Photorealistic Real-Time Outdoor Light Scattering Some of the most striking aspects of outdoor scenes

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1 naty hoffman & arcot j. preetham Photorealitic Real-Time Outdoor Light Scattering Some of the mot triking apect of outdoor cene are the reult of light interacting with the atmophere: hade of blue in a clear noon ky; the red and gold color of unet; the purple tint of ditant hill; the gray, wahed-out look of a foggy day. In thi article, we will explain the baic principle of cattering phyic, and ue them to derive a cattering model. We will then how how to implement thi model with a vertex hader, o that thee effect can be generated and changed in real time. Scattering Fundamental We will tart with ome fundamental concept a a backgrounder: Radiant flux (φ) meaure a quantity of light through a urface (through all point and in all direction). Radiant flux i power, which i meaured in watt. Solid angle (w) meaure a heaf of direction in 3D, like an angle meaure a heaf of direction in 2D. While angle are defined a arc on a circle and meaured in radian (2π in a complete circle), olid angle are defined a patche on a phere and meaured in teradian (4π in a complete phere). Radiant intenity (I) meaure a quantity of light through all point in a urface going in a ingle direction. Radiant intenity i power over olid angle, and i meaured in watt per teradian. Radiance (L) meaure a quantity of light in a ingle ray (through a ingle point in a ingle direction). Radiance i power over (area time olid angle), and i meaured in watt per teradian per meter quared. The pixel value of the final rendered image are derived from the radiance value for ray going through each pixel into the camera. Pixel value are RGB triple. However, radiance i ditributed along a continuou range of frequencie. There are two way to derive RGB value from a et of wavelength-dependent equation. The fat way (commonly ued for real-time graphic) i to plug three ample frequencie into the equation, reulting in an RGB triple. The more precie way (often ued for offline rendering) i to ue everal dozen ample evenly ditributed throughout the viible pectrum. The reulting erie of number i converted to an RGB triple via perceptual weighting and integration. Image by Solomon Srinivaan NATY HOFFMAN Naty ha been leading the development of the Earth & Beyond graphic engine at Wetwood Studio ince Previouly he worked at Intel a the lead microproceor architect for the Pentium with MMX chip and contributed to the MMX, SSE, and SSE 2 intruction et. Contact him at naty@wetwood.com. 32 ARCOT J. PREETHAM Preetham (preetham@ati.com) i a oftware engineer working on variou rendering technique for next-generation graphic hardware at ATI Reearch. Prior to thi, he developed 3D modeling and revere-engineering oftware at Paraform, and worked on rendering atmopheric effect for flight imulator at Evan & Sutherland. augut 2002 game developer

2 We will ue the fat method, but at a cot. Figure 1 how the pectral enitivity for the three kind of cone in the human retina. We can ee that no matter which three ampling frequencie we pick, we will loe information on the pectral tructure between them, which introduce inaccuracie. FIGURE 3. The atmophere remove ome unlight before it reache the eye by out-cattering an aborption. The cattering of blue light make the ky appear red when the un i near the horizon and it light mut travel farther through the atmophere to reach the eye. FIGURE 1. Spectral enitivity curve for cone in the retina. Atmopheric Light Scattering There are three type of interaction that can occur between a photon and a particle (an atom, molecule, dut peck, water droplet, and o on). The particle may catter the photon into the line of ight (in-cattering), it may catter it out of the line of ight (out-cattering), or it may aborb the photon altogether (aborption). color of the remaining unlight to hift toward yellow and red. When the un i near the horizon, unlight travel a much larger ditance through air than when it i at the zenith. Thi explain why thi effect i tronget at unrie and unet. Aerial perpective caue ditant object to hift in color. In Figure 4 we can ee how the atmophere attenuate the light from ditant object via out-cattering and aborption, and add new light via in-cattering. Since motly blue light i involved, thi caue ditant dark object to appear blue and ditant bright object to appear reddih. FIGURE 4. Aerial perpective caue ditant bright object to appear reddih and ditant dark object bluih. FIGURE 2. Skylight i cattered toward the eye by the atmophere. Becaue blue light catter more than red light, the ky uually appear blue. Atmopheric light cattering (we include both cattering and aborption under thi term) i reponible for many varied viual effect in outdoor cene, but for the purpoe of thi article we will concentrate on three: the ky, unlight, and aerial perpective. Firt we will dicu the ky. When looking at a clear ky you would ee nothing but black if atmopheric light cattering wa not preent. In Figure 2 we can ee how the atmophere catter unlight toward the eye. Since blue light tend to catter more than red light (more on thi later), the ky uually appear blue. In Figure 3 we can ee how the atmophere, via out-cattering and aborption, remove part of the unlight before it reache the eye. Again, motly blue light i affected, which caue the We can divide the cattering phenomena into two group: thoe which remove light from a ray (aborption and out-cattering, which we will combine under the term extinction ), and thoe which add it (in-cattering). Let compare the radiance in a ray before and after it i affected by atmopheric light cattering (L 0 and L cattering, repectively). Extinction ha a multiplicative effect on L 0, which we can expre a a dimenionle factor F ex. In-cattering ha an additive effect on L 0, which we can expre a a radiance value L in. Thi give u Equation 1: Lcattering = Fex L0 + Lin The ret of thi article will focu on how to calculate F ex and L in for all object in a cene in real time. Aborption Eq. 1 The aborption cro ection meaure how well a ingle particle aborb light around it. In Figure 5 we ee a ingle aborbent particle. The firt aumption we will make i that 33

3 the particle interact with light in an iotropic manner, that i, it doen t matter from which direction the light come. Given thi, we will look only at the light coming from a ingle direction and ignore light coming from other direction. In thi example, the light from thi direction ha a contant radiance L. The particle will aborb a certain amount of total radiant flux φ ab, however we only care about the aborbed flux coming from one direction (the aborbed radiant intenity, I ab ). In Figure 6, we ee a thin lab of aborbent medium with depth d and area A, through which photon are paing in a perpendicular direction. The total aborption cro ection of the lab i σ ab multiplied by the number of particle in the lab. The number of particle i equal to ρ ab multiplied by the lab volume, which i equal to Ad. Thi give u a total aborption area of A ab = σ ab ρ ab Ad for the lab. The probability P ab that any given photon will be aborbed i equal to the ratio of the total aborption area to the lab area, which i: FIGURE 5. A ingle particle aborbing light. We define the aborption cro ection a the aborbed radiant intenity per unit incident radiance, or I ab /L (thi i equivalent to the commonly ued definition of aborbed flux per unit irradiance and i eaier to explain). If we aume that the particle i a olid aborbent phere, then for the purpoe of aborbing light from thi direction we can treat it a a flat dic perpendicular to the light. Each point in thi dic aborb an amount of radiance. Integrating over the dic area A give u I ab =AL, o σ ab = A. If the particle i very large compared to the light wavelength, then it aborption cro ection i equal to it geometric cro ection. Smaller particle cannot really be treated a phere (or a having any hape at all), but fortunately we don t need to care σ ab capture everything we need to know about how well they aborb light. Note that σ ab varie a a function of R G B wavelength, o it i actually an RGB triple:,,. P ab = A ab /A = σ ab ρ ab d = ab d Eq. 2 σ c So the ignificance of ab i that it relate the ditance a photon travel through the medium to it chance of being aborbed. Thi explain why it ha unit of invere length ab time ditance equal probability, a dimenionle number. Another way to look at ab i that it relate the ditance a ray of light travel through the medium to the degree by which it radiance i attenuated by aborption. With thi in mind, we can rewrite Equation 2 a a differential equation: σ ab σ ab σ ab dl = ab L d We will aume that ab i contant along the ray path. Then we can olve thi equation to get the radiance of a ray (with tarting radiance L 0 ) after traveling a ditance through the medium: L ()= Le ab 0 Thi i a imple exponential decay formula. If ab i not contant, the olution i more complicated (ee Hoffman and Preetham in For More Information). Note that if we have different type of aborbent particle in the medium we can jut add their aborption coefficient together and ue the um a the total aborption coefficient. L(), L 0, and ab are RGB triple. d FIGURE 6. A thin lab of aborbent medium. Out-Scattering The derivation for out-cattering i imilar to that for aborption. We have a cattering cro ection σ c (ee Figure 7), cattering coefficient c = σ c ρ c and the equation for radiance attenuation due to out-cattering in a contant medium i: L ()= Le c 0 Undertanding how a ingle particle aborb light i all well and good, but how i light affected by paing through an aborbent medium containing many uch particle? We will characterize thi with a new quantity: the medium aborption coefficient, defined a the particle denity multiplied by. Since denity i meaured in meter 3 (particle per cubic meter), the unit of work out to be meter 1, or invere length. Thi eem a bit odd at firt but will make perfect ene in a moment. FIGURE 7. A ingle particle cattering light. σ c 34 augut 2002 game developer

4 We can alo add up the cattering coefficient for different particle type and ue the um a the total cattering coefficient. Extinction Since both aborption and out-cattering caue attenuation of light, we can um the aborption and cattering coefficient to get the extinction coefficient: ex = ab + c Then the total attenuation due to extinction i: F e ex ()= ex In-Scattering Light i cattered into the view ray from all direction; we will only handle in-cattering from the un. The cattering coefficient tell u how much light i cattered but not in which direction. For thi we define the cattering phae function f(θ,ϕ). Thi i a denity function for the probability of a photon being cattered in the direction θ,ϕ. We aume that f(θ,ϕ) depend only on the angle θ between the incoming direction and the catter direction (f(θ,ϕ) = f(θ)), and that it i not wavelength-dependent. Both aumption are reaonably accurate for mot clae of atmopheric particle. The phae function unit are invere olid angle, and integrating it over the phere yield a reult of 1.0. FIGURE 8. A ingle in-cattering event. Eq. 3 In Figure 8 we can ee that the view ray interect the cattering cro ection of the particle, o an in-cattering event i happening (a well a an out-cattering event, but that in t relevant to the preent dicuion). How much radiance i cattered into the view ray by thi event? Firt we need to integrate f(θ) over the un olid angle to get the probability that radiance i incattered from there. Since the un cover a mall cone (about half a degree acro), we can aume f(θ) doe not vary within it. In thi cae the in-cattering probability i equal to f(θ)ω un. To find the amount of radiance in-cattered by thi event, we multiply the in-cattering probability by the un radiance, L un, to get f(θ)ω un L un. We define a new contant E un = ω un L un, which expree the total illumination intenity of the un and i imilar to intenity value ued in point light ource lighting equation (there the intenity i queezed into a zero olid angle, which implie an infinite radiance point light ource don t really exit). Then the radiance added by a ingle in-cattering event i E un f(θ). To get the total in-cattered radiance over a hort ditance d, we need to multiply E un f(θ) by the probability of an in-cattering event, which i c d. The reult i E un f(θ) c d. We define the angular cattering coefficient c (θ) a equal to c f(θ). Then the in-cattered radiance over the ditance d i E un c (θ)d. Thi give u another differential equation: dl = Eunc( θ) d Unfortunately, we can t olve thi equation without taking extinction into account, ince in-cattered light undergoe extinction before it reache the eye. Adding extinction give u the following differential equation: dl = Eunc( θ) exl d If we aume E un, c (θ), and ex are contant along the path, then the olution to thi equation i fairly imple (otherwie the olution i much more involved, ee Hoffman and Preetham in For More Information). It i eentially Equation 3, plu a new in-cattering factor: 1 ex Lin(,θ) = Eunc( θ) ( 1 e ) ex Eq. 4 The in-cattered radiance i a function of (the ditance from the eye) and θ (the angle between the viewing ray and the un). Equation 1, 3, and 4 together decribe the complete cattering equation. Filling in the Parameter Now that we have the complete cattering equation, we R need to determine the parameter value to plug into it: ex, G B R G B ex, ex, c, c, c, E un, and f(θ). E un i itelf dependent on extinction we will take care of it in the implementation ection. In the next two ection we will look at two kind of particle and determine the coefficient and phae function for each. We will um ex for the two type to get the total ex, and the two c (θ) function will be added to get the total c (θ). Air Molecule and Rayleigh Scattering Firt we will look at particle much maller than the wavelength of viible light, uch a air molecule. Thee particle do not aborb light, o we will look only at cattering. The cattering coefficient for thee particle were dicovered by Lord Rayleigh around 1870, o thi type of cattering i called Rayleigh cattering (ee For More Information). For air we ue the following cattering coefficient: cair = 2 ( ) π n 1 4 3Nλ 6 3p n 6 7p n 35

5 Where n i the refractive index of air (a dimenionle quantity, equal to in the viible pectrum), N i the number of molecule per cubic meter (equal to for air at 0 C and 1 atmophere) and p n i the depolarization factor (a dimenionle quantity, equal to for air). Plugging in the value for air, together with the R, G, and B ample frequencie (650, 570, and 475 nm repectively) yield the following number: If your game ha ignificantly different condition (for example, high altitude or a planet with very high air preure), you can work out new value. The important thing to note here i that Rayleigh cattering ha a very trong preference for horter wavelength, o blue i cattered much more than red. The Rayleigh phae function for air cattering i: 3 f Air ( θ) = ( 1 + θ) 16π co 2 Figure 9 i the polar plot for thi function. We can ee that Rayleigh cattering i weakly directional and include equal amount of forward and backward cattering. Haze Particle and Mie Scattering Particle much larger than air molecule (oot, dut, water vapor, ice crytal, and o on) are called haze particle. The abhaze coefficient can vary from 0 to about m 1 ; it i uually negligible unle there i a lot of pollution preent ( abhaze uually ha no trong wavelength dependence). A theoretical model which cover cattering for thee particle wa publihed by Gutav Mie in 1908, o thi type of cattering i called Mie cattering. Mie equation are very complex and highly dependent on particle ize. Haze particle ize ditribution in the real world are alo highly varied, and it i difficult to model chaze and f haze (θ) analytically. Fortunately, many empirical meaurement are available. The phae function can be approximated by the Henyey-Greentein phae function (ee For More Information): 2 ( 1 g) fhg( θ) = 32 / 2 4π 1 g 2gco θ 36 R cair G cair B cair = m 1 = m 1 = m 1 FIGURE 9. Rayleigh phae function polar plot. ( + ( )) 0.75 FIGURE 10. Henyey-Greentein phae function polar plot. The equation may look cary, but from Figure 10 we can ee that thi i imply the polar form of an ellipe, where i the eccentricity parameter (and alo control whether the ellipe point forward or backward). For mot haze ditribution, hould be negative. A chaze increae, g increae in magnitude and chaze become more monochromatic. A group of typical value derived from empirical meaurement can be een in Table 1. TABLE 1. Typical haze parameter value. Decription Light haze Heavy haze Light fog Heavy fog 0.5 Thee value are for normal real-world environment. For more unuual environment, almot any value can be ued; feel free to have a trongly colored aborption coefficient, or even a red-colored cattering coefficient (thoe even happen in the real world on rare occaion, thu the expreion once in a blue moon ). Aerial Perpective R G B chaze chaze chaze g Aerial perpective i caued by both extinction and in-cattering. Since the viewing ray are cloe to the ground, the contant denity atmopheric model i a reaonable aumption and all the equation hold up. We treat the original (without cattering) color of the object a L 0 and multiply with F ex () and add L in (,θ) to get the final color. Thee factor can be precalculated into texture and rendered uing function of and θ a texture coordinate, calculated per-vertex, or calculated perpixel either on the fly or in a pot-proceing pa. In our implementation we choe to calculate them per-vertex in a vertex hader or vertex program. Thi approach make good ue of modern hardware capabilitie, enable changing parameter on-the-fly efficiently, and hould work reaonably well even on older hardware with oftware vertex proceing. Our implementation happen to ue a DirectX 8 pixel hader for combining the factor with the origiaugut 2002 game developer

6 nal color, but advanced fragment proceing i not necearily required depending on what ele i happening in that pa, it could be poible imply to tore the factor in the diffue and pecular vertex color and combine them uing the tandard fragment pipeline. We can ee L 0, F ex (), and L in (,θ) being combined in Figure 11. i horter for them). For other direction the length follow Equation 5 (ee Iqbal in For More Information): ( ) = l θ lzenith co( θ ) ( θ ). Note that in thi equation θ i in degree. Since we are uing two different value of (for air molecule and for haze), the equation for F ex look a little different: Eq. 5 ( ) = + exair Air exhaze Haze F, e ( ) ex Air Haze Sky Color FIGURE 11. Aerial perpective in action. Sunlight The un intenity factor E un i ued for lighting the cene and for in-cattering calculation. We calculate it once a frame by applying an extinction factor F ex () to the un intenity in 0 0 outer pace E un. We could get an exact value for E un in watt per meter quared for R, G, and B, but we would till need to convert the reulting radiance value at the end to pixel value we can handle. Since thi factor cale every radiance value in the cene, we will imply et it to the larget illumination value we can handle. On older ytem thi may be 1,1,1 (perhap 2,2,2 with ome careful ue of overbrightening technique), but on newer hardware and graphic engine we hould be able to ue larger value. We will ue the ame parameter and model to calculate F ex () a we ued for aerial perpective. However, it i not clear what to ue for, and the atmopheric denity along the path i far from contant. We olve both problem at the ame time by uing optical length for. Optical length i a ditance defined a the integrated denity along the ray divided by the denity at ground level. So if we ue our contant-denity atmophere model and ue optical length for, we will get the right extinction reult. The optical length of the atmophere for air molecule i about 8.4 kilometer at the zenith (traight up). The exact optical length for haze particle depend on variou factor, but a reaonable value to ue i 1.25 kilometer (haze particle thin out fater with height than air molecule, o the optical length The ky color in all direction i the reult of in-cattering. An accurate model would take multiple cattering into account and would be quite complex and expenive to evaluate, epecially ince it need to be evaluated for many point every frame. In thi cae we go for conitency over accuracy and ue the ame cattering model for the ky a we ued for the un and other object. A ky meh i created, reaonably well teellated, which conform in ize to the air molecule optical length value in all direction. Note that the ky meh i alway centered on the camera. The ky meh need to be rendered with a imilar vertex hader or vertex program to that ued for the aerial perpective. The main difference i that here we have different value of for the two particle type. Fortunately, the ratio between the two i a contant, o we can ize the meh to the air molecule optical length and then pa in the ratio a an additional parameter to the vertex hader. The vertex hader can then internally generate Haze from the vertex ditance and the ratio contant. We aume that Air > Haze, o we can treat the atmophere a two hell: the inner hell contain both air and haze and the outer contain only air. The reulting equation for L in i: ( ) = L,,θ E in Air Haze un cair ( )+ ( ) cair θ chaze θ exair + exhaze Haze e exair + ( 1 ( ) ) exhaze exair ( ) θ ( ) ( + ) ( 1 e ) e exair Air Haze exair exhaze Haze Since the ky ue a different vertex hader, we can alo take advantage of the fact that the tarting color i black (outer pace) and kip the extinction factor calculation. We can alo ue a impler fragment pipeline etup that jut copie the interpolated in-cattered color. 37

7 Vertex Shader In our implementation, we ued a Direct3D 8.1 vertex hader. It hould be traightforward to implement thi in OpenGL a well, given acce to the appropriate extenion. The vertex hader compute F ex () and L in (,θ), then write them into od0 and od1. The input to the vertex hader are vertex poition, tranformation matrice, unlight intenity factor E un, the un direction (for computing coθ), the variou extinction and cattering coefficient, and the Henyey-Greentein aymmetry factor g. The equation to compute F ex () and L in (,θ) are the ame one preented earlier in thi article, and the vertex hader i a traightforward implementation of thee equation. Our current verion (which ha not yet been thoroughly optimized) ue 33 intruction (not including macro expanion) and eight temporary regiter. A Reult and Sample Demo We can ee ome reult in Figure 12a c. Figure 12a how a cene with a low concentration of haze particle, o Rayleigh cattering i predominant. The un i high in the ky. Figure 12b how a cene with a high concentration of haze particle (Mie cattering i predominant) and a high un angle. In Figure 12c there i an intermediate haze concentration, and the un angle i low. Thee image were rendered on a 600MHz Pentium III with an ATI Radeon 8500 at about 60 fp. The ample demo (available at require graphic hardware that upport Direct3D pixel and vertex hader and include hader ource code. The application ha ome lider which control the variou parameter and a flythrough demo mode. Getting Light Right With the right implification and aumption, a full model of the interaction of light with the atmophere can be expreed with a few reaonably imple equation. Thee equation can be evaluated in real time to add light cattering and aborption to any outdoor cene without unduly impacting performance. For future work we would like to make the ky color model more accurate, and alo handle cloud within a phyical cattering framework. q ACKNOWLEDGEMENTS We would like to thank Kenny Mitchell for providing the terrain-rendering and lighting engine ued a the bai for the demo, and Solomon Srinivaan for help with the fly-through mode. FOR MORE INFORMATION B C FIGURES 12A 12C. Reult of low haze with Rayleigh cattering (12a), high haze with Mie cattering (12b), and intermediate haze with a low un angle (12c). Blinn, J. F. Light Reflection Function for Simulation of Cloud and Duty Surface. Computer Graphic, 16(3): 21 29, July Henyey, L. G., and J. L. Greentein. Diffue Reflection in the Galaxy. Atrophyical Journal, vol. 93: 70, Hoffman, N., and A. J. Preetham. Rendering Outdoor Light Scattering in Real Time. Proceeding of the Game Developer Conference, March Iqbal, M. An Introduction to Solar Radiation. Academic Pre, Klaen, R. V. Modeling the Effect of the Atmophere on Light. ACM Tranaction on Graphic, 6(3): , July Mie, G. Bietage zur Optik truber Medien Speziell Kolloidaler Metalloungen. Annallen der Phyik, 25(3): 377, Preetham, A. J., P. Shirley, and B. E. Smit. A Practical Analytic Model for Daylight. Computer Graphic (Proceeding of SIG- GRAPH 1999): , Augut Strutt, J. W. (Lord Rayleigh). On the Light from the Sky, It Polarization and Colour. Philoophical Magazine, vol. 41: , , April augut 2002 game developer