Temporal behavior of thermal images

Nissim Ben-Yosef and Kalman Wilner

The temporal behavior of the contrast in a thermal image is related to the temporal change of the radiancestatistics over such an image. It is shown that, due to the statistical distribution of the thermal propertiesover an object, the statistics of the radiance vary with time during temporal changes in the heat balance. Inspecific cases the contrast of a thermal image can obtain a transient maxima larger than the steady-statevalues.

1. Introduction

The quality of a thermal image depends on the dy-namic range of the radiance over the object. This dy-namic range depends in turn on the local temperatureand emissivity variations. Experimental findings ofthermal mapping of natural terrain in a given timepointl- 3 show the statistical and spatial properties ofsuch an image, see Ref. 4. The time variations of theaverage properties of such a scene were studied byWatson5 in relation to the geological application ofthermal imaging. Besides the temporal variations ofthe average properties, one can expect a temporalvariation of the radiance standard deviation over theimage (proportional to the image contrast).

The radiance from each resolvable element of theimage depends on the local temperature and emissivity.During a temporal change (due, for example, to thetemporal variation of the heat source) these propertieswill change with time according to the local heat balanceproperties. As these properties vary from point to pointover the image one would expect that the radiance sta-tistics over the entire image will change with time.

II. Simple Model

For small temperature variations one can assume thatthe heat loss of each spatial element is proportional toits temperature. This assumption is, of course, correctfor conduction but needs linearization of the radiativeterm.5 Under this assumption the heat balance for eachelement can be described by

The authors are with Hebrew University of Jerusalem, School ofApplied Science & Technology, Applied Physics Division, Jerusalem,Israel.

Received 7 June 1984.0003-6935/85/020284-03$02.00/0.© 1985 Optical Society of America.

dT + x T = Y F(t),dt z -Ft) (1)

where x is proportional to the heat conductivity of eachspatial element, z is proportional to its heat capacity,and y is proportional to the coupling between the ele-ment and the external heat source, F(t), which is timedependent. It is assumed that x, y, and z are uncorre-lated and that adjacent elements are not too stronglycoupled. This assumption will allow us to regard x, y,and z over the entire scene as statistical ensembles ofuncorrelated variables.

In a steady-state situation, i.e., F(t) = F, the as-ymptotic solution is immediate; T = (y/x)FO, and theradiance from each element relative to a given referencewill be R = AT = A (y/x)Fo, where A is a constant andlinearization of the Planck law is used. The varianceof the radiance over the entire scene can be obtained inthe quadratic approximations using the variance of thevarious parameters as follows:

Ox + 'dy xoyo'zo,eo Y

+d aRxoa0ozocJ2 .0 + de Xo'Yo'z0J ,U*2' (2)(OR 2 (OR

where xoyozoe 0 are the mean values (over the image)of the parameters. By calculating the derivatives andrearranging the terms one obtains the normalizedvariance of the radiance over the entire scene:

2R 22 a2

R2(xo,yozoco) X 2 + 2

It is observed that in the steady-state situation thevariance of z (the specific heat of the image element)does not affect the normalized variance. A quantitativeupper limit for the normalized variance in a steady-statesituation can be obtained from the experimental results.The radiance statistics over real scenes are reported in

284 APPLIED OPTICS / Vol. 24, No. 2 / 15 January 1985

Refs. 1-3, where the radiance is expressed in tempera-ture units with (R) = 300 K and uR < 7 K. It followsthat

a2 a2 a2 CT

_ -< 6 X 10-4. ~(3)R2 Xo 2 o 2 o

The situation is different if the heat source changes withtime, daily variation or a faster change like a movingcloud covering the sun. In this case the contributionof a, will not vanish and the normalized radiance vari-ance will exceed this limit, i.e., the contrast over theimage will be greater due to the temporal change. Toanalyze this possibility an exact solution for the stepfunction will be given. Let

F(t) Fo t < F1 t>O;

the solution of Eq. (1) is given immediately:

T(t) = Y F1 + F2 exp - t)]

for t > 0 and F2 = F0 - F1 so that

R(t) E F + F2 exp -- t)* (4)

This function represents the temporal behavior of theradiance of the spatial element having the thermalproperties {xyz1. As x, y, and z are distributed overthe image R (t) will be distributed over the image as well.So that in the quadratic approximation we can calculatethe statistical average of the radiance over the entireimage (which, of course, will be time dependent) and thevariance of the radiance which is time dependent aswell. By calculating the statistical average (see Ref. 6)one finds that practically

(R (t)) - R(t) 10,yo,,,bc. (5)

Calculating the variance using Eq. (2) one obtains

2 (e~ 2 rlae\2 F2 12 Iax12a = f F,(-J [1 +F1 exp( r) +

x [1 + (1 + T) exp(-T)I

+ t)2 [1 + F2 exp(e-r)2

+ I F[]exp(-T)12i (6)

where r = (xo/yo)t.As the changes of terrain temperatures are not very

large (say L30 K around 300 K) the term IF2/F11 < 0.2.It means that the first three terms are still bounded byEq. (3). In addition, the time behavior of these termsis monotonic with <50% variation. The fourth term,representing the z contribution to the radiance variance,has a transient behavior, it vanishes asymptotically andobtains a maxima at r = 1. This behavior means thatin a transient regime the radiance variance (propor-tional to the image contrast) can obtain a transientmaxima larger than the steady-state values. This be-havior means that features in the terrain can exist withlow contrast in a steady-state situation and therefore

C I I I I I0

. 470

a 440 _

U)

0 2 4 6 8 10 12Time [normalized units I

Fig. 1. Behavior of the radiance standard deviation during heatingand cooling. In both cases IF 2 /F1I = 0.2.

(1) o2 )= (TxO2 (z)2=0

(2) lor-)2 + A)~~ = 1, (y-) = 1, A),~ = 5,ro 0 XO ZO

() ()2 + ()2 =1 (ax2 1 (a)2 15()2 + (Ž=,(r i(~21(4)(a~ + = 2=1, (ax2 J21

(4) 15Yo

are unobservable, which appear with higher contrast fora finite period during transient heating (or cooling).The functional behavior of Eq. (6) is shown in Fig. 1 forthe heating and cooling step source.

111. Experimental

The hypothesis that the standard deviation of theradiance can show a transient increase was checkedexperimentally in a preliminary experiment. The im-ager used has 64 X 64 resolvable elements NEAT = 0.2K, a frame time shorter than 3 msec and operating at awavelength of 10 ,jm. To simulate the previous calcu-lations, i.e., step function response, an artificial scenewas constructed. A sandbox with different objects init, gravel, rocks, metal, glass, and plastic pieces, servedas the object. The step function was simulated usinga movable sunshade. In this configuration the sunserved as a constant source and the sunshade served asthe step function. The two parameters measured as afunction of time were the average scene radiance and theradiance standard deviation over the image. Themeasured radiance was in arbitrary units so that theexperimental values are proportional to the radiance.To eliminate from the experimental data the effects ofdc shifts and the unknown constant of proportionalitybetween radiance and measured signal the data werereduced as follows: if h(t) is the measured value, theparameter

g(t) = log h(t) - hminhmax - hmin

is calculated. Compared with Eq. (5) one observes thatfor the theoretical average radiance one will get a linearbehavior. Compared with Eq. (6) one observes that,when (/zo) is not negligible, one will get a curve above

15 January 1985 / Vol. 24, No. 2 / APPLIED OPTICS 285

C

1.4 -

- 1.2

1.0

0.81 1 8 I | | 00 200 400 600 800 1000

Time sec

Fig. 2. Average radiance and radiance standard deviation duringa cooling transient; open circles, average radiance, solid circles, ra-

diance standard deviation. For the solid lines see text.

the curve of the average. The experimental results areshown in Fig. 2. The continuous curves are the theo-retical curves obtained with

(-)2+(t)2= ()2= 1,and() =2.eo Xo IOJ

One definite conclusion can be drawn from this figure,the variance of the radiance does not follow the averageradiance during transient cooling, the decrease is slowerand can be explained by Eq. (6).

IV. Discussion

The prediction that the standard deviation can obtainmaxima during transient heating or cooling is based onthe assumption that the three thermal parameters,x,y,z, heat conductivity, coupling to the heat source, and

heat capacity, respectively, are statistically uncorre-lated. In cases where this assumption is incorrect,different results can be obtained. In Eq. (6) additionalterms, including the cross correlation between the pa-rameters, will be added. In such cases the standarddeviation time behavior can be below the average ra-diance curve. A numerical example of such a situationcan be observed in Fig. 1 of Ref. 5.

In this paper it was shown that the study of the tem-poral behavior of the thermal image contrast can serveas a tool to study the thermal properties of the object.Additional conclusions can be obtained for the caseswhere a standard deviation maximum exists; there arethermal scenes in which a transient increase in contrastcan be observed if transient heating or cooling exists.

References1. Y. Itakura, S. Tsutsumi, and T. Takagi, "Statistical Properties of

the Background Noise for the Atmospheric Windows in the In-termediate Infrared Region," Infrared Phys. 14, 10 (1974).

2. A. J. Larocca and J. R. Maxwell, "Statistical Analysis of TerrainData," Report ERIM-132300-2-F (Environmental Research In-stitute of Michigan, Ann Arbor, 1978).

3. J. R. Maxwell, "Statistical Analysis of Selected Terrain and WaterBackground Measurement Data," Report ERIM-132300-I-F(Environmental Research Institute of Michigan, Ann Arbor,1978).

4. N. Ben-Yosef, B. Rahat and A. Feigin, "Simulation of IR Imagesof Natural Backgrounds," Appl. Opt. 22, 190 (1983).

5. K. Watson, "Geological Applications of Thermal Infrared Images,"Proc. IEEE 63, 128 (1975).

6. A. Papoulis, Probability, Random Variables and StochasticProcesses (McGraw-Hill Kogahusha, Tokyo, 1965, Chap. 7.

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286 APPLIED OPTICS / Vol. 24, No. 2 / 15 January 1985