ICONICS–THE SCIENCE OF IMAGES

Operational thresholds of the spatial resolution of the visual system and the contrastperception of objects

L. N. Aksyutov

Scientific Research Institute for Comprehensive Testing of Optoelectronic Devices, Sosnovy Bor,Leningrad Oblast~Submitted January 10, 2002!Opticheski� Zhurnal69, 36–42~August 2002!

Based on a model for predicting the probability of visually distinguishing/recognizing objectsfrom an image and on the data of an experimental study, a method has been developedfor analytically determining the operational thresholds of the spatial resolution of the visual systemand of the contrast perception of a stimulus when the duration of its exposure does notaffect perception. Previously unknown relationships that connect these quantities are establishedfor wide ranges of the angular size of the stimulus and the adaptation luminance. Thecharacteristic of the receptive field of the visual system corresponding to these relationships isobtained. ©2002 Optical Society of America

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The threshold characteristics of visual perception at vous background luminances are of interest for psychophyand the physiology of vision,1 for solving applied problemsfor instance, in illumination engineering,2 and in estimatingthe possibilities of visually distinguishing small, low-contraobjects of a scene.

There are a large number of papers in the literatwhose authors studied threshold quantities—the limiting dtinguishable contrast and angular size of visual stimulivarious background luminances, with a description ofresults in the form of an analytic function. The fact that thefunctions, given in Ref. 3, are so diverse and so numerraises the question of how reliable they are. This questioassociated with the fact that the limiting threshold valuesthese quantities are not invariant, but vary within rather wlimits in different observers.4 Such smearing of the limitingthresholds of perception is explained by the fact that thdepend not only on the psychophysiological features buton the functional state of the subject of the activity.5 There-fore, to increase the reliability of the description of the funtions of visual perception, it is necessary to use the optional thresholds of visually perceptible quantities.

Reference 1 defines the threshold intensity of a stimuof any modality asoperationalwhen it provides a level ofperception sufficient for a 50% probability that it candistinguished. This level is characterized by the balancefavorable and unfavorable factors that influence the discrination of a stimulus, which corresponds to the maximumthe distribution of instantaneous fluctuations of the threshof perception. Therefore, the operational threshold of a qutity that characterizes the perception of a stimulus determthe half-saturation level of theS-shaped psychometric function of its discrimination, regardless of the features of tobserver’s psychophysiology. For visual stimuli, suchquantity can be, for example, the contrast or the angular s

The goal of this paper is to create a method for anal

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cally determining the operational thresholds of the visuaperceptible contrast of stimuli~objects! and of the angularresolution of the visual system in a wide range of adaptatluminance for an exposure time of the stimuli that preventfrom affecting the perception.

To achieve this goal, we chose the data of an experimtal study carried out by Blackwell,6 which is taken asclassical,7 and the main analytical relationships of a modfor predicting the probability of the discriminationrecognition of objects from an image~the PRP model of Ref.8!, obtained theoretically in Ref. 9 on the basis of the mpsychophysical law of perception~Fechner’s law!, retrievaltheory, and the law of constancy of perception. These rtionships are represented by

Pp5120.5C2hn when C.1,

Pp50.5 when C51, ~1!

Pp50.5Chn when C,1,

where C is an invariant of the information content of aimage, which equals the ratio of any quantity that determithe intensity of a visual sensation~impression! from a per-ceptible stimulus to its operational threshold;hn is the con-fidence index of discrimination:hn52 ln Pfa , wherePfa isthe false-alarm probability when solving a visual problewith probabilityPp , as well as the formula that connects thoperational threshold of angular resolution~OTAR! d t,L ,expressed in milliradians, of the visual system with the aaptation luminanceLad:

d t,L5 ln 2/~ logLad12.5!. ~2!

Equation ~2! can be used for the range22.089< logLad<3.535, the limit of which, logLad522.089, vir-tually coincides with the limit of variation of the characteistic of the threshold illumination of the pupil from back

552552-06$18.00 © 2002 The Optical Society of America

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TABLE I. Operational threshold of the angular resolution of the visual system (d t,L) from Blackwell’s data6 and their discrepancy~Dd t,L ,%! with the calculated values from Eqs.~2! and ~4!.

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ground luminance~see Ref. 2, Fig. 1.49!. However, we havenot checked it against experimental data, because thereno such data in the literature.

The problem investigated in Ref. 6 was to determineminimum contrast that ensures that a stimulus will be vially distinguished with probabilityPp50.5. According towhat was said above, this minimum contrast, which wenote by the symbolKt , corresponds to theoperationalthreshold of contrast perception~OTCP!.

Table VIII of Ref. 6 gives numerical values of logKt forstimulus sizes of 3608, 1218, 55.28, 18.28, 9.688, 3.68, and0.5958 for nine values of adaptation luminanceLad from therange24.464< logLad<3.535 that differ by an by order omagnitude.

According to Blackwell, the problem of the investigatiois adequately solved within the limits of experimental erof 65% when the time of perception of stimuli with positivand negative contrast is equal to 15 sec. He establishedregardless of the observation conditions, which are demined by the adaptation luminanceLad, the contrastKs ~Ks

is the differential or physiological contrast!, and the time ofperception of the stimuli, the psychometric functionPp

5 f (Z5Ks /Kt) of distinguishing the stimuli, constructefrom the experimental data, is satisfactorily described byintegral curve of a normal distribution.

This curve corresponds to Eqs.~1! with the followingvalues of the parameters:hn523.2 for C5Z.1 and hn

51.8 for C5Z,1. For example, forC51.62, the discrep-ancy between the calculated discrimination probabilityPp

50.893 and the probabilityPp50.90 indicated in Ref. 6 forZ51.62 is less than one percent. With the same values ohn parameters, the analogous function obtained in Ref.for a perception time of visual stimuli of 0.3 sec is describand this indicates that it is independent of the exposure tiThe substantial difference of the quantitative descriptionthe functionPp5 f (Z5Ks /Kt) by an integral curve of thenormal distribution in Refs. 6 and 10 and by Eqs.~1! withthe indicated values of parameterhn consists only of the facthat the calculated values ofPp.0.9 in the latter case converge to unity more slowly asC5Z increases.

Thus it is possible to use Eqs.~1! to predict by a calcu-lational method the probability of visually distinguishing

J. Opt. Technol. 69 (8), August 2002

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stimulus with known contrast and size for any given obsvation conditions. However, to implement the method, dare needed on the OTAR of the visual system. Therefore,present the experimental values of the OTCP not in the foof the dependence on the angular size of the stimuli, whiccharacteristic of earlier studies of the perception of thcontrast,2,3,11 but in the form of the dependence on the rat

do5ds /d t,L , ~3!

whereds is the angular size of the stimulus, andd t,L is theOTAR, calculated from Eq.~2! for the adaptation luminanceLad of the visual system.

For adaptation luminances from the range24.5< logLad,22.089, the thresholdd t,L of the visual system isdetermined from

d t,L5exp10@2~0.3logLad10.4!#, ~4!

which we obtained on the basis of Blackwell’s data~seeTable I!.

Since the quantity logLad for the visual system hasfunctional value,12 we present the data of Table VIII fromRef. 6 in Fig. 1 in the form of the dependence logKt

5f(logdo ,logLad), a feature of which is that it contains linear and nonlinear sections.

The linear relation of the quantities logKt and logdo ,clearly expressed in Fig. 1 for the adaptation luminance frthe indicated range for values of logdo< logdo* , correspondsto the equality

logKt52 logdo2. ~5!

We established that the quantity logdo* for an adaptationluminance from the range22.089< logLad<1.535 is deter-mined by Eqs.~1! for the invariant of the information contenof an imageC5 logLad13.5/2.75 and with the same valueof parameterhn that correspond to the function considerabove,Pp5 f (Ks /Kt):

logdo* 5121

2 S logLad13.5

2.75 D 23.2

,

if 20.75< logLad<1.535, ~6!

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The value of logdo*50.928 calculated from Eq.~6! forlogLad51.535 is also valid for adaptation luminances frothe range 1.535, logLad<3.535, for which the experimentavalues of logKt correspond to the dependence logKt

5f(logdo) shown by curve7 in Fig. 1.For logLad,22.089, we used the results of an analy

of Blackwell’s data to obtain the expression

logdo* 50.14 logLad10.442,

if 24.465< logLad,22.089. ~8!

The formulaKtdo251 that corresponds to Eq.~5! when

logdo<logdo* is substantially different from Wald’sformula:13

LKdx5k for 0,x<2,

which, for a probability of 0.5 of visually distinguishing aobject, connects its angular sized and contrastK with back-ground luminanceL. It is emphasized in Ref. 13 that the uof Wald’s formula, a particular case of which forx52 andd,78 is Ricco’s law, assumes that it is necessary to makpreliminary measurement of parametersx and k under spe-cific observation conditions.

Thus, Eq.~5!, which assumes only that the adaptatiluminance of the visual system is measured while an ob

FIG. 1. Operational thresholdKt of visual contrast perception of stimuli athe function logKt5f(log do ,log Lad) from the data of Ref. 6:Lad53.4331025 ~1!, 3.4331024 ~2!, 3.4331023 ~3!, 3.4331022 ~4!, 3.4331021

~5!, 3.433100 ~6!, 3.433101– 3.433103 ~7! cd/m2.

554 J. Opt. Technol. 69 (8), August 2002

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with a known angular size is being observed, has obviadvantages over Wald’s formula. One of them is as follow

It is well known that the spatial resolution of any imagformation system, which it is customary to estimate assquare of its angular resolution,~see, for example, Ref. 11!determines the area of a resolution element~a pixel! of thissystem in the picture plane. Therefore, the value ofdo

2 in Eq.~5! is equivalent to the number of pixels of the visual systeper area of the visual projection of an object. In general, tnumber determines the information content of an image ptern of the object when solving the problem of recognizingfrom the attributes of the parts and elements of the patthat can be visually discriminated.8,9 Therefore, Eq.~5!makes it possible to determine the OTCP that providegiven probability of recognizing the category to which aobserved object belongs.

For values ofdo that satisfy inequality logdo,logdo* ,we used Eqs.~5! and ~3! and theKt values obtained in Ref6 to calculate the OTAR of the visual system and the dcrepancyDd t,L of its value from that calculated from Eqs~2! and ~4!. The results of the calculation are given in TabI.

Table I illustrates that the results of the calculation usthe theoretically obtained dependence of Eq.~2! are in goodagreement with the data of the independent experimenRef. 6, the more so that thed t,L values were not directlymeasured in this experiment. This fact is evidence ofreliability of Eq. ~2! as well as of Eqs.~1!, which are con-nected with it. Moreover, the data of Table I make it possito regard the measurement of the OTCP of stimuli as a nfairly accurate method of determining the operational threold of spatial resolution of the visual system.

Analysis of the data of Table I shows that the factor thlimits the spatial resolution during visual discriminationthe objects of natural scenes is not the scattering of lightthe optical medium of the eye,14 and not the physical size othe photoreceptor~Ref. 15, page 100!, but the effective areaof a pixel of the visual system that determines the receptphotoresponse, determined by the excitation–inhibition pcesses of the neurons, forming the receptive fields ofvisual system at the actual adaptation luminance.12

Now let us consider the nonlinear section of the logKt

5f(logdo ,logLad) dependence when logdo.logdo* andlogKt,0 ~see Fig. 1!. To analyze it, we introduce the following notation:

Y5 log~Kt* /Kt! and X5 log~do /do* !. ~9!

These coordinates are used in Fig. 2 to replot the data of1.

Figure 2 displays a functional relationship that was uknown earlier: TheY5 f (X) dependence is described bysingle curve not only for photopic levels of adaptation lumnance (logLad>1.5), but also for its scotopic level(logLad<21.46), at which rod vision predominates. For itermediate, mesopic levels of adaptation luminance, theY5 f (X) function is determined by the numerical valuelogLad.

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The family of curves in Fig. 2 is described by the folowing relationships: When the inequality 0,X<0.20 is sat-isfied for the adaptation luminance from the range24.465<Lad<3.535, the value ofY is determined from the equalit

Y51.45X. ~10!

For X.0.20, the values ofXrel5X/Xm and Yrel5Y/Ym arecomputed, where

H Xm51.4020.78 logLad, if 1.46< logLad<20.11,

Xm51.4820.48 logLad, if 0.11, logLad<1.5,

Ym50.75Xm20.32. ~11!

The resulting value ofXrel is used to determine theYrel

value from

Yrel5@A~Xrel20.05!#1/2

5@~1.610.23 logLad!~Xrel20.05!#1/2, ~12!

if 0 ,Xrel<0.56;

Yrel5@12B~12X!#1/2

5@12~0.4220.26 logLad!~12Xrel!#1/2, ~13!

if 0.56,Xrel,1.0.

This can be combined with theYm value to calculate theOTCP:

logKt5 logKt* 2YrelYm . ~14!

In calculating logKt from Eqs.~9!–~14!, the parametersXm , Ym , A, andB were rounded to the second figure aftthe decimal point~see Table II!. For adaptation luminancefrom the range24.465< logLad,21.46, the calculationuses the values of the parameters from the first row of TaII, whereas the fourth row is used for adaptation luminanfrom the range 1.5, logLad<3.535.

Table III shows the discrepancyD5@(Kt,B /K)21#3100% between the values of the operational thresholdKt,B

from Table VIII of Ref. 6 and the threshold valueKt calcu-lated from the relationships given above, using Eq.~14!. The

FIG. 2. The dependenceY5 log (Kt* /Kt) vs X5 log (do /do* ) in the range ofadaptation luminanceLad53.4331025– 3.433103 cd/m2: the points showthe data of Ref. 6~symbols the same as in Fig. 1!, and the curves show thecalculated values, using Eqs.~10!–~14! for Lad53.4331025– 3.4731022

~1!, 3.4331021 ~2!, 3.433100 ~3!, 3.163101– 3.433103 ~4! cd/m2.

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discrepanciesD in % enclosed by a frame in the lower partTable III refer to theKt values calculated from Eq.~5!.

It can be seen from Table III that, for most of the calclated values ofKt , the discrepancy from the analogous daof Blackwell lies within the limits of the stated error65%.However, for the adaptation luminance logLad521.465, thecalculatedKt values proved to be systematically underesmated for all stimulus sizesds with respect to the valuesobtained in experiment.

Considering the overall duration~up to 2.5 yr! of thestudy carried out by Blackwell, the reason of the discrepaD.65% can probably be explained by the difference in tadaptation luminance indicated in Ref. 6 from its actuvalue, which corresponds to the experimental value ofOTCP. In order to avoid this, we calculated the valuelogKt for logLad521.556, which, according to Eq.~2!, cor-responds to the mean valued t,L50.734 mrad obtained forlogLad521.465 in Table I. The discrepancy between theKt

values calculated for logLad521.556 in Table III andBlackwell’s data for logLad521.465 can be regarded asconfirmation of our assumption.

The results given in Table III make it possible to coclude that reliable values for the OTCPKt calculated by themethod that we proposed are obtained for adaptation lunances Lad in kd/m2 from the range 24.465< logLad

<3.535, and to recommend this method for practical apcation.

We now turn to the fact that theXm and Ym values foreach adaptation luminance correspond to logdo,m and logKt

values, with the latter remaining constant when logdo

.logdo,m ~see Fig. 2!. This fact agrees with the theoreticallconfirmed relationship of Ref. 12, according to which icreasing the area of thereceptive field~RF! of the visualsystem reduces the spatial summation of the responses oneurons that forms this field, minimizing the thresholdperceptible contrast. Consequently, the value of logdo,m cor-responds to the angular size of the RF in which spatial sumation completely terminates at the existing adaptationminance.

It is established from the results of these calculatiothat the angular sizedRF of the receptive field of the visuasystem in milliradians, corresponding to the total terminatof spatial summation, is described by anS-shaped responslogdRFc5 f (logLad), which in its representation by Eqs.~1!has the form

TABLE II. Parameters for calculating the value of logKt from Eqs.~9!–~14!in the range of adaptation luminance24.465< log Lad<3.535.

log Lad Xm Ym A B

21.46 2.54 1.59 1.26 0.8020.465 1.76 1.00 1.49 0.54

0.535 1.44 0.76 1.72 0.281.5 1.36 0.70 1.72 0.28

555L. N. Aksyutov

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TABLE III. The discrepancy~D, %! between the operational threshold of contrast perception of visual stimuli from Blackwell’s data6 andthe values calculated from Eqs.~5! and ~9!–~14!.

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logLad19.5D23.5G11.3, if

logLad,21.2,

logdRFc51.15S 8.3

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logLad.21.2.~15!

Table IV shows the values ofXm , logdo,m, logdo* , d t,L ,and logdRF obtained from Eqs.~2!–~11! for the range ofadaptation luminance considered here and the discrepD5@(dRF/dRFc)21#3100% of the values ofdRF anddRFc.

The values in Table IV illustrate how the quantities thcharacterize the perception of visual stimuli depend onadaptation luminance:

a! The value of logdo* is directly proportional and thevalues of logdt,L and logdRF are inversely proportional tologLad.

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b! The value of logdo,m depends in a complex way ologLad: It is constant for photopic levels of adaptation lumnance, increases for logLad,1.5, reaching a maximum alogLad'21.46, and decreases as the radiance decreasether.

Using Eqs.~15!, it is easy to determine, for examplethat, for an adaptation luminanceLad5500 cd/m2, corre-sponding to daytime illumination with a clear sky, the opetional threshold of spatial resolution isd t,L50.133 mrad andthe angular size of the RF of the visual system that is respsible for this value is 25.5 mrad or about 1.5°, i.e., somewmore than the angular size of the central field of vision.important practical conclusion follows from this fact: In oder to reliably estimate the probability of visual detectionrecognition of small, low-contrast objects on natural bacgrounds, it is necessary to adequately estimate the adaptluminance, which determines the actual value of the opetional thresholds of spatial resolution of the visual syst

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TABLE IV. The quantities that characterize the operational threshold of the angular resolution and the corresponding receptive fievisual system.

556L. N. Aksyutov

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and of the perceptible contrast of objects in the zones owhich the focal attention of the observer is distributed. Tangular sizes of these zones along the horizon are 10°,and 1°, which ensure clear visibility of a scene, of segmeof a scene, and of detailed differences of the segmerespectively.16

In conclusion, the following conclusions can be formlated:

1! Methods have been proposed for determining the optional thresholds of perceptible contrast of objects andthe spatial resolution of the visual system in the lumnance range from 3.433103 to 3.4331025 cd/m2. Thequantities to be determined are reliable when adequvalues of the adaptation luminance and the angularof the observed object are measured.

2! A new approach to the analysis of the experimental dis used to develop the proposed methods, making it psible to establish previously unknown relationshipsthe perception of visual stimuli, which are representedthis article by analytical dependences.

3! The results of this work showed that the main formuof a model for predicting the probability of thdiscrimination/recognition of objects~the PRP model!completely correspond to the relationships of visual pception, and this is evidence that the model is reliab

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557L. N. Aksyutov