to find the smallest difference ATbetween the temperature of the sample and of thebackground radiation which will still allow us to detectthe target. From Eq. (2.17 W.) we computeAT= 0.98X 1-2 deg. Thus an ideal detector whichcounts every liberated electron with no sources of error,should be able to discriminate a temperature differenceof about 0.01 deg. This is a reasonable result when we

compare it with the experimental minimum detectabletemperature difference of 0.03 deg mentioned by

7T. S. Huang, 0. J. Tretiak, B. Prasada, and Y. Yamaguchi,Proc. IEEE 55, 331 (1967).

s W. F. Schreiber, Proc. IEEE 55, 320 (1967).9 N. Ginsburg, W. R. Fredrickson, and R. Paulson, J. Opt. Soc.

Am. 50, 1176 (1960).

99LE LE N HIS I S I) VlA L

Ginsburg, Fredrickson, and Paulson. The discriminationstep of the number of pulses is, according to Eq.(2.10 W.), An12=1.34X 105. The maximum samplenoise-to-signal ratio which allows us to distinguish thethermally emitting sample from the background radia-tion incident upon it is, according to Eq. (3.1 W.),1-3.46X 10-4= 0.999654. The maximum allowable pathnoise-to-signal ratio is found from Eq. (3.2 W.) to be7.53X10'. Here we have used (ni2)= 3 .50X10' as inRef. 4 for path noise, assuming that this noise is due tothermal emission from a mirror with reflectance 0.9;(ni2) is therefore reduced by 10% compared with thesample-noise case.

In the following, we apply the discrimination-stepformula and the formula for the number of steps to thehuman eye. We must assume that the eye is adapted toa constant luminance. Although many factors compli-cate the performance of the eye, it should be interestingto see how far a theory which assumes that the retinaldetectors work like photoemissive cells can explainvisual phenomena. As remarked by Rose,10 we shouldexpect a theory like the present one to give the bestrepresentation of the experimental data at the low-light end of the light range over which the eye operates.It is here that fluctuation limitations may be expectedto be the dominant factor. At very high luminances,other limitations set in, as for example, the limitedtraffic-carrying capacity of the optic-nerve fibers.According to Rose, a theory based on fluctuationlimitations should be applicable to a light range ofthe order of magnitude of 10-a-10+2 millilamberts.Of course there are no such limitations in the case ofartificial photoemissive detectors, assuming that thepulse due to every liberated electron is counted.

We start with a comparison of our theory with thetheory and experimental data of Rose's paper10 . Rose'stheory is very similar to ours in the Wien limit. Accord-ing to Rose, Eq. (la), AN= kNI. Rose's AN is the dis-crimination step, either at threshold or at greaterluminances; N is the same as our (INi2>m at threshold,and the same as our n2 for greater luminances; le iscalled the signal-to-noise ratio by Rose. For An82/,2<<1,it is equal to our c=6.8, Eqs. (2.8), (2.10 W.). For thethreshold signal we obtain from Eq. (1.2) kz=( (i2)m/

(iti2)m-1 = (2re) 1=4.1. Rose does not derive a theoretical

value for ke but finds experimentally that it lies in theneighborhood of 5, in good agreement with our range of4.1 to 6.8.

According to Rose's theory, Ref. 10, Eq. (4a), thethreshold contrast or relative discrimination stepC= AN/N is proportional to 1/a for a given sceneluminance, where a is the angular size of the test object.This is confirmed to quite a good degree of approxi-miation by his own experiments and by the plots of thedata of other experimenters in Rose's Figs. 7-10.According to our Eq. (2.10 W.), A1z 2/n1 2 is also propor-

1o A. Rose, J. Opt. Soc. Am. 38, 196 (1948).

tional to 1/nt2'i or to the angular size of the test object,if we assume that the eye adds up the numbers ofpulses from individual detectors so that they act as ifthey were a single detector. The proportionality ofAN/N and 1/a is approximately confirmed experi-mentally, according to Rose, for values of a from 2' to100'. This should indicate, that when the eye sees asample of uniform luminance it adds up the numbers ofpulses from single detectors for sample sizes up to 100'.Such addition means, according to Eq. (2.10 W.), thatthe relative discrimination step decreases as the size ofthe test object increases for a given luminance. Thisresult also agrees with observations mentioned byWright (Ref. 1, p. 192), and a summation of the totalamount of light by the eye is given as a possible explana-tion. The same explanation may also hold for the effectobserved by Huang' in connection with PCM trans-mission of TV pictures. He found that the quality ofthe picture of a face, with rather large areas of uniformluminance, improved noticeably when the number ofluminance levels for each picture element was raisedfrom 16 to 64. For the detailed picture of a crowd,where the luminance varies rapidly from one pictureelement to the next, there was, on the whole, no im-provement when the number of levels was raisedabove 16.

As an illustration for the numerical value of the dis-crimination step, we choose a luminance of 700 nits,which is the highest level of a commercial TV tube.(700 nits is not the luminance of an average picturewhen the brightness adjustment is at its maximum. Itis rather the luminance of the screen as a whole whenall picture elements send out the highest attainableluminance). We assume that the eye is adapted to thismaximum luminance. The pupil diameter correspond-ing to 700 nits is 2.45 mm." Further, we assume aresolving time for the eye of 0.2 sec10, and the equiva-lence 1 lumen of white light=1.3X106 quanta/sec. 10

For a sample subtending a solid angle of one squareminute, so that its image covers approximately oneretinal detector, and using a quantum efficiency of0.5% for the eye"' we obtain n2=3630. Eq. (2.10 W.)then gives An2/n 2 = 11%. For a sample subtendingten minutes square we obtain An,2/n 2= 1.1%, and fora one-hundred-minute-square sample An2/n2=0.11%,assuming that the numbers of pulses from the retinaldetectors are added up. Comparison of measurements ofAm)12/nt2 for a given luminance and for different samplesizes with Eq. (2.10W.) should give an indication ofthe range of angular sample sizes for which such sum-mation occurs. We would expect that when the samplesize exceeds a certain limit, there will be no furtherdecrease of An,2/n2.

The average information which a picture element of aTV screen can transmit to the eye is determined by themaximum luminance of the TV screen, assuming that

"1 P. Reeves, J. Opt. Soc. Am. 4, 35 (1920).

990 Sol. 58

DISCRIMINATION STEP

this is equal to or lower than the luminance of theoriginal scene. With (n7i2=3630 as before, we obtainfrom Eq. (4.9 W.) for the average information perpicture element i 2= 3.2 nats= 4.6 bits for a picture withcontinuous luminance levels.

Using this value of i2 we obtain from Eq. (4.8 W.) or(4.10 W.) for the number of discrimination steps perpicture element Sn: 18 for a picture with continuousluminance levels. If the picture has ten-minutes-squareareas of uniform luminance, and if we assume that theeye responds to the total number of pulses from suchan area, then (ni2) must be multiplied by 100 and Sbecomes 180 according to Eq. (4.8 W.). In the case of acontinuous TV picture, the increased number of dis-crimination steps S for a larger field is due to the actionof the detector or eye, which is assumed to perform asummation of the signals from single detectors. It hasnothing to do with the TV picture itself, for which wehave assumed a given maximum luminance. For apicture whose luminance levels are quantized accordingto Eq. (2.10 W.) and with the same maximum lumi-nance as above, we have of course the same number ofdiscrimination steps, namely 18 steps for one-minute-square areas, and 180 steps for ten-minute-square areasof uniform luminance. This means, however, that for aquantized system we must choose a number of levelsaccording to the size of the areas of uniform luminancewhich we may expect in the picture. This agrees withthe experimental results of Huang.6 The information perone-minute-square picture element is found from Eq.(4.13 W.) or (4.14 W.) to be 2.7 nats=3.9 bits, i.e.,less than the 4.6 bits which we found for a continuouspicture. If we assume that the picture is of such acharacter that only ten-minute-square areas need beresolved, we obtain from S=180 and Eq. (4.14 W.),i2 = 5 nats= 7.2 bits per ten-minute-square area. How-ever, we need now only 1/100th as many picture ele-ments as for the detailed picture with one-minutesquare areas, so the total number of bits necessary fortransmission of the picture is reduced.

The theoretically computed limits for the necessarynumber of quantization steps, S= 18 to 180, agree wellwith those which are usually assumed necessary forgood TV reproduction. Schreiber8 mentions that 16-256

brightness levels are required to obtain an apparentlycontinuous tone scale. According to Huang et al.,7 32to 128 brightness levels are required for good reproduc-tion with uniform quantization. If instead of usingEq. (2.10 W.) for A1n2, we had assumed that the dis-crimination step is equal to the rms fluctuation of thenumber of photons, we would have obtained the limits6.8X18 to 6.8X180 or 122 to 1220 for the necessarynumber of quantization steps. These limits are certainlytoo high compared with the experimentally acceptedones.

Equation (4.8 W.) for the number of discriminationsteps agrees with the relation used in the originalMunsell system for the number of uniform lightnesssteps between black and a given gray as a function ofthe reflectance of the gray sample [Ref. 12, Eq. (2.16)],if we assume that equal lightness steps contain equalnumbers of discrimination steps. Munsell's relationapplies best to observations with a white background,"which agrees with our requirement, that the eye mustbe adapted to a constant luminance. The logarithmicWeber-Fechner law is often used to define a uniformlightness scale. According to Ref. 12, p. 266, this lawis satisfactory, provided that the visual appraisal ofeach small difference is made by an observer completelyadapted to an intermediate luminance within the givenstep. When observing a TV picture, the observer's eyeis adapted to a constant luminance. It therefore seemsreasonable that the Munsell law is preferable to theWeber-Fechner law for the purpose of determininglightness-quantization steps for a TV screen.

In conclusion we believe it is fair to say that there issatisfactory agreement between the experimental datafor the discrimination step and the values computedfrom the theoretical formulas derived in this paper.There is some indication that the eye adds up pulsesfrom neighboring retinal detectors for areas of uniformluminance. The original Munsell formula for uniformlightness steps agrees with such a theory for the eye,assuming that retinal detectors act like photoelectriccells.

1 2 D. B. Judd and G. Wyszecki, Color in Business, Science, andIndustry (John Wiley & Sons, Inc., New York, 1963), 2nd ed.

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