reconstruction of the net emission distribution from the total radiance distribution on a reflecting...

216 J. Opt. Soc. Am. A/Vol. 18, No. 1 /January 2001 Wim Ruyten

Reconstruction of the net emission distributionfrom the total

radiance distribution on a reflecting surface

Wim Ruyten

Sverdrup Technology, Inc./AEDC Group, Arnold Air Force Base, Tullahoma, Tennessee 37389-4300

Received December 22, 1999; revised manuscript received June 13, 2000; accepted August 9, 2000

The problem is considered of reconstructing, from a measurement of the total radiance distribution on an emit-ting surface, the radiance distribution that would be observed in the absence of reflected radiation. An ex-plicit solution of the implied inverse problem is derived for the case in which the reflective properties of thesurface are given in terms of a bidirectional reflection distribution function. Also considered are the limitingcases of diffuse and specular reflection. Practical considerations are discussed for application of the theory tothe nonintrusive and remote measurement of temperatures and pressures on concave surfaces, either by tra-ditional radiometry or by the use of thermographic phosphors and temperature- and pressure-sensitive paints.© 2001 Optical Society of America

OCIS codes: 100.3190, 120.5700, 120.5630, 260.3800.

1. INTRODUCTIONSeveral engineering disciplines rely on the use of imagingtechniques for mapping, nonintrusively and remotely,surface-temperature distributions of objects. In tradi-tional radiometry the detected radiation is the thermalemission from the object being studied.1,2 There is alarge volume of literature dealing with the associatedproblem of remote sensing of temperature, ranging fromsimple enclosures3 to canopy models in environmentalscience.4 Recently, the versatility of such temperaturemeasurements has been extended to thermographicphosphors5 (TPs) and so-called temperature-sensitivepaints6–8 (TSP’s). Also, pressure-sensitive paints (PSP’s)are being developed for use in wind-tunnel testing.8–12

Like TP’s and TSP’s, PSP’s must be excited by some lightsource and have an emission yield that is engineered todepend on the parameter of interest, in this case, air pres-sure (or, to be more precise, the density of the quenchingspecies—oxygen—at the surface).

One of the issues faced by all these measurements isthat an incorrect temperature or pressure may result inthe presence of reflected radiation.13–15 Some efforts tocorrect for such errors, using a combination of experimen-tal and numerical techniques, have been reported.16–18

In particular, in Refs. 19 and 20 the task was undertakenof correcting for self-illumination effects for the specialcase of a surface that acts as a Lambertian emissionsource and as a diffuse reflector, in the sense that the re-flected radiance distribution is Lambertian and indepen-dent of the angle of the incident radiation. It is the pur-pose of the present paper to extend the theory of Ref. 19 tosurfaces with more general emission and reflection char-acteristics. That is, the problem is studied of how to re-construct, from measurement of the total radiance distri-bution on an emitting surface, the radiation distributionthat would have been obtained in the absence of reflec-tions. The implicit premise henceforth is that, given this

0740-3232/2001/010216-08$15.00 ©

net emission distribution, it is possible to obtain correcttemperatures and pressures on a concave, reflecting sur-face with the techniques mentioned.

2. PROBLEM STATEMENTIn its generic form the task undertaken here can be for-mulated without specific reference to traditional radiom-etry, TP’s, TSP’s, or PSP’s. Let S be a surface that is be-ing imaged by one or more imaging cameras. Forexample, S may be the surface of a scale model of an air-plane in a wind tunnel (Fig. 1). In fact, S may consist ofa number of topologically distinct macroscopic surfaces,including, for example, the enclosure of the object beingstudied. For simplicity we shall refer simply to the sur-face S throughout this paper. Furthermore, let S pro-duce emission by radiation and let S reflect part of thisemitted radiation back onto itself and toward the imagingcameras. Then our task is to reconstruct, from a mea-surement of the total radiation distribution on surface S,the radiation distribution as it would appear in the ab-sence of reflections. That is, our task is to perform a self-illumination correction whereby the net emission distri-bution on the surface S is recovered from a measurementof the total radiance distribution. To cast this problem intractable form, assume that the surface S comprises Nsurface elements DSi and that for each element DSi atleast one measurement is available of the total radianceLic in the direction of an imaging camera. In fact, toavoid the problem of conflict resolution in the case of mul-tiple measurements per surface element, we assume thatprecisely one measurement per surface element is avail-able. It is also assumed implicitly that a resectionmethod is available whereby image data can be convertedto a set of values Lic that are associated with a particularset of surface elements DSi .21 Then our task is to recon-struct, from the N measured total radiances Lic , the as-

2001 Optical Society of America

Wim Ruyten Vol. 18, No. 1 /January 2001 /J. Opt. Soc. Am. A 217

sociated net emission radiances Lic(0) that would be ob-

tained in the absence of reflected radiation. In Section 3it is shown that this problem can be formulated as a lin-ear inverse problem if the following additional assump-tions are made:

1. The geometry of surface S is known.2. The initial emission process has a known angular

dependence that (in the case of TP’s, TSP’s, or PSP’s) isindependent of the direction of the incident excitation ra-diation.

3. The reflective properties of each surface elementDSi are known.

4. The positions of the imaging cameras are knownrelative to the surface S.

5. A surface element DSi is sufficiently small that theassociated radiance may be treated as a constant acrossthe extent of the surface element.

6. A surface elements DSi is flat enough that it doesnot radiate onto itself.

7. The cameras may be treated as point objects in thatno integration of emitted and reflected radiation over theextent of the camera optics is required.

8. The only radiation that is seen by the imaging cam-eras is that which is generated by the surface S by emis-sion. Specifically, it is assumed that the object is not ex-posed to radiation from its environment.

Most of these assumptions may be relaxed at the ex-pense of a more complex model. Finally, it is assumedthat the radiation can be characterized fully by total en-ergy without regard to wavelength and polarization de-pendence. If desired, a more elaborate model may be for-mulated that does not have these restrictions. In thatcase, changes in spectrum and polarization content mustbe modeled for the reflection process.

On one count the above assumptions are more restric-tive than those of Ref. 19. Namely, by assumption (8), ef-fects of spectral leakage are not considered in the presenttreatment. When thermographic phosphors or lumines-cent paints are used, such spectral leakage manifests it-self if the radiation sources that are used to excite thepaint emit radiation at the paint’s emission wavelengthor if the imaging cameras are sensitive to the short-wavelength radiation that produces the emission. Inboth cases, reflected radiation is produced that is mis-taken for emission. Such spectral leakage can be in-cluded in the present model, but would require specifica-

Fig. 1. Schematic of the eight-camera imaging system used forluminescent-paint measurements in the 16-ft Transonic WindTunnel at the Arnold Engineering Development Center. Cam-eras are mounted approximately in a plane.

tion of the positions and angular intensity distributions ofthe light sources that are used to excite the paint.

Another specific feature of luminescent-paint measure-ments that is not emphasized in the present treatment isthe ratioing of image data that are obtained at both mea-surement and reference conditions. This ratioing is nec-essary to remove effects of nonuniform illumination of ex-citation radiation on a test article.10,11 From the presentperspective, this means merely that corrections for re-flected emission radiation must be performed to the datafrom both the measurement and the reference conditions.

Finally, radiances Lic and Lic(0) need not be calibrated in

an absolute sense. In this regard, the theory that is de-veloped in the following sections applies equally well toquantities that are merely proportional to radiance, forexample, measured signal values in an imaging system.The question addressed in the following can then be in-terpreted alternatively as follows: Given a set of mea-sured signal values, how can the signal values be ob-tained that would have been measured in the absence ofreflections?

3. FORMULATION OF INVERSE PROBLEMAs our starting point, let us write the total radiance of asurface element DSi in some direction a as

Lia 5 Lia~0 ! 1 Lia,refl , (1)

where the two terms on the right are the contributions tothe radiance of DSi in the direction a by emission and re-flection, respectively. The direction a is defined either byanother surface element DSj on the surface S or by one ofthe imaging cameras (Fig. 2). Formally, we distinguishbetween these two cases by setting in Eq. (1) a 5 j in theformer case and setting a 5 c in the latter. Althoughthere are N intrasurface radiances Lij for each surface el-ement DSi , only one measured radiance Lic is consideredfor each surface element DSi per camera. The notationLic is thus to be interpreted as the radiance of the surfaceelement DSi in the direction of the one and only camera cthat is imaging the surface element DSi .

From assumption (8) of Section 2, the reflected radi-ance term in Eq. (1) is due entirely to contributions fromother surface elements DSh on the surface S. Thus thereflected term in Eq. (1) may be written as

Lia,refl 5 (h51

N

DLhia , (2)

Fig. 2. Definition of angles of emission or reflection from a sur-face element DSi . The angles are defined either by another sur-face element DSj or by the camera c 5 ci by which DSi is viewed.


where DLhia is the partial radiance of the surface elementDSi in direction a that arises as a result of the reflectionof radiation from the surface element DSh . We assumethat the characteristics of the reflection process areknown in terms of a bidirectional reflection distributionfunction (BRDF). From the definition of the BRDF thepartial reflected radiance DLhia can then be expressedas22–24

DLhia 5 RhiaDEi←h , (3)

where DEi←h is the irradiance of the surface element DSiproduced by the radiation that is received by DSi from an-other surface element DSh , and Rhia is the BRDF of thesurface element DSi for the incident angle q ih and the re-flected angles q ia and whia (Fig. 3) that are defined by thesurface elements DSh , DSi , and the direction a. It is as-sumed implicitly that, for a given h, i, and a, the BRDFvalue Rhia can be obtained, for example, by calculatingthe incident and the reflected angles and looking up orcalculating the associated value Rhia .

Calculation of the irradiance term DEi←h in Eq. (3)may be found in most texts on radiative transfer. The re-sult may be expressed as1,2,24

DEi←h 5 pDFihLhi , (4)

where Lhi is the total radiance of the surface element DShin the direction of the surface element DSi , and DFih isthe configuration factor (also known as shape factor, formfactor, and view factor) for the surface elements DSi andDSh . It is given by

DFih 51

pcos q ihDV ih , (5)

where (Fig. 4) q ih is the angle between the surface normalof DSi and the line of sight between surface elements DSiand DSh , and DV ih is the solid angle constituted by thesurface element DSh from the vantage point of DSi . Ifthe line of sight from DSi and DSh is obstructed, the fac-tor DFih should be set to zero. Likewise DFih is set tozero if h 5 i. That is, DSi does not radiate onto itself inaccordance with assumption (6) in Section 2.

Strictly speaking, the factors p in Eqs. (4) and (5) carryunits of steradian. Following convention, these units are

Fig. 3. Definition of incident angle q ih and reflected angles q iaand whia for specification of the bidirectional reflection coefficientof a surface element DSi being irradiated by a surface elementDSh . The reflected direction a is defined either by a camera oranother surface element (see Fig. 2).

not indicated explicitly. We can then combine Eqs. (1)–(4) to express the total radiance of the surface elementDSi in some direction a in terms of the net emitted radi-ance from DSi in this same direction and the total radi-ance distribution on the surface S:

Lia 5 Lia~0 ! 1 (

h51

N

pRhiaDFihLhi . (6)

The order of the subscripts in Eq. (6) is that of the direc-tion of the radiation that they represent (i to a, h to i toa), with the exception of the configuration factor DFih ,which is written in the notation that is standard for thisquantity.

To proceed, we separate the magnitude and direction-ality of the net emission terms Lia

(0) in Eq. (6) (in typicalunits of W m22 sr21) by writing them in terms of a to-be-recovered net radiant exitance Mi

(0) (in units of watts persquare meter) and an angular distribution function fia

(0)

(in units of inverse steradians) as

Lia~0 ! 5 Mi

~0 !f ia~0 ! . (7)

From assumption (2) in Section 2 the angular dependenceof the net emission process is known, typically, as a func-tion of the emission angle q ia (Fig. 2). Thus the factorsf ia

(0) can be calculated for all viewing directions a, be theydefined by another surface element DSj on the surface S(with a 5 j) or by a camera (with a 5 c).

By the very nature of the separation of the magnitudeand directionality of the net emission process in Eq. (7),the net emission exitances Mi

(0) carry only a single sub-script. Likewise, we might define a total radiant exi-tance Mi for each surface element DSi . However, thisquantity does not play a role in our analysis. On theother hand, it will prove useful to introduce an exitance-like quantity that is not particularly meaningful as aphysical quantity. This is the radiant exitance Mia of thesurface element DSi that would be calculated from the to-tal radiance Lia of DSi in the direction a if it were as-sumed (falsely) that the angular distribution of the totalradiated energy from DSi is the same as that for the netemission process. That is, it will prove useful to definethe pseudoexitances,

Mia [ Lia /f ia~0 ! , (8)

where f ia(0) is the same angular distribution function that

appears in Eq. (7) for the net emission process.

Fig. 4. Azimuth angle q ih and solid angle DV ih associated withthe definition of the configuration factor DFih between surface el-ements DSi and DSh .


If we substitute Eqs. (7) and (8) into Eq. (6) [using Eq.(8) also for the radiances Lhi in Eq. (6) by an appropriatechange of indices], we arrive at the following expressionfor the pseudoexitances Mia :

Mia 5 Mi~0 ! 1 (

h51

N

Bhia Mhi , (9)

where the scaled BRDF value Bhia is defined as

Bhia [ pRhiaDFih

f hi~0 !

f ia~0 !

. (10)

From assumptions (1)–(5) of Section 2, the scaled BRDFvalues Bhia can be calculated for any combination of indi-ces h, i, and a. With respect to finding the emission fac-tors f hi

(0) and f ia(0) , this involves finding first the associated

emission angles q ih and q ia and next the calculation ofthe corresponding factors, i.e., f hi

(0) 5 f (0)(q ih) and f ia(0)

5 f (0)(q ia). Likewise, calculation of the BRDF valuesRhia involves, in addition, calculation of the forward azi-muthal angle whia (Fig. 3) and subsequent calculation ofthe BRDF value, i.e., Rhia 5 R(q ih ; q ia , whia).

The result from Eq. (9) will serve as the basic buildingblock of the remainder of the analysis. Regarding thescaled BRDF values Bhia , it is worthwhile to distinguishbetween the two cases with a 5 j (i.e., the reflected angleis defined by another surface element DSj) and a 5 c (thereflected angle is defined by a camera). Although thereare N3 values Bhij , there are only N2 values Bhic . Toavoid confusion, we write the latter set of coefficients as

Bhi* [ Bhic . (11)

A similar distinction applies to the pseudoexitances Miafrom Eqs. (8) and (9). On the one hand, these quantitiesmay be defined between any two surface elements DSiand DSj on the surface S, yielding N2 values Mij . Butalso, for any surface element DSi on S, a pseudoexitanceMic may be calculated for the specific viewing directiona 5 c that is associated with the camera that is viewingthe surface element DSi . Of the latter variety, there areprecisely N values that we denote as

Mi* [ Mic 5 Lic /f ic~0 ! , (12)

where f ic(0) is the angular emission factor of the surface el-

ement DSi in the direction of the camera that is viewingDSi . From assumption (7) in Section 2, this is a singledirection that does not require integration over the extentof the camera lens. Our task now is to relate the N ex-perimental quantities Mi* to the N to-be-recovered netemission exitances Mi

(0) . To do this, we evaluate Eq. (9)for the camera viewing direction given by a 5 c and per-form recursion on the terms Mhi , using Eq. (9) with anappropriate change of indices. This leads to the follow-ing explicit expression for the pseudo-exitances Mi* interms of the net emission exitances Mi

(0) :

Mi* 5 Mi~0 ! 1 (

h51

N

Bhi* Mh~0 ! 1 (

g51

N

(h51

N

Bghi Bhi* Mg~0 !

1 (f51

N

(g51

N

(h51

N

Bfgh Bghi Bhi* Mf~0 ! 1 ... . (13)

This result is easy to interpret (Fig. 5). It states that thepseudoexitance Mi* (a scaled value of the experimentallydetermined radiance Lic for surface element DSi) is thesum of an infinite number of terms. The first term rep-resents the net emitted radiance produced by DSi in thedirection of the camera that is looking at DSi . The sec-ond term represents radiation that is generated by emis-sion elsewhere on the surface S and is then reflected onceby DSi toward the camera that is looking at DSi . Thethird term represents radiation that is generated some-where on the surface S, is reflected once before reachingthe surface element DSi , and is then reflected a secondtime in the direction of the camera that is looking at DSi ,and so on for higher-order terms.

By changing summation indices, we can rewrite Eq.(13) in standard linear form as

Mi* 5 (j51

N

Cij Mj~0 ! , (14)

where the elements Cij are given by

Cij 5 d ij 1 Bji* 1 (k51

N

Bjki Bki* 1 (k51

N

(l51

N

Bjkl Bkli Bli*

1 ..., (15)

with d ij 5 1 for j 5 i and zero otherwise. The result ofEqs. (14) and (15) holds for each of the N surface elementson the surface S. If we consider the individual termsMi

(0) and Mi* to be elements of the N-component vectorsM(0) and M* , respectively, Eq. (14) can be written as

M* 5 CM~0 !, (16)

where C is an N 3 N matrix that is given by an infiniteseries over all reflection orders. That is,

C 5 1 1 C~1 ! 1 C~2 ! 1 C~3 ! 1 ..., (17)

with elements

Cij~1 ! 5 Bji* , (18a)

Cij~2 ! 5 (

k51

N

BjkiBki* , (18b)

Cij~3 ! 5 (

k51

N

(l51

N

BjklBkliBli* , (18c)

and so on.This completes the formulation of the inverse problem.

Given N measured total radiances Lic on a surface S, theproblem of reconstructing the radiances Lic

(0) that would

Fig. 5. Illustration of total radiance as an infinite sum of suc-cessive reflection orders: solid dots, emission; open dots, reflec-tion.


be observed in the absence of reflections on the surface Scan now be solved by performing the following threetasks:

1. Convert the N measured radiances Lic to N scaledpseudoexitances Mi* , using Eq. (12).

2. Find the N net emission exitances Mi(0) from the N

pseudoexitances Mi* by solving the linear system posedby Eq. (16).

3. Convert the resulting N net emission exitances Mi(0)

to the associated N net radiances Lic(0) in the directions of

the imaging cameras, using Eq. (7) with a 5 c. That is,calculate the N reconstructed net emission terms

Lic~0 ! [ Mi

~0 !f ic~0 ! . (19)

The first and third tasks are simple scaling operations.The main challenge is thus to find an efficient solution tothe linear system of equations posed by Eq. (16). Thistask is undertaken in Section 4.

4. EXPLICIT SOLUTION OF THE INVERSEPROBLEMIn principle, the solution of the inverse problem posed byEq. (16) can be found by brute-force computation. Thiswould involve, first, calculation of the matrix elements Cijfrom Eq. (15)—truncating the infinite expansion in Eq.(15) at some finite number of terms—and, second, solvingthe linear system of Eq. (16) by using some standard lin-ear algebra package. The results of Ref. 19 for the spe-cial case of Lambertian emission and diffuse reflectionsuggest that this may not be the best approach. Namely,in this case, it is possible to sum the infinite series for thematrix C in Eq. (17) analytically and perform the result-ing inversion of the linear system from Eq. (16) explicitly.This approach does not seem possible for the generalproblem considered here. However, we shall see that theessence of the solution for the diffuse reflection case canbe retained for the general case if we invert the infiniteexpansion of the matrix C in Eq. (17) explicitly. To dothis, we first obtain a recurrence relation for the pseudo-exitances Mhi from Eq. (9) that does not involve the netemission exitances Mi

(0) . This is accomplished by sub-tracting Eq. (9) from itself, with a 5 j one time anda 5 c the other time. This gives

Mij 5 Mi* 1 (h51

N

DBhij* Mhi , (20)

where the quantity DBhij* is defined as

DBhij* [ Bhij 2 Bhi* . (21)

That is, DBhij* is the difference of two scaled BRDF val-ues. Both are associated with the incident direction h onthe surface element DSi , but they correspond to differentreflected directions a, namely, those defined by anothersurface element DSj on the surface S and by the camerathat is viewing DSi . There are precisely N3 valuesDBhij* , because all three subscripts can range freely overthe N surface elements on the surface S.

We are now in a position to derive an infinite expansionfor the net emission exitances Mi

(0) in terms of the scaled

experimental values Mi* . To do this, we substitute Eq.(20) into Eq. (9) recursively with a 5 c. This gives

Mi~0 ! 5 Mi* 2 (

h51

N

Bhi* Mh* 2 (g51

N

(h51

N

DBghi* Bhi* Mg*

2 (f51

N

(g51

N

(h51

N

DBfgh* DBghi* Bhi* Mf* 2 ... .

(22)

The structure of Eq. (22) is strikingly similar to that ofEq. (13) for the inverse problem. Clearly, the individualterms on the right-hand side are associated with reflec-tion orders of the radiation field. Other than this, how-ever, the interpretation of Eq. (22) is not as straightfor-ward as that of Eq. (13), in which each term can beassociated with a ray as it travels from an initial emissionpoint to a camera, being reflected some number of timesalong the way.

As was done for Eq. (13), the result from Eq. (22) can berewritten in standard linear form, namely,

M~0 ! 5 DM* , (23)

where D is an N 3 N matrix that is the inverse of thematrix C in Eqs. (16) and (18). According to Eq. (22), itsexpansion is given by the infinite series

D 5 1 2 D~1 ! 2 D~2 ! 2 D~3 ! 2 ... (24)

and has elements given by

Dij~1 ! 5 Bji* , (25a)

Dij~2 ! 5 (

k51

N

DBjki* Bki* , (25b)

Dij~3 ! 5 (

k51

N

(l51

N

DBjkl* DBkli* Bli* , (25c)

and so on. As a check, it may be verified that the productof the series expansions for C in Eq. (17) and for D in Eq.(24) is indeed unity. This can be done by grouping termsformed in the product by reflection order and verifyingthat, successively, C(1) 5 D(1), C(2) 5 C(1)D(1) 1 D(2),C(3) 5 C(2)D(1) 1 C(1)D(2) 1 D(3), and so on.

5. LIMITING CASESThe obvious advantage of the explicit solution in Section 4to the reflected radiation problem is that it does not re-quire the solution of a large, linear system of equations.At the same time, calculation of matrix D still involves aninfinite expansion. In practice, the usefulness of the im-plied reflection-correction algorithm will depend on trun-cating this expansion efficiently. Many considerationscome into play in this regard. It is not within the scopeof this discussion to address them in detail. However,two limiting cases are considered that should prove usefulas a starting point for developing practical correction al-gorithms for the general case: the cases of a diffuse anda specular surface.


A. Diffuse SurfaceThe case of a surface that is Lambertian (i.e., diffuse)with regard to both emission and reflection has alreadybeen treated in Ref. 19. For completeness, the case isconsidered here once more in terms of the current nota-tion.

For a Lambertian surface the radiance distribution isindependent of the direction of observation. Thus, in thedefinition of the scaled BRDF terms Bhia from Eq. (10),angular emission factors f hi

(0) and f ia(0) are both equal to

the constant value p21 sr21. Furthermore, the BRDFproduct pRhia in Eq. (10) is equal to the reflectivity r i ofthe surface element DSi . The scaled BRDF values Bhiafrom Eq. (10) may thus be written as

Bhia,Lamb 5 r iDFih [ Aih . (26)

These values are independent of the index a, which is as-sociated with the direction of the reflected radiation. Butthen, by Eqs. (11) and (21), all the BRDF difference termsDBhij* vanish. By Eqs. (25), all terms beyond first orderin the expansion of the matrix D in Eq. (24) then vanishas well, so that D is given by

DLamb 5 1 2 D~1 ! 5 1 2 A, (27)

where A is a matrix with elements as defined in Eq. (26).This result is derived in Ref. 19 for the case of no spectralleakage in three steps: (1) by calculating the terms C(n)

from Eqs. (20) in terms of the matrix A, yielding C(n)

5 An for all n . 0; (2) by explicitly summing the result-ing expansion for C in Eq. (17), yielding C 5 1 1 A1 A2 1 ... 5 (1 2 A)21 ; (3) by inverting the resultingexpression for C, yielding D 5 C21 5 1 2 A.

The attractive feature of the Lambertian case is that asimple, exact reflection-correction algorithm results.Specifically, in terms of the experimental radiances Lic ,the radiances in the absence of reflected radiation arefound to be given by

Lic~0 ! 5 Lic 2 (

j51

N

Aij Ljc . (28)

The removal of the reflected radiation component from aset of N total radiance terms Lic thus involves, for everysurface element DSi , at most N multiplications. In fact,many of the terms Aij may be zero owing to occlusions be-tween the associated surface elements.

B. Specular SurfaceThe extreme opposite of the Lambertian surface fromSubsection 5.A is the purely specular surface. In thiscase the reflected radiance from a surface element DSidue to radiation from another point is zero, except at thespecular angle. At the specular angle, the radiance ofthe reflected radiation is just the radiance of the surfaceelement that is producing the radiation, multiplied by thespecular reflectivity of the element DSi . If we denotethe latter quantity by s i , the reflected radiance termsDLhia in Eqs. (2) and (3) may be written as

DLhia 5 H s iLhi if a 5 aspec~i, h !

0 otherwise. (29)

Here aspec(i, h) defines the direction of specular reflectionfor the surface elements DSi and DSh . Strictly speak-ing, this specular direction is defined by the requirements(Fig. 3) that q ia 5 q ih and whia 5 0. However, for dis-crete surface elements, only discrete directions a are de-fined. A given direction a may then be considered to sat-isfy the specular condition if it deviates less than aspecified amount from the true specular direction. WithEq. (29) the scaled BRDF values from Eq. (10) are givenby

Bhia,spec 5 H s if hi~0 !/f ia

~0 ! if a 5 aspec~i, h !

0, otherwise. (30)

Consider now the calculation of the net emission exi-tances Mi

(0) from the measured pseudoexitances Mi* inEq. (22). The sums over h in Eq. (22) reduce to a singleterm at most, because the terms Bhi* in Eq. (22) (writtenin full as Bhic) are nonzero only if there exists a surfaceelement DSh that forms a specular triplet with the ele-ment DSi and the camera ci that is viewing DSi . (A sub-script is added to the camera index c to emphasize thefact that different surface elements may be viewed by dif-ferent cameras.) The sums over h in Eq. (22) thus reduceto a single term at most, with h 5 hspec(ci , i). Likewise,the sums over g in Eq. (22) reduce to two terms at most,because the terms DBghi* in Eq. (22) (written in full asBghi 2 Bghc , with c 5 ch) are nonzero only if there existseither a surface element DSg for which g 5 g1[ gspec(i, h) or g 5 g2 [ gspec(ch , h), with h5 hspec(ci , i) in both cases. Continuing in this manner,one sees that the sums over f in Eq. (22) reduce to at mostfour terms, with f 5 f11 [ fspec(h, g1), f 5 f12[ fspec(cg1

, g1), f 5 f21 [ fspec(h, g2), f 5 f22

[ fspec(cg2, g2), and so on.

The surface elements hspec(ci , i), g1 , g2 , and so on canbe found by a process of inverse ray tracing, as illustratedin Fig. 6. As an illustration, consider the case of a specu-larly reflecting surface with Lambertian emission charac-teristics. The ratio of the angular emission factors f hi

(0)

and f ia(0) in Eq. (30) is then unity, and the pseudoexitances

in Eq. (22) may be replaced by the actual radiances. Wethen have, to second order in reflection,

Lic~0 ! 5 Lic 2 s iLhc 2 s ish~Lg1c 2 Lg2c!2..., (31)

where h, g1 , and g2 are the specular indices associatedwith the inverse ray traces from Fig. 6. Radiances Lic ,Lhc , Lg1c, and Lg2c are those measured by the camerasthat are viewing the associated surface elements, that is,the cameras with indices c 5 ci , c 5 ch , etc.

Fig. 6. Identification of surface elements along specular rays byinverse ray tracing. Natural truncation of the paths occurs if aray trace fails to intersect a surface element on the surface S.


Natural truncation of the result from Eq. (31) may oc-cur after a few reflection orders if the inverse ray tracesfrom Fig. 6 fail to intersect the surface being studied.Otherwise, the effects of increasing reflection orders inEq. (31) decreases as a power series in the reflectivities s,which are less than unity. Either way, it should be pos-sible to truncate the infinite series implied by Eq. (31) af-ter a few terms for most problems of practical interest.We thus see that the computational effort associated withperforming the specular reflection correction from Eq.(31) may be significantly smaller than performing the dif-fuse reflection correction from Eq. (28), which may involvea sum over many hundreds of surface elements.

6. DISCUSSIONAn explicit solution has been obtained for the problem ofreconstructing, from a measurement of the total radiancedistribution on a reflecting surface, the emitted radiancedistribution as it would appear in the absence of reflec-tions. For the most general case, in which the reflectiveproperties of the surface are known in terms of a BRDF,this solution is embodied by Eq. (22) [although a practicalalgorithm would be more efficiently based on the recur-rence relations from Eqs. (9) and (20)]. For the limitingcases of diffuse reflection and specular reflection, the so-lutions to the inverse reflection problem are given by Eqs.(28) and (31), respectively. It is not within the scope ofthe present paper to discuss in detail the implementa-tions of these various solutions for practical problems in-volving temperature measurement or pressure measure-ment using imaging techniques on a surface. However,some guiding principles can be established.

First, implementations of the derived correction algo-rithms involve a trade-off between accuracy and speed.In those cases in which the reflection correction involvesan infinite expansion over all reflected orders of the radia-tion field, this trade-off can be controlled by the number ofreflection orders that are included in the correction. Letn be this number, and let, as above, N denote the numberof surface elements, which may be in the range 102 –105.Then, for the general solution from Eq. (22), the compu-tational cost of performing the correction on the entiresurface being studied is roughly proportional to N2 1 (n2 1)N3. [For n . 3 this presumes use of the recur-rence formula from Eq. (20) rather than the straightfor-ward evaluation of Eq. (22).] For the general reflectioncase there is thus a big premium for going beyond a first-order correction (cf. computational efforts proportional toN3 for n 5 2 versus N2 for n 5 1). In fact, little may begained when going past n 5 1 if the surface behaves, ap-proximately, as a diffuse reflector. This follows from thefact that, for a perfectly diffuse surface, terms of order n. 1 in Eq. (22) vanish identically. By comparison, forspecular surfaces, even the inclusion of several reflectionorders does not result in a significant computational ef-fort, as the operation count in this case is roughly (2n

2 1)N. Even for moderate values of n (say, n < 3), thisnumber should be vastly smaller than either the order N2

or the order N3 efforts for the diffuse and the generalmodels. An attractive solution may be to approximatethe actual BRDF as a sum of a nearly diffuse component

and a specular component. This would guarantee thatthe computational effort is no larger than order N2.

A second guiding principle for practical implementa-tions of the proposed reflection-correction algorithms isthat efficient procedures must be established for calculat-ing the necessary correction coefficients. For the diffuse-reflection problem these are the coefficients Aij from Eq.(28), which are essentially scaled values of the configura-tion factors DFih from Eq. (26). For complex geometricsurfaces, calculation of these factors is nontrivial as a re-sult of the need for occlusion detection. However, vari-ous methods have been developed to deal with this issue,especially in the context of computer rendering of complexarchitectural scenes.25–27 In particular, various modelshave been developed to model the forward problem of cal-culating reflected illumination distributions given somesource distribution. These include both radiosity andray-trace methods.26–29 Configuration factors also needto be calculated for the general case of the BRDF-basedmodel, so that the factors Bhi* , DBghi* , etc., in Eq. (22) canbe calculated with Eqs. (21), (11), and (10). Additionalpractical considerations in this regard concern storage ofthe calculated coefficients. For example, if the surface Sis represented by 104 surface elements, there are a totalof 1012 coefficients DBghi* . An on-the-fly calculation ofthese coefficients may thus be desirable to avoid excessivestorage requirements. Even for the pure specular model(which does not require the calculation of configurationfactors and does not pose significant storage problems)geometric complexities are introduced through the needfor inverse ray tracing. This also involves occlusion de-tection.

Finally, if desired, the theoretical model that is usedhere can be adapted if the need arises. In particular,several of the basic assumptions stated in Section 2 maybe relaxed, at the expense of a more cumbersome theoret-ical model. For example, there is no fundamental barrierto the inclusion of the polarization dependence of the re-flection process in the present theory, for example,through the use of Mueller matrices.30 As far as themeasurement of temperatures and pressures with imag-ing systems is concerned, it is anticipated that such re-finements will be not be required in the immediate future.

ACKNOWLEDGMENTDiscussions with my colleagues Charles Fisher, KarlKneile, and Sid Steely are greatly appreciated.

REFERENCES1. F. Grum and R. J. Becherer, Optical Radiation Measure-

ments, Vol. 1: Radiometry (Academic, New York, 1979).2. R. W. Boyd, Radiometry and the Detection of Optical Radia-

tion (Wiley, New York, 1983).3. E. F. Thacher and P. S. Giannola, ‘‘Radiometric measure-

ment of temperature distributions in simple enclosures,’’Exp. Heat Transfer 5, 217–236 (1992).

4. J. A. Smith and S. M. Goltz, ‘‘Updated thermal model usingsimplified short-wave radiosity calculations,’’ Remote Sens.Environ. 47, 167–175 (1994).


5. S. W. Allison and G. T Gillies, ‘‘Remote thermometry withthermographic phosphors: instrumentation and applica-tions,’’ Rev. Sci. Instrum. 68, 2615–2650 (1997), with 281references.

6. L. N. Cattefesta III, T. Liu, and J. P. Sullivan, ‘‘Uncertaintyestimates for temperature-sensitive paint measurementswith charged-coupled device cameras,’’ AIAA J. 36, 2102–2108 (1998).

7. K. Asai, H. Kanda, T. Kunimasu, T. Liu, and J. P. Sullivan,‘‘Boundary-layer transition detection in a cryogenic windtunnel using luminescent paint,’’ J. Aircr. 34, 34–42 (1997).

8. T. Liu, B. T. Campbell, S. P. Burns, and J. P. Sullivan,‘‘Temperature- and pressure-sensitive luminescent paintsin aerodynamics,’’ Appl. Mech. Rev. 50, 227–246 (1997).

9. J. Kavandi, J. B. Callis, M. P. Gouterman, G. Khalil, D.Wright, E. Green, D. Burns, and B. McLachlan, ‘‘Lumines-cent barometry in wind tunnels,’’ Rev. Sci. Instrum. 61,3340–3347 (1990).

10. M. J. Morris, J. F. Donovan, J. T. Schwab, S. D. Levy, andR. C. Crites, ‘‘Aerodynamics applications of pressure-sensitive paint,’’ AIAA J. 31, 419–425 (1993).

11. B. G. McLachlan and J. H. Bell, ‘‘Pressure-sensitive paintin aerodynamic testing,’’ Exp. Therm. Fluid Sci. 10, 470–485 (1995).

12. J. W. Holmes, ‘‘Analysis of radiometric, lifetime, and fluo-rescent lifetime imaging for pressure sensitive paint,’’ Aero-naut. J., 189–194 (April 1998).

13. J. L. Cogan, ‘‘Remote sensing of surface and near surfacetemperature from remotely piloted aircraft,’’ Appl. Opt. 24,1030–1036 (1985).

14. R. R. Corwin and A. Rodenburgh II, ‘‘Temperature error inradiation thermometry caused by emissivity and reflec-tance measurement error,’’ Appl. Opt. 33, 1950–1957(1994).

15. W. L. Wolfe, ‘‘Effects of reflected background radiation onradiometric temperature measurements,’’ in Infrared Imag-ing Systems Technology, W. L. Wolfe and J. Zimmerman,eds., Proc. SPIE 226, 133–135 (1980).

16. Y.-W. Zhang, C.-G. Zhang, and V. Klemas, ‘‘Quantitativemeasurements of ambient radiation, emissivity, and truthtemperature of a greybody: methods and experimental re-sults,’’ Appl. Opt. 25, 3683–3689 (1986).

17. A. G. Jackson, ‘‘Infrared temperature measurements in thepresence of reflected radiation,’’ Rep. AEDC-TR-90-12 (Ar-

nold Engineering Development Center, Tullahoma, Tenn.,1990).

18. A. P. Cracknell, ‘‘Method for the correction of sea tempera-ture derived from satellite thermal infrared data in an areaof sunglint,’’ Int. J. Remote Sens. 14, 3–8 (1993).

19. W. M. Ruyten, ‘‘Correction of self-illumination and spectral-leakage effects in luminescent-paint measurements,’’ Appl.Opt. 36, 3079–3085 (1997).

20. W. M. Ruyten, ‘‘Self-illumination calibration technique forluminescent-paint measurements,’’ Rev. Sci. Instrum. 68,3452–3457 (1997).

21. J. H. Bell and B. G. McLachlan, ‘‘Image registration forpressure-sensitive paint applications,’’ Exp. Fluids 22,78–86 (1996).

22. F. E. Nicodemus, ‘‘Directional reflectance and emissivity ofan opaque surface,’’ Appl. Opt. 4, 767–773 (1965).

23. C. Asmail, ‘‘Bidirectional scattering distribution function(BSDF): a systematized bibliography,’’ J. Res. Natl. Inst.Stand. Technol. 96, 215–223 (1991).

24. R. S. Siegel and J. R. Howell, Thermal Radiation HeatTransfer, 2nd ed. (McGraw-Hill, New York, 1981), Sect.3–5.

25. A. F. Emery, O. Johansson, M. Lobo, and A. Abrous, ‘‘Acomparative study of methods for computing the diffuse ra-diation viewfactors for complex structures,’’ J. Heat Trans-fer 113, 413–422 (1991).

26. J. D. Foley, A. van Dam, S. K. Feiner, and J. F. Hughes,Computer Graphics: Principles and Practice, 2nd ed.(Addison-Wesley, Reading, Mass., 1990), Sect. 16–13.

27. F. X. Sillion and C. Puech, Radiosity and Global Illumina-tion (Morgan Kaufmann, San Francisco, 1994), Chap. 3.

28. D. Hart, P. Dutre, and D. P. Greenberg, ‘‘Direct illumina-tion with lazy visibility evaluation,’’ in Proceedings of theACM SIGGRAPH Conference on Computer Graphics (Asso-ciation for Computing Machinery, New York, 1999), pp.147–154.

29. J. E. Mason, J. R. Schott, C. Salvaggio, and J. D. Sirianni,‘‘Validation of contrast and phenomenology in the DigitalImaging and Remote Sensing (DIRS) lab’s image genera-tion (DIRSIG) model,’’ in Infrared Technology XX, B. F.Andresen, ed., Proc. SPIE 2269, 622–633 (1994).

30. W. S. Bickel and W. M. Bailey, ‘‘Stokes vectors, Mueller ma-trices, and polarized scattered light,’’ Am. J. Phys. 53, 468–478 (1985).

reconstruction of the net emission distribution from the total radiance distribution on a reflecting...

Documents