2007 Peer-Reviewed Conference Paper

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1 - University of Canberra Research Pubication Coection Facuty of Information Sciences & Engineering 007 Peer-Reviewed Conference Paper Citation: Lise, Ian & Huang, Tracy(007), Agorithms for spherica harmonic ighting, GRAPHITE 007: Proceedings of the 5th Internationa Conference on Computer Graphics and Interactive Techniques in Austraia and Southeast Asia, 1-4 December 007, doi: / Find this item in the UC Research Repository: Copyright: 01 ACM Version: This is the authors peer reviewed version of a work that was accepted for pubication in the ACM Digita Library. The definitive version is avaiabe at Changes resuting from the pubishing process may not be refected in this document. University of Canberra, ACT 601 Austraia, Switchboard The University of Canberra is ocated on Ngunnawa Country. CRICOS number: University of Canberra / University of Canberra Coege #001K.

2 Agorithms for Spherica Harmonic Lighting Ian G. Lise S.-L. Tracy Huang Schoo of Information Sciences & Engineering University of Canberra Abstract Spherica harmonic (SH) ighting modes require efficient and genera ibraries for evauation of SH functions and of Wigner matrices for rotation. We introduce an efficient agebraic recurrence for evauation of SH functions, and aso impement SH rotation via Wigner matrices constructed for the rea SH basis by a recurrence. Using these agorithms, we provide a freey distributabe C / OpenGL impementation for SH diffuse unshadowed, shadowed and interrefected modes. Our impementation aows fexibe switching of scene, ight probe, SH degree and ighting mode at run time. CR Categories: I.3.7 [Computing Methodoogies]: Computer Graphics Three-Dimensiona Graphics and Reaism; I.3.6 [Computing Methodoogies]: Computer Graphics Methodoogy and Techniques Keywords: recurrence 1 Introduction spherica harmonic ighting, Wigner matrix, rotation, Spherica harmonic (SH) functions have found increasing use in computer graphics over the ast decade, in ighting, BRDF, shape recognition and other areas. Most notaby, precomputed radiance transfer (PRT) methods for ighting [Kautz et a. 00; Soan et a. 00] make heavy use of spherica harmonics. In outine, PRT methods work as foows [Soan et a. 00; Green 003]: 1. In a preprocessing pass, expand the ighting environment L f, scene geometry / visibiity V G and (possiby) BRDF in SH series. Ray tracing is required to expand V G. SH coefficients of V G are recorded at each vertex.. At runtime, rotation of the scene is transferred into the SH domain by computation of Wigner matrices once per frame. The Wigner matrix is appied to rotate the ighting SH coefficients. 3. At each vertex and each frame, accumuation of a dot product of SH coefficients of V G with those of the rotated ight environment L f gives an approximation to scene ighting. This paper is the resut of buiding a C/OpenGL impementation of PRT methods with our bare hands. As our impementation evoved, it became apparent that a significant part of the effort had to be directed not at the graphics, but at underying mathematica issues. PRT methods are impemented in DirectX, but ony an API is avaiabe: the associated papers [Kautz et a. 00; Soan et a. 00] are rather dense, provide no source code and ony sketch the e-mai: Ian.Lise@canberra.edu.au e-mai:tracy.huang@canberra.edu.au agorithms. Green [003] in his Gritty Detais paper undertook to fi the gaps, providing C source for many aspects of the method. However, as a tutoria, the paper presents code for SH function evauation that is extremey inefficient. Aso, the most mathematicay chaenging part of PRT methods is Wigner rotation, but the discussion in [Green 003] is incompete, with no source code. As a consequence, any researcher seeking to work in the area of PRT or other SH methods must first batte through some unfamiiar advanced mathematics to write a ibrary of functions that impements SH evauation, Wigner matrix construction, and rotation of SH coefficients. Wigner matrices are we enough known in areas such as quantum mechanics [Edmonds 1957], but transfer of these resuts to graphics is hampered by a number of factors. First, physicists customariy use a compex vaued SH basis which eads to forma convenience but is not we suited to the numerica demands of graphics cacuations. Secondy, there is a variety of conventions for choice of phase and normaisation of SH functions (see for instance the discussion of SH conventions on the SHToos site wieczor/shtools/shtools.htm). Finay, it can be difficut to disentange mathematica and agorithmic issues from the physics of the originating fied. The resut of a this is that transfer of SH resuts from other fieds is confusing and time consuming. In this paper we present detaied agorithms for cacuating SH functions and Wigner matrices, and methods for expanding functions that occur in ighting. By coecting these resuts here we hope to imit further reinventions of the whee on these matters. Agorithms.1 Spherica Harmonics Spherica harmonic functions are eigenfunctions of the Lapace operator on the sphere S [Beecker and Csordas 199]. SH Basis A convenient rea-vaued basis for SH functions is [Green 003] for each = 0, 1,,... and for m = 1,,..., Y,0 (θ, φ) = P (cos θ) Y,m (θ, φ) = cos mφ P m (cos θ) Y, m (θ, φ) = sin mφ P m (cos θ) (1) where P is a Legendre poynomia, P m are the associated Legendre functions (and P 0 P ) [Abramowitz and Stegun 1964], and (θ, φ) are coatitude and azimuth spherica coordinates. These spherica harmonic functions Y m are unnormaised; we denote the corresponding normaised SH functions by y m : Z y m = K m Y m ; such that ym = 1 () S The normaising constants K m [Green 003, p.1] are found by a simpe recurrence.

3 Agebraic SH Recurrence Because SH functions are so heaviy used in computer graphics now, it is important to have an efficient evauation procedure for them. Various high quaity SH ibraries impementing SH functions exist, but inconvenienty use a compex SH basis ŷ m (θ, φ) = N m P m (cos θ)e imφ (3) For the rea basis (1), a recurrence for P m based on Numerica Recipes [Press et a. 199] is found in e.g. the Gritty Detais [Green 003], athough the impementation there is not particuary efficient. (For instance it is better to generate a P m at once than to evauate them one at a time.) The P m recurrence is agebraic (i.e. poynomia) in the quantities cos θ, sin θ, so invoves ony mutipication and addition. The trigonometric evauations cos mφ, sin mφ apparenty needed in (1) can simiary be generated by an agebraic recurrence from cos φ, sin φ [Banco et a. 1997]. However, instead of specifying a point u S in spherica poars (θ, φ), it is more convenient to give u = (x, y, z) as a unit vector. (For instance ray directions wi aready be in this form.) SH theory [Beecker and Csordas 199] shows that spherica harmonics Y m are homogeneous poynomias of degree in (x, y, z). By combining recurrences eqs.(9 1) and (13 14) of Banco et a. [1997] one obtains a purey agebraic recurrence for Y m : BASE OF RECURRENCE: Y 0,0 = 1, Y 1, 1 = y, Y 1,1 = x, Y 1,0 = z EDGES: For =, 3,... Y, = ( 1)` xy 1, 1 + yy 1, ( 1) Y, = ( 1)` yy 1, 1 xy 1, ( 1) INTERIOR: For =, 3,... and m = 0, 1,,..., 1 Y,m = 1 `( 1) zy 1,m ( + m 1) Y,m m Y, m = 1 `( 1) zy 1, m ( + m 1) Y, m m For m = 1, invaid terms Y,±( 1) appear in the recurrence: these terms are to be set to 0. This recurrence avoids conversion to spherica poar coordinates by computing directy in terms of (x, y, z): there are no trigonometric evauations. Normaisation is done as a fina step using the precomputed factors K m (). Data Structure There are ( + 1) spherica harmonics of degree up to, and they can be stored in an array of foats, with Y,m ocated at position ( + 1) + m in the array. However, to impement the recurrence of.1 one woud ike to use the syntax Y[][m] even for m negative. Modifying a common trick for mutidimensiona arrays [Oiveira and Stewart 006, 8.3], this is neaty achieved in C by defining a data type: typedef struct { GLfoat **Y; } ShCoeffs; GLfoat *YVaues; The array of (+1) foats in the YVaues fied hods SH vaues or coefficients as per [Green 003], whie the array of ( + 1) pointers is initiaised so that Y[] points to the vaue Y 0.. Rotations: Wigner Matrices A remarkabe feature of PRT methods is that a scene can be rotated reative to the ighting environment at run time. For a very modest computationa cost one gets soft shadows that adjust dynamicay to changed ighting. To perform this rotation requires computation of Wigner matrices reative to the chosen SH basis. Wigner matrices correspond to the odd-dimensiona irreducibe representations of the rotation group SO(3). If R SO(3) and if f is a square integrabe function on the sphere S, the eft reguar representation of SO(3) is the action (R f)(u) = f`r 1 (u). Invariance of the -th SH subspace impies the existence of coefficients Dmn(R) such that R y m = Dnm(R) y n (4) n= For a given, the entries D mn form a (+1) (+1) orthogona matrix; the probem is to construct D mn from R. Let f have SH coefficients {a m }. Then if R f has SH coefficients {a m}, the Wigner matrix entries connect the two via a m = Dmn(R) a n (5) n= By appying this rotation to SH coefficients per frame at draw time, a scene can be rotated reative to its ighting environment. Wigner matrix entries Dmn(R) can be cacuated in many ways. They depend on the choice of SH basis, so the comments in 1 about confusing basis conventions sti appy. The confusion is worsened by two further factors. First, an agorithm for Dmn(R) must choose a parametrisation of the rotation group SO(3): via a matrix, or a quaternion, or one of the many versions of Euer anges. Secondy, there are different choices for the (, m, n) to be used in a recurrence for Dmn(R). Recurrence for Wigner Matrices Mathematica compexity is a deterrent in cacuating Dmn, and and there does not appear to be a detaied code in the pubic domain for carrying through the computations with respect to the basis (1). Ivanic and Ruedenberg [1996] gave a recurrence for Wigner matrix entries in the rea spherica harmonic basis, whie Choi et a. [1999] use a recurrence based on the compex basis (3), but the method beow uses a different recurrence. Soan, et a. [00] describe the method in outine, but do not provide detaied formuas. Green [003] constructs some Wigner matrices expicity but his method is not genera and cannot easiy go to arbitrary SH degree. Beow we provide a detaied soution by a method that permits cacuation of Wigner matrix eements D mn(r) to arbitrary degree. The input to the agorithm is a rotation matrix R SO(3). Step 1. Extract Z-Y -Z Euer anges from the rotation matrix R. A rotation R is decomposed as R = R z(γ)r y(β)r z(α). The Wigner matrices D (R) can then constructed by D (R) = D `R z(γ) D `R y(β) D `R z(α) (6) Step. Wigner matrix for Y -rotation in rea SH basis. This is by far the most difficut step. For a Y -rotation through ange β we sha use d mn(β) = D mn`ry(β) to denote Wigner matrix entries with respect to rea SH basis (1). Based on the compex recurrence in [Kosteec and Rockmore 003] we derive a rea basis recurrence for d mn(β): BASE: d 0 = [1], d 1 = cos β sin β5 0 sin β cos β

4 EDGES: For =, 3,... d, = 1 cos β d 1 1, d 1 +1, +1 d, = 1 d 1 1, cos β d 1 +1, +1 and for m = 0,..., 1 d m, = d m, = q ( 1/) m sin β d 1 m, 1 q ( 1/) sin β d 1 m m, +1 INTERIOR: For =, 3,... and for m, n = 0,..., 1 d m,n = d m, n = ( 1) ( m )( n ) cos β d 1 m,n mn ( 1) d 1 m, n «(( 1) m )(( 1) n ) d ( 1)( 1) m,n ( 1) mn ( 1) d 1 m,n + cos β d 1 m, n ( m )( n ) (( 1) m )(( 1) n ) d ( 1)( 1) m,n The eements d m,n are zero for m = 1,...,, n = 0,...,, as are d m, n for m = 0,...,, n = 1,...,. Some computation is aso saved by using the symmetry d n,m = ( 1) m n d m,n. Step 3. Compose with Wigner matrices for Z-rotations Once d mn(β) are known they can be pre- and post-mutipied by Z-rotation Wigners as described by Green [003] to finay give the vaues D mn(r) (6). The above method aows generation of d mn to arbitrary degree. The method is reasonaby efficient, but further optimisation shoud be possibe by eiminating the use of Euer anges. In the first instance PRT [Soan et a. 00] treats the ighting environment to be infinitey distant, so that it is the same at each point in the scene. In this case, rotation need be done to the SH ighting coefficients a m once per frame, so efficiency is not a pressing issue. Data Structure To code the above recurrence we buid on the idea of the SH data structure of.1 and define a data type typedef struct { foat ***D; foat **DL; foat *DVaues; } WignerMatrix; If w is a WignerMatrix, then with a suitabe constructor function, w.d[][m][n] is the vaue D mn associated with w. The DVaues fied is an array of foats hoding the Wigner matrix entries; the DL fied is an array of pointers to the rows of the Wigner matrices; and the D fied is an array of pointers-to-pointers such that D[] can be thought of as the -th Wigner matrix. 3 Circuary Symmetric Function A function f on the sphere S is SH-expanded f P,m a my m by evauating integras over the sphere: a m = R S fy m. A commony arising case is where f is circuary symmetric about axis n. In this case one can be more expicit. First we state Theorem 3.1 (The Addition Theorem). [Bayis 1999, p.38]: Let {y m } be normaised spherica harmonic basis functions, and et «u, n S be points on the sphere (unit vectors). Then P (u n) = 4π + 1 m= y m (u)y m (n) It is then straightforward to show the foowing: Proposition 3.. Let f be a function on the sphere that is circuary symmetric about axis n. Then (i) There is a function h : [ 1, 1] R such that f(u) = h(n u) (ii) The SH coefficients b m of f P,m b my m are b m = π h(z)p (z) dz y m (n) (7) 1 The proof of (ii) is by expanding the function h of (i) in a series of Legendre poynomias P then appying the Addition Theorem Appication to Diffuse Unshadowed Lighting The diffuse unshadowed SH ighting mode deveoped by Ramamoorthi and Hanrahan [001a] is a oca iumination mode requiring ony surface normas n to compute shading from a ighting environment. Dot product ighting is mediated by the circuary symmetric geometry function G : S R defined by G(u) = H(u n) where H(z) = ( z 0 z z 0 Its SH expansion G P,m b my m foows immediatey from Proposition 3. as b m = πh y m (n), where H = 0 (8) zp (z) dz (9) The coefficients H were given by Ramamoorthi and Hanrahan [001b], who showed that H 0 = 1, H 1 = and that for 3 3 odd the H vanish. Their eq.(8) gives a formua for H n, but these but are more convenienty computed by Abramowitz and Stegun [1964,.13.8] H n = ( 1)n Γ(n 1 ) Γ( 1 )Γ(n + ) The H n can then be evauated once and for a from the recurrence H 0 = 1, H n+ = 3. Appication to Circuar Light Source 1 n Hn (10) + n Another appication of Proposition 3. is to SH expanding a circuar ight window that is a circuar patch on the sphere of radius ζ radians that emits ight of constant intensity. Such ight sources provide good test environments and can exhibit various artifacts associated with SH ighting. Let the ight function L f be centred on vector n and chosen so that the irradiance from L f is a constant 4π for a apertures ζ. Then L f (u) = h(u n), where h(z) = ( 1 cos ζ if z > cos ζ 0 otherwise

5 If the SH expansion of L f is L f P,m a my m, then appying Proposition 3. shows that 4π a m = 1 cos ζ cos ζ P (z) dz y m (n) R 1 1 Letting f (z) = P 1 z z (t)dt and appying the formuas of [Abramowitz and Stegun 1964, ch.], we derive the recurrence f 0(z) = 1, f 1(z) = 1 (1 + z), f +1 (z) = 1 `( + 1) zf (z) ( 1)f 1 (z) + (11) This gives the SH expansion of such a ight window with very itte effort in particuar without ray samping. 4 Impementation We have impemented three diffuse ighting modes in the PRT framework: unshadowed, shadowed and interrefected transfer. Our code uses C and OpenGL, so can buid on a variety of patforms. Because of the number of vertex attributes required, our impementation uses CPU for ighting cacuations. (We aso impemented unshadowed SH ighting in GLSL using the method of 3.1.) The code fexiby demonstrates the attributes of SH diffuse ighting modes: changing ighting mode, scene, ight probe and SH degree at run time. As we as ight probes we impement the circuar ight window of 3. with dynamicay variabe aperture. 4.1 Comparison Because DirectX provides a back box impementation of PRT it is difficut to find a compete impementation of PRT modes in open source. Green [003] gives the most compete detais but does not provide code for SH rotation. Some SH ighting demonstrations (e.g. Dempski and Viae [005]) do not even attempt rotation, negating one of the principa advantages of the PRT method. SH ighting is impemented in some student projects, but none has a genera impementation of Wigner matrices. We make our source code pubicy avaiabe ( au/dispay/shlight/): we beieve it to be the first pubicy avaiabe genera source for PRT SH rotation. The diffuse unshadowed mode is subject to some confusion in the iterature. Ramamoorthi and Hanrahan [001b] give the exact formua, but because Green s [003] sampe code cacuated the SH coefficients by ray casting, this practice has been foowed by other impementations. Such a cacuation takes around 10 sec, yet the method of 3.1 can do it (exacty) in some 0.0 sec. Combined with the agebraic SH recurrence of.1, the cacuation is fast enough to be done per vertex in a shader, as described by Ramamoorthi and Hanrahan [001b] to SH degree. This method is not as efficient as rotating via a Wigner matrix, but is much simper to code. The agebraic recurrence of.1 is of interest independent of ighting cacuations. It is easy to code, is much more efficient than the sampe code from the Gritty Detais [Green 003], and is sighty faster than the method of [Banco et a. 1997]. Benchmarking 10 6 SH evauations to degree 6 on a.13 GHz Inte Core Duo 6400, the recurrence of.1 takes 0.34 sec, or sec after unroing oops. This compares favouraby with D3DXSHEvaDirection at 0.17 sec. (The Gritty Detais code takes over 10 sec.) based on Ravi Ramamoorthi s prefiter.c code. References ABRAMOWITZ, M., AND STEGUN, I Handbook of Mathematics Functions. Dover Pubications, Inc., New York. BAYLIS, W. E Eectrodynamics: A Modern Geometric Approach. Birkhäuser, Boston. BLANCO, M. A., FLÓREZ, M., AND BERMEJO, M Evauation of the rotation matrices in the basis of rea spherica harmonics. Journa of Moecuar Structure (Theochem) 419, BLEECKER, D., AND CSORDAS, G Basic Partia Differentia Equations. Van Nostrand Reinhod, New York. CHOI, C. H., IVANIC, J., GORDON, M. S., AND RUEDENBERG, K Rapid and stabe determination of rotation matrices between spherica harmonics by direct recursion. Journa of Chemica Physics 111, DEMPSKI, K., AND VIALE, E Advanced Lighting and Materias with Shaders. Wordware Pubishing. EDMONDS, A. R Anguar Momentum in Quantum Mechanics. Princeton University Press, Princeton, N.J. GREEN, R Spherica harmonic ighting: The gritty detais. In Game Deveopers Conference. IVANIC, J., AND RUEDENBERG, K Rotation matrices for rea spherica harmonics. Direct determination by recursion. Journa of Physica Chemistry 100, KAUTZ, J., SLOAN, P.-P., AND SNYDER, J. 00. Fast, arbitrary BRDF shading for ow-frequency ighting using spherica harmonics. In proceedings of the 1th Eurographics Workshop on Rendering, P. Debevec and S. Gibson, Eds., KOSTELEC, P. J., AND ROCKMORE, D. N FFTs on the rotation group. Tech. Rep , Santa Fe Institute. OLIVEIRA, S., AND STEWART, D Writing Scientific Software: A guide to good stye. Cambridge University Press, Cambridge, UK. PRESS, W., TEUKOLSKY, S., VETTERLING, W., AND FLAN- NERY, B Numerica Recipes in C: The art of scientific computing, nd ed. Cambridge University Press, Cambridge, UK. RAMAMOORTHI, R., AND HANRAHAN, P On the reationship between radiance and irradiance: Determining the iumination from images of a convex Lambertian object. Journa of the Optica Society of America A 18, RAMAMOORTHI, R., AND HANRAHAN, P An efficient representation for irradiance environment maps. In Proceedings of ACM SIGGRAPH 001, ACM Press / ACM SIGGRAPH, New York, E. Fiume, Ed., Computer graphics Proceedings, Annua Conference Series, ACM, SLOAN, P.-P., KAUTZ, J., AND SNYDER, J. 00. Precomputed radiance transfer for rea-time rendering in dynamic, owfrequency ighting environments. ACM Transactions on Graphics 1, 3, Acknowedgements We used sampe ight probe images by Pau Debevec Our ight probe processing is