# On the relationship between radiance and irradiance: determining the illumination from images of a convex Lambertian object

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4 R. Ramamoorthi and P. Hanrahan Vo. 18, No. 10/October 2001/J. Opt. Soc. Am. A 2451 E,, L(R,, i, i )A i d. (7) Note that this equation is essentiay a convoution, athough we have a rotation operator rather than a transation. The irradiance can be viewed as a convoution of the incident iumination L and the transfer function A cos i. Different observations of the irradiance E, at points on the object surface with different orientations, correspond to different rotations of the transfer function (since the oca upper hemisphere is rotated), which can aso be thought of as different rotations of the incident ight fied. In Section 4, we wi see that this integra becomes a simpe product when transformed to spherica harmonics, further stressing the anaogy with convoution. Our goa wi then be to deconvove the irradiance in order to recover the incident iumination. 4. SPHERICAL HARMONIC REPRESENTATION We now proceed to construct a cosed-form description of the irradiance in terms of spherica harmonic coefficients. Spherica harmonics are the anaog on the sphere to the Fourier basis on the ine or the circe. The spherica harmonic Y,m is given by Y,m, N,m P,m cos expim, (8) where N,m is a normaization factor. In the above equation, the azimutha dependence is expanded in terms of Fourier basis functions. The dependence is expanded in terms of the associated Legendre functions P,m. The indices obey 0 and m. Thus there are 2 1 basis functions for given order. Inui et a. 18 is a good reference for background on spherica harmonics and their reationship to rotations. We begin by expanding the ighting in goba coordinates in terms of spherica harmonics: L i, i 0 L,m Y,m i, i. (9) m We wi then transform to oca coordinates by appying the appropriate rotations. A. Rotation of Spherica Harmonics Let us now buid up the rotation operator on the spherica harmonics. We wi use notation of the form R()Y,m () to stand for a rotation by the rotation operator R of the parameters of the spherica harmonic Y,m. For instance, R z Y,m i, i Y,m (R z i, i ). (10) First, from the form of the spherica harmonics, rotation about z is simpe. Specificay, R z Y,m i, i Y,m i, i expimy,m i, i. (11) Rotation about y is more compicated and is given by R y Y,m i, i m D m,m Y,m i, i, (12) where D isa(2 1) (2 1) matrix that tes us how a spherica harmonic transforms under rotation about the Y axis, i.e., how to rewrite a rotated spherica harmonic as a inear combination of a the spherica harmonics of the same order. The important thing to note here is that the m indices are mixed a spherica harmonic after rotation must be expressed as a combination of other spherica harmonics with different m indices. However, the indices are not mixed; rotations of spherica harmonics with order are composed entirey of other spherica harmonics with order. Finay, we can combine the above two equations, with a simiar resut for the z rotation by, to derive the required rotation formua: R,, Y,m i, i D m,m R z R y R z Y,m i, i m D,, Y m,m,m i, i,,, D m,m expimexpim. (13) In terms of group theory, the matrix D can be viewed as the 2 1-dimensiona representation of the rotation group SO(3), with the interesting dependence being encapsuated by D. Equation (13) is simpy the standard rotation formua for spherica harmonics. Since the transfer function A cos i has no azimutha dependence, terms with m 0 wi vanish when we perform the integration in Eq. (7). Therefore we wi be most interested in the coefficient of the term with m 0, i.e., D m,0 D m,0 ()exp(im). It can be shown that this is simpy equa to 4/(2 1)Y,m (, ). This resut wi be assumed in Eq. (23). Another way to derive the resut just stated without appeaing to the properties of the matrix D is to reaize that we simpy want the coefficient of the term with no azimutha dependence. At i 0, the rotated function is determined ony by the term with m 0. In fact, it can be shown (Jackson, 19 Eq. 3.59) that, with D m,0 equa to the desired coefficient, we have R,, Y,m 0, i 2 1 1/2 D m,0,,. 4 (14) Noting that R,, Y,m (0, i ) corresponds to evauating Y,m at the oca Z axis or norma, i.e., goba coordinates of (, ), we see that D m,0 4/(2 1) 1/2 Y,m (, ). B. Representing the Transfer Function We switch our attention now to representing the transfer function A( i ) cos i. Since an object refects ony the upper hemisphere, A( i ) is nonzero ony when cos i 0. The transfer function A( i ) 0 over the

5 2452 J. Opt. Soc. Am. A/ Vo. 18, No. 10/ October 2001 R. Ramamoorthi and P. Hanrahan ower hemisphere where cos i 0. We may refer to the transfer function as the camped-cosine function, since it is equa to the cosine function over the upper hemisphere but is camped to 0 when cos i 0 over the ower hemisphere. We wi need to use many formuas for representing integras of spherica harmonics, for which a reference 20 wi be usefu. First, we expand the transfer function in terms of spherica harmonics without azimutha dependence: A i cos i n0 A n Y n,0 i. (15) The coefficients are given by /2 A n Y n,0 i cos i sin i d i. (16) 20 The factor of 2 comes from integrating 1 over the azimutha dependence. It is important to note that the imits of the integra range from 0 to /2 and not, because we are considering ony the upper hemisphere. The expression above may be simpified by writing in terms of Legendre poynomias P(cos i ). Putting u cos i in the above integra and noting that P 1 (u) u and that Y n,0 ( i ) (2n 1)/(4) 1/2 P n (cos i ), we obtain A n 2 2n 1 1/2 1 P n up 1 udu. (17) 4 0 To gain further insight, we need some facts regarding the Legendre poynomias. P n is odd if n is odd and even if n is even. The Legendre poynomias are orthogona over the domain [1, 1], with the orthogonaity reationship being given by 1 2 P a up b u 1 2a 1 a,b. (18) From this, we can estabish some resuts about Eq. (17). When n is equa to 1, the integra evauates to haf the norm above, i.e., 1/3. When n is odd but greater than 1, the integra in Eq. (17) vanishes. This is because for a n and b 1, we can break the eft-hand side of Eq. (18), by using the oddness of a and b, into two equa integras over [1, 0] and [0, 1]. Therefore both of these integras must vanish, and the atter integra is the righthand integra in Eq. (17). When n is even, the required formua is given by manipuating Eq. (20) in Chap. 5 of MacRobert. 20 Putting it a together, we have A n n 2. A pot of A n for the first few terms is shown in Fig. 3, and approximation of the camped cosine by spherica harmonic terms as n increases is shown in Fig. 4. C. Spherica Harmonic Version of the Refection Equation We now have the necessary toos to write Eq. (7) in terms of spherica harmonics. Substituting Eqs. (13) and (15), we obtain E,, n0 T n,,m i 0 0 m m L,m A n D m,m expimexpimt n,,m, 2 i 0 Y,m i, i Y n,0 i, i sin i d i d i. (20) Note that we have summed over a indices and have removed the restriction on the integra to the upper hemisphere, because that restriction has now aready been foded into the coefficients A n. By orthonormaity of the spherica harmonics, 2 i 0 i 0 Y,m i, i Y n,0 i, i sin i d i d i,n m,0. (21) Therefore terms that do not satisfy n, m 0 wi vanish. Making these substitutions in Eq. (20), we obtain E,, 0 m1 L,m A D m,0 expim. As noted in Subsection 4.B, it can be shown that 4 D m,0 expim 1/2 2 1 (22) Y,m,. (23) n 1: A n /3, n 1, odd: A n 0, n even: A n 2 2n 1 4 1/2 1 n/21 n 2n 1 n! 2 n n!/2 2. (19) We can determine the asymptotic behavior of A n for arge even n by using Stiring s formua. The bracketed term goes as n 1/2, which cances the term in the square root. Therefore the asymptotic behavior for even terms is Fig. 3. The soid curve is a pot of A n versus n. It can be seen that odd terms with n 1 have A n 0. Aso, as n increases, the coefficients rapidy decay.

8 R. Ramamoorthi and P. Hanrahan Vo. 18, No. 10/October 2001/J. Opt. Soc. Am. A 2455 of coroaries for various specia cases, where we show that increasingy compex ighting conditions admit no nonzero norm perturbation. Coroary 1. Aowabe perturbations L( i, i ) must be 0 at points where the true ighting vaue L( i, i ) and the true ighting vaue for the antipoda point L( i, i ) are both 0. In particuar, if the true ighting L is everywhere 0, there is no aowabe perturbation. This foows triviay from reation (33). Note that if the true ighting is a singe directiona source, this condition forces the aowabe perturbation to be 0 everywhere except at the source and its antipoda point. Coroary 2. If the true ighting is a singe directiona source, there is no nonzero norm aowabe perturbation, and therefore the true radiance can be recovered from the irradiance distribution at a surface orientations. From the discussion beow Coroary 1, the aowabe perturbation can have a nonzero (negative) vaue ony at the source with a corresponding negation (positive vaue) at the antipoda point. A perturbation L at the source and L at the antipoda point yieds a nonzero moment with respect to one or both of cos i and sin i the order-1 modes. In other words, since we are on a sphere, one of x, y, orz has to be nonzero, so the inear moments cannot vanish. Note that by the second required property, these moments must vanish in an aowabe perturbation. Therefore a perturbation consisting of a negative spike with an equa positive spike at the antipoda point cannot be constructed by using ony oddorder spherica harmonics with order 1. In particuar, this aows Coroary 2 to be extended to the case of two antipoda sources, since an aowabe perturbation must have the same form a negative spike at one of the sources aong with an equa positive spike at the other source. A much stronger statement is found beow. Coroary 3. If the true ighting consists of two distinct directiona sources or a nondegenerate arrangement of three directiona sources, there is no aowabe nonzero norm perturbation. From the above discussions, we know that aowabe perturbations can be nonzero ony at the sources and their antipoda points. The condition of vanishing moments with respect to the inear order-1 modes eads to a set of three simutaneous equations requiring us to choose perturbation intensities at the sources so that the moments in x, y, and z vanish. This is not possibe for nondegenerate configurations of three sources uness a perturbation intensities are 0. It is aso not possibe for any two distinct sources, since we can reparameterize so that one of the sources is on the Z axis. To make a inear moments vanish, the other source woud need to have x y 0, i.e., be antipoda at Z, in which case, by the argument beow Coroary 2, there is no aowabe perturbation. The coroary does not extend to more than three directiona sources. For four directiona sources, we may set the (negative) perturbation arbitrariy at one of the sources to fix the scae. We can then sove the vanishing moment equations for the three remaining perturbations, and the condition for a nonzero norm perturbation is that a three perturbations at the sources are negative, as required by reation (32). With more than four directiona sources, we have much more freedom in choosing the perturbations, and it is ikey that there aways exist aowabe perturbations if the sources are far enough apart, i.e., do not satisfy the conditions of Coroary 4 beow. Our primary consideration in this paper has been continuous ighting distributions or area sources. For these sources (which can be thought of as made up of an infinite number of directiona sources), Coroary 3 ceary does not appy. Coroary 4. If there exists a hemispherica region in which the true ighting L 0 everywhere, i.e., a the sources are stricty confined to one hemisphere, there is no aowabe nonzero norm perturbation. We can aways reparameterize so that a the sources ie in Z for instance, and since L 0 over this hemisphere (since L at the antipode is 0), the inear moment over Z must be negative, vioating the condition that the inear moments of the perturbation must vanish. This coroary may have impications for natura ighting when the upper hemisphere is the major contributor. Note that the sources must be stricty confined to one hemisphere a coection of sources at the equator or any other great circe does not satisfy the requirements of the coroary. In Section 6, we wi consider practica issues with ighting recovery. It wi be shown that regardess of the theoretica resuts derived here, inverse ighting is in practice poory conditioned, so even in cases where a unique soution exists, a numerica agorithm is unikey to find it. Nevertheess, the resuts of this set of coroaries tes us that it wi be hepfu to try to expicity ensure positivity of the recovered soutions in any numerica agorithm; this makes possibe the soution of some cases or reduces the norm of the maximum aowabe perturbation that are otherwise ambiguous by Theorems 1 and PRACTICAL CONSIDERATIONS This section briefy discusses some practica considerations with respect to the radiance-from-irradiance probem and describes a simpe experiment verifying the formuas derived. Equation (26) and the discussion in Subsection 4.E considering asymptotic forms show that the coefficients of the irradiance fa off rapidy, i.e., E,m 5/2 L,m. This indicates that in practice the inverse ighting probem is very poory conditioned. In fact, we can expicity write out numericay the first few terms for the irradiance: E 0, L 0,0, E 1,m 2.094L 1,m, E 2,m 0.785L 2,m, E 3,m 0,

10 R. Ramamoorthi and P. Hanrahan Vo. 18, No. 10/October 2001/J. Opt. Soc. Am. A 2457 Fig. 5. (A) One of the photographs of the mirror sphere (eft) and the Tefon sphere (right), (B) the rea ighting as recovered by using the mirrored sphere, (C) the ighting obtained by considering ony the first nine coefficients of (B), i.e., up to order 2, (D) the recovered ighting obtained by cacuating the first nine coefficients of the ight from the radiant exitance of the Tefon sphere, (E) the rea ighting up to order 4, (F) an attempt to recover the ighting up to order 4 by aso cacuating the nine order-4 modes. Images (B) (F) are visuaizations obtained by unwrapping spherica coordinates of the ighting. i ranges over [0, ] uniformy from top to bottom, and i ranges over [0, 2] uniformy from eft to right. The zero of the ighting is the gray coor used for the background of image (B). Tabe 1. Comparison of Recovered and Rea Lighting Coefficients (,m) E,m 4/(2 1) 1/2 A L,m (Rec.) L,m (Rea) (0,0) (1,1) (1,0) (1,1) (2,2) (2,1) (2,0) (2,1) (2,2) (3,3) (3,2) (3,1) (3,0) (3,1) (3,2) (3,3) the irradiance coefficients E,m for 3. According to the theory, these coefficients shoud be identicay 0, since A 3 0. We see that the experimenta vaues are indeed very cose to CONCLUSIONS We have presented a theoretica anaysis of the reationship between radiance and irradiance. We have shown that the operation of refection is anaogous to convoution of the iumination and a camped-cosine function and have derived a simpe cosed-form formua in terms of spherica harmonic coefficients. We have further demonstrated that the camped cosine or, equivaenty, the Lambertian BRDF acts as a very ow-pass fiter, making deconvoution to recover the ighting difficut. In fact, we have demonstrated that odd-order modes of the incident iumination with order 1 cannot be recovered from the irradiance, i.e., from observation of a Lambertian surface. In other words, the radiance-from-irradiance probem is i posed or ambiguous. We have aso presented evidence showing that in practica terms, the refected ight fied from a Lambertian surface is characterized ony by spherica harmonic modes up to order 2, and we can

12 R. Ramamoorthi and P. Hanrahan Vo. 18, No. 10/October 2001/J. Opt. Soc. Am. A A. Gershun, The ight fied, J. Math. Phys. 18, (1939) (transated by P. Moon and G. Timoshenko). 13. R. McCuney, Introduction to Radiometry and Photometry (Artech House, Norwood, Mass., 1994). 14. S. Gorter, R. Grzeszczuk, R. Szeiski, and M. Cohen, The umigraph, in SIGGRAPH 96, Computer Graphics Proceedings, Annua Conference Series, H. Rushmeier, ed. (Association for Computing Machinery, New York, 1996), pp M. Levoy and P. Hanrahan, Light fied rendering, in SIG- GRAPH 96, Computer Graphics Proceedings, Annua Conference Series, H. Rushmeier, ed. (Association for Computing Machinery, New York, 1996), pp D. Wood, D. Azuma, K. Adinger, B. Curess, T. Duchamp, D. Saesin, and W. Stuetze, Surface ight fieds for 3D photography, in SIGGRAPH 2000, Computer Graphics Proceedings, Annua Conference Series, K. Akeey, ed. (Association for Computing Machinery, New York, 2000), pp R. Ramamoorthi and P. Hanrahan, Anaysis of panar ight fieds from homogeneous convex curved surfaces under distant iumination, panarf. 18. T. Inui, Y. Tanabe, and Y. Onodera, Group Theory and its Appications in Physics (Springer Verag, New York, 1990). 19. J. D. Jackson, Cassica Eectrodynamics (Wiey, New York, 1975). 20. T. M. MacRobert, Spherica Harmonics; An Eementary Treatise on Harmonic Functions, with Appications (Dover, New York, 1948). 21. M. D Zmura, Shading ambiguity: refectance and iumination, in Computationa Modes of Visua Processing, M. S. Landy and J. A. Movshon, eds. (MIT Press, Cambridge, Mass. 1991), pp B. Cabra, N. Max, and R. Springmeyer, Bidirectiona refection functions from surface bump maps, Comput. Graph. 21(4), (1987). 23. S. Westin, J. Arvo, and K. Torrance, Predicting refectance functions from compex surfaces, Comput. Graph. 26(2), (1992). 24. F. Siion, J. Arvo, S. Westin, and D. Greenberg, A goba iumination soution for genera refectance distributions, Comput. Graph. 25(4), (1991). 25. J. Koenderink and A. van Doorn, Phenomenoogica description of bidirectiona surface refection, J. Opt. Soc. Am. A 15, (1998).

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