Generalization of Lambert s Reflectance Model

Transcription

1 Genealzaton of Lambet s Reflectance Model Mchael Oen and Shee K. Naya Depatment of Compute Scence, Columba Unvesty Ne Yok, NY 007 Abstact Lambet s model fo body eflecton s dely used n compute gaphcs. It s used extensvely by endeng technques such as adosty and ay tacng. Fo seveal eal-old objects, hoeve, Lambet s model can pove to be a vey naccuate appoxmaton to the body eflectance. Whle the bghtness of a Lambetan suface s ndependent of veng decton, that of a ough suface nceases as the veng decton appoaches the lght souce decton. In ths pape, a compehensve model s developed that pedcts body eflectance fom ough sufaces. The suface s modeled as a collecton of Lambetan facets. It s shon that such a suface s nheently non-lambetan due to the foeshotenng of the suface facets. Futhe, the model accounts fo complex geometc and adometc phenomena such as maskng, shadong, and nteeflectons beteen facets. Seveal expements have been conducted on samples of ough dffuse sufaces, such as, plaste, sand, clay, and cloth. All these sufaces demonstate sgnfcant devaton fom Lambetan behavo. The eflectance measuements obtaned ae n stong ageement th the eflectance pedcted by the model. CR Descptos: I.3.7 [Compute Gaphcs]: Thee- Dmensonal Gaphcs and Realsm; I.3.3 [Compute Gaphcs]: Pctue/Image Geneaton; J. [Physcal Scences and Engneeng]: Physcs. Addtonal Key Wods: eflecton models, Lambet s model, BRDF, ough sufaces, moon eflectance. Intoducton An actve aea of eseach n compute gaphcs nvolves the ceaton of ealstc mages. Images ae endeed usng one of to ell-knon technques, namely, ay tacng [36] o adosty [7]. The qualty of a endeed mage depends to a geat extent on the accuacy of the eflectance model used. In the past decade, compute gaphcs has tnessed the applcaton of seveal physcallybased eflectance models fo mage endeng (see [8], [7], [0], [4]). Reflecton fom a suface can be boadly classfed nto to categoes: suface eflectance hch takes place at the nteface beteen to meda th dffeent efactve ndces and body eflectance hch s due to subsuface scatteng. Most of the pevous ok on physcally-based endeng has focused on accuate modelng of suface eflectance. They pedct deal specula eflecton fom smooth sufaces as ell as de dectonal lobes fom oughe sufaces [4]. In contast, the body component has most often been assumed to be Lambetan. A Lambetan suface appeas equally bght fom all dectons. Ths model as advanced by Lambet [0] moe than 00 yeas ago and emans one of the most dely used models n compute gaphcs. Fo seveal eal-old objects, hoeve, the Lambetan model can pove to be a poo and nadequate appoxmaton to body eflecton. Fgue (a) shos a eal mage of a clay vase obtaned usng a CCD camea. The vase s llumnated by a sngle dstant lght souce n the same decton as the senso. Fgue (b) shos a endeed mage of a vase th the same shape as the one shon n Fgue (a). Ths mage s endeed usng Lambet s model, and the same llumnaton decton as n the case of the eal vase. As (a) Fgue : (a) Real mage of a cylndcal clay vase. (b) Image of the vase endeed usng thelambetaneflectancemodel. In both cases,llumnaton s fom the veng decton. expected, Lambet s model pedcts that the bghtness of the cylndcal vase ll decease as e appoach the occludng boundaes on both sdes. Hoeve, the eal vase s vey flat n appeaance th mage bghtness emanng almost constant ove the ente suface. The vase s clealy not Lambetan. Ths devaton fom Lambetan behavo can be sgnfcant fo a vaety of eal-old mateals, such as, concete, sand, and cloth. An accuate model that descbes body eflecton fom such commonplace sufaces s mpeatve fo ealstc mage endeng. What makes the vase shon n Fgue (a) non-lambetan? We sho that the pmay cause fo ths devaton s the oughness of the suface. Fgue llustates the elatonshp beteen magnfcaton and eflectance (also see [7]). The eflectng suface may be veed as a collecton of plana facets. At hgh magnfcaton, each pctue element (endeed pxel) ncludes a sngle facet. At loe magnfcaton, each pxel can nclude a lage numbe of facets. Though the Lambetan assumpton s often easonable hen look- (b) Note that the eal vase does not have any sgnfcant specula component,n hch case, a vetcal hghlght ould have appeaed n the mddle of the vase.

2 ng at a sngle plana facet, the eflectance s not Lambetan hen a collecton of facets s maged onto a sngle pxel. Ths devaton s sgnfcant fo vey ough sufaces, and nceases th the angle of ncdence. In ths pape, e develop a compehensve model that pedcts body eflectance fom ough sufaces, and povde expemental esults that suppot the model. Lambet s model s an nstance, o lmt, of the poposed model. pxel Fgue : The oughness of a suface causes ts eflectance popetes to vay th mage magnfcaton. The topc of ough sufaces has been extensvely studed n the aeas of appled physcs, geophyscs and engneeng. The follong s a bef summay of pevous esults on the subject. In 94, Opk [5] desgned an empcal model to descbe the non- Lambetan behavo of the moon. In 94, Mnnaet [] modfed Opk s model to obtan the follong eflectance functon: pxel f = k (cos cos )(k;) (0 k ) hee, and ae the pola angles of ncdence and eflecton, and k s a measue of suface oughness. Ths functon as desgned to obey Helmholtz s ecpocty pncple [] but s not based on any theoetcal foundaton. It assumes that the adance s symmetcal th espect to the suface nomal. It ll be shon n ths pape that ths assumpton s ncoect. Hapke and van Hon [3] also obtaned eflectance measuements fom ough sufaces by vayng the souce decton fo a fxed senso decton. They found the peak of the adance functon to be shfted fom the peak poston expected fo a Lambetan suface. They ntepeted ths as a mno dscepancy and concluded the Lambetan model to be a easonable appoxmaton. Ou on measuements demonstate that ths non-lambetan behavo s clealy notceable and sgnfcant hen vee decton s vaed athe than souce decton. The studes cted above ee attempts to desgn eflectance models based on measued eflectance data. In contast, Smth [30] and Buhl et al. [4] attempted to develop theoetcal models fo eflecton fom ough sufaces. These effots ee motvated pmaly by eflectance chaactestcs of the moon. Vsble and nfaed emssons fom the moon ee ecoded by a numbe of eseaches (fo examples, see [6] and [9]). These measuements ndcate that the moon s suface eflects moe lght back n the decton of the souce (the sun) than n the nomal decton (lke Lambetan sufaces) o n the foad decton (lke specula sufaces). Ths phenomenon s efeed to as backscatteng. Smth modeled the oughness of the moon as a andom pocess and assumed each pont on the suface to be Lambetan n eflectance. Smth s analyss, hoeve, as confned to the plane of ncdence and s not easly extensble to eflectons outsde ths plane. Moeove, Smth s model does not account fo nteeflecton effects. A dffeent backscatteng mechansm, called etoeflecton o opposton effect, poduces a shap peak close to the souce decton (see [3, 9, 3, 4, 8, ]). Ths s not the mechansm dscussed n ths pape. Buhl et al. [4] modeled the suface as a collecton of sphecal cavtes. They analyzed nteeflectons usng ths suface model, but dd not pesent a complete model that accounts fo maskng and shadong effects fo abtay angles of eflecton and ncdence. Subsequently, Heng and Smth [5] deved a detaled themal emsson model fo sufaces modeled as a collecton of V-cavtes. Hoeve, all cavtes ae assumed to be dentcal and algned n the same decton, namely, pependcula to the souce-vee plane. Futhe, the model s lmted to the plane of ncdence. Moe ecently, body eflecton has emeged as a topc of nteest n the gaphcs communty. Pouln and Foune [7] deved a eflectance functon fo ansotopc sufaces modeled as a collecton of paallel cylndcal sectons. Addessng a dffeent cause fo non-lambetan eflectance fom the one dscussed hee, Hanahan and Kuege [] used lnea tanspot theoy to analyze subsuface scatteng fom a mult-layeed suface. Othe eseaches n gaphcs have numecally pe-computed faly complex eflectance functons and stoed the esults n the fom of look-up tables o coeffcents of sphecal hamonc expanson (fo examples, see [5] [7][35]). Ths appoach, though pactcal n many nstances, does not eplace the need fo accuate analytcal eflectance models. The eflectance model developed hee can be appled to sotopc as ell as ansotopc ough sufaces, and can handle abtay souce and vee dectons. Futhe, t takes nto account complex geometcal effects such as maskng, shadong, and nteeflectons beteen ponts on the suface. We begn by modelng the suface as a collecton of long symmetc V-cavtes. Each V- cavty has to opposng facets and each facet s assumed to be much lage than the avelength of ncdent lght. Ths suface model as used by Toance and Spao [3] to descbe ncoheent dectonal component of suface eflecton fom ough sufaces. Hee, e assume the facets to be Lambetan 3. Fst, e develop a eflectance model fo ansotopc sufaces th one type (facetslope) of V-cavtes, th all cavtes algned n the same decton on the suface plane. Next, ths esult s used to develop a model fo the moe geneal case of sotopc sufaces that have nomal facet dstbutons th zeo mean and abtay standad devaton. The standad devaton paametezes the macoscopc oughness of the suface. The fundamental esult of ou ok s that the body eflectance fom ough sufaces s not unfom but nceases as the vee moves toad the souce decton. Ths devaton fom Lambet s la s not pedcted by any pevous eflectance model. We pesent seveal expemental esults that demonstate the accuacy of ou model. The expements ee conducted on eal samples such as sand, plaste and cloth. In all cases, eflectance pedcted by the model as found to be n stong ageement th measuements. The deved model has been mplemented as a shadng functon n RendeMan [33]. We conclude by compang eal and endeed mages of a vaety of objects. These esults demonstate to ponts that ae fundamental to compute gaphcs: (a) Seveal eal-old objects have body eflecton components that ae sgnfcantly non-lambetan. (b) The model pesented n ths pape can be used to ceate ealstc mages of a vaety of eal-old objects. Radometc Defntons In ths secton, e defne adometc concepts that ae used n the emande of ths pape. These concepts ae dscussed n detal n [3]. Fgue 3 shos a suface element da llumnated fom the decton ŝ =( ) and veed by a senso (mage pxel) n the decton ˆv =( ). We use to denote pola angles and to 3 Ths assumpton does not lmt the mplcatons of the eflectance model pesented hee. The non-lambetanbehavoepotedhee s expectedfo a de ange of local body eflectance models (see [6], fo example) snce suface oughness s shon to play a domnant ole.

3 s^ -φ x^ da z^ φ dω Fgue 3: Geomety used to defne adometc tems. denote azmuth angles. The senso subtends an nfntesmal sold angle d! fom any pont on the suface. The lght enegy eflected by the suface patch s popotonal to the lght ncdent on the patch. Iadance s defned as the lght flux ncdent pe unt aea of the suface: E( ) = dφ ( ) da Ths s the dectonal adance of the suface as t epesents lght enegy ncdent fom the decton ( ). The total adance of the suface s the flux ncdent fom all dectons and may be denoted smply as E. The bghtness measued by the senso s popotonal to the adance of the suface patch n the decton ( ). Suface adance s defned as: y^ v^ () L ( ; ) = d Φ ( ; ) da cos d! () It s the flux adated by the suface pe unt sold angle, pe unt foeshotened aea. It depends on the decton of llumnaton and the senso decton. The elatonshp beteen adance and adance of a suface s detemned by ts eflectance popetes. The b-dectonal eflectance dstbuton functon (BRDF) s defned as the ato of adance to adance: f ( ; ) = dl( ; ) de( ) (3) All the above defntons ae geneal, n that, they ae vald fo sufaces th any eflectance chaactestcs. Fo an sotopc suface, adance and BRDF do not change f the suface s otated about ts nomal vecto. Fo such sufaces, the BRDF s smply: f ( ; ) = dl( ; ) de( ) (4) A specal type of eflectance that s dely used fo mage endeng s Lambetan eflectance. A Lambetan suface s an deal dffuse hose adance s ndependent of the veng decton of the senso; t appeas equally bght fom all dectons. Its BRDF s f = hee s the albedo of the suface and epesents the facton of ncdent enegy that s eflected by the suface. 3 Suface Roughness Model Thee ae seveal ays of modelng suface oughness. The geneal appoach s to select a model that s capable of epesentng eal sufaces and at the same tme easy to use dung the mathematcal development of the eflectance model. All suface models found n appled physcs and geophyscs lteatue can be dvded nto to boad categoes. In the fst case, the suface s modeled as a andom pocess (see [, 34, 30]). Usng ths appoach, t s dffcult to deve a eflectance model fo abtay souce and vee dectons as ell as to analyze nteeflectons. In the second categoy, sufaces ae assumed to be composed of seveal elements th some pmtve shape, fo example, sphecal cavtes, V-cavtes, holes, etc (see [4, 3]). As shon n ths pape, the effects of shadong, maskng, and nteeflectons need to be modeled to obtan an accuate eflectance model. To acheve ths, e use the oughness model poposed by Toance and Spao [3] that assumes the suface to be composed of long symmetc V-cavtes (see Fgue 4) th the uppe edges n the same plane. Each cavty conssts of to plana facets. The dth of each facet s assumed to be small compaed to ts length. The oughness of the suface s specfed usng a pobablty functon fo the dstbuton of facet slopes. a^ a da da ^n Fgue 4: Suface modeled as a collecton of V-cavtes. The V-cavty oughness model can be used to descbe sufaces th both sotopc as ell as ansotopc (dectonal) oughness. We assume each facet aea da s small compaed to the aea da of the suface patch that s maged by a sngle senso pxel. Hence, each pxel ncludes a vey lage numbe of facets. Futhe, the facet aea s lage compaed to the avelength of ncdent lght and theefoe geometcal optcs can be used to deve the eflectance model. The above assumptons can be summazed as: da da (5) The facets could be elatvely small as n the case of sand and plaste, o lage as n the case of outdoo scenes of tean. Slope-Aea Pobablty Dstbuton: We denote the slope and oentaton of each facet n the V-cavty model as ( a a). Toance and Spao have assumed all facets to have equal aea da. They use the dstbuton N ( a a) 4 to epesent the numbe of facets pe unt suface aea that have the nomal â =( a a). Hee, e use a pobablty dstbuton to epesent the facton of the suface aea that s occuped by facets th a gven nomal. Ths s efeed to as the slope-aea dstbuton P ( a a). The facet-numbe dstbuton and the slope-aea dstbuton ae elated as follos: P ( a a)==n (a a) da cos a (6) The slope-aea dstbuton s ease to use than the facet-numbe dstbuton n the follong model devaton. Fo sotopc sufaces, N ( a a) = N (a) and P (a a) =P (a), snce the dstbutons ae otatonally symmetc th espect to the global suface nomal ˆn (Fgue 4). 4 In [3], N(a a) s denoted by p() hee a = and a = 0.

4 4 Reflectance Model In ths secton, e deve a eflectance model fo body eflectance fom ough sufaces. The V-cavty model s used to descbe suface geomety and each facet on the suface s assumed to be Lambetan n eflectance. The follong thee types of sufaces th dffeent slope-aea dstbutons ae examned. (a) Un-dectonal Sngle- Slope Dstbuton: Ths dstbuton esults n a non-sotopc suface hee all facets have the same slope and all cavtes ae algned n the same decton. (b) Isotopc Sngle-Slope Dstbuton: Hee, all facets have the same slope but they ae unfomly dstbuted n oentaton on the suface plane. (c) Gaussan Dstbuton: Ths s the most geneal case examned hee the slopeaea dstbuton s assumed to be nomal th zeo mean. The oughness of the suface s detemned by the standad devaton of the nomal dstbuton. The eflectance model obtaned fo each of the above suface types s used to deve the succeedng one. Effect of Roughness on Body Reflectance: Befoe e poceed to deve eflectance models fo the abovementoned suface types, a bef llustaton of the effect of oughness on body eflecton ould be useful. Consde, fo the pupose of dscusson, the sngle V-cavty shon n Fgue 5. Both facets of the cavty ae fully llumnated by a dstant souce on the ght sde. If the facets ae Lambetan th equal albedo, the left facet appeas bghte than the ght one as t eceves moe ncdent lght. If the V-cavty s veed fom the left sde by a dstant obseve, a lage facton of the foeshotened cavty aea s dak and a smalle facton s bght. As the obseve moves to the ght, toads the souce decton, the facton of bghte aea nceases hle that of the dake aea deceases. Consequently, the total bghtness, o adance, of the cavty nceases as the obseve appoaches the souce decton. Note that ths esults fom the bghtness dspaty beteen the to facets hch nceases th the angle of ncdence. Ths effect s n contast to Lambetan sufaces hose bghtness does not vay th the veng decton. The above llustaton demonstates that ough dffuse sufaces ae nheently non-lambetan n eflectance. The adance nceases as the vee appoaches the souce decton. No e pesent a fomal teatment of the above effects. Vee decton Bghte Souce decton Dake Vee decton Bghte Dake Bghte Dake Fgue 5: The adance of the V-cavty nceases as the vee moves toads the llumnaton decton. The Pojected Radance: Consde suface aea da that s maged by a sngle senso element n the decton ˆv =( ) and llumnated by a dstant pont lght souce n the decton ŝ =( ). The aea da s composed of a vey lage numbe of symmetc V-cavtes. Each V-cavty s composed of to facets th the same slope but facng n opposte dectons. Ou appoach s to compute the adance contbuton of each facet on the suface. Then, the total adance of the suface patch can be detemned as an aggegate of the contbutons of all facets. Consde the flux eflected by a facet th aea da and nomal â =( a a). The pojected aea on the suface occuped by the facet s da cos a (see Fgue 4). Hence, hle computng the contbuton of the facet to the adance of the suface patch, e need to use the pojected aea da cos a and not the actual facet aea da. The adance contbuton thus detemned s hat e call the pojected adance of the facet: L p(a a)= d Φ (a a) (da cos a) cos d! (7) Fo ease of descpton, e have dopped the souce and veng dectons fom the notatons fo pojected adance and flux. Total Radance: No consde the slope-aea dstbuton of facets gven by P ( a a). The total adance of the suface can be obtaned as the aveage of L p(a a) of all facets on the suface: L ( ; )= (8) Z a=0 Z P ( a a) Lp(a a) sn a d a d a a=0 Thus, e have decomposed the poblem of computng the adance of any ough suface to one of computng the pojected adance fo each facet on the suface. The total adance of the suface s then obtaned by ntegatng the poduct of the pojected adance and the slope-aea dstbuton functon ove all facet nomals. 4. Model fo Un-dectonal Sngle-Slope Dstbuton The fst suface type e consde has all facets th the same slope a. Futhe, all V-cavtes ae algned n the same decton; azmuth angles of all facets ae ethe a o a. The esults obtaned fo ths ansotopc suface ll be used late n the analyss of sotopc sufaces. Radance fom a Lambetan Facet: Consde a Lambetan facet that s fully llumnated (no shadong) and s completely vsble (no maskng) fom the senso decton. The adance of the facet s popotonal to ts adance and s equal to E(a a). The adance of the facet s E( a a) = E0<ŝ â>, hee, E 0 s the adance hen the facet s llumnated head-on (.e. ŝ =â), and < >denotes the dot poduct beteen to vectos. Usng the defnton of adance, the flux eflected by the facet n the senso decton s obtaned as: d Φ = E0 <ŝ â><ˆv â>dad! (9) Substtutng the above eflected flux n (7), e get: L p(a a) = E0 <ŝ â>< ˆv â> <â ˆn><ˆv ˆn> (0) Ths expesson clealy ndcates that the pojected adance of a tlted Lambetan facet s not equal n all veng dectons. Consequently, a ough suface compsed of tlted Lambetan facets s non-lambetan; ts adance vaes th the veng decton. Ths phenomenons obsevedeven n the absenceof maskng, shadong, and nteeflecton effects.

5 4.. Geometc Attenuaton Facto If the suface s llumnated and veed fom the nomal decton (ŝ = ˆv = ˆn), all facets ae fully llumnated and vsble. Fo lage angles of ncdence and eflecton, hoeve, facets ae shadoed and masked by adjacent facets (see Fgue 6). In the case of shadong, a facet s only patally llumnated as the adjacent facet on the V-cavty casts a shado on t. In the case of maskng, the facet s only patally vsble to the senso as ts adjacent facet occludes t. Both these geometcal phenomena affect the pojected adance of the facet and hence must be taken nto account. The esult s a geometcal attenuaton facto (GAF) that les beteen zeo and unty (also see [3][3]). It s the educton n the pojected adance of a facet due to maskng and shadong effects; t equals the ato of the facet aea that s both vsble and llumnated, to the total facet aea. The detals of the devaton of the GAFae gven n appendx A The fnal esult can be compactly epesented as: GAF = Mn Max 0 <ŝ ˆn><â ˆn> <ŝ â> < ˆv ˆn><â ˆn> < ˆv â> The above GAF s vald fo any facet nomal, â, not necessaly the bsecto of the angle beteen the souce and the senso decton. () nteeflectons and gnoe subsequent bounces. Smulatons of the nteeflecton pocess ee used to vefy that ths appoxmaton s good. In the follong dscusson, e efe to suface adance due to dect llumnaton by the souce as L and adance due to nteeflectons as L. We ll use the same supescpts fo pojected adance. The to-bounce nteeflecton component fo a Lambetan facet can be expessed as [9][8][9] []: L ZZ (~x) = K(~x ~y)l (~y)d~y () hee ~x s a pont on the facet hose nteeflecton component s detemned as an ntegal of the adance of all ponts ~y on the adjacent facet. K(~x ~y) s the kenel and epesents the geometcal elatonshp beteen ~x and ~y. Snce the V-cavty s long compaed to ts dth, t can be veed as a one-dmensonal shape th tanslatonal symmety. Fo such shapes, the nteeflecton component can be detemned as an ntegal ove the one-dmensonal cosssecton of the shape. The above nteeflecton equaton s theefoe educed to: L Z (x) = K 0 (x y)l (y)dy (3) ^s n ^z v ^ n ^z hee x and y ae the shotest dstances of ponts ~x and ~y fom the ntesecton of the to facets (see Fgue 7). K 0 s the kenel fo the tanslatonal symmety case and s deved n [6] and [9] to be: K 0 (x y) = sn ( a) xy (x xy cos ( a)y ) 3= (4) a m s x^ a m v ^x We kno that the nomal of the consdeed facet s â =( a a) (a) (b) Fgue 6: (a) Shadong and (b) maskng n a V-cavty. s^ ^ z v^ Pojected Radance and GAF: The pojected adance of a Lambetan facet s obtaned by smply multplyng the geometc attenuaton facto th the pojected adance (0) deved unde the assumpton of no maskng and shadong. Table detals the GAF and the coespondng pojected adance fo all cases of shadong and maskng. Note that the pojected adance s denoted as L p; the supescpt s used to ndcate that the adance s due to dect llumnaton by the souce. In the next secton, e ll use L p to denote adance due to nteeflectons. 4.. Inteeflecton Facto In ou eflectance model, e also account fo nteeflectons; lght ays bouncng beteen adjacent facets. These effects ae sgnfcant fo ough sufaces th elatvely hgh albedo values. When the suface s llumnated fom lage angles ( ) and veed fom the opposte sde at lage angles ( ), none of the facets that ae vsble to the senso ae llumnated by the souce. If nteeflectons ae not consdeed, the adance of the suface ould be zeo n ths case. Hoeve, the vsble facets eceve lght fom the adjacent facets that face the souce and hence ae llumnated. These nteeflectons esult n non-zeo suface adance. Ou analyss and expemental esults suggest that the contbuton due to nteeflectons can be sgnfcant and cannot n geneal be gnoed. We have the task of modelng nteeflectons n the pesence of maskng and shadong effects. In the case of Lambetan sufaces, the enegy n an ncdent lght ay dmnshes apdly th each nteeflecton bounce. Theefoe, e model only to-bounce x y Fgue 7: Inteeflectons n a V-cavty. and the nomal of the adjacent facet s â 0 = ( a a ). The lmts of the ntegal n the nteeflecton equaton ae detemned by the maskng and shadong of these facets. As befoe, let m v be the dth of the facet hch s vsble to the vee. Let m s be the dth of the adjacent facet that s llumnated. As n Secton 4.., expessons can be obtaned fo the vsble and llumnated sectons: h h m v = Max 0 Mn ; <â0 ˆv> (5) <â ˆv> m s h h = Max <â ŝ> 0 Mn ; <â 0 (6) ŝ> Fom the defnton of pojected adance (7) and expesson (3) e have: L p = l<â ˆv> da <â ˆn>< ˆv ˆn> Z ( ) E 0 l<â 0 ŝ><â ˆv> da <â ˆn>< ˆv ˆn> x=mv Z x=mv ^ x L (x) dx = (7) Z y=m s K 0 (x y)dy dx

6 GAF L p( a a) No Maskng <ŝ â>< ˆv â> E0 <â ˆn>< ˆv ˆn> = o E0 cos cos a tan tan a cos ( ; a) Shadong tan tan a cos ( ; a) Maskng Shadong < ˆv ˆn><â ˆn> < ˆv â> <ŝ ˆn><â ˆn> <ŝ â> E0 <ŝ â> = E0 cos cos a tan tan a cos ( ; a) <ŝ ˆn>< ˆv â> E0 = < ˆv ˆn> E0 cos cos a tan tan a cos ( ; a) Table : Pojected adance of a facet fo dffeent maskng/shadong condtons. Usng the follong change of vaables: t = x ; = y, the adance due to to-bounce nteeflectons gven by (7) can be tten as: (8) L Z <â 0 Z ŝ><â ˆv> p =( ) E 0 K 0 (t )d dt <â ˆn>< ˆv ˆn> t= mv = m s Usng (4), the above ntegal s evaluated as: Z t= mv hee: Z = m s K 0 ( t)d dt = (9) d( mv ms ms )d( ) ; d( mv ) ; d( ) d(x y) =p x xy cos ( a)y (0) We efe to the ght hand sde of equaton (9) as the nteeflecton facto (IF). The total pojected adance of the facet s the sum of the pojected adance due to souce llumnaton (gven n Table ) and the above nteeflecton component: L p(a a) = L p ( a a) L p ( a a) () The un-dectonal sngle-slope suface e have consdeed n ths secton has only to types of facets th nomals ( a a) and ( a a ). Hence, the adance of the suface fo any gven souce decton and senso decton s smply the aveage of the pojected adances of the to facet types: L p(a a)lp(a a ) L ( ; ; a a)= () 4. Model fo Isotopc Sngle-Slope Dstbuton We no consde a suface th V-cavtes that have facets th the same slope ( a), but unfomly dstbuted n oentaton (a) n the plane of the suface. The esult s a suface th sotopc oughness. The eflectance model deved fo ths suface s based on the esults obtaned n the pevous secton fo the sngle-slope suface. The esults obtaned n ths secton ae mpotant as they can be used to deve a eflectance model fo any sotopc suface. Fom the pevous secton, e kno the adance L p( a a) of a facet th nomal â = ( a a). Theefoe, the adance of the sngle-slope sotopc suface due to dect souce llumnaton s detemned as an ntegal of the pojected adance ove a: L p( a) = Z L p( a a)da (3) a=0 Gven souce decton ( ) and senso decton ( ), e fst need to fnd the anges of facet oentaton a fo hch the facets ae masked, shadoed, masked and shadoed, and nethe masked no shadoed 5. The adance fo each ange s gven n Table. The poblem then s to decompose the above ntegal nto dffeent pats, each coespondng to a dffeent maskng/shadong ange. We efe the nteested eade to Appendx B. fo detals on the evaluaton of ntegal (3). The fnal expesson fo suface adance s found to be: L p( a)= E0 cos cos a " (4) cos ( ; ) A (; a) tan A( ; ; a) ( ;jcos ( ; )j)a3( ; a) # hee, = Max[ ] and = Mn[ ]. The expessons fo the coeffcents A, A, and A 3 ae gven n Appendx B.. Note that the above pojected adance s the same as the total adance of the suface (L ( ; ; a)=lp ( a)) snce all facets on the suface have dentcal slope, a. The devaton n Appendx B. does not consde multple eflectons, as the nteeflecton component (9) s dffcult to ntegate ove all cavty oentatons a.in Appendx B., an appoxmaton to the nteeflecton component (L ( ; ; a)=lp ( a)) s gven. Once agan, t s mpotant to note that the adance of the ough suface consdeed hee s not constant th espect to the veng decton ( ); t s non-lambetan. We ll study ths behavo moe closely n the follong secton. 4.3 Model fo Gaussan Slope-Aea Dstbuton The suface consdeedabove conssts of V-cavtes th equal facet slope. Realstc sufaces can be modeled only f the slope-aea dstbuton P ( a a) ncludes a vaety of dffeent facet slopes. If the suface oughness s sotopc, the slope-aea dstbuton can be descbed usng a sngle paamete namely a snce the facets ae unfomly dstbuted n a. The adance of any sotopc suface can theefoe be detemned as: Z L ( ; )= P ( a)lp(a) sn a d a (5) 0 hee the souce llumnaton (no nteeflecton) component of L p(a) s gven by (4). We assume the sotopc dstbuton to be Gaussan th mean and standad devaton,.e. P ( a; ). 5 Imagne a V-cavty otated about the global suface nomal fo any gven souce and senso decton. Vaous maskng/shadongscenaos can be vsualzed.

7 Reasonably ough sufaces can be descbed usng a zeo mean ( = 0) Gaussan dstbuton: hee the nomalzaton constant c s: =c = P ( a) = c e ; a (6) Z a=0 Z a=0 e ; a sn a d a d a The eflectance model s to be obtaned by substtutng the adance L p( a) gven by (4) and the Gaussan dstbuton P ( a; 0) n ntegal (5). The esultng ntegal cannot be easly evaluated. Theefoe, e pusued a functonal appoxmaton to the ntegal that s accuate fo abtay suface oughnessand angles of ncdence and eflecton. In devng ths appoxmaton, e caefully studed the functonal fom of L p( a) gven by (4). Ths enabled us to dentfy bass functons that can be used n the appoxmaton. Then, e conducted a lage set of numecal evaluatons of the ntegal n (5) by vayng suface oughness, the angles of ncdence ( ) and eflecton ( ). These smulatons and the dentfed bass functons ee used to ave at an accuate functonal appoxmaton fo suface adance. Ths pocedue as appled ndependently to the dect llumnaton component as ell as the nteeflecton component. The fnal appoxmaton esults ae gven belo. Once agan, let = Max[ ] and = Mn[ ]. The dect llumnaton component of adance of a suface th oughness s: If the suface s extemely ough, causng the zeo-mean Gaussan model to be an naccuate appoxmaton, an addtonal paamete can be used to eght the nteeflecton component. Ou smulatons sho that ths enables the model to stetch a bt beyond ts theoetcal lmts. Fnally, the BRDF of the suface s obtaned fom ts adance and adance as f ( ; ; ) = L ( ; ; ) =E 0 cos. It s mpotant to note that the abovemodel obeyshelmholtz s ecpocty pncple (see []). Also note that the model educes to the Lambetan model 6 hen = 0. Note that by substtutng the albedo as functon of the avelength, (), the dependency of the model on the avelength comes out explctly. In the next secton, e pesent seveal expemental esults that vefy the deved eflectance model. Hee, e gve a bef llustaton of the man chaactestcs of the model. Fgue 8 shos the eflectance pedcted by the model fo a vey ough suface th = 30 and = 0:9. The adance L n the plane of ncdence ( = ) s plotted as a functon of the eflecton angle fo ncdence angle = 75. To cuves ae shon n the fgue, both obtaned by numecal evaluaton of the ntegal n (5). Shotly, e shall examne the accuacy of the functonal appoxmaton. The fst cuve (sold lne) ncludes both dect llumnaton and nteeflecton components of adance, hle the second (thn lne) s only the dect llumnaton component. Notce that these 5 L stong maskng L ( ; ; ) = E 0 cos "C () (7) 5 C C tan cos ( ; )C (; ; ; ; ) tan ;jcos ( ; )j C 3 (; ; ) tan hee the coeffcents ae: C = ; 0:5 0:33 C = 8 < : C 3 = 0:5 # 0:45 0:09 sn f cos ( ; ) 0 0:45 sn ; ( 0:09 )3 othese 0:09 4 Usng a smla appoach, an appoxmaton to the nteeflecton component as also deved. In ths case, the nteeflecton component fo the sngle-slope sotopc suface (Appendx B.) as used to guess the bass functons. The fnal appoxmaton to the nteeflecton component of adance fo a suface th oughness s: L ( ; ; ) = (8) " 0:7 E0 cos # ; cos ( ; ) 0:3 The to components ae combned to obtan the total suface adance: L ( ; ; ) = (9) L ( ; ; ) L ( ; ; ) Lambetan stong nteeflecton = foad backad Fgue 8: Dffuse eflectance n the plane of ncdence fo a suface th = 30, = 0:90, and ncdence angle = 75. The thn lne s adance due to dect llumnaton (thout nteeflectons). adance plots devate substantally fom Lambetan eflectance. Suface adance nceases as the veng decton appoachesthe souce decton. The cuves can be dvded nto thee sectons. In the backad (souce) decton, the adance s maxmum and gets cut-off due to stong maskng effects hen exceeds. Ths cut-off occus exactly at = and s ndependent of oughness. In the mddle secton of the plot, adance vaes appoxmately as a scaled tan functon th constant offset. Fnally, nteeflectons domnate n the foad decton hee most facets ae self-shadoed and the vsble facets eceve lght pmaly fom adjacent facets. Ths s llustated by the dffeence beteen the to cuves. Fgue 9 shos the effect of vayng suface oughness. When = 0, the model pedcts exactly Lambetan eflectance. The devaton fom Lambetan behavo nceases damatcally th oughness. In Fgue 0, the effect of vayng the ncdence angle s shon. Hee e have chosen to plot BRDF athe than adance to bette llustate the effect of vayng. It s nteestng 6 When = 0, C =, C = 0, and C 3 = 0, yeldng L = E0 cos.

8 L σ = 40 σ = 0 σ = 0 σ = 0 = L φ φ φ φ =0 φ =60 φ = Fgue 9: Effect of oughness on suface adance ( = 0:9). = 75 and Fgue : Radance outsde the plane of ncdence. = 40 and = 0: f =75 =60 =45 =0 Fgue 0: BRDF fo dffeent angles of ncdence. = 40 and = 0:9. to note that the model pedcts nea-lambetan behavo fo vey small ncdence angles ( 0). Ths esults fom both facets of a V-cavty havng nealy equal adance fo small angles of ncdence. As the ncdence angle nceases, the backscatte phenomenon begns to domnate. Fgue shos the effect of placng the senso outsde the plane of ncdence. When the senso-nomal plane s pependcula to the souce-nomal plane, the ough suface agan exhbts nea-lambetan chaactestcs. Fgue shos compasons beteen adance values computed by numecal evaluaton of (5) (thck lne) and the functonal appoxmaton (thn lne) gven by (7) and (8). Once agan, adance s measued n the plane of ncdence ( = ). In all cases, the functonal appoxmaton poves to be vey accuate. 4.4 Qualtatve Model In ths secton, e popose a futhe smplfcaton to the eflectance model pesented n the pevous secton. In ode to obtan ths smplfcaton, a slght sacfce n accuacy must be made. In etun, some computatons can be saved dung mage endeng. The follong smplfed model as aved at by studyng, though numeous smulatons, the elatve sgnfcance of vaous tems n the functonal appoxmaton gven by (7). The smulatons shoed that coeffcent C 3 makes a elatvely small contbuton to the total adance. A smple model s thus obtaned by dscadng C 3 and gnong nteeflectons: L ( ; ; ) = (30) E0 cos (A BMax 0 cos ( ; ) sn tan ) A = :0 ; 0:5 0:33 B = 0:45 0:09 The to coeffcents A and B ae obtaned dectly fom C and C, espectvely. Note that the qualtatve model also educes to the Lambetan model hen = 0. In Fgue 3, e have compaed the qualtatve model th the numecal evaluaton of the model 7. Ths model can be of sgnfcant pactcal value n applcatons hee hgh accuacy s not ctcal. 5 Expemental Vefcaton We have conducted seveal expements to vefy the accuacy of the eflectance model. The expemental set-up used to measue the adance of samples s shon n Fgue 4. In the case of outdoo scenes, each senso element (pxel) typcally ncludes a lage suface aea (seveal nches n dmensons and often moe). Commecally avalable eflectance measuement devces ae applcable only to small samples. Consequently, e developed ou on measuement devce. Each sample s appoxmately nches. It s maged usng a 5480 pxel CCD camea that s mounted at the end of a 6 foot long beam. The othe end of the beam s attached to a otay stage to facltate pecse vaaton of the veng angle. The sample s llumnated usng a 300 Watt ncandescent lght souce. The sold angles subtended by the senso and souce fom the sample ae appoxmately d! = 0:003 steadans and d! = 0:0009 steadans, espectvely. The llumnaton decton ( ) s adjusted manually. Images of the sample ae dgtzed and the adances computed as the aveage bghtness ove all pxels thn an mage ndo that les on the sample. The mage ndo sze s vaed as a functon of senso decton to ensue that the same aea on the sample s alays used. Fgue 5 shos esults obtaned fo a sample of all plaste. The sample has matte local eflectance popetes but s vey ough; t s exactly the type of suface that ou eflectance model chaactezes. Reflectance s epesented by the nomalzed adance L ( )=L(0) hee L(0) s the adance measued hen the sample s veed fom the nomal decton. The nomalzed 7 Dscepances caused by the lack of the nteeflecton componentn the qualtatve model can be patally compensated by eplacng the constant 0:33 n coeffcent A th 0:57.

9 L L (a) = 0 (b) = L L (c) = (d) = 85 Fgue : Compason beteen numecal evaluaton of the model (thck lne) and functonal appoxmaton (thn lne) fo a suface th = 30 and = 0:90. adance s also equal to the nomalzed BRDF f ()=f(0). The adance of each sample s plotted as a functon of the senso decton fo dffeent angles of ncdence. These measuementsae made n the plane of ncdence ( = = 0). The dots epesent measued adance values hle the sold lnes ae pedctons obtaned usng the eflectance model fo Gaussan suface oughness. In these ntal expements, as empcally selected to obtan the best match beteen measued and pedcted eflectance. Hee, e have used the numecal evaluaton of the model (equaton 5). Ths as done to demonstate not only the accuacy of the model but also the valdty of all assumptons made hle developng the model. Smla esults ae pesented n Fgues 6 and 7 fo sample B (panted sand pape) and sample C (hte sand). Fo all thee samples, adance nceases as the veng decton appoaches the souce decton (backad eflecton). Ths s n contast to the behavo of ough specula sufaces that eflect moe n the foad decton, o Lambetan sufaces hee adance does not vay th veng decton. Fo all thee samples, the model pedctons and expemental measuements match emakably ell. In all cases, a small peak s notced nea the souce decton. Ths phenomenon as dscussed eale n the pape and s dffeent fom the one descbed by ou model; t s the backscatte peak studed by seveal eseaches [3][4] [][9][3] and dscussed n the context of gaphcs endeng by [8]. Some of the dscepances beteen the model and measued data n the foad decton can be attbuted to the long V-cavty assumpton. In the case of sample C (sand), e see a small specula component n the foad decton. Ths s due to the specula chaactestcs of ndvdual sand patcles. In Fgue 8, e have shon measuements obtaned outsde the plane of ncdence ( 6= 0) fo sample C. These measuements ae a ctcal measue of the accuacy of any eflectance model but ae seldom found n eflectance lteatue. Once agan, the model and measued data ae n stong ageement. Fgues 9 though sho esults fo samples that have not only a body eflectance component but also a sgnfcant suface eflecton component. These samples ae ncluded to sho that sufaces th a suface eflecton component can also exhbt the backscatteng phenomenon that the ne model descbes. In these expements, the eflectance model used s a lnea combnaton of ne model and the Toance-Spao model [3] that descbes the ncoheent dectonal component of suface of eflecton. We selected ths model as t s based on exactly the same suface oughness assumptons (symmetc, long, V-cavtes) as the pesent model. The adance pedcted by the model s: L s F (n)gaf = e ; a cos cos (3) a hee, F s the Fesnel eflecton coeffcent, and n s the efactve ndex of the suface medum. The cos a n the denomnato esults fom usng the slope-aea dstbuton nstead of the facet-numbe dstbuton (see Secton 3). Ths model pedcts a peak n the foad decton (close to the specula decton) and the dstbuton of the eflected adance gets de th ncease n suface oughness. The total adance s expessed as a lnea combnaton of the body and suface components: L = kb L b k s L s (3) hee the body eflectance component L b s pedcted by the model poposed n ths pape. In these expements, e used the functonal

10 L 5 sample (a) ; = 0 camea (a) lght souce Fgue 4: (a) Sketch and (b) photogaph of the set-up used to measue eflectance. (b) L (b) ; = 45 Fgue 3: Compason beteen numecal evaluaton (thck lne) and the qualtatve model (thn lne): (a) n the plane of ncdence ( ; = 0 ), and (b) outsde the plane of ncdence. In both cases, = 30, = 0:90, and = 75. appoxmaton (7) nstead of the numecal evaluaton of ntegal (5). Moeove, the eflectance paametes,, k b, and k s ee estmated by fttng (usng non-lnea optmzaton) the above model to measued data. Snce t s dffcult to obtan meanngful estmates of n fo the synthetc samples e have used, the effect of the Fesnel coeffcent as gnoed by assumng F =. Note that fo all thee samples, the poposed model does vey ell n descbng the ncease n adance as the vee appoaches the souce, as ell as the cut-off n adance that takes place n the souce decton. Ths despte the fact that the thee samples have oughness chaactestcs that dffe fom the V-cavty model. Fo the foam sample, the suface eflectance component (adance ncease n the foad decton) s descbed ell by the Toance-Spao model. The Toance-Spao model does only easonably ell fo the cloth toel and not vey ell fo the ood-shavng sample. The man eason s the follong: hle the Gaussan oughness model appeas explctly n the Toance-Spao model, t s ntegated ove all facet oentatons n ou case. As a esult, ou model s less senstve to the actual suface oughness dstbuton than the Toance-Spao model. 6 Implcatons fo Gaphcs In ths secton, e descbe the mplcatons of the poposed model fo ealstc endeng. Fgue (a) shos a eal mage of the ough cylndcal clay vase dscussed n the ntoducton. Fgue (b) shos a endeed mage of the vase usng the Lambetan model and ts knon geomety. Clealy, ths endeed mage does not match the eal mage of the vase. On the othe hand, the appeaance of the endeed vase usng the poposed eflectance model, shon n Fgue (c), closely esembles the eal vase. The model paametes = 0:7 and = 40 ee chosen empcally to obtan the best ft to the measued bghtness values. Fgue 3(a) compaes bghtness values along the coss-secton of the thee dffeent vase mages n Fgue. It s nteestng to note that the bghtness of the eal vase emans nealy constant ove most of the coss-secton and dops quckly to zeo vey close to the lmbs. The poposed model does vey ell n pedctng ths behavo, hle the Lambetan model poduces lage bghtness eos. Fgue 3(b) shos smla plots fo llumnaton fom 0 to the ght of the senso. In ths case, bghtness vaaton on the eal vase s asymmetc. Once agan, the poposed model closely matches the eal mage. Hoeve, the Lambetan model foces the bghtness close to the ght lmb of the vase to dop much faste than n the eal mage. As a esult, the bghtness peak pedcted by the Lambetan model s sgnfcantly aay fom the actual peak. The functonal appoxmaton, gven by equaton (7), and the qualtatve model, gven by (30), ae easly used fo ealstc endeng. We have mplemented the functonal appoxmaton as a shade usng the RendeMan shadng language [33]. The endeed fgues ae povded as addtonal tff fles. The fst mage (n the fle sphees.tf ) shos sphees endeed usng the shade. In all fou cases, the sphee s llumnated fom the vee decton. In the fst case (the leftmost sphee), = 0, and hence the sphee appeas Lambetan. The oughness paametes of the othe sphees to ght ae: = 0, = 0 and = 40, n that ode. As the oughness nceases, the sphee begns to appea flatte. In the exteme oughness case, as n the ghtmost sphee, the sphee appeas lke a flat dsc th nea constant bghtness. Ths phenomenon has been dely obseved and epoted n the case of the full moon ([6],[9]). Fnally, the fou mages named scene (n the fles named: scene..tf,:::, scene.4.tf ) sho endeed mages of a scene th thee matte objects, a vase, cylndcal block and a cube. In the fst mage, (fle scene..tf ), all thee objects have zeo macoscopc oughness,.e. they ae Lambetan. Illumnaton n ths case s fom the vee decton. Note that the vase and the cylnde have stong bghtness vaatons, and the thee vsble faces of the cube have dstnctly dffeent bghtness values. In the second mage (fle scene..tf ), the scene s agan llumnated fom the vee decton, but the thee objects have oughness = 30. Consequently, the shadng ove the vase and the cylnde s dmnshed consdeably. Futhemoe, the contast beteen the flat and cuved sectons of the cylndcal block and also the contast beteen the thee faces of the cube ae educed substantally. It s mpotant to note that the modeate shadng s acheved thout any ambent component n the llumnaton, but athe fom modelng of oughness effects. In the thd and the fouth mages (fles scene.3.tf and scene.4.tf ), the thee objects have the same

11 Wall Plaste L = 30 = 45 = 60 = 60 φ = L 5 Fgue 5: Reflectance measuement and eflectance model (usng = 30, = 0:90) plots fo all plaste (sample A). Radance s plotted as a functon of senso decton ( ) fo dffeent angles of ncdence ( = ). Sand Pape L = 45 = 60 = 75 = 75 φ = 60 4 (a) L 0.04 Fgue 6: Reflectance measuement and eflectance model (usng = 40, = 0:80) plots fo panted sand-pape (sample B). L Sand = 45 = 60 = (b) Fgue 8: Reflectance measuement and eflectance model plots fo sample C. These measuements ee obtaned fo senso dectons outsde the plane of ncdence: (a) = 60 and = 45 ; and (b) = 75 and = 60. Appendx A Devaton of the Geometc Attenuaton Facto Fgue 7: Reflectance measuement and eflectance model (usng = 35, = 0:80) plots fo hte sand (sample C). oughness values as n the fst and second mages espectvely, but the scene s llumnated fom 0 degees to the ght of the veng decton. Agan, shadng dffeences ae seen n the to mages, though less than hen the scene s llumnated fom the veng decton. 7 Summay In concluson, e have developed a compehensve model fo body eflectance fom sufaces th macoscopc oughness. A model as fst deved fo ansotopc sufaces that have facets th only one slope. Ths esult as used to develop a model fo sotopc sufaces th Gaussan slope-aea dstbuton. We have also pesented a qualtatve model fo dffuse eflecton that has a smple functonal fom. Numeous expements ee conducted to vefy the eflectance mechansm descbed n ths pape. Real and endeed mages of dffuse objects ee compaed to demonstate that the poposed model has mpotant mplcatons fo compute gaphcs. In ths appendx e pesent the detals of the devaton of the geometc attenuaton facto fo abtay souce, vee and facet nomal dectons. GAF fo Pependcula V-Cavtes: We fst estct ouselves to V-cavtes that ae oented pependcula to the senso-souce plane. Late, the analyss s extended to abtay senso and souce dectons. Fgue 6 llustates the maskng and shadong phenomena fo the case of pependcula V-cavtes. Ou objectve s to detemne, fo a gven souce decton ŝ and senso decton ˆv the facton of facet aea that s llumnated and vsble. If the vsble aea s smalle than the llumnated aea, maskng domnates. Lkese, f the llumnated aea s smalle than the vsble aea, shadong domnates. We denote the length (extent on the suface plane) and dth of the facet by l and, espectvely. Futhe, m s and m v ae sectons of the facet that ae shadoed and masked, espectvely. The aea of a facet that s both llumnated and vsble s l Mn[ ; m s ; m v]. The GAF s obtaned by dvdng ths expesson by the aea lof the facet: h GAF = Mn ; ms ; mv (33) We ould lke to expess the GAF n tems of the angles of ncdence (souce) and eflecton (senso). Fom the tangle (; m s; n)

12 L Foam 0. 5 = 45 = 60 = 75 Fgue 9: Reflectance measuement and eflectance model ( = 0, = 0:8, k s=k b = 0:09) plots fo foam (sample D). Cloth 0. 5 L = 45 = 60 = 75 (a) Image (b) Lambetan (c) Model Fgue : Real mage of a cylndcal clay vase compaed th mages endeed usng the Lambetan and poposed models. Illumnaton s fom the camea decton. Lambetan Model Bghtness Measuements X Fgue 0: Reflectance measuement and eflectance model ( = 4, = 0:75, k s=k b = 0:085) plots fo a cotton toel (sample E). L Wood Shavng 0. 5 = 45 = 60 (a) = 0 Bghtness = 75 (b) = 0 X Fgue : Reflectance measuement and eflectance model (usng = 6, = 0:7, k s=k b = 0:043) plots fo fne ood shavngs (sample F). n Fgue 6, e have: n sn = ms cos a cos a (34) n cos = ;ms sn a sn a By multplyng the fst expesson by cos and the second by ; sn and addng the esults e get: m s o ; ms = ; cos (a ) cos ( a ; ) (35) = cos a cos cos ( a ; ) In the above expesson, the angles and ae postve n the counte-clockse decton and negatve n the clockse decton. It can be easly shon that thee s no shadong hen j a j and j a; j ms,.e. ;. On the othe hand, the ente facet s shadoed f j ; a j ms,.e. ; 0. A smla esult s obtaned fo maskng. All these condtons ae ncluded n the follong GAF expesson fo pependcula V-cavtes: cos cos a cos cos a GAF = Mn Max 0 (36) cos ( ; a) cos ( ; a) Fgue 3: Compason beteen mage bghtness along the coss-secton of the eal vase, and vases endeed usng the Lambetan and poposed models. GAF fo the Geneal Case: In the geneal case,souce and senso dectons ae abtay and can le outsde the plane pependcula to the V-cavty. To make the maskng/shadong calculatons tactable, e nvoke the assumpton that the length of facets s lage compaed to the dth 8,.e. l. Then, the analyss of maskng and shadong s educed to the pependcula V-cavty case by pojectng the souce decton ŝ and senso decton ˆv onto the plane pependcula to the cavty. These pojectons ae done usng basc tgonomety as shon n Fgue 4. The pojected angles ae then substtuted nto (36), n place of and, to obtan the geneal GAF expesson: GAF = Mn Max 0 (37) cos cos a cos cos a sn sn a cos ( ; a) 8 When facet length s much lage than facet dth, the exact shape of the cast shado at the to ends of the facet can be gnoed.

13 z^ ^ x p φ ^ y sn p = sn cos p cos sn cos cos p = cos p cos sn cos tan p = tan cos Fgue 4: Relatonshp beteen pojected and actual angles. cos cos a cos cos a sn sn a cos ( ; a) Altenatvely, the GAF can be expessed n tems of the souce, senso, facet nomal, and suface nomal vectos: <ŝ ˆn><â ˆn> < ˆv ˆn><â ˆn> GAF = Mn Max 0 (38) <ŝ â> < ˆv â> B Radance of Isotopc Suface th Sngle-Slope Dstbuton In ths appendx, e outlne devatons fo the dect llumnaton and nteeflecton components of pojected adance fo the sotopc suface dscussed n Secton 4.. These esults ae used n Secton 4.3 to deve the eflectance model fo a suface th Gaussan slope-aea dstbuton. B. Radance due to Dect Illumnaton Ou objectve hee s to evaluate the ntegal n (3). Fo any gven souce decton ( ) and senso decton ( ), facets on the sotopc suface could be masked, shadoed, masked and shadoed, o nethe masked no shadoed. The adance fo each of these cases s gven n Table. The poblem theefoe s to decompose the ntegal n (3) nto pats, each coespondng to a dffeent maskng/shadong ange. Usng basc geomety, e have dentfed the lmts of the ntegals coespondng to dffeent anges of shadong/maskng. These lmts ae epesented by the ctcal angles c (fo shadong) and c (fo maskng). The ctcal angle c s elated to the slope a of suface facets: cos ; ( c = tan a tan ) f (tan a tan ) > (39) 0 othese The angle c s detemned usng the same expesson by eplacng th. These ctcal angles ae elated to the maskng/shadong anges as shon n Table. Usng the above ctcal angle expessons, Table, and Table, e decompose (3) nto the sum of seveal ntegals. Each ntegal can be evaluated fo any fxed vee decton. Hoeve, fo abtay dectons seveal cases ase and the esults ae not easy to use n pactce. Theefoe, e have chosen to expess the adance of the suface fo any abtay veng decton ( ) as a eghted sum of the adance L p k n the plane of ncdence ( = ), and the adance L p? n the pependcula plane ( = ). Radance n the Plane of Incdence: Thee ae to cases to consde. In the fst, =. Wthout loss of genealty, e can assume = = 0. When, adance s obtaned as: L p k ( a)= E0 cos cos a (40) Z c ( tan a tan cos a)da ; c Z ; c ( tan a tan cos a) c ( tan a tan cos a)da = E0 cos cos sn a c tan a tan tan a tan tan ( ; c sn ( c) ) When, the senso and souce dectons ae smply stched n the expesson nsde the squae backets. In the second case, =. Agan, thout loss of genealty, e can assume = 0 =. When,e get: L p k ( a)= E0 cos cos a (4) Z c ( tan a tan cos ( ; a))da c Z ; c ( tan a tan cos a) c ( tan a tan cos ( ; a))da = E0 cos cos a ; c sn c ; sn c tan a tan ; tan a tan tan ( ; c sn ( c) )