Background Data for Validation of the He-Torrance Model

Transcription

1 Backgound Data fo Valdaton of the He-Toance Model Hongsong L Kenneth E. Toance TR-PCG-5- Apl 5, 5 We pesent an expemental study of the angula dstbuton of lght scatteed fom seveal ough metallc sufaces, whch cove a ange of oughness condtons. A BRDF model based on the Kchhoff appoxmaton was used to demonstate the elaton between suface-heght statstcs and the angula dstbuton of the scatteed lght. To do ths, the angula dstbutons calculated wth the BRDF model wee ft to the measuements; the suface-heght statstcs wee computed wth a composte oughness model, and wee used as nputs to obtan the BRDF pedctons. We show that the Kchhoff appoxmaton can be appled to ough metallc sufaces that have multple scales of oughness and nea-, but not pefect, Gaussan suface-heght dstbutons.

2 . Intoducton The compute gaphcs (CG) communty needs effcent analytcal models to accuately descbe the angula dstbuton of lght scatteed fom a ough suface. Physcally-based models have been developed fo ths pupose, but the models need to be vefed by expements. To use a physcally-based model, we also need a pactcal method to estmate the suface-oughness nput paametes, based on suface heght analyses o measuements on a suface. In ths wok, we wll examne the usefulness of an exstng eflectance model, whch s based on the Kchhoff appoxmaton,,3 fo oughened metallc sufaces and popose a method of suface-heght chaactezaton. A. Analytcal Models of Suface Bdectonal Reflectance fo Compute Gaphcs Applcatons The lght scatteng fom ough sufaces has long been a subject of nteest fo the compute gaphcs (CG) communty. The goal s to smulate the appeaance of thee-dmensonal scenes n the eal wold. To do ths, analytcal models ae used to descbe the dectonal and spectal dependence of the eflectance ove the ente angula doman and vsble spectum. The dectonal popetes of the suface eflectance ae usually chaactezed by the bdectonal eflectance dstbuton functon (BRDF). 4 Fo CG applcatons, a BRDF model should be effcent, accuate, and easy to use. Fst, the calculaton of the BRDF values has to be effcent, because mllons of BRDF evaluatons ae equed to get qualty mages even fo a scene wth smple geomety and lghtng condtons. We pefe analytcal models that demand less computatonal powe and have elatvely smple

3 analytcal expessons. Second, the BRDF model should cove the ente ncdent and eflecton hemsphees, and ove the ente vsble wavelength ange, wth good accuacy. Ths equement s necessay to ende the appeaance of sufaces unde abtay oentatons and lghtng condtons, and fo coect colo epoducton. Thd, the nput paametes fo the BRDF model should be ntutve and easy to detemne. Lmted by computatonal powe, the ealy BRDF models n CG emphaszed effcency moe than accuacy and applcablty, followng an empcal appoach. 5~7 To mpove the models, advancements fom optcs wee ntoduced. 8- Among these, BRDF models employng the Kchhoff appoxmaton wee pomsng due to the elatve smplcty, analytcal fom, ease of computaton, accuacy, and statstcal epesentaton of ough sufaces. The scala fom of the Kchhoff appoxmaton was developed by P. Beckmann and a moe geneal vecto fom by A. Stogyn 3. The latte allows fo a full Fesnel eflectvty and polazaton. Based on the foegong, X. He et. al. poposed a compehensve BRDF model (He-Toance model heeafte) fo CG applcatons. The model ncopoates the vecto fom, 3 the maskng/shadowng facto developed by B. G. Smth, and apples fo fst-suface eflectons. Any multple scatteng on the suface, o any possble subsuface scatteng, wee modeled wth a constant Lambetan tem. (Moe detals on the He-Toance model and the basc equatons ae avalable n Appendx A.) Snce the He-Toance model s based on the Kchhoff appoxmaton, the valdty of that appoxmaton wll dectly affect any pedctons. The lmtatons and accuacy of the Kchhoff appoxmaton have been studed theoetcally and expementally. Geneally, the Kchhoff appoxmaton s accuate when the adus of suface 3

4 cuvatue (due to oughness elements, fo example) s lage compaed wth the llumnatng wavelength and the angle of ncdence s not lage. In addton, the accuacy may be lmted by othe assumptons of the theoetcal model, such as a Gaussan dstbuton of suface heghts, a sngle length scale of oughness, o a constant eflecton coeffcent. B. Expemental Valdaton of Suface Bdectonal Reflectance Models In the CG communty, expemental tests of BRDF models ae somewhat lmted. Seveal BRDF nstuments wee bult and used to valdate models. 9,3-5 Pehaps the ealest BRDF measuements used by the CG communty wee publshed by K. E. Toance and E. M. Spaow. 6 Those measuements wee adopted to develop what s now known n the feld as the Cook-Toance model 8 and the He-Toance model. Unfotunately, none of the measuements povded ample angula and spectal coveage. Fo valdatng the He-Toance model, the expemental studes ncluded only lmted measuements of the suface-heghts of the samples. In the optcs communty, extensve expemental nvestgatons of lght scatteng have been caed out to valdate vaous BRDF models. One classc study s that of K. A. O Donnell and E. R. Mendez, 9 who took geat effots to fabcate pefectly Gaussan-dstbuted sufaces wth a sngle scale of oughness. The sufaces wee fabcated by a lthogaphy technque, whch makes t possble to contol the heght statstcs of the fabcated sufaces. To valdate the Kchhoff appoxmaton, the suface-heghts of these sufaces wee measued and analyzed. Next, the BRDFs of these sufaces wee measued. Calculated dectonal eflectances based on the suface statstcs ageed well wth the measued eflectances. Ths success nsped moe expemental wok to study sufaces fabcated wth smla technques. -4 Although these pefect samples led to good ageement between the theoy and the expemental esults, the 4

5 applcablty of the Kchhoff appoxmaton fo eal-wold sufaces was stll not vefed, snce such sufaces may not have nea-gaussan suface-heght dstbutons o may have multple scales of oughness. Expemental studes of lght scatteng and suface-heght statstcs wee also caed out fo less pefect samples, such as alumnzed gound glass, 5 magnesum oxde ceamc, 6 fused polycystallne alumnum oxde, 7 hand-lapped stanless steel, 8 damond-tuned bass/nckel, 9 and epoxy coatngs. 3 The ageement between theoy and measuement was less satsfactoy, compaed wth pefect sufaces. E. Max and T. V. Vobuge 8 studed the elaton between the suface-heghts of hand-lapped stanless steel (measued wth a stylus poflomete) and the angula dstbuton of lght scatteng (measued wth a lase scatteomete). A theoetcal model, whch s based on the scala fom of the Kchhoff appoxmaton, was used to compute the scatteng ntensty wth the heght statstcs; on the othe hand, the model was also ft to the measuements to obtan the heght statstcs. Multple scales of oughness wee aely consdeed, a fact that may patally explan the dscepances between measued and calculated esults. One appoach to descbe the lght scatteng fom ough sufaces wth multple oughness scales s to use composte oughness models. 6-8 Such models dvde a ough suface nto two ndependent components: a hgh-fequency small-scale oughness component dng on a lowfequency lage-scale oughness component. The lght scatteed by these two components s teated sepaately by usng ethe a petubaton o a Kchhoff appoach, dependng on the heght statstcs. Howeve, t s mpactcal to dectly use the composte models fo CG because 5

6 of the complexty. Anothe appoach s to measue the suface-heghts and to solate the oughness scale (o the spatal bandwdth) that domnates scatteng behavo. E. Max and T. V. Vobuge 8 ealzed the mpotance of the spatal bandwdth selecton. They flteed the suface-heght data to attenuate spatal wavelengths > 5µm. The oughness statstcs of the flteed data ageed bette wth that nfeed by fttng a BRDF model to the optcal measuements. Howeve, ths flteng technque dd not have a theoetcal bass. The fst pupose of the pesent pape s to povde measued data on lght scatteng wth suffcent angula and spectal coveage fo CG applcatons. Second, we want to examne the usefulness of the He-Toance model fo some eal-wold metallc sufaces. Thd, a pactcal method s poposed to chaacteze such sufaces and to geneate nput oughness paametes fo the He-Toance model. The evsed suface-heght statstcs can help to pedct the angula dstbuton of scatteed lght fo vsual appeaance smulatons. Fo smplcty, we lmt ou effot to metallc sufaces and the vsble lght spectum. The pape stats n Secton wth the fabcaton and chaactezaton of the test sufaces. In Secton 3 we descbe the eflecton measuements. In Secton 4 we pesent suface-heght statstcs and descbe the elaton to values needed fo the He-Toance model. In Secton 5 we dscuss expemental and pedcted lght scatteng dstbutons.. Suface Pepaaton and Chaactezaton Fve sotopc metallc sufaces wee fabcated wth thee pepaaton methods, as descbed n Secton 3A. In Secton 3B, we pesent the suface-heght measuements and a composte 6

7 oughness model to obtan heght statstcs that ae meanngful to ou applcatons. Tables and povde a lst of the samples and some measued and nfeed oughness statstcs. A. Pepaaton of Rough Sufaces To detemne how to fabcate test sufaces, we consdeed seveal factos. Fst, the fnshed suface of a fomatve pocess should epesent a class of eal-wold sufaces. Second, we should undestand the mechansm of the fomatve pocesses that ceate these test sufaces so that we can contol the heght statstcs of the fnshed sufaces to some extent. Thd, the fomatve pocess should be able to ceate a lage (lage than mm mm), unfom (less than ±% non-unfomty), and sotopc test suface, because at gazng angles of eflecton the detecto vews a lage aea on the sample suface. We chose thee methods of suface pepaaton: bead-blastng, gndng, and etchng. Wth bead-blastng, vey small, pefectly ound beads ae popelled by hgh pessue a to emove/defom the top laye of a suface. Ths s a numeous, andom, and cumulatve pocess esultng n a Gaussan suface. An ndustal bead-blaste was used (wth glass beads of.5mm damete). The a pessue was adjusted to fabcate sufaces wth dffeent oughness. Sufaces BBQAL and BBQAL wee pocessed wth a at 69kPa and 38kPa, espectvely. The substates of these test sufaces ae Q-panels (R-46), a efeence steel panel (mm x 5mm) used fo the pant and manufactung ndustes. Moe than mnutes of blastng guaanteed the andomness of the pocess. The second method, abasve gndng, usually esults n non-gaussan sufaces. Snce the suface-heghts geneated by ths pocess ae detemned by the lagest gndng gt, ths pocess 7

8 s andom but not cumulatve. We used a BUEHLER NELSON-ZIMMER system to gnd two peces of 5mm 5mm wndow glass. Gndng powdes wth two gt szes, gt and 4gt, wee used to fabcate test samples GGAL and GGAL4, espectvely. Moe than fve hous of gndng guaanteed the andomness and unfomty. A mcoscope was used to nspect the fnshed sufaces; the gound glass sufaces look jagged. The thd method s etchng, whch may o may not esult n a Gaussan suface. The uncetanty ases because the etchng pocess s nfluenced by the mateal's mcostuctue. Theefoe, etchng s not completely andom. The test sample, NGGAL, s commecal, acd-etched, nonglae glass (5mm 5mm). A thn flm of pue alumnum was deposted on each test suface wth a CHA RAP-6 Themal Evapoato. The flm has a thckness of appoxmately 5nm, so that no lght enegy tansmtted though the flm. Thus, subsuface lght scatteng can be neglected. B. Chaactezaton of Rough Sufaces A suface measuement nstument, a Tenco Alpha Step 5 Suface Poflomete, was used to measue the suface-heghts (o pofles) of the test samples. Ths s a stylus-tavesng nstument that povdes z(x) heght data wth -Angstom vetcal and.µm hozontal esolutons. Fo each test suface, nne measuements wee taken n andomly chosen egons and n andom dectons. Each heght measuement conssts of 4 data ponts, whch epesents a 4µm tavese. 8

9 The heght data wee analyzed statstcally. Fo a andom ough suface, the vetcal and hozontal heght dstbutons can be espectvely descbed by the heght pobablty dstbuton functon (PDF) and the spatal autocoelaton functon (ACF). Though not nomalzed, the hstogam of the heght data can be used as the PDF. The ACF ρ(β) s gven by: z( x) z( x + β ) dx ρ ( β ) = lm () L L L z( x) dx whee L s the stylus tavese length. When both the PDF and the ACF confom to a Gaussan (nomal) dstbuton, the suface s sad to be Gaussan. The PDF and ACF ae qualtatve descptons; we need both to evaluate a depatue fom Gaussan behavo. Quanttatve measues of the PDF and ACF ae espectvely povded by the oot mean squae (RMS) oughness heght σ and the hozontal autocoelaton length τ. Both ae e-foldng lengths assocated wth exponentals. The RMS oughness σ can also be obtaned fom: L σ = z( x) dx () L The slope of a ough suface s the ato of the RMS oughness to the autocoelaton length, namely, σ/τ. These quanttes ae used as nputs fo the He-Toance and othe BRDF models. Cae s equed n ntepetng these statstcal measues, as eal-wold sufaces may have a wde ange of oughness scales. Among them, only a lmted ange (of spatal bandwdths) ae elevant hee; n patcula, only those domnatng the dectonal lght scatteng. 9

10 Usng computatonal studes of the lght scatteed fom sufaces wth multple oughness length scales, Y. Yang and R. O. Buckus 3 detemned that the oughness doman wth the lagest slope domnated the dectonal scatteng dstbutons. Ths povdes the bass fo ou oughness model. We assume that ou measued suface-heghts can be dvded nto two components: a hgh-fequency small-scale component and a low-fequency lage-scale component. Fo ou data, the hgh-fequency small-scale component tuned out to have the hghest slopes. We thus used a hgh-pass flte on ou measued oughness data fo each suface to flte out the lagescale components. The lengths of the lage-scale components that wee flteed out wee about 5~4µm, vayng fom suface to suface. The shotest flte, 5µm, s about 5 tmes the ncdent wavelength. Ou flte length scales ae close to the flte length used by E. Max and T. V. Vobuge (5µm). 8 The foegong s llustated gaphcally n Fgue. The esults of one measuement of z(x) fo suface GGAL was dvded nto two components by flteng. Snce the hgh-fequency small-scale component s expected to domnate the scatteed lght dectonal dstbutons, we focus on the statstcs of that component. The low-fequency lage-scale component wll be neglected. Although ths smplfed appoach wll bng eos to the statstcal esults, we beleve that the eo s acceptable fo ou applcatons. Fo each test sample we dd smla sepaatons. We calculated the RMS oughness σ and the autocoelaton length τ fo the nne suface-heght taveses, fo the hgh-fequency small-scale component only (esults ae avalable n Appendx B). We also calculated the aveage and the standad devaton of σ to assess spatal unfomty. Smlaly, we also calculated σ and τ fom

11 the aw heght data. Results fo σ, τ and the slope σ /τ appea n Table. We selected the heght measuements neaest the cente of the suface to obtan the PDF and ACF gaphs that ae dsplayed late. 3. Reflectance Measuements We measued BRDFs wth a gonoeflectomete and dectonal-hemsphecal eflectances wth an Optoncs OL-75 dffuse eflectomete. The pocedues ae descbed n Sectons 3A and 3B. To llustate some measued esults, we wll use Fgues to 4 whch coespond to the bead-blasted steel suface BBQAL. A. Measung BRDF The BRDF s the ato of the adance eflected fom a suface n the decton (θ,φ ) to the adance onto the suface fom the decton (θ,φ ). dl ( λ, θ, φ, θ, φ ) f ( λ, θ, φ, θ, φ ) = (3) de ( λ, θ, φ ) whee λ s the wavelength, θ and φ ae the zenth and azmuthal angles of the adance, and θ and φ ae the zenth and azmuthal angles of the eflected adance. The coodnate system s shown n Appendx A. Fo an sotopc suface, only thee angles ae needed. The eflected adance dl and adance de have unts of W/m s and W/m, espectvely. An automated, thee-axs BRDF measuement nstument, called a gonoeflectomete, was bult n the Optcal Measuement Laboatoy at the Conell Pogam of Compute Gaphcs The gonoeflectomete pefoms apd measuements of the BRDF of flat, sotopc sample sufaces ove the vsble spectum and ove most of the ncdent and eflecton hemsphees. Specal

12 desgn was appled to enable BRDF measuements at gazng angles up to 85. The spectoadometc detecto takes 3 convolved wavelength samples coveng the vsble spectum (4nm-7nm) n one snapshot. Fo the measuements on ou ough metallc sufaces, the BRDF values wee obtaned by a elatve method, whch compaed the eflected sgnal fom ou test sufaces wth that fom a efeence suface (Spectalon) wth a known BRDF. The flexble samplng mechansm of the gonoeflectomete enables us to sample the BRDFs n the plane of ncdence, as well ove the ente eflecton hemsphee. Fgue shows BRDF measuements n the plane of ncdence fo vaous llumnaton angles θ. We also measued the BRDFs at unfomly-dstbuted samplng postons ove the eflecton hemsphee. Fo an ncdent decton of and a wavelength of 55nm, the BRDF n the eflecton hemsphee s shown n Fgue 3. The vetcal axs s the BRDF; the left and ght othogonal axes map the sphecal coodnates above a suface. The plane of ncdence coesponds to on the left axs. The ponts ae plotted n a unfom paametezaton of the hemsphee such that each gd element n the plot epesents a egon of the hemsphee wth the same sold angle. 35 B. Measung Dectonal-hemsphecal Reflectance The dectonal-hemsphecal eflectance ρ dh s the ato of the adant powe de eflected to the hemsphee to the adant powe de ncdent onto a suface, both n W/m ; ρ dh can also be deved by ntegatng the BRDF ove the eflecton hemsphee: de ρ dh ( λ, θ, φ ) = = f ( λ, θ, φ, θ, φ ) cosθ dω (4) de Ω whee the ntegal s ove the eflecton hemsphee Ω. The ntegal s bounded by unty due to enegy consevaton.

13 The OL75 system dectly measued the dectonal-hemsphecal eflectance ρ dh (the left sde of (4)) fo a flat suface at an ncdence angle of wth a ated eo of less than %. On the othe hand, the BRDFs measued by the gonoeflectomete at an ncdent decton of wee numecally ntegated ove the eflecton hemsphee (the ght sde of (4)) to obtan the dectonal-hemsphecal eflectances. Fgue 4 shows a compason of the ntegated and dectly-measued ρ dh fo the suface BBQAL. (Note: both nstuments used Spectalon as the common efeence suface.) In Fgue 4, ove the ente vsble spectum, the dectonalhemsphecal eflectance obtaned fom the ntegated BRDF measuements s wthn % of the compason values obtaned by dect measuement wth the OL-75 system. The only excepton s at shot wavelengths, below 4nm. The compason suggests that the BRDF measuements ae accuate n both magntude and spectum, wth less than % ntegated eo. Fo all the test sufaces, we measued the BRDFs n the plane of ncdence and ove the eflecton hemsphee. Wth the OL75 measuement data, we confmed that the accuacy of the BRDF measuements s geneally wthn %. Fo CG applcatons, the measuements povde suffcent accuacy as well as ample angula and spectal coveage. Subsequently, the BRDF measuements ae used to study the elaton between suface oughness statstcs and lght scatteng behavo. Measued esults fo all fve ough metallc samples ae pesented n Secton 5. Lmted by space, only epesentatve esults ae shown. 4. Relatng Suface-heght Statstcs to the Angula Dstbuton of Scatteed Lght 3

14 The elaton between suface-heght statstcs and the angula dstbuton of lght eflected fom a suface leads to nvese and dect scatteng poblems. The nvese poblem s to ft a model (n ths case, the He-Toance model) to the BRDF measuements to obtan the suface-heght statstcs; the dect poblem s to pedct the BRDF of a suface fom a gven set of sufaceheght statstcs. We stat wth the nvese poblem. The pefomance of the data fttng helps us to evaluate the applcablty of the model. Fo a metallc suface, the He-Toance model contans fve paametes: wavelength λ, RMS oughness σ, autocoelaton length τ, ndex of efacton n(λ), and a unfom-dffuse tem a(λ) (whch allows fo multple suface scatteng and/o subsuface scatteng). Two of these paametes ae ethe ndependently specfed (λ) o found fom tables (n(λ)). 37 Fo the oughness ange of ou metallc sufaces, the BRDF s detemned pedomnantly by the oughness slope paamete (σ/τ). Thus, we need to fnd only the values of the slope (σ/τ) and unfom-dffuse (a(λ)) paametes to acheve a ft. As an uppe-bound fttng constant, we used the dectonal-hemsphecal eflectance value (equaton (4)) found by ntegatng the Fesnel eflectvty fo polshed, pue alumnum. We used the nonlnea optmzaton functon of Matlab whch vaes model paametes to mnmze the fttng eo between model and measued data. We used an enegy-weghted oot mean squae fttng eo: σ fttng = {[ f, mod el ( λ, θ, θ, φ ) f, measuements ( λ, θ, θ, φ )] cos( θ )} (5) N 4

15 The functons f (model and measued) depend on fou ndependent vaables (λ,θ I,θ,φ ) whch span the measuement space. Fo smplcty, we elected to ft only at the mdpont of the vsble spectum (λ = 55nm). The cosne tem n (5) gves smalle weghts to BRDF values at lage gazng angles of eflecton, whee the absolute BRDF values can be seveal odes of magntude lage than those n othe pats of the eflecton hemsphee. Othewse, the lage-magntude dffeences n the BRDF values can bas a fttng. The fttng method appeas to be obust and was ndependent of the ntal values. The dect poblem s solved by employng the suface heght statstcs found by mechancal measuement n Secton. Of couse, thee ae ntnsc eos n those measuements and also some vaatons ove the suface due to suface non-unfomtes. We allow fo ths and late povde uppe and lowe bounds fo dectly-calculated BRDFs fom the model. It s geneally complcated to do goous data fttng fo a nonlnea functon lke the BRDF. Howeve, fo ou applcatons n CG we ae allowed to elax some of those concens, due to lmtatons n human vson, cameas, and dsplay devces, such as polazaton nsenstvty and lmted dynamc ange. As a esult, some effects lke naow eto-eflectons and depolazaton can often be neglected; and the elatve eos n BRDF pedctons at vey low o vey hgh absolute magntudes may not be peceptble to the eye. Howeve, human vson s senstve to suface glossness, whch affects the lateal shape and angula postons of the eflectance lobe, so these featues should be accuately pedcted by a eflectance model. 5. Expemental Results 5

16 Results fo the alumnzed bead-blasted steel panels, the alumnzed gound glass, and the alumnzed acd-etched glass ae gven n the followng sectons. Snce the eflectng metal, alumnum, has an almost flat eflectance spectum ove the vsble egon, we pesent esults only fo a sngle wavelength, λ = 55nm. A. Bead-blasted Steel Panels wth Alumnum Coatng Fo the two bead-blasted steel sufaces, BBQAL (oughest) and BBQAL (smoothest), the nomalzed heght hstogams and the autocoelaton functons (ACF) fo the hgh-fequency small-scale component of the oughness ae shown n Fgue 5. The sold lnes ae the ft of Gaussan cuves. Fo the hstogams, the cuves agee modeately well wth the data, suggestng that the hgh-fequency small-scale component of the oughness s nealy Gaussan. Fo the autocoelaton functons ACF, the ftted cuves agee vey well wth the data. Consdeng the andom chaactestcs of the bead-blastng pocess, these esults ae undestandable. Reflectance measuements fo the two bead-blasted steel sufaces, BBQAL and BBQAL, ae pesented n Fgues 6 to ; of these, Fgues 6, 7, 9, and ae ncdence-plane BRDFs, wheeas Fgue 8 dsplays BRDFs n the hemsphee. In the ncdence-plane fgues, the cuves ae esults fom the model. In Fgues 6 and 7, the cuves ae the best-ft cuves; n Fgues 9 and, the same measued data ae shown, but the cuves ae calculated usng the measued slopes (σ/τ) fo the small-scale oughness as nputs to the model. The uppe/lowe bounds wee found by vayng the nput slopes by +/- amounts coespondng to the measued pecentage nonunfomty n the oughness of the suface (see Table ). 6

17 Results fo the oughest (σ=.7µm) and smoothest (σ=.54µm) bead-blasted metal sufaces ae shown, espectvely, n Fgues 6 and 7. The oughness to wavelength atos ae of ode one, so a wave optcs model s appopate. Note the good ageement between the model and the measuements. A smla compason ove the eflecton hemsphee s shown n Fgue 8 fo the oughe suface. Fo small angles of ncdence ( and 3 ), the measuements shown n Fgues 6 and 7 ae Gaussan-lke cuves wth maxma nea o just beyond the specula angle. Fo 6 ncdence, off-specula maxma ae appaent. Best-ft slope values fo Fgues 6 and 7 ae.44 and.8, espectvely. On the othe hand, measued values fo the hgh-fequency small-scale suface oughness component yeld.9 and.84, espectvely. The fome s % low and the latte s 4% hgh. If we also compae to slope values fom the aw heght data (befoe scale sepaatons), we have slope values of.95 and.4, whch ae espectvely 3% low and 5% low as compaed to the best-ft values. The aw data values eflect the nfluence of the gentle slopes of the lage-scale oughness components. The close ageement of the best-ft and mechancally-measued hgh-fequency small-scale oughness slopes stongly suggests the value of scale sepaaton when tyng to use mechancal oughness measuements as nput to a model. Fo the bead-blasted steel sufaces, an altenatve dsplay of the measued data (n Fgues 6 and 7) s pesented as a senstvty study n Fgues 9 and. As noted eale, the cuves ae calculated fom the eflecton model usng slope values (σ/τ) that bound the vaablty n the measued small-scale oughness slopes (σ/τ). Accodngly, the measued slope of.9 fo the 7

18 oughest suface was vaed by ± 3.9% n Fgue 9 to get the uppe and lowe cuves. Smlaly, the measued slope of.84 fo the smoothest suface was vaed by ± 6.8% n Fgue. In Fgue 9, the model esults fall below the measuements. Indeed, the best ft nfeed slope fo ths suface s about % hghe. In Fgue, on the othe hand, uppe/lowe model esults tend to backet the measuements. The He-Toance model appeas to wok vey well fo the bead-blasted steel sufaces. Ths s not a supse snce the sufaces have heght dstbutons and autocoelaton functons close to Gaussan. Moeove, the small-scale oughness statstcs appea to be moe easonable to use than those of the aw heght data. B. Gound Glass wth Alumnum Coatng Fo the two gound glass sufaces, GGAL (oughest) and GGAL4 (smoothest), the hstogams and the autocoelaton functons (ACF) fo the hgh fequency small-scale component of the oughness ae shown n Fgue. The ftted Gaussan cuves agee modeately well suggestng that the hgh-fequency small-scale oughness components ae nealy Gaussan. Howeve, the gndng pocess tself mples that the oughness should not be Gaussan. To explan ths, the heght hstogam and ACF fo the aw heght data of suface GGAL ae shown n Fgue. The assocated hstogam s nealy Gaussan but slghtly skewed to the ght towad hghe heghts, consstent wth a gndng pocess that emoves moe peaks than valleys. On the othe hand, the assocated autocoelaton functon s athe moe exponentally dstbuted than Gaussan. Ths s consstent wth the jagged appeaance obseved unde a mcoscope. The statstcal data mply that a gound glass suface conssts of a hgh-fequency, small-scale Gaussan oughness component (statstcs n Fgue ) supeposed 8

19 on a low-fequency lage-scale jagged oughness component. The small-scale oughness s expected to domnate the lght scatteng. Incdence-plane BRDF measuements and data fttng esults ae shown fo the oughest (σ=.86µm) and smoothest (σ=.6µm) ough glass sufaces, GGAL and GGAL, espectvely, n Fgues 3 and 4. A smla compason ove the eflecton hemsphee s shown n Fgue 5 fo the oughe suface. Incdence-plane BRDF measuements and esults fom the model usng the measued small-scale oughness slopes ae shown n Fgues 6 and 7; uppe/lowe estmates eflect the measued non-unfomty of the suface. Results n Fgues 3 and 4 ae nealy dentcal because the hgh-fequency small-scale oughness values ae vey smla fo the two sufaces (Table ). The BRDFs show a Gaussanlke cuve only fo ncdence. An off-specula peak s appaent fo hghe angles of ncdence. The fttngs of the He-Toance model to the eflectance data ae geneally good except that the model oveestmates the magntude of the off-specula peaks at ncdence angles of 6 and hghe. The eos ae manly due to shadowng/maskng effects fom one tem n the model. A bette shadowng/maskng facto could educe those eos. The jagged lage-scale oughness only makes the effects moe appaent. The best-ft slope value fom the data fttng n Fgue 3 s.66, about 8% hghe than the measued value of.54 (fo the hgh-fequency small-scale oughness component). The best-ft slope value fo Fgue 4 s.58, about % hghe than the measued slope of.3. Fo the two fgues, the slopes calculated fom the aw heght data ae.64 and.68, espectvely, fo Fgues 3 and 4, even though the aw heght data ndcate vey dffeent suface oughness (σ=.66µm vs. σ=.63µm). 9

20 Thus, the BRDF measuements fo the two gound glass samples ae supsngly close, even though the aw heght suface oughnesses ae vey dffeent. The smlaty of the measuements suggests that the small-scale stuctues of the two sufaces ae qute smla. That s confmed by the data fttng, snce ealy equal small-scale slopes wee nfeed. The lagescale oughness of the two sufaces, though qute dffeent, has lttle nfluence on the angula dstbuton of the scatteed lght. Fo completeness, the effects of suface non-unfomty may be nfeed fom Fgues 6 and 7, whee t s clea that the measued slopes (σ/τ) yeld BRDF pedctons that fall somewhat above the best-ft pedctons n Fgues 3 and 4. The He-Toance model has a elatvely good pefomance wth the gound-glass sufaces. Although the fttng esults ae not as satsfactoy as those fo the bead-blasted sufaces, the model pedcts the off-specula peaks wth modeate magntude eos. C. Acd-etched Glass wth Alumnum Coatng The alumnzed acd-etched glass s much smoothe than the othe test sufaces (Table ). On the othe hand, the suface s suffcently ough so that no mo eflecton can be obseved except at hgh gazng angles of ncdence. The small-scale oughness s vey close to Gaussan, as shown n Fgue 8. Theefoe, good compasons wth the He-Toance model ae expected. Incdence-plane BRDF measuements and data fttng ae shown n Fgues 9 and. The latte fgue uses a logathmc odnate to bette dsplay the dynamc ange of the BRDF, snce thee s vtually no dffusely scatteed lght. Also, no off-specula peak s obseved due to the gentle slope of the suface-heghts. The fttngs of the data to the He-Toance model ae vey good,

21 except that the model slghtly undeestmates the maxmum eflectance n the specula decton fo lage angles of ncdence (moe than 6 ). The slope value obtaned fom the data fttng s., about 5% lowe than the measued value of.4 (fo the small-scale oughness component). The slope calculated fom the aw heghts data s appoxmately.8, whch s vey close to the slope obtaned wth the scale sepaaton. By dong a senstvty study, and vayng the mechancally-measued slope (σ/λ=.4) by the obseved suface non-unfomty (5.3%), the esultng uppe and lowe bounds essentally backet the data n Fgue. The He-Toance model woks well fo ths acd-etched glass suface. The postons and the lateal shapes of the eflecton lobes ae accuate. The undeestmaton of the maxmum BRDF values s acceptable fo ou applcatons due to the lmted dynamc anges of human vson and dsplay devces. Ou suface chaactezaton method s less necessay n ths case. Although the measued slope s close to the fttng esult, the mpovement s not sgnfcant. The heght statstcs suggest that ths suface may have only a sngle scale of oughness. 6. Concluson In ths wok, we expementally study the elaton between the suface-heght statstcs and the angula dstbuton of the fst-suface lght scatteng fom seveal ough metal sufaces. The suface-heghts wee measued wth a suface poflomete; the BRDFs of the sufaces wee measued wth a gonoeflectomete. An analytcal model based on the Kchhoff appoxmaton, the He-Toance model, was ft to the measued BRDFs of the ough metal sufaces. The fttngs show that the model s able to descbe the fst-suface scatteng of ough metallc sufaces that have multple scales of

22 oughness and nea-gaussan suface-heght dstbutons, wth an accuacy suffcent fo compute gaphcs applcatons. The good ageement between the model and measuements mples that only one scale of oughness domnates the lght scatteng behavo. The lght scatteng dstbutons ae pmaly attbutable to the steepest-slope sufaceoughness components. 3 We popose a composte oughness model, and wth a flte, we extact the steepest-slope components fom the suface poflomete measuements. Usng the statstcs of that component (.e., the oughness slope σ/τ) as nput to the He-Toance model, satsfactoy ageement wth the measued BRDFs was obseved fo ou ough metal sufaces. The appoach povdes compute gaphcs pacttones wth a pactcal method to obtan the nput oughness data fo use wth a physcal-optcs-based lght eflecton model. Acknowledgements The authos would lke to thank Pofesso Donald Geenbeg, Steve Westn, and Huf Sheldon of the Conell Pogam of Compute Gaphcs fo the encouagement and techncal suppot. Equpment was donated by the Imagng Scence Dvson of Eastman Kodak (Lay Iwan) and the Hewlett-Packad Company. Ths eseach was suppoted by the NSF Infomaton Technology Reseach Pogam (ACI-385), the NSF Scence and Technology Cente fo Compute Gaphcs and Scentfc Vsualzaton (ASC-899), the NSF Conell Cente fo Mateals Reseach (DMR-7999) and the NSF Themal Systems Pogam (CTS-9383).

23 Refeences. D. P. Geenbeg, K. E. Toance, F. X. Sllon, J. Avo, J. A. Feweda, S. Patanak, E. P. F. Lafotune, B. Walte, S. C. Foo, and B. Tumboe, A famewok fo ealstc mage synthess, Comput. Gaph. Poc., Annual Confeence Sees (SIGGRAPH95), (995).. P. Beckmann and A. Spzzchno, The scatteng of electomagnetc waves fom ough sufaces, Pegamon Pess, Oxfod, UK, A. Stogyn, Electomagnetc scatteng fom ough, fntely conductng sufaces, Rado Sc. (4), 45-48, F. E. Ncodemus, Reflectance nomenclatue and dectonal eflectance and emssvty, Appl. Opt. 9, , B. T. Phong, Illumnaton fo compute geneated pctues, Communcatons of the ACM 8(6), 3-37, J. F. Blnn, Models of lght eflecton fo compute syntheszed pctues, Comput. Gaph. (SIGGRAPH77), 9-98, E. P. F. Lafotune, S. C. Foo, K. E. Toance, and D. P. Geenbeg, Non-lnea appoxmaton of eflectance functons, Comput. Gaph. Poc., Annual Confeence Sees (SIGGRAPH97), 7-6 (997). 8. R. L. Cook and K. E. Toance, A eflectance model fo compute gaphcs, ACM Tans. Gaph., 7-4, G. J. Wad, Measung and modelng ansotopc eflecton, Comput. Gaph. 6 (SIGGRAPH9), 65-7, 99. 3

24 . X. D. He, K. E. Toance, F. X. Sllon, and D. P. Geenbeg, A compehensve physcal model fo lght eflecton, Comput. Gaph. 5 (SIGGRAPH9), 75-86, 99.. B. G. Smth, Geometcal shadowng of a andom ough suface, IEEE Tans. Ant. Pop. AP-5, , E. I. Thosos, The valdty of Kchhoff appoxmaton fo ough suface scatteng usng a Gaussan oughness spectum, J. Acoust. Soc. Am. 83, 78-9, S. Maschne, S. Westn, E. Lafotune, and K. Toance, Image-based measuement of the bdectonal eflectance dstbuton functon, Appled Optcs 39, 6, 59-6,. 4. K. J. Dana, BRDF/BTF measuement devce, n Poceedngs of the Intenatonal Confeence on Compute Vson (IEEE Compute Socety Pess, Los Alamtos, Calf., ), ,. 5. W. Matusk, H. P. Pfste, M. Band, and L. McMllan, A data-dven eflectance model, ACM Tansactons on Gaphcs, v., 3, , G. R. Valenzuela, Scatteng of electomagnetc waves fom a tlted slghtly ough suface, Rado Sc. 3, 57-66, G. S. Bown, Backscatteng fom a Gaussan-dstbuted pefectly conductng ough suface, IEEE Tans. Ant. Pop. AP-6, 47-48, S. T. McDanel and A. D. Goman, An examnaton of the composte-oughness scatteng model, J. Acoust. Soc. Am. 73, , K. A. O Donnell and E.R. Mendez, Expemental study of scatteng fom chaactezed andom sufaces, J. Opt. Soc. Am. A 4, 94-5, P. F. Gay, A method of fomng optcal dffuses of smple known statstcal popetes, Opt. Acta 5, (978). 4

25 . M. J. Km, J. C. Danty, A. T. Fbeg, and A. J. Sant, Expemental study of enhanced backscatteng fom one- and two-dmentonal andom ough sufaces, J. Opt. Soc. Am. A. 7, , 99.. M. E. Knotts, and K. A. O Donnell, Measuement of lght scatteng by a sees of conductng sufaces wth one-dmensonal oughness, J. Opt. Soc. Am. A., 697-7, J. Q. Lu, J. A. Sánchez-Gl, E. R. Méndez, Z.-H. Gu, and A. A. Maadudn, Scatteng of lght fom a ough delectc flm on a eflectng substate: dffuse fnges. J. Opt. Soc. Am. A 5, 85-95, E. I. Chakna, A. G. Navaete, E. R. Méndez, A. Matínez, and A. A. Maadudn, Coheent scatteng by one-dmensonal andomly ough metallc sufaces, Appl. Opt. 37, -, H. E. Bennett, Specula eflectance of alumnzed gound glass and the heght dstbuton of suface egulattes, J. Opt. Soc. Am. 53, , K. E. Toance and E. M. Spaow, Theoy fo off-specula eflecton fom oughened sufaces, J. Opt. Soc. Am. 57, 5-4 (967). 7. D. H. Hensle, Lght scatteng fom fused polycystallne alumnum oxde sufaces, Appl. Opt., 5-58, E. Max and T. V. Vobuge, Dect and nvese poblems fo lght scatteed by ough sufaces, Appl. Opt. 9, , L. X. Cao, T. V. Vobuge, A. G. Lebeman, and T. R. Lette, Lght-scatteng measuement of the ms slopes of ough sufaces, Appl. Opt. 3, 3-37, 99. 5

26 3. M. E. McKnght, T. V. Vobuge, E. Max, M. E. Nadal, P. Y. Banes, and M. A. Galle, Measuements and pedctons of lght scatteng by clea coatngs, Appl. Opt. 4, 59-68,. 3. T. R. Thomas, Rough sufaces, Longman Pess, London UK; New Yok, USA, Y. Yang and R. O. Buckus, Suface length scale contbutons to the dectonal and hemsphecal emssvty and eflectvty, Jounal of Themophyscs and heat tansfe 9(4), , S. C. Foo, A gonoeflectomete fo measung the bdectonal eflectance of mateal fo use n llumnaton computaton, M.S. thess (Conell Unvesty, Ithaca, NY, 997). 34. H. L and K. E. Toance, Valdaton of the Gonoeflectomete, Techncal Repot PCG- 3- (Pogam of Compute Gaphcs, Conell Unvesty, Ithaca, NY, 3). 35. H. L, S. C. Foo, K. E. Toance, and S. E. Westn, Automated thee-axs gonoeflectomete fo compute gaphcs applcatons, submtted to SPIE P. Shley and K. Chu, Notes on adaptve quadatue on the hemsphee, Techncal Repot 44 (Depatment of Compute Scence, Indana Unvesty, Bloomngton, IN, 994). 37. E. D. Palk, Handbook of optcal constants of solds, Academc Pess, London, UK,

27 Appendx A: He-Toance BRDF Model The He-Toance BRDF model s based on the vecto fom of the Kchhoff appoxmaton, descbng the specula, dectonally dffuse, and unfomly dffuse scatteng behavos. The angula dstbutons of the scatteed lght depend on wavelength, ncdent angles, two oughness paametes, and ndces of efacton. The model apples fo a wde ange of mateals and suface fnshes. The coodnate system s shown n the followng fgue. Coodnate system Whee A s the pojected aea of the suface; kˆ s unt vecto n wave decton; sˆ, pˆ ae s and p polazaton unt vectos espectvely. Assumptons of the model: Local adus of cuvatue of the suface s lage compaed to the llumnatng wavelength; Both suface-heghts and autocoelaton functon ae Gaussan dstbuted; A sngle oughness length scale; Plane, unfom, and sotopc sufaces. 7

28 Equatons: ρ bd = ρbd ( λ, σ, τ, n ( λ), a( λ)) = ρbd, sp + ρbd, dd + ρbd, ud (6) The specula eflecton tem: ρ s ρbd, sp = ; (7) cosθ dω The dectonal dffuse tem: ρ bd, dd F G S D = ; (8) π cosθ cosθ The unfomly dffuse tem: ρbd, ud = a( λ) ; (9) g ρ = F e S () s, f n specula cone =, othewse () F = ( Fs + Fp ) = f ( θ, θ n( λ)) () λ v v G = ( ) π v z kˆ kˆ 4 [(ˆ s kˆ ) + ( pˆ ˆ k ) ] [(ˆ s kˆ ) + ( pˆ kˆ ) ] (3) S = S θ ) S ( θ ) (4) ( τ cotθ S ( θ ) = ( efc( )) /( Λ(cotθ ) + ) (5) σ τ cotθ S ( θ ) = ( efc( )) /( Λ(cotθ ) + ) (6) σ σ τ cotθ τ cotθ Λ (cotθ ) = { exp[ ( ) ] efc( )} (7) / π τ cotθ σ σ π τ D = 4λ m= m g e m! m g vxyτ exp( ) 4m (8) g πσ λ = [ (cosθ + cosθ )] (9) 8

29 / ] ) ( [ + = σ σ σ z () ) exp( ) ( 4 σ σ π z K K z + = () ) cot ( tan efc K θ σ τ θ =, ) cot ( tan efc K θ σ τ θ = () z v y v x v k k v z y x ˆ ˆ ˆ ) ˆ ˆ ( + + = = λ π, y x xy v v + = v (3) n k n k s ˆ ˆ ˆ ˆ ˆ =, (4) k s p ˆ ˆ ˆ = n k n k s ˆ ˆ ˆ ˆ ˆ =, (5) k s p ˆ ˆ ˆ = whee n s the ndex of efacton, ρ s s the specula eflectvty, s a delta functon, F s the Fesnel eflectvty fo unpolazed lght evaluated at the bsectng angle gven by ) / ( cos k k ) ), G s a geometcal facto, S s the shadowng/maskng facto, and D s a dstbuton functon fo the dectonal-dffuse eflecton tem. 9

30 Appendx B: The Statstcs of Hgh-fequency, Small-scale Roughness Component Fo each test suface, nne mechancal suface scans wee caed out n andomly chosen egons and n andom dectons. Fo each scan, we extacted the hgh-fequency, small-scale oughness components of the suface heght data wth a hgh-pass flte. The statstcs of the small-scale components ae summazed n Table 3~7. 3

31 Table : Summay of Measued Heght Statstcs Suface Raw heght data Small-scale oughness σ [µm] τ [µm] σ/τ σ [µm] τ [µm] σ/τ Nonunfomty BBQAL, Alumnzed Bead-blasted Steel % BBQAL, Alumnzed Bead-blasted % Steel GGAL, Alumnzed Gound % Glass, gt GGAL4, Alumnzed Gound % Glass, 4gt NGGAL, Alumnzed Acd-etched % Glass Table : Summay of Data Fttng Results Suface Slope σ/τ Unfom dffusve tem, a BBQAL, Alumnzed Bead-blasted Steel.44.7 BBQAL, Alumnzed Bead-blasted Steel.8.37 GGAL, Alumnzed Gound Glass, gt GGAL4, Alumnzed Gound Glass, 4 gt NGGAL, Alumnzed Acd-etched Glass.. 3

32 Table 3: Statstcs of small scale oughness component fo BBQAL, alumnzed beadblasted steel suface; flte sze was 3µm Scan # RMS Roughness σ [µm] Autocoelaton Length τ [µm] Slope σ/τ AVERAGE STDEV Non-unfomty.39 Table 4: Statstcs of small scale oughness component fo BBQAL, alumnzed beadblasted steel suface; flte sze was 6µm Scan # RMS Roughness σ [µm] Autocoelaton Length τ [µm] Slope σ/τ AVERAGE STDEV Non-unfomty.68 3

33 Table 5: Statstcs of small scale oughness component fo GGAL, alumnzed gound glass; flte sze was 34µm Scan # RMS Roughness σ [µm] Autocoelaton Length τ [µm] Slope σ/τ AVERAGE STDEV Non-unfomty.45 Table 6: Statstcs of small scale oughness component fo GGAL4, alumnzed gound glass; flte sze was 8µm Scan # RMS Roughness σ [µm] Autocoelaton Length τ [µm] Slope σ/τ AVERAGE STDEV Non-unfomty.4 33

34 Table 7: Statstcs of small scale oughness component fo NGGAL, alumnzed acd-etched glass; flte sze was 34µm Scan # RMS Roughness σ [µm] Autocoelaton Length τ [µm] Slope σ/τ AVERAGE STDEV.7.. Non-unfomty.53 34

35 Lst of Fgues: Fgue Sepaaton of a mechancally-measued suface-oughness heght scan nto hghfequency (small-scale) and low-fequency (lage-scale) components fo the alumnzed gound glass suface GGAL Fgue Measued ncdence-plane BRDFs fo alumnzed bead-blasted steel suface BBQAL (the smoothest steel suface) fo seveal ncdence angles θ ; λ=55nm Fgue 3 Measued BRDF ove the mapped eflecton hemsphee fo alumnzed bead-blasted steel suface BBQAL (the smoothest steel suface, slope σ/τ=.84) fo θ = and λ=55nm Fgue 4 Compason of dectly-measued (OL75) and BRDF-ntegated (gonoeflectomete) dectonal-hemsphecal eflectances fo alumnzed bead-blasted steel suface BBQAL (smoothest steel, slope σ/λ=.84), θ = Fgue 5 Nomalzed hstogam and autocoelaton functon of the small-scale oughness component fo alumnzed bead-blasted steel sufaces BBQAL (oughest, slope σ/τ=.9) and BBQAL (smoothest, slope σ/τ=.84). The cuves ae Gaussan dstbutons... 4 Fgue 6 Compason of measued BRDFs (symbols) and data fttng esults (cuves) fo alumnzed bead-blasted steel suface BBQAL (oughest, slope σ/τ=.9), n the plane of ncdence, fo seveal ncdence angles θ ; λ=55nm... 4 Fgue 7 Compason of measued BRDFs (symbols) and data fttng esults (cuves) fo alumnzed bead-blasted steel suface BBQAL (smoothest, slope σ/τ=.84), n the plane of ncdence, fo seveal ncdence angles θ ; λ=55nm... 4 Fgue 8 Compason of measued BRDFs (uppe ow) and data fttng esults (lowe ow) fo alumnzed bead-blasted steel suface BBQAL (smoothest, slope σ/τ=.84), ove the ente eflecton hemsphee, fo seveal ncdence angles θ ; λ=55nm

36 Fgue 9 Compason of measued and pedcted BRDFs fo alumnzed bead-blasted steel suface BBQAL (oughest, slope σ/τ=.9), n the plane of ncdence, fo seveal ncdence angles θ ; λ=55nm. Mechancally-measued small-scale oughness values wee used fo the pedctons; uppe/lowe bounds coespond to the obseved oughness vaaton ove the suface... 4 Fgue Compason of measued and pedcted BRDFs fo alumnzed bead-blasted steel suface BBQAL (smoothest, slope σ/τ=.84), n the plane of ncdence, fo seveal ncdence angles θ ; λ=55nm. Mechancally-measued small-scale oughness values wee used fo the pedctons; uppe/lowe bounds coespond to the obseved oughness vaaton ove the suface Fgue Nomalzed hstogam and autocoelaton functon of the small-scale oughness component fo alumnzed gound glass sufaces GGAL (oughest, slope σ/τ=.54) and GGAL4 (smoothest, slope σ/τ=.3) The cuves ae Gaussan dstbutons Fgue Nomalzed hstogam and autocoelaton functon of aw, mechancally-measued, heght data fo alumnzed gound glass GGAL (oughest, slope σ/τ=.54). The aw data was late sepaated nto small-scale (Fgue ) and lage-scale components. The sold lnes ae Gaussan dstbutons; the dashed lne s an exponental dstbuton Fgue 3 Compason of measued BRDFs (symbols) and data fttng esults (cuves) fo alumnzed gound glass GGAL (oughest, slope σ/τ=.54), n the plane of ncdence, fo seveal ncdence angles θ ; λ=55nm Fgue 4 Compason of measued BRDFs (symbols) and data fttng esults (cuves) fo alumnzed gound glass GGAL4 (smoothest, slope σ/τ=.3), n the plane of ncdence, fo seveal ncdence angles θ ; λ=55nm Fgue 5 Compason of measued BRDFs (uppe ow) and data fttng esults (lowe ow) fo alumnzed gound glass GGAL (oughest, slope σ/τ=.54), ove the ente eflecton hemsphee, fo seveal ncdence angles θ ; λ=55nm

37 Fgue 6 Compason of measued and pedcted BRDFs fo alumnzed gound glass GGAL (oughest, slope σ/τ=.54) n the plane of ncdence, fo seveal ncdence angles θ ; λ=55nm. Mechancally-measued small-scale oughness values wee used fo the pedctons; uppe/lowe bounds coespond to the obseved oughness vaaton ove the suface Fgue 7 Compason of measued and pedcted BRDFs fo alumnzed gound glass GGAL4 (smoothest, slope σ/τ=.3) n the plane of ncdence, fo seveal ncdence angles θ ; λ=55nm. Mechancally-measued small-scale oughness values wee used fo the pedctons; uppe/lowe bounds coespond to the obseved oughness vaaton ove the suface Fgue 8 nomalzed hstogam and autocoelaton functon of the small-scale oughness component fo alumnzed acd-etched glass NGGAL (slope σ/τ=.4). The cuves ae Gaussan dstbutons Fgue 9 Compason of measued BRDFs (symbols) and data fttng esults (cuves) fo alumnzed acd-etched glass NGGAL (slope σ/τ=.4), n the plane of ncdence, fo seveal ncdence angles θ ; λ=55nm Fgue Results fom Fgue 9 eplotted wth a logathmc odnate: compason of measued BRDFs (symbols) and data fttng esults (cuves) fo alumnzed acd-etched glass NGGAL (slope σ/τ=.4), n the plane of ncdence, fo seveal ncdence angles θ ; λ=55nm Fgue Compason of measued and pedcted BRDFs fo alumnzed acd-etched glass NGGAL (slope σ/τ=.4), n the plane of ncdence, fo seveal ncdence angles θ ; λ=55nm. Mechancally-measued small-scale oughness values wee used fo the pedctons; uppe/lowe bounds coespond to the obseved oughness vaaton ove the suface

38 heght functon (mcon) 6 4 Raw data x (mcon) Small scale oughness Lage scale oughness x (mcon) Fgue Sepaaton of a mechancally-measued suface-oughness heght scan nto hghfequency (small-scale) and low-fequency (lage-scale) components fo the alumnzed gound glass suface GGAL 5.5 Absolute BRDF (s - ) Angle of eflecton Fgue Measued ncdence-plane BRDFs fo alumnzed bead-blasted steel suface BBQAL (the smoothest steel suface) fo seveal ncdence angles θ ; λ=55nm 38

39 .4. BRDF (s - ) Fgue 3 Measued BRDF ove the mapped eflecton hemsphee fo alumnzed bead-blasted steel suface BBQAL (the smoothest steel suface, slope σ/τ=.84) fo θ = and λ=55nm Dectonal Hemsphecal Reflectance (%) ) OL75 Gonoeflectomete Wavelength (nm) Fgue 4 Compason of dectly-measued (OL75) and BRDF-ntegated (gonoeflectomete) dectonal-hemsphecal eflectances fo alumnzed bead-blasted steel suface BBQAL (smoothest steel, slope σ/λ=.84), θ = 39

40 BBQAL BBQAL Nomalzed Hstogam - - z(um) - - z(um) ACF x(um) x(um) Fgue 5 Nomalzed hstogam and autocoelaton functon of the small-scale oughness component fo alumnzed bead-blasted steel sufaces BBQAL (oughest, slope σ/τ=.9) and BBQAL (smoothest, slope σ/τ=.84). The cuves ae Gaussan dstbutons θ =6 Absolute BRDF (s - ) θ = θ = θ (Degees) Fgue 6 Compason of measued BRDFs (symbols) and data fttng esults (cuves) fo alumnzed bead-blasted steel suface BBQAL (oughest, slope σ/τ=.9), n the plane of ncdence, fo seveal ncdence angles θ ; λ=55nm 4

41 Absolute BRDF (s - ) θ =6 θ =3 θ = θ (Degees) Fgue 7 Compason of measued BRDFs (symbols) and data fttng esults (cuves) fo alumnzed bead-blasted steel suface BBQAL (smoothest, slope σ/τ=.84), n the plane of ncdence, fo seveal ncdence angles θ ; λ=55nm θ = θ =4 θ = Measuement Data Data Fttng to He-Toance Model Fgue 8 Compason of measued BRDFs (uppe ow) and data fttng esults (lowe ow) fo alumnzed bead-blasted steel suface BBQAL (smoothest, slope σ/τ=.84), ove the ente eflecton hemsphee, fo seveal ncdence angles θ ; λ=55nm 4

42 4 Absolute BRDF (s - ) Uppe Bound Lowe Bound Measuement dat θ = θ =3 θ = θ (Degees) Fgue 9 Compason of measued and pedcted BRDFs fo alumnzed bead-blasted steel suface BBQAL (oughest, slope σ/τ=.9), n the plane of ncdence, fo seveal ncdence angles θ ; λ=55nm. Mechancally-measued small-scale oughness values wee used fo the pedctons; uppe/lowe bounds coespond to the obseved oughness vaaton ove the suface. 4