DIFFRACTION SHADING MODELS FOR IRIDESCENT SURFACES

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1 DIFFRACTION SHADING MODELS FOR IRIDESCENT SURFACES Emmanuel Agu Department of Computer Scence Worcester Polytechnc Insttute, Worcester, MA 01609, USA Francs S.Hll Jr Department of Electrcal and Computer Engneerng, Unversty of Massachusetts, Amherst, MA 0100, USA Abstract Dffracton and nterference are optcal phenomena whch splt lght nto ts component wavelengths, hence producng a full spectrum of rdescent colors. Ths paper develops comput er graphcs models for rdescent colors produced by dffractve meda. Dffracton gratngs, certan anmal skns and the crystal structure of some precous stones are known to produce dffracton. Several technques can be employed to derve solutons to the dffracton problem ncludng: (1)Electromagnetc boundary value methods ()Applyng the Huygens-Fresnel prncple (3)Applyng the Krchoff-Fresnel theorem (4)Fourer optcs. Prevous work n developng dffracton models for computer graphcs has used boundary value methods and Fourer optcs but no models usng Huygens- Fresnel prncple have been publshed. Ths paper derves a set of dffracton solutons based on the Huygens -Fresnel prncple, whch are then used to extend well-known llumnaton models and are ncorporated nto a ray tracer. Key Words: dffracton, rdescence, llumnaton models, raytracer. 1. Introducton Humans lve n a world flled wth beautful colors. Irdescent colors refer to the dfferent colors that are gven off by some surfaces at dfferent lght source or vewer angles. New colors whch were nether vsble n the ncdent lght nor the object beng observed appear to have been created. These surfaces are sometmes sad to shmmer as they are rotated. Common sources of rdescent colors nclude ranbows, shny CD-ROM surfaces, opals, hummngbrd wngs, some snake skns, ol slcks and soap bubbles. Four mechansms are known to produce rdescent color, namely, dspersve refracton, scatterng, nterference and dffracton. Dffracton occurs when lght encounters some obstacle or aperture of a dmenson comparable wth ts wavelength. For example, a dffracton gratng has a seres of fnely ruled parallel grooves. Dfferent wavelengths are dffracted at dfferent angles; hence dfferent colors are produced at dfferent angles, leadng to the phenomenon of rdescence. Common sources of rdescent colors nclude dffracton gratngs, opals and some lqud crystals, hummngbrd wnds and some snakeskns. Dffracton, also referred to as a wavefront splttng phenomenon s dstngushed from nterference, an ampltude-splttng phenomenon []. A common problem n optcs s that of determnng the outgong lght ntensty, wavelength and color, gven that lght s ncdent on the dffracton surface at a certan angle and ntensty and s comprsed of specfed wavelengths. In partcular, a closed-form expresson relatng ncomng and outgong lght permts dffracton surfaces to be elegantly modelled n computer graphcs. Several technques have been employed n the optcs lterature to derve solutons to ths problem [9] ncludng (1) Solutons derved by applyng electromagnetc boundary value methods ()Solutons based on Huygens-Fresnel prncple (3)Krchoff -Fresnel based solutons (4) Solutons usng Fourer optcs. Two man bodes of work have been dentfed whch attempt to model dffracton n computer-generated magery. Thorman [10] frst developed a smple computer graphcs model for dffracton by usng the gratng equaton derved by applyng electromagnetc boundary value methods. Stroke [9] derves the geometrcal condtons for rdescence. Thorman [10] then uses Stroke's results and addresses the specfc ssue of modellng rdescent color s produced by dffracton n computer graphcs. Thorman's work focuses on determnng drectons n whch the gratng equaton would produce peak responses, and hence render mages accordngly. Specfcally, the gratng equaton used by Thorman s not a contnuous functon but only gves the drectons for perfect constructve nterference. No nformaton s gven on the behavor of the reflected lght at angles whch are away from those of perfect constructve nterference. Also, Thorman erroneously assumes that all peak ntensty values are equal. A contnuous functon needs to be derved whch gves the behavor of lght n all outgong drectons. Stam [8] has publshed work usng Fourer analyss. Fourer optcs solutons are beleved to gve the most accurate but also most complex solutons. However, detaled nformaton about both the scene confguraton

2 and dffracton surface profle are vtal before Fourer solutons can be evaluated and n fact, each soluton s vald only for one confguraton. In order to arrve at a Fourer optcs soluton, Stam makes assumptons about the scene confguraton, as well as the profle and dstrbuton of the gratng surface. The fact that the actual form and nature of Fourer solutons are confguraton-dependent ncreases ther complexty. In what follows, we wll apply the Huygens -Fresnel prncple to derve a contnuous functon that defnes the behavor of lght n all outgong drectons and then nclude t n a complete llumnaton model, showng clearly how to use the model to render pctures of surfaces wth dffracton.. Our Optcs Model The Huygens-Fresnel prncple states that every unobstructed pont of a wavefront, at a gven nstant n tme, serves as a source of sphercal secondary wavelets (wth the same frequency as that of the prmary wave). The ampltude of the optcal feld at any pont beyond s the superposton of all these wavelets (consderng ther ampltudes and relatve phases). Ths prncple arses because all vbratng partcles exert a force on ther neghbors and thus act as pont sources. It explans why when a wave passes through an aperture or obstructon, t always spreads to some extents to regons that were not drectly exposed to the oncomng waves and can be used to derve useful approxmate solutons to the dffracton problem. In optcs, dstncton s also made between near-feld or Fresnel and far-feld or Fraunhofer dffracton. In Fresnel dffracton, the pont of observaton s so close to the aperture that the mage formed bears a close resemblance to the aperture, the emergent waves are sphercal and ntenstes receved at a gven pont vary as one travels along the aperture wdth. For Fraunhofer dffracton on the other hand, the pont of observaton s so far that the mage formed bears almost no resemblance to the actual aperture, the emergent waveforms can be approxmated as planar waves wth unform ntenstes from any pont on the aperture wdth. We deal only wth Fraunhofer dffracton n ths paper snce t encompasses almost all confguratons of practcal nterest n computer graphcs. We shall now apply the Huygens-Fresnel prncple to derve an expresson for the rradance from an N-slt dffracton gratng where N s suffcently large and the pont of observaton s suffcently far from the gratng surface. Note that the followng dervatons apply to gratngs wth slts, transmssve and reflecton gratngs. Applyng the Huygens-Fresnel prncple, each slt or gratng edge of a wdth much less than, now acts as a secondary source. The number of slts, N (typcally n the thousands per nch of dffracton gratng) s tremendously large and ther separaton s small. Fgure 1 s an example of such a gratng. Only a few slts of the gratng are shown n the fgure for ease of llustraton. a b S Fgure 1: Dffracton System wth N Slts. We wsh to derve a closed form expresson for the net contrbutons of these N slots of a dffracton gratng at an arbtrary locaton n space P. The fnal closed form dervaton wll be approached n a smple two-stage process. Frst we derve the effect of one of these N slts at the pont P, followng whch we shall consder the net effect of vector addton of several of these slts at the pont P. b o ds q D q Fgure : Sngle Slt. R j R y Fgure shows a sngle slt of the dffracton gratng. ds s the elemental wdth of the wave front n the plane of the slt, at a dstance s from the center O, whch we shall refer to as the orgn. The wavelet emtted by the element ds, observed at the pont P, wll be proportonal to the length of ds and nversely proportonal to the dstance x [4], [5], [6]. The general equaton for a sphercal wave can be wrtten as: a y = sn ( wt kr) 1 r where a s the ampltude at a unt dstance from the source and r s the dstance of the observaton pont from the source. Hence, from Fgure, the nfntesmal dsplacement produced at the pont P by the nfntesmal element ds can be wrtten as ads dy o = sn ( wt kx) x The dsplacement vares n phase as the poston of ds changes by a factor, due to the dfferent path lengths to P 0, whch can be expressed as = s. snθ. So, at a gven pont s below the orgn, the contrbuton wll be q j x q P P 0 x P

3 ads dy = sn[ wt-kx-ks sn θ ] 3 s x Integratng equaton 3 from one edge of the slt to the other, we get ab sn β y = sn( wt kx) 4 x β Hence, the resultant vbraton s a smple harmonc one, the ampltude of whch vares wth poston P. Thus, the ntensty at the screen due to one slt s then sn β I A 0 5 β Now, consderng a dffracton gratng system of N slts, wth apertures of wdth a, and wdth b of the opaque porton separatng apertures (See fgure 3). Next, we shall determne the vector sum of several of these sngle slts at the arbtrary observaton pont P. The ncdent lght s stll at a normal angle of ncdence and the phase dfference, δ, between dsturbances from correspondng strps of adjacent apertures s agan π δ = d sn θ 6 ( a b) δ πd sn θ π + snθ α = = = 10 Normalng our result, we get 1 sn β sn Nα I = Io β N sn α 11 here I 0 s the ntensty of the ncomng lght ray n the θ = 0 drecton. Equaton 11 s our fnal expresson for rradance from an arbtrary N-slt dffracton gratng. Fnally, we modfy our results to take nto account oblque ncomng and outgong angles. Consder the followng dagram: q q P where d = a + b specfes the dstance between smlar ponts n adjacent apertures as shown n fgure 3 below: d snq d snq Fgure 4: Oblque Incdence. Fgure 3: Dagram showng a and b n adjacent slts. Expressed as a complex quantty, the phase dfference between dsturbances from correspondng strps of δ e adj acent apertures s. Hence, addng the net contrbutons of multple slts of ampltude sn β 7 β b as determned n the dervaton for a sngle slt and observed at a pont P n space, the complex ampltude of the resultant s gven by sn β δ [ δ δ ( N 1) 1 e e = e ] 8 β whch gves sn β sn Nα = 9 β sn α where a In the case of oblque angles, the general expresson for rradance stays the same. However, the phase dfference between dsturbances (contrbutons) from successve slots s gven by and π δ= ( a + b)( sn θ sn θ ) 1 ( sn θ sn θ ) ( sn θ sn θ ) kb β = πb = 13 ( sn θ sn θ ) k( a + b) α= πd = ( sn θ snθ ) 14 πd Snce, α= ( sn θ sn θ ), d ( θ snθ ) = m sn 15 where m = 0, ± 1, ±,... Equaton 15 s wdely known as the gratng equaton and gves the locatons of maxma. We can see from equaton 15 that dfferent wavelengths (and hence dfferent colors) wll peak at dfferent angles wth dfferent (snθ - snθ ) as shown n fgure 5.

4 spectrum has wavelengths n the 380nm to 700nm range. Fgure 5: Whte Lght Incdent Perpendcular to the Surfaces of a Gratng and the Frst Order Dffracton Spectrum on Each Sde. m = 0 reflected m = 1 dffracted m = -1 dffracted Incdent ray In our llumnaton model of equaton 16, we have replaced the specular term that was used prevously n the Phong model by a more general dffracton component that also ncorporates the Phong (or Cook- Torrance) specular component. In renderng the model, postons of peak wavelengths for each poston are precalculated and care taken to nclude the peak ntensty n the rendered mage. 4. Renderng Our Models In ths secton, we dscuss how to use our llumnaton model to render pctures wth dffractve surfaces. A ray tracer s used as our renderng system. However, frst, we shall outlne ssues whch need to be taken nto consderaton before these models can be used. Fgure 6: Vew n the Dffracton Plane of a Monochromatc Incdent Ray and the Dffracted Rays of Frst Order on Each Sde. Lkewse, we can see from fgure 6 that accordng to equaton 15, dfferent modes peak at dfferent angles. 3. Our Illumnaton Model In ths secton, we shall outlne our new llumnaton models whch are based on the Huygens -Frsenel prncple, nclude dffracton and can render rdescent surfaces. We can express our dffracton llumnaton model as I = Ambent + Dffuse + Dffracton 16 We ntroduce a new dffracton component n equaton 16 above to account for both the drectonal specular and dffracton effects. The ambent and dffuse components n equaton 16 are the same as those used n the Phong model [7]. The dffracton component n equaton 16, I(θ) s expressed as (see equaton 11) I ( θ ) 1 = I o, N sn β sn N α β sn α 17 Thus far, our expressons for our llumnaton model have not reflected the fact that the gratng or dffractve surface may be transformed nto new arbtrary 3D postons of the user s choce. A convenent way of ncorporatng transformatons and ncludng the 3D case, s by expressng our llumnaton model usng the half vector. It s nterestng to note that n the case where the dffracton gratng s concdent wth the - axs, the term snθ + snθ n our dffracton expresson s equal to the -component of the normaled halfway vector, H. Hence, we can smplfy equaton 17 by replacng snθ - snθ wth H.. We can thus re-wrte equatons 1-14 as and π π δ = d H. = ( a + b) H. 18 H. kb β = πb = H. 19 ( a + b) H. k α = πd = H. 0 Ths sngle substtuton s very powerful and greatly smplfes our expresson and eases manpulaton wthn a renderng system. It s also possble to create patterns usng by ntroducng an arbtrary twst n the gratng drecton or by creatng checkerboard patterns. where equaton 17 s a summaton of equaton 11 over a dscrete set of wavelengths and α and β equals the values expressed n equatons 13 and 14 and Io, s the lght ntensty at a summaton wavelength,. The summaton above s over a chosen set of dscrete s, [ ] n the vsble range. We recall that the vsble Color for dsplay n computer graphcs s usually specfed by relatve amounts of a set of prmary colors (e.g. Red, Green and Blue or ther RGB values) whch they contan. Snce, the models for rdescence whch were descrbed n the precedng sectons as well as the accompanyng trgonometry, were specfed on a wavelength bass, converson from wavelength to RGB

5 becomes necessary. Furthermore, followng converson, some colors whch are readly specfed n wavelengths may fall outsde the gamut of RGB tuples whch the CRT can dsplay. In such a case, these colors need to be systematcally converted or transformed to trples whch can be dsplayed by the CRT. The underlyng process of truncaton or transformaton s known as color clppng. In our ray tracer, we have nvestgated and mplemented three alternatve polces for color clppng. These are clampng, ntensty scalng and constant ntensty scalng. Hall [3] has a complete dscusson of these three scalng technques. In a ray tracer, the llumnaton models are evaluated per pxel and t s suffcent to descrbe what steps are taken to render a pxel. For each pxel that hts a dffracton surface, the followng steps are taken: Fgure 8: Scene showng a cube and cylnder wth dffractve sdes on a wood textured surface 1. Determne ht pont: determne the frst object the eye sees through ths pxel whle lookng nto the scene to the ray traced.. Compute ambent and dffuse components 3. Buld lght and eye vectors and compute normaled Half Vector, H 4. Transform gratng drecton vector, normal vector, twst gratng vector 5. Replace ( snθ + snθ ) wth H. (half vector component n the gratng drecton) n equaton Search for all vs ble modes and correspondng wavelengths that peak at ths angle. If no modes or wavelengths are vsble, return dffuse + ambent color. If mode ero s vsble, return specular component. 7. Evaluate equaton 17 at peak wavelengths. 8. Convert peak wavelengths to RGB colors as descrbed n secton 4 and evaluate rdescent color. Fgure 9: Scene showng a checkerboard dffracton gratng pattern on a wood textured surface Whle steps 1-3 above are the same as prevous ray tracers, steps 4-8 are related to our dffracton models. 5. Results Fgure 10: Scene wth checkerboard dffracton pattern cube and cylnder wth alternated groove twst angle. Fgure 7: Scene showng a dffracton gratng and a reflectve cube on a wood textured surface Fgures 7 through 11 are all mages ncorporatng our models and show some rdescent colors. We have demonstrated a basc dffracton gratng wth and wthout an arbtrary twst n the gratng drecton. We have also demonstrated varants of the checkerboard pattern alternatng wth a dffuse surface, as well as alternatng the twst angles. The patterns created were extremely colorful and resembled real lfe rdescent dffracton patterns. Addtonally, we have shown other geometres, such as the cube and cylnder that have been made from these dffractve gratngs or patterns. Fnally, we have also rendered a CD-ROM surface exhbtng rdescence. Smple anmatons were also

6 produced to llustrate the color varance as vewer and surface orentatons were altered. Fgure 11: Scene showng a CD-ROM whose surface demonstrates rdescence, and an mage-mapped can on a wood textured surface 6. Concluson In ths paper, we have defned the concept of angledependent rdescent coloraton of certan materals, due to the optcal phenomenon of dffracton. We have ntroduced other optcal phenomena such as nterference, dspersve refracton and scatterng that also produce rdescent colors. Two earler attempts by Thorman and Stam, to develop dffracton llumnaton models have been revewed and the shortcomngs of these attempts clearly stated. We have developed dffracton shadng models for computer graphcs. Our models were developed n two dstnct phases; frst, we developed an optcs model, whch descrbes the nteracton of ncdent lght wth our dffractve surface. Next, we ncluded our optcs model n our complete llumnaton model. In developng our optcs model, we have appled the Huygens -Fresnel prncple. In our dervatons, we have made assumptons whch make sense for computer graphcs applcatons. These nclude assumptons that ncdent lght s non-polared, that the gratng s several wavelengths away from the pont of observaton such that emergent waves can be approxmated as plane waves (Fraunhofer dffracton). We have ncorporated our optcs model nto a complete llumnaton model for computer graphcs by addng dffuse and ambent terms, smlar to those used n the Phong and Cook-Torrance llumnaton models. We have rendered these models usng a ray tracer and practcal ssues encountered dscussed. Photorealstc scenes wth dffractve surfaces, ncludng dffracton gratngs, checkerboard patterns and CD-ROMs, have been produced. Possble areas for future research nclude: t would be worthwhle to nvestgate radostybased solutons whch can effcently render the dffracton models descrbed n ths paper, especally because one of the advantages of radosty s ts ablty to track and render surface nter -reflectons. Dfferent lght sources: Snce dffracton s smply a separaton of wavelengths n a lght source, dfferent lght source geometres and consttuent wavelengths have an effect on mages produced and wll need to be studed n the future. Three-dmensonal gratngs: Our dffracton models deal only wth two-dmensonal gratngs. It would be desrable to derve a smlar closed-form optcs model for three-dmensonal dffracton n crystals. Dffracton n anmals: It would be mpressve to model anmals such as snakeskns and butterfles whch exhbt rdescence. It s our hope that the present work may be of use n attackng these more complex problems. 7. References [1] E. Agu, Dffracton shadng models n computer graphcs, (PhD Dssertaton, Unversty of Massachusetts-Amherst, September 001). [] J. S. Gondek, G. W. Meyer and J. G. Newman, Wavelength Dependent Reflectance Functons, Proc. SIGGRAPH 1994, pp 13-0, proc. [3] R. Hall, Illumnaton and Color n Computer Generated Imagery, (Sprnger-Verlag, 1989). [4] E. Hecht, Optcs, Second Edton, (Addson-Wesley Publshng Company, 1987). [5] F. A. Jenkns and H. E. Whte, Fundamentals of Optcs, Thrd Edton, (McGraw -Hll, 1957). [6] R. S. Longhurst, Geometrcal and Physcal Optcs, Second Edton, John Wley and Sons, [7] B. Phong, Illumnaton for Computer-Generated Pctures, Communcatons of the ACM, 18(6), pp , [8] J. Stam, Dffracton Shaders Proc. ACM Computer Graphcs (SIGGRAPH '99), pp [9] G. W. Stroke, Dffracton Gratngs pp , Handbuck Der Physk (Encyclopaeda of Physcs), (Sprnger-Verlag, 1967). [10] S. Thorman, Dffracton based models for rdescent colors n computer generated magery (PhD Dssertaton, U. of Massachusetts-Amherst, 1996). Effcent, radosty-based soluton: Snce several dffracton surfaces are shny and hghly reflectve,

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