Advances. Photometric Stereo

Transcription

1 Advances n Photometrc tereo uggero Pntus Tutor Prof. assmo Vanz

2 Contents ) ntroducton g.3 ) mage formaton and reflectance ma g.5 3) Changng from two to four lghts g. 4) Basc theory gradent extracton g. 5) ays correcton g.3 6) Gradent ntegraton g.4 7) Alcatons g.55 8) Aendx A g.7 9) eferences g.77

3 . ntroducton Photometrc tereo s a fundamental art of the entre rocess concernng comuter vson []. Ths ssue s very mortant n ndustral alcatons artfcal ntellgence researches and so on. t s based on an mage rocessng rocedure that analyses the scene and tres to recover ts three-dmensonal surface. There are a lot of technues to 3D ma a sngle obect or an entre scene and each of them has dfferent features regardng the aroach to the roblem. The most used s laser scannng technues snce t s able to 3D ma searately and most of all ndeendently each sngle ont of a samle. nstead almost all other methods need to analyse a xel cluster or even all xels n N mages to recover surface heghts. bvously ths reresents a drawbac snce the heght value of one ont (whose data s suosed noseless) wll be based by the nose n the adacent xels. For ths reason all of these methods are carefully used and n most cases are merged wth other more accurate methods []. However deendng on the nature of samles and mage acuston systems sometmes we are forced to mae use of only one technue. For nstance n secton 7 we show the 3D ma of obect observed wth scannng electron mcroscoe (E); we beleve that t s really dffcult to use a laser scannng nsde the E don t you? n ths sense we have focused our efforts n solvng some trouble concernng both the management of nondealtes (wth resect to the mathematcal model) and the numercal algorthm to enhance ts erformances n terms of accuracy and comutatonal tme. Before showng our wor we want to lst some reference about revous wors n Photometrc tereo and generally n hae from X (e.g. shae from shadng) algorthms. Photometrc tereo was conceved by Woodham [3] that taes the theory of shae from shadng whch reconstructs the heghts usng only one mage and extends t to a number N mages. Ths technue uses a reflectance ma as a mathematcal model or a loou table whch decreases the comutatonal tme n some smle alcatons. The loou tables are bult wth a calbraton shere of nown shae and materal behavour. Deendng on used algorthm the shere should be of the same materal of obects to be 3D maed. n the other hand the algorthms that reconstruct also albedo arameters don t need ths constrant. Note that f there s no need to fnd albedo t s better to acure only gray-scale mages to mnmze used memory whle colour mages are obvously necessary f we need chromatc nformaton. Generally the vast maorty of methods use gray-scale mages. For Lambertan surface as we ll see n next sectons hotometrc euatons are lnear and only three mages are enough to reconstruct both gradent and albedo. Wth more than three mages some aers use the surlus nformaton to fnd other obect art as outlers [4] or to recover unnown llumnaton drectons and strengths [5]. Whle some methods start from a gven reflectance ma and then fnd the heghts other technues focus ther attenton to estmate reflectance arameters. 3

4 ome nstances are Nayar [6] who uses the so-called hybrd reflectance model and Tagare and de Fgueredo [7] who develoed a hotometrc stereo for the class of m-lobe reflectance mas (they called t so). As we ll also see n ths wor many aers deal wth the ossblty to consder a surface as Lambertan even f t has non-dealtes as shadows or hghlghts. The trc s to consder a mathematcal model that defne hghlghts and shadows as devatons from Lambertan law; n ths way the algorthm can manage them and reconstruct surfaces roerly. ne of these technues was roosed by Coleman and Jan [8]. n the other hand colour mages contan a lot of redundant nformaton regards the gradent extracton. f one could have ths data he should use ths nformaton whether to mae the algorthm more robust or to comute more than heghts. Chrstensen and haro [9] ntroduced the method of colour hotometrc stereo (CP) for surfaces wth an arbtrary reflectance ma. A drawbac of ths technue s that t needs a calbraton ste whle (other aers already do that) we ll demonstrate n the followng sectons that t s not necessary. ther aers (as []) searate the Lambertan sgnal to the others wth an mage rocessng tool whch ermts to avod any non-dealty bas. Before startng to face our ssue we lst here the sectons of ths wor. ecton descrbes the bascs n hotometrc stereo hyscal model and fnds the most used reflectance ma euaton for Lambertan surfaces. ecton shows the frst stes we had done studyng P and esecally the changng from usng two to four lght sources and lays the bases of next sectons. Then ecton 3 llustrates the theory of gradent and albedo extracton. ecton 4 deals wth an algorthm we have develoed to avod llumnaton bases. n ecton 5 we exlan our gradent ntegraton algorthm and comare t wth exstng ones. We also study ts erformances n terms of comutatonal tme and robustness. ecton 6 s made by all alcatons we are contended wth and we ll show there our algorthm otental. The last two sectons are the Aendx A n whch we demonstrate the mathematcal detals n choosng the best lghtng condtons used to lght the analytcal P samle (the vase) and references. 4

5 . mage formaton and reflectance ma n ths secton we deal wth the mage formaton rocess and we fnd a mathematcal model that relates obect and lght geometry wth xels ntensty. Frst we ntroduce some notons as radometry rradance radance mage brghtness and bdrectonal reflectance dstrbuton functon. The latter s the most mortant functon that together wth nown lght sources oston leads to reflectance ma formulaton. Followng theory sums u the bascs of hotometrc stereo as dscussed n []. To understand how the xels ntensty n acured mages s determned we should study some radometry concets. Frst the rradance s the amount of lght fallng on a surface and t s the ower er unt area ncdent on t ( W m ). The radance nstead s the amount of lght radated from a surface and t s measured as the ower radated from the surface er unt area er unt sold angle ( W m ). emember that the sold angle s defned as: sr Acosϑ Ω (.) where A s surface area ϑ s the angle between surface normal and a lne connectng the atch to the orgn and s the length of ths lne. tartng from these defntons we now dscuss the mage formaton rocess. Let s consder Fg.. where there are one obect a lens and an mage lane. Fg.. mage formaton system 5

6 As easy to see f the rays assng through the lens are not deflected the sold angle concernng the obect s eual to the sold angle regardng the mage lane. These sold angles are: δ cos f cos δ cosϑ ( z ) cos (.) and f they are eual t holds δ δ cos cosϑ f z (.3) Now we should nvestgate how much of the lght emtted from the surface atch go through the lens. The sold angle subtended by the lens as seen from surface atch s π d cos π d 3 Ω cos 4 ( z ) 4 z cos (.4) where d s dameter lens. The ower of the lght emtted from the surface atch and fallng on mage lane s π d 3 δp LΩδ cosϑ Lδ cos cosϑ (.5) 4 z where L s the radance of the surface n the drecton toward lens. The rradance of the mage lane atch wll be δp δ π d 3 E L cos cosϑ (.6) δ δ 4 z 6

7 Usng (.3) we fnd ths relatonsh π d 4 E L cos (.7) 4 f Thus mage rradance s roortonal to scene radance L and to the angle alha between the ray connectng the center of the lens and the atch and the otcal axs. Ths s the frst mortant result n the rocedure that leads to reflectance ma. We saw how mage rradance deends on scene radance but what does radance deend on? t deends on the lght that falls on the obect and the fracton of ths lght that s reflected due to many reasons as obect geometry or obect reflecton roertes. The euaton that descrbe what we have ust sad s called bdrectonal reflectance dstrbuton functon (BDF) f ( ϑ ϕ ; ϑ ϕ ) ( ϑe ϕe ) ( ϑ ϕ ) δl e e (.8) δe where ( ϑ e ϕ e ) and ( ϕ ) ϑ are the vewng and lghtng drecton (Fg..). Fg.. BDF geometry 7

8 ne mortant roerty of ths functon conssts n Helmholtz recrocty condton f ( ϑ ϕ ; ϑ ϕ ) f ( ϑ ϕ ; ϑ ϕ ) (.9) e e e e Generally we don t have a sngle secfc drecton whch the lght comes from but we have a sy wth dfferent radance that deends on the sy regon. Consder such a sy as a hemshere and a art of t of sze ( δϑ δϕ ). uosed that a atch s n the hemshere center and then t subtends a sold angle (.) δω snϑ δϑ δϕ The rradance receved from ths art of the sy wll be E ( ϑ ϕ ) sn ϑδϑ δϕ (.) o the total rradance s π π π ( ϑ ϕ ) E E sn ϑ cosϑδϑδϕ (.) where cos accounts for the foreshortenng of surface seen from the drecton ( ) ϑ o the radance of the surface wll be ϑ. ϕ π π e e e e sn ϑ cos π ( ϑ ϕ ) f ( ϑ ϕ ; ϑ ϕ ) E( ϑ ϕ ) L ϑ δϑ δϕ (.3) The same aroach has to be consdered f we want to comute the radance of a Lambertan surface. n an deal Lambertan surface one atch aears eually brght from all vewng drectons and doesn t absorb any radaton ower. o the BDF functon must be constant. To fnd that value we have to use the revous ntegrals settng that the total rradance s eual to radance ntegral over all drectons so 8

9 π π π f ( ϑ ϕ ; ϑe ϕe ) E cosϑ sn ϑe cosϑeδϑeδϕ e E cos ϑ (.4) whch becomes π πf sn ϑ cosϑ δϑ (.5) e e e Ths leads to π f (.6) o L E E cosϑ π π (.7) We have found the relaton that correlates the drecton and ntensty of llumnaton wth the surface radance whch the xels brghtness deends on. Now we have only to wrte ths relaton n terms of surface dervatves along x and y. We can easly demonstrate that a normal vector could be wrtten as r n ( ) T (.8) n the same way and for usng smlar notaton we could defne the llumnaton vector as 9

10 r l ( ) T (.9) Now usng these euatons cosϑ (.) nce ths euaton s the unue term that really nfluence the xels brghtness whle the other terms are only scale factors usually we defne a normalzed reflectance ma ( ) ( ) (.) As we shall see n next secton ths nd of reflectance ma doesn t care about the surface albedo. Ths surface roerty should be added as an mortant scale factor that deends on the nature of each sngle area atch corresondng wth one mage xel. Before movng on wth the next secton Photometrc tereo bascs are now descrbed. Consder now the ( ) sace and an acured mage of a surface wth unform albedo. Let us consder also a sngle xel wth value E. Then we have ( ) E (.) Ths euaton s a curve n the ( ) lane. We can t comute the ( ) value matchng E snce there are nfnte values of ths gradent ar. We need much more nformaton. We need more mages wth dfferent lght condton (dfferent l r vector)! ther mages reresent other curves and the ntersecton ont of these curves s the gradent of the analysed ont. Fg..3 shows reflectance ma curves for some E values settng and. 5. Ths s the startng ont for the develoment of all P algorthms.

11 Fg..3 eflectance ma curves for ten values of E

12 3. Changng from two to four lghts When we started to aly P algorthm to real surfaces we used a artcular form of the reflectance ma seen n revous chaters to comute the gradent feld and we laced only two lghts one on the rght and the other on the left of the camera z z tan cos β senβ ( )cos x y ρ x y (3.) z z x y where s the acured obect mage whch deends on the scene geometry and on obect reflectance roertes s the ntensty of the lght ( x y) 3.). ρ s the albedo and ( β ) defnes the oston of each lght (Fg. Fg. 3. ystem geometry Wth ths model we used to comute the dervatves n such an easy way: n the case of dervatve along x we consder β and β π and easly fnd

13 3 tan tan )cos ( tan )cos ( x z y z x z x z y x y z x z x z y x ρ ρ (3.) and n the same way choosng π β and π β tan y z (3.3) However ths leads to a bg roblem snce wth two lghts could obtan only a art of the gradent feld that s absolutely necessary for comutng the deth ma. Wth one dervatve could only now the heght ma of one row along x a tme (or column along y) but can t establsh a connecton wth the heght nformaton of adacent rows (or column). For ths reason the frst 3D reconstructons are very oor and there s a great roblem wth nose effects on them. The frst soluton s to use the same formulas but four lghts the frst two ones for the x dervatve and the latter ones for the y dervatve. t s obvous that ths aroach ncreases enormously the 3D reconstructon algorthm ossbltes. Let s see an examle of ths frst smle aroach at the hotometrc stereo ssue. n every art of ths thess we ll use always one artcular P samle whch ermts us to comare dfferent methods whether a gradent extracton or an ntegraton algorthm s erformed. Fg.3.3 shows ths obect to be 3D maed. ne bg ssue n hotometrc stereo and shae from shadng algorthms s accuracy estmaton snce f a real samle s chosen rarely t s ossble to numercally comare the real geometry wth the reconstructed one. o n addton to a real samle we have to consder durng ths wor also an analytcal one. Tycally there are some P and F numercal surface (Fg.3.) []. We choose the vase and the reasons of ths wll be clear when the ntegraton algorthms wll be develoed and analysed.

14 a) b) c) d) e) f) Fg. 3. tycal P and F test mages 4

15 Fg. 3.3 the P samle n ths wor to test used algorthms Fg. 3.4 The crcle wth the mnmum ray that ncludes all surface( ) ars was dslayed n red and t corresonds to the maxmum alha value that doesn t allow shadows n gray scale mages 5

16 n the case of the vase we have to choose the best llumnaton condton. eferrng to aendx A we should analyse the gradent feld n the ( ) sace to solve ths tas. Precsely consderng all ossble beta values we have to fnd the alha value that doesn t n any case mae the shadow lne ntersect the mnmum crcle that ncludes all surface onts lotted n the ( ) lane. Fg.3.4 shows vase gradent ars lotted n ( ) crcle that contans them. The coordnates of the most dstant ( ) we could fnd that.79rad max lane and the mnmum ar are(.7.4). Usng ths arameter n (8.9) n the aendx Usng the euatons above we could comute the dervatves along x and y of both shell and vase. tartng wth the x one we have to consder β and β π. Fg. 3.5 and 3.6 show ( ) of shell and vase and ther x dervatve. nstead for the y dervatve we ut π β and β π. Fg. 3.7 and 3.8 show ( ) of shell and vase and ther y dervatve. Note that n shell case we don t now the numercal exact value of alha (assumng arallel rays see secton 5) but t doesn t matter because t wll only affect scale 3D reconstructon. 6

17 a) b) c) Fg. 3.5 a) and b) show shell mages lt resectvely from rght and left whle c) surface dervatve along x 7

18 a) b) c) d) Fg. 3.6 a) and b) show vase mages lt resectvely from rght and left whle c) surface dervatve along x. d) s the error between the orgnal x dervatve and the comuted one 8

19 a) b) c) Fg. 3.7 a) and b) show shell mages lt resectvely from the usde and downsde whle c) surface dervatve along y. N.B. n real mages the y axs has the ooste drecton than y axs n analytcal mage (vase) 9

20 a) b) c) d) Fg. 3.8 a) and b) show vase mages lt resectvely from the usde and downsde whle c) surface dervatve along y

21 n ths sense the gradent recovery assumes a mnor mortance than the second ste the gradent ntegraton whch s a fundamental ssue to reach the heght ma startng from the gradent feld. However s the four lghts set the best soluton and f yes n whch terms? Actually the choce of a secfc lghts set deends on many factors. For nstance f we want to decrease acuston tme we could acure only three color mages whch are the mnmum data set that ermts to comute the gradent feld and the albedo. However f the surface s away from the smoothness condton more than four mages are robably needed to avod shadows dstortons n gradent and albedo. o t s obvous that the more mages we have the more accurate wll be the 3D recovered ma but the more s also the acuston tme. n any case we need to fnd a trade-off between these ssues. For examle n real-tme alcatons we are bound by hardware acuston rate and we should choose the best soluton n terms of mages number and mages resoluton. The latter deends on whch obect we have to deal wth. n the next secton the basc gradent extracton theory and formulas were exlaned [].

22 4. Basc theory gradent extracton n ths secton we dscuss the theory of gradent comutaton. n order to defne a roer mathematcal model we consder a Lambertan surface behavour. any wors demonstrate that t s ossble under some assumtons to searate ths secal cue to the others (hghlght shadows an so on) []. n such a way many non-deal surfaces can be transformed to be treated as Lambertan. For ths reason the followng theory s usable n almost every hotometrc stereo exerment. The reflectance functon n Lambertan case s the scalar roduct of two vectors the lghtng one and the surface normal: (4.) ( ) ( ) ( ) Now to smlfy notaton consder ths vectors: l n ( ) ( ) (4.) f we have K N lght sources we can wrte: l n (4.3) Ths formula doesn t care about surface chromatcty (albedo). But where could albedo occur and what does ts aearance deend on? Whle we are sure that t occurs n surface mcrofacets (whch are roected onto xels n mage lane) as ts absolute roerty so-called body colour ts aearance could deend on the lght colour. Consderng ths latter factor we have to rewrte the revous euaton n ths way:

23 3 ( ) ( ) n l L µ µ (4.4) where [ ] T B G µ µ µ µ f we consder the searated GB sgnals. Now from (4.3) and (4.4) we obtan ( ) ( ) ( ) ( ) ( ) ( ) n L B G T B G µ µ µ µ µ µ µ (4.5) Let now deal wth the body colour. As the lght colour s a GB vector multled by the lght drecton vector n the same way the body colour s a GB vector multled by surface normal. (4.4) becomes ( ) ( ) n N l L ρ ρ µ µ (4.6) where [ ] T B G ρ ρ ρ ρ. n ths case the colour acured mage s ( ) ( ) ( ) ( ) ( ) ( ) N L B B G G B B G G T B G B G µ ρ µ ρ µ ρ µ ρ µ ρ µ ρ ρ ρ ρ µ µ µ (4.7)

24 Generally snce the concet of colour s relatve to a fxed colour whch s consdered the reference one we could consder system normalzed regardng the lght colour. n such a way the mage euaton s: ( L n) ρ (4.8) Note that n the deal case t doesn t matter usng gray-scale or colour mages and t s obvous after consderng last euaton. However f there s some nose n acured mage the analyss of three cues (GB) could allow us to understand the orgnal colour vector n the GB cube [][] or at least to obtan most accurate 3D ma. Consderng (4.8) t s obvous that we need at least three mages for havng a soluton. Wth three mages ( 3 ) we have ths lnear system: [ L] ρ n (4.9) 3 where [ L] [ L L L ] T 3 and [ ] T nverse matrx:. As well nown f > 3 we should aly the oore-penrose T T ([ L] [ L] ) [ L] ρ n (4.) That s all regardng the basc theory of gradent extracton. ore comlex ssues wll be treated n next sectons. For the moment we are nterested n comarng ths nd of gradent comutaton and the artcular method based on four mage acustons seen n revous secton. We analytcally analyse euaton form usng oore-penrose matrx and n the same lght set condton than before. Wth the four lght sources used n secton 3 where we have 4

25 5 4 3 π β π β π β β (4.) the L matrx wll be cos sn cos sn cos sn cos sn L (4.) so [ ] [ ] ( ) [ ] 4 3 sec sec sec sec csc csc csc csc 4 L L L n T T ρ (4.3) Ths leads to ( ) sec 4 cot cot ρ (4.4) n the GB case we could wrte these euatons n such a way

26 6 ( ) B B B B G G G G B G B G B G B G B G B G B G B G B G B G B G B G B G B G B G sec 4 cot cot ρ ρ ρ (4.5) n absence of nose each colour cue roduces the same gradent. However t s very useful because we could use ths behavour to mae the algorthm more robust n resence of nose. Euatons (4.4) comared wth euatons (3.) and (3.3) tell us that f there aren t any non-dealtes 4 3 (4.6) Ths wll be a smle way to detect non-dealtes such shadows hghlghts or non-arallel rays. Now let comare the gradent feld of the two used mages sets (shell and vase) comuted wth the latter euatons and wth (3.) and (3.3). We should exect the same result wth the vase due to the fact that t s an analytcal deal surface whle we should have a very dfferent result n the shell case most of all n shadows areas. Fg.4. shows what we exected.e. the two vase comuted gradents are numercally the same. nstead Fg.4. states that the two shell gradent felds are dfferent. Precsely they dffer more n the zone n whch there are non-dealtes. For ths reason Fg. 4.3 shows an mage where blac means less dfference; as exected shadows zone are whte.

27 a) b) Fg. 4. a) and b) are the dfferences between the vase x and y dervatves comuted wth (4.4) and (3.) wth (3.3) 7

28 a) b) Fg. 4. a) and b) are the dfferences between the shell x and y dervatves comuted wth (4.4) and (3.) wth (3.3) 8

29 a) b) Fg. 4.3 a) and b) are the dfference mages between the shell x and y dervatves comuted wth (4.4) and (3.) wth (3.3) 9

30 5. ays correcton o far we have seen the basc theory of gradent extracton. However we made some assumtons that could not ft the real condtons. f we tae care about consderng surface non-dealtes there could be also lght behavours that roduce gradent and 3D reconstructon bases. n ths sense and under artcular condtons we have develoed a method to correct ths structured nose. n ths secton we study the smle case of ont lght source near the obect and startng from ths artcular stuaton we fnd that t s ossble to aly the founded fnal euatons to any case. We start ths analyss assumng that a ont lght source s lghtng a lane so for all xels. f the lght s near the lane we easly note that two surface onts wth the same ( ) values wll be lt wth dfferent alha. o the corresondng xels n lane mage have dfferent gray level Fg. 5.. Ths s n contrast to (4.8). Fg. 5. wth a ont lght source to surface atches wth the same gradent are lt wth dfferent lght drectons and then have dfferent xel gray levels ne thng s clear: for a lane have a dfferent alha value for each surface ont. We solve ths roblem (for now n the lane case) usng three stes. The frst s consderng a lane wth nown albedo (or we could consder ts albedo as the reference colour value). Then an average alha value s found. The last ste s to buld a correcton matrx for the secfc lght source. After these stes we are able to 3D ma and to recover albedo of any lane at the same z 3

31 value. t s not really a bg result but after descrbng ths rocedure we demonstrate that t s ossble to mrove t under artcular but not too bndng condtons. nly n ths case we use a dfferent samle nstead of shell (t would be van to consder an deal surface such the vase) due to the fact that what we are gong to exlan s more vsble wth t rather than the shell. Let consder a whte lane a whte lght source and gray-scale mages (for now we don t care about random nose correcton). For ths surface as state above for all atches so we can wrte: L N cos ( ) ( ) T (5.) f rays are not arallel we should mae a dstncton between all xels so n matrx terms ~ cos (5.) f we now the value of alha we also now the alha bases n each real mage xel. Now how we could state the real value of alha f the lght s a ont lght source? ne method should be to consder the alha between the ray whch lns lght oston and the ntersecton between lane and vew drecton. However t would be a good method only for ont lght source and we should choose another aroach wth a casual lght dstrbuton (not only regardng lght drecton but also regardng lght ntensty). nstead nowng the albedo and gradent of the lane we could measure the average alha value. Fg. 5. shows the same lane lt from four dfferent drectons (the same of the revous secton). As s easy to see nstead of the deal case the gray-level dstrbuton s not unform n a sngle mage and t s not eual comarng mages one another. 3

32 a) b) c) d) Fg. 5. a)b)c) and d) show the whte lane lt resectvely from rght left usde and downsde o f we consder a Nx matrx the mean alha value s K N arccos (5.3) KN 3

33 33 These two methods are the aroach boundares. The frst s the same as the second wth a Drac functon aled n the mage center. A reasonable trade-off s addng a weght deendng on the dstance between the xel and mage center. The euaton becomes N K N w K w arccos (5.4) (N.B. we ut K out of the summary because the four weghted matrces are eual each other) where N N w (5.5) nce we have comuted alha we buld a correcton matrx n such a way ~ (5.6) so cos cos ~ (5.7) We have founded a smle euaton for the correcton matrx ~ cos (5.8)

34 Fg. 5.3 shows lane mages after alyng the correcton matrces on them. The comuted alha value s.45rad 6. Chosen these correcton matrces obvously t doesn t matter f we ll deal wth a lane wth dfferent albedo because t s only a gray-level xel scale factor. n ths case we only have a dfferent euaton ~ ρ cos ρ cos (5.9) o K N w ρ arccos N (5.) K w and ρ cos ~ (5.) As we sad before ths s not a really great result snce f move the lane forwards or bacwards along z axs the alha correcton at the same xel changes. For nstance f ut an obect on the lane and want to recover ts gradent should consder a dfferent alha dstorton than the lane case. How can go on to mae ths result be worth effort? n Fg. 5. we have seen that two xels nvolve dfferent lght drectons f the lght s not far enough to mae us consder arallel rays. But f the lght s not too near the lane we could demonstrate that for a lttle dfference n z axs the same correcton matrx s vald yet. The only thng we have to study s ths z range n whch we could generalze the artcular lane correcton matrces and use them for any obect. Let start to see the geometrc roblem (Fg. 5.4). have to fnd the z that ermts me to consder alha varaton numercally zero n the case of ont lght source. ndeendently of the surface sloe the alha varaton nvolve a varaton n l term. We would want that each element of lght vector doesn t change the value out of a secfc range. 34

35 a) b) c) d) Fg. 5.3 a)b)c) and d) show the whte lane lt resectvely from rght left usde and downsde after alyng the correcton matrx on t 35

36 Fg. 5.4 For alyng correcton matrces to any obect beng 3D maed should fnd the varaton of z that ermts to consder numercally zero the varaton of alha The nvolved vector s [ sn cos β sn sn β cos ] l (5.) We want that l max l l % (5.3) Ths condton leads to sn sn. ε sn cos cos. ε cos (5.4) Usng Fg. 5.4 we can wrte 36

37 εb δ ε z (5.5) Ths result means that f the lane n whch the lght sources le s one meter and a half far from the lane where could ut any samle on could reconstruct a gradent feld of a surface that has a z ma n almost 3 cm range. Ths s a very smlar condton comarng wth our exerments so we could aly correcton matrces even f there s an obect on the orgnal whte lane. Fg. 5.5 shows four mages of an archaeologcal samle. Now we aly on them the correcton matrces (Fg. 5.6) and we show the results (Fg. 5.7). Even f we ddn t menton the ntegraton algorthm yet n order to understand the effect of ths correcton aroach we also show the 3D reconstructons usng the orgnal mages and the corrected ones (Fg. 5.8). As s easy to see the 3D ma usng orgnal mages s based: the lane should be a flat surface nstead of sem-shercal. n the other hand the surface comuted after usng correcton matrces aear no based and the lane under the samle s flat as t should be. 37

38 a) b) c) d) Fg. 5.5 a)b)c) and d) show the archaeologcal samle lt resectvely from rght left usde and downsde before alyng the correcton matrx on t 38

39 a) b) c) d) Fg. 5.6 a)b)c) and d) show the archaeologcal samle lt resectvely from rght left usde and downsde after alyng the correcton matrx on t 39

40 a) b) Fg. 5.7 a) b) show the archaeologcal samle gradent comuted after alyng the correcton matrx on t 4

41 a) b) Fg. 5.8 the archaeologcal samle 3D ma comuted a) wthout and b) after alyng the correcton matrx on t 4

42 6. Gradent ntegraton n revous secton we have analysed the gradent comutaton rocess. Now we start to deal wth the second man ste.e. the ntegraton ste whch ermts to erform a transformaton between the gradent feld and the corresondng heghts ma. n the ast the frst test conssts n alyng the emann one dmensonal ntegraton snce we had only two hotos and one dervatve (x or y). The results are not really good most of all n the real case. ndeed the emann aroach s not too bad n the deal case but t s very nose senstve so t becomes unusable wth real surfaces. Concernng the vase for examle the 3D ma s ute good n the case of x dervatve whle s really bad wth the y dervatve (Fg. 6.). Ths deends on the fact that emann ntegraton erforms only one ntegraton er row (or column) and doesn t mae any correlaton between columns (or rows). Furthermore f t doesn t have the boundary condton as ntal cue the frst row (or column) starts from zero value. t s not worth develong a emann based algorthm that cares about the boundares. For ths reason an deal surface wth the frst column eual to zero (as the vase) has a good reconstructon wth the emann ntegral aled on the x dervatve (Fg. 6.). Ths reresents the frst tral and for obvous reasons we mmedately abandon ths nd of aroach. We have read a lot of wors and tred some nown algorthms concernng ths ssue and there s a lot of dfferent ways to recover z ma from the gradent feld for nstance usng Fourer transforms [3] or Green s formulas but so far we beleve that the best tradeoff between the reconstructon accuracy and comutatonal tme s the bconugate gradent method (BG) aroach [4]. n ths secton we re gong to dscuss a artcular algorthm we have develoed based on BG tang care of both surface and boundary ntegrals. Frst we ll tal about the matrx roblem nvolved n two-dmensonal ntegraton rocess (Posson matrx) then we ll show the theory concernng BG and fnally we ll llustrate our modfed algorthm. 6. Posson matrx Posson euaton s the rototycal elltc euaton x z y z ρ ( x y) (6.) where ( x y) ρ s the nown term. Usng the fnte-dfference and matrx reresentaton t becomes 4

43 43 z z z z z z ρ (6.) where s the dstance n ( ) y x lane between two surface atch centers along x (or y) suosed they are eual. Now let us consder a surface that conssts n a Nx matrx. We could wrte z z z z z 4 ρ (6.3) for N K and K. Ths leads to a matrx formulaton whch nvolves the so-called trdagonal wth frnges matrx. For examle f ( ) ( ) 66 N we ll have a 6x6 Posson matrx x L L L L L L L L L L L L L L L L L L (6.4)

44 a) b) Fg. 6. 3D ma comuted alyng emann ntegral to a) x and b) y dervatve The unnown term s [ z z z z z z z z z z z z z z z ] T z (6.5) nstead the gven term s buld by ρ and the boundary condtons n such a way 44

45 [ ρ ρ 3 ρ 4 ρ 5 ρ 3 ρ33 ρ 34 ρ35 ρ 4 ρ 43 ρ 44 ρ 45 ρ 5 ρ 53 ρ 54 ρ55 ] T [ z z6 z3 z36 z4 z 46 z5 z56 ] [ z z z z z z z z ] T T (6.6) Note that the four xels at the matrx edge are not nvolved n the surface ntegraton. o we could generalze these matrces usng a generc Nx matrx and usng ths reresentaton Az b (6.7) where A and : KN : : A 4 n K N 3 : n K : n K N 3 : n : A : A ( N ) ( N ) A A (6.8) b ρ z bc z br (6.9) where z bc n( ) ( ) n KN 3 : zbc n KN : n zbc z n KN 3 n KN : n z n n ( ) n ( ) z bc (6.) z br : K z : : br z ( )( N 3) K( )( N ) zbr z N ( )( N 3) K ( )( N 3) K( )( N ) z br (6.) 45

46 46 For solvng (6.7) euaton t s ossble to use a lot of numercal methods but we should narrow down the ossble algorthm tang care that most of all nvolve A nverse matrx comutaton. nce A matrx has ( )( ) ( )( ) ( ) N N dmenson even wth small value of N and the calculaton of the nverse could be comutatonally very hard. The BG avods the nverse comutaton but nvolves the orgnal sarse matrx storage. For the moment we shall exlan ths algorthm and then we ll modfy t to avod also ts storage n hardware memory ncreasng comutatonal tme. 6. BG The conugate gradent algorthm s based on the mnmzaton of ths formula [4]: ( ) z b z A z z f (6.) t s mnmzed when b z A f (6.3) o t s s mnmzed for the z values that solves (6.7). The generc numercal algorthm (nown as bconugate gradent method) starts sulyng ntal vectors r and r (whch have the same z dmenson) the frst guess z and set r and r. Then t carres out ths recurrence T z z r r r r r r A r r A r r A r r β β β (6.4)

47 n our case A s symmetrc so we can set r r and. Furthermore we choose z whch means that the resdual r b. The recurrence becomes r z r r A r r β r z r r r A β (6.5) 6.3 ur algorthm o far we have seen the two-dmensonal ntegraton roblem and the numercal method to solve t. However n real hotometrc stereo case we haven t any gven boundary condton and we really don t want to store a bg matrx even f t s sarse. n fact f we have a common 5x5 mage the A matrx wll be 644x644 matrx wth more than mllon non-zero elements. We mrove ths algorthm erformng two tass frst we use the conugate method to execute a boundary ntegraton and then we avod storng the sarse matrx wth a smle algorthm that comute A Boundary ntegraton uosed the dervatves along x and y are two NxN matrces. A artcular case of Posson s formula s used to comute the deth ma n the matrx contour. Frst of all the boundary s consdered as a lnear vector [ z z L z z z L z z z L z z z z z ] T zc N N N N N N N N N N N N L Then the euaton (6.) s rewrtten consderng only one dervatve along the drecton of the contour vector (6.6) z z C u z C z C ρ (6.7) 47

48 48 where ρ deends on the surface dervatves along x and y and ts value s N N N K N N N N K N N N N K N N N K K K N N K N K N N N N N N K N K N N K K K K K K K K K K ρ (6.8) Note that the frst and the last elements of C z vector are the same and snce an ntegraton nvolves an addtve constant we could set z. Ths rocedure leads to an A matrx wth the dagonal element eual to - and for each row the element value before and after - s urface ntegraton Usng revous results as BG algorthm cues we have to analyse only the nown term of (6.7). Ths s the sum of the second dervatves but n our exerments we have the frst dervatve. Usng fnte-dfference aroxmaton we ut y y z x x z (6.9) tate ths we could wrte the nown term as br z bc z b ρ where bc z and br z are the same as before and ρ s ρ (6.)

49 The last thng we have to dscuss s the rocedure that ermts us to avod A matrx storage. We sad that we erfectly now the A matrx and excet lttle dfferences the n row s eual to the shfted n row. f we manage these lttle dfferences we could erform the roduct only wth a for loo wthout storng the sarse matrx. These dfferences are the zero values that nterrut the dagonal wth "" values adacent to matrx dagonal. These values occur wth a regular recurrence whch deends on the number of columns n the orgnal acured mages. o every change s redctable and there s no roblem to erform ths oeraton. n ths way we could reach an algorthm erformance better than common algorthm used n rofessonal software for comutaton. Now we show the result of ths rocedure n the analytcal and real test cases (vase and shell) (Fg. 6. and 6.3). As we exlan n secton 4 the algorthm recovers the albedo nformaton too. Fg. 6.3 shows the shell reconstructed surface wth ts aled albedo. Then we nvestgate algorthm changed verson to mrove ts erformances and among many tral we found that the teraton number (hence the comutatonal tme) could be decreased usng a 9 xels ernel nstead of Posson s 5 ernel. nce the dervatves along all drecton could be wrtten as a lnear combnaton of x and y dervatves f we consder also the dervatves along the dagonals of a surface matrx we ll have a Posson s euaton that nclude all xels n a 3x3 ernel centered on ( ) coordnates. Fg. 6.4 shows a comarson between 5-ernel and 9-ernel algorthm erformances n terms of number of teraton and comutatonal tme. 49

50 a) b) c) Fg. 6. a) Vase 3D ma comuted alyng BG ntegral b) the orgnal surface and c) the heght error %. 5

51 a) b) c) Fg. 6.3 a) b) shell 3D ma comuted alyng BG ntegral c) the reconstructed surface wth aled albedo ma. 5

52 a) b) Fg. 6.4 a) b) show the erformance dfferences between our algorthm based on Posson s euaton (5 ernel) and the modfed 9 ernel algorthm resectvely regardng the number of teratons and comutatonal tme. n abscssas there s the dmenson N of a NxN surface matrx. 5

53 Fg. 6.5 A comarson between our algorthm and a famous calculus software n solvng BG teraton rocess. n abscssas there s the dmenson N of a NxN surface matrx. These mrovements mae our algorthm worng better than commercal software n solvng Posson s and modfed Posson s algorthms. We fnd that our algorthm taes less tme to comute the bconugate gradent rocess than a famous software (that we are not gong to menton) (Fg.6.5). There s another way to mrove the comutatonal tme regardng the 3D ma of a surface. any aers deal wth Photometrc stereo ssue but most of all tae as ntal cue the boundary condtons whch s a really bndng thng. We are able to fnd a 3D ma wthout any nformaton but the gradent and ths s a secal feature that allows us to more decrease the tme taen to have surface heghts. We have only to tae not the entre mage n a sngle ste but art of t. After comutng the boundary we reconstruct the boundary of a surface regon (e.g. the uer left uarter); startng from the nown boundary we have to comute only two edges of ths subsurface. Then ts surface ntegral s erformed and we could contnue fndng the 3D ma of the other regons. Note that after ths ste f we want to reconstruct the uer rght uarter we already now the heghts of three of ts edge and so on. The ont of fact s that the tme needed to wor the entre surface out s more than the tme taen for calculatng four sub-surfaces. Let us consder a NxN matrx wth N 6. ur algorthm taes more than seven mnutes (about seven mnutes and a half) for the entre surface whle t taes about one second for a nxn sub-matrx wth n. o wth ths dvde et mera aroach we send about 4 seconds nstead of 45. f we consder a nxn wth n 5 the entre surface wll be reconstructed n about 3 seconds. We decrease the comutatonal tme by 94%. Ths aroach could be very useful regardng the real-tme ssue whch could clear the way for new nnovatve alcatons. 53

54 The ossblty of havng an algorthm that comute boundary of any regon wth any toologcal (twodmensonal) features allows us to solve one of the most mortant roblems: the bases of shadow regon. There are some aers [] that state the caablty of dentfyng shadow zones and robustly comutng the real gradent wthout any dstorton. But the shadow zone contans also a lot of nformaton n the case of a non-deal surface most of all n the resence of dscontnutes whch are the devl n hotometrc stereo. n such cases shadows can hel us frst to recognze dscontnutes and then to reconstruct the rght gradent and the rght surface. Consder a lt cube lyng on a lanar surface. t roduces a cast shadow on the lane. f we aled one algorthm to remove the shadows effect n gradent feld we ll have an eual to zero gradent and the reconstructed surface wll be a lane. Ths s one nstance whch shows both the lac of many algorthms n managng shadows and the great nformaton contaned n them. As an examle of alyng a smle algorthm to rght reconstruct shadow zones wthout losng the nformaton held n them let see Fg. 6.6 where we use the caabltes of the curvlnear algorthm (one-dmensonal Posson s aroach) to reconstruct a self-shadow n a facal 3D ma. a) b) Fg. 6.6 Comarson between facal reconstructon a) wthout shadow managng and b) removng one of the cast shadows due to the nose 54

55 7. Alcatons The most wonderful feature n ths nd of research s that P could be aled to a large number of samles and t wors not only for near Lambertan surface and usng common lghts but also wth dfferent nd of mages and wth surfaces that s non-lambertan. For nstance we reconstruct obects acurng ther scannng electron mcroscoe (E) mages. Another examle s face 3D ma; whle the sn s aroxmately Lambertan the eye s rght reconstructed even f we couldn t consder t Lambertan at all. These are to examles that demonstrate the adatablty of P to many real cases even f they are really far from the deal condtons. Ths fact roves the develoed algorthm s robust regardng most of the dstortons causes. Due to ths robustness we have aled t to many research and alcaton felds and n the followng secton we searately analyse each of them: - archaeology - bometrcs - grahology - huge and far obects 3D ma - olce nury - E samles - bology - olynomal texture ma (PT) samles Furthermore durng these exerments we need a software to manage some data. o we had develoed a vsualzaton tool and a software that controls the acuston of E mages (secton 7.). We ll broadly show them and ther behavour. 7. tcal alcatons Archaeology s the frst alcaton feld we have dealt wth durng our research. The frst samle was a double of a umeran tablet wth a great number of ncsons on t. We were surrsed to note that the 3D ma s erfectly readable even f n the frst exerments we use a surface matrx wth low resoluton (Fg. 7. and Fg. 7.). The effect of the heghts wth albedo searaton s evdent f we comare the 3D ma wth and wthout the chromatc nformaton. 55

56 a) b) c) Fg. 7. a) one of umeran tablet (front) orgnal mages b) 3D ma and c) 3D ma wth albedo 56

57 a) b) c) Fg. 7. a) one of umeran tablet (rear) orgnal mages b) 3D ma and c) 3D ma wth albedo Concernng archaeologcal ssues a very mortant alcaton s stratgrahy. Usng our algorthm archaeologsts can 3D ma every ste of stratgrahc rocess avodng the needed drawng tas. Ths maes ths rocedure more accurate and relable. Fg. 7.3 shows a smulaton of a stratgrahc oeraton suosng we have three surfaces. 57

58 a) b) c) Fg. 7.3 These are three stratgrahc surfaces. a) s the frst layer b) the frst and the second layers and c) all layers. 58

59 ne feature of our algorthm s the searated extracton of heghts and chromatc nformaton. Ths roves to be very useful n archaeology when an nscrton s llegble due to ts drtness. Fg. 7.4 shows a headstone comletely mossble to read whle Fg. 7.5 demonstrates that removng the albedo wth a 3D ma we are able to read the orgnal oman nscrtons. Fg. 7.4 A comletely llegble headstone 59

60 a) b) Fg. 7.5 a) An mage of a art of the headstone and b) ts 3D ma n whch we are able to read the orgnal nscrtons Another mortant alcaton s bometrcs. n ths secton we are gong to show the caabltes of P n 3D mang art of human body useful for nstance for securty alcaton. The frst samles we llustrate are facal reconstructons (N.B. we have human gunea-g leave to ublc her hoto). n Fg. 7.6 we have a face wth oen eyes. Ths demonstrates that even non-deal surfaces could be consdered almost Lambertan due to the algorthm robustness regardng non-dealtes. 6

61 Fg. 7.6 A face 3D ma wth rght reconstructon of eyes zone n ths samle we could see that the face s reconstructed roerly but a lot of dstortons aear n the har. Deendng on the secfc alcaton (for nstance securty) we could consder a art of mage n whch we are able to manage all non-deal elements and to avod surface bases. The followng fgure (Fg. 7.7) roves that t s ossble to use our algorthm to 3D ma fngerrnts. Ths feature s really mortant because generally fngerrnts dentfcaton s erformed wth two dmensonal data. Possblty to use three-dmensonal nformaton mroves the dentfcaton accuracy and decreases a lot otental errors. Fg. 7.7 A fngerrnt 3D ma 6

62 Grahology s the scence of human dentfcaton by hs handwrtng. Broadly t uses letter shaes and n ntensty to dentfy the author of any wrtng; all of these nstruments are two-dmensonal. Even n ths case havng three-dmensonal nformaton can hel n dentfyng tas. We have tred our algorthm reconstructng two suermosed drawn lnes (Fg. 7.8). Usng ths nd of nformaton we mmedately see that one lne s deeer than the other so the man who wrtes n ths aer has used dfferent effort drawng the lnes. Ths could be a good nformaton deendng on what somebody s searchng for. Fg. 7.8 A 3D ma of two suermosed drawn lnes Fnally the last nd of alcaton concernng otcal doman s the 3D ma usng solar lght of obect at a great dstance from the vewer. n Fg. 7.9 we show the 3D reconstructon of a moon crater. nstead Fg. 7. reresents a summarse of the caabltes of our method. 6

63 a) b) Fg. 7.9 A moon crater. rgnal mage a) and reconstructed detal b) by -source (sunlght at mdnght of two successve days) Photometrc tereo. a) b) c) Fg. 7. Pure albedo a) ure geometry b) and rendered 3D reconstructon c) of a shell by Photometrc tereo. These are the general results after alyng our algorthm to a samle n otcal doman 63

64 7. cannng Electron croscoe o far we have dealt wth otcal mages and we have consdered surfaces behavour as Lambertan. There s another tye of mages that even f t s hyscally comletely dfferent from common mages ermts to consder the surface n the same Lambertan way. They are scannng electron mcroscoe (E) mages. bvously t ncreases the ossble alcaton feld and as we demonstrate that t s ossble to reconstruct huge and far obect usng for nstance solar lght now we can say we are able to nvestgate the mcroscoc world. n the followng we ll see many exemlars of tycal E samles but before dong ths we should llustrate the rocedure to acured E mages for 3D reconstructon algorthm. Even f the mage acuston system s very dfferent from many ont of vew we need the same mages set. o not consderng the nature of E sgnal the unue dfference s that E mages are gray-level matrces and the concet of albedo n ths data s not the real chromatc roerty. n scannng electron mcroscoe albedo varatons reresent materal varatons along the surface. Gvng a closer loo to the P-based 3D recovery method t conssts of the acuston of 4 Bac-cattered (B) mages whch allow us to comute surface gradent and snce we have to rotate the samle under observaton due to the fact that we have only one lght source (n E the corresondng lght source s the electron detector) and snce the rotaton s not erfectly erformed by the mcro mechancal system n E we need to algn acured mages to gve our algorthm them. tated ths to acure mages we need to erform the followng rocedure due to the ntrnsc mcroscoe behavour. a) b) Fg. 7. a) B and b) mages of the same frame scan. Frst of all the subect s framed magnfed and focused. Then both brghtness and contrast are searately set for the two detectors to be used.e. the off-axs Baccattered (B) electron detector and the comlete crcular axal 64

65 B detector whch roduce sotroc shadng (we ndcated them as mages). n Fg. 7. you can see the B and mages related to the frst mage acuston. o after framng these two mages of the obect we want to 3D ma we start ths seuence ) Acuston of a ar B and mages as n fg. 7.. ) mechancal rotaton of the secmen by 9 3) Acuston of a reduced mage n the central area that s frst counter-rotated by 9 and then Fg. 7. reduced mage counter-clocwse rotated by 9 4) cross-correlate wth the revous mage. Ths gves the ( x y) that we use to re-algn the mage n E acuston dslay Fg. 7.3 cross-correlate wth the revous mage 5) The new mage center of the rotated secmen s mechancally brought bac by ( x y) as close as ossble to the revous centre 6) tes -4 are reeated 3 tmes At the end of the rocess (whch taes few mnutes ncludng the mage acuston) a twn set of 44 mages s stored (Fg. 7.4). 65

66 Fg. 7.4 tored mages at the end of E acuston rocess. B mages n the frst row and mages n the second. The automatc algnment durng the acuston rocess s not really accurate for a good 3D reconstructon so we have to erform also a software mage rocessng to refne the algnment. Then we have the four mages (Fg. 7.5) that wll be used by the P algorthm to fnd surface heghts (Fg. 7.6). Fg. 7.5 The mages after the mage rocessng algnment. These mages are ready to be rocessed by P algorthm Fg. 7.6 The 3D ma of ths E samle Followng fgures shows other ossble nd of E samles that are useful to be 3D reconstructed. 66

67 Fg. 7.7 A set of B mages (uer row) the corresondng twn set (central row) and some 3D vews (lower row) of the reconstructed surface of a damaged sold-state electron devce. a) b) Fg. 7.8 a) ne of the B E mages of a con detal and b) ts 3D reconstructon. a) b) c) Fg. 7.9 a) tcal mage and b) E detal of the ercusson surface of an exloded bullet-case; c) reveals the 3D shae. 67

68 7.3 Polynomal Texture a The last alcaton we want to descrbe nvolves an algorthm and software develoed n HP labs at Palo Alto (CA). Ths algorthm called Polynomal Texture ang (PT) taes N mages of an obect and nterolates ths cue data to vrtually render the surface as lt from any chosen drectons [5]. nce we can choose the drecton of llumnaton of a PT samle t s evdent that we can lght the surface from the drectons we need to erform P algorthm and to fnd surface 3D ma. For ths reason we download from h-labs web ste the PT software and some samles and we fnd the heghts of them. Next fgures demonstrate the advantages of mergng our and ther algorthm. a) b) c) Fg. 7. a) one of PT (umeran tablet) orgnal mages b) 3D ma and c) 3D ma wth albedo 68

69 a) b) c) Fg. 7. a) one of PT (a front of an ancent con) orgnal mages b) 3D ma and c) 3D ma wth albedo 69

70 a) b) c) Fg. 7. a) one of PT (a rear of an ancent con) orgnal mages b) 3D ma and c) 3D ma wth albedo 7

71 8. Aendx A n order to have the best P test mages a artcular ssue has to be faced: the lght set-u. nce we have an analytcal surface ma (wth real obect t s more comlcated and we don t deal wth t here) t s necessary that the roduced mages set s the best to be aled to P algorthms. n few words we should avod n the mages every nondeal element such as cast and self shadows hghlghts and nose. Frst of all we should note that the smoothness of surface must be sutable for P whch s not able enough to recognze strong surface dscontnutes and we should not have any nose or hghlghts; ths s due to the fact that the surface s not a real obect and we could choose the roer materal behavour. The last ssue s the resence of shadows (cast and/or self). To solve ths roblem we have to analyse the gven analytcal surface gradent feld and fnd a rule to decde the llumnaton arameters whch don t roduces any shadow. Let s mae an examle n one dmenson. uose we have a yramd wth the base n the x-y lane and let z-axs be ts axs. Consder a secton on the x-z lane (Fg. 8.). Before movng on note that shadows n mages (both cast or self) nvolve the resence of a negatve scalar roduct between the llumnaton drecton and surface normal n at least one mage xel (or one obect mcrofacet). Fg. 8. one dmensonal geometry for best lght set-u estmaton We now that the reflectance functon stated that 7

72 (8.) ( ) ( ) ( ) where the term wth the s-ndces s the llumnaton drecton vector whle the other term s the surface normal. f we want to avod shadows we should choose a roer llumnaton vector whch assures us that n every obect mcrofacet the functon s non-negatve. Ths tas s really smle f we now surface gradent feld. As we can easly see n one dmenson f shadows along the entre surface wll be π (and ths s a necessary condton n P) then the maxmum alha value to avod π π z max γ tan (8.) x max Now let consder the usual two dmensonal condton. Whle n the revous case we deal wth alha snce the dervatve could be only n x drecton now the analyss of beta s needed. n other words we should study all ossble drectons of lght rays to choose the maxmum alha value that does not roduce any shadow ndeendently of beta value. n ths sense frst of all note that usually the P algorthms consder a contnuous surface made by a fnte number of mcrofacets whch begn xels n mage doman. Furthermore we consder the xel as an mage of a suare obect regon wth untary edge. tated ths the deth varaton (Fg. 8.) wthn a xel s δ z δx δy (8.3) 7

73 Fg. 8. The change n heght s the sum of the roducts between dervatve and the ste along each drecton Fg. 8.3 mage gray level value deends on the lght drecton vector and the normal of the z z segment that les on ntersecton lane 73