Abstract. Calculation of mirror profile The theory underlying the calculation of the mirror profiles is described below.

Transcription

1 A New Clss of Mirrors for Wide-Angle Imging Mm V. Srinivsn Centre for Visul Sienes, Reserh Shool of Biologil Sienes, Austrlin Ntionl Universit, PO Bo 475, Cnerr, ACT 261, Austrli Astrt Conventionl mirrors for pnormi imging usull pture irulr imges. As these imges re diffiult to interpret visull, the re often rempped digitll into retngulr imge in whih one is represents imuth the other elevtion. This pper desries lss of mirrors tht perform the pture s well s the rempping, thus eliminting the need for omputtionl resoures. The provide uniform resolution in imuth elevtion, n e designed to mke full use of mer s imging surfe. Introdution Over the pst five ers or so there hs een onsiderle interest in the design development of refletive surfes for pnormi imging. Suh surfes n e used in vriet of pplitions inluding surveillne, seurit, rootis, video onferening, we-sed dvertising tourism. Refletive surfes re often more suitle for suh pplitions thn onventionl wide-ngle lenses euse the former re lighter, less epensive over greter field of view. In ddition, s we shll see elow, refletive surfes n e redil tilored to imge the environment in vriet of presried ws. The most ommon lss of refletive mirrors used for pnormi imging onsists of mirror in the shpe of generlied one. The mer, ligned long the optil is of the one, ptures irulr imge of the environment, in whih the rdil distne of piel from the enter of the imge is funtion of the ngle of elevtion of the point in the environment tht it represents, the imuthl diretion of the piel orresponds to the imuthl ering of this point. Mirrors of this kind hve een designed to suit vriet of speifi requirements, suh s (i ensuring tht the effetive point from whih the environment is viewed is onstnt, independent of the elevtion or the imuth of the viewing diretion [1]; (ii ensuring tht piel resolution of elevtion is onstnt, irrespetive of the diretion of view [2, 3]; (iii using omintion of refrtive refletive optis to hieve requirement (i [4]. While this lss of mirrors is ver effetive in imging lrge setions of the environment, the irulr imges tht the pture re diffiult to interpret the humn visul sstem. To filitte visulition, the imge must e digitll re-mpped, or unwrped, to produe retngulr imge in whih the siss represents imuth, for emple, the ordinte elevtion. The unwrped imge is more menle to inspetion, interprettion understing of the sene. However, the unwrping requires omputtionl resoures. A few studies hve eplored the design of mirrors tht hieve wide-ngle imge pture s well s unwrping, optill. One stud presents mirror design for imging fronto-prllel plne in this w [5]. Another hieves mpping of limited setion of the environment in elevtion imuth [6]. Here we desrie new lss of mirrors tht mp lrge setions of the environment produe redil interpretle, retngulr imges of the environment in oordintes of imuth elevtion. The mirrors hieve wide-ngle imge pture s well s unwrping, thus eliminting the need to use omputer for the unwrping proess. Orgnition of the pper This pper is strutured s follows. First, we present the theor underling the derivtion of the mirror profiles. We then onsider two lsses of mirror profiles: one in whih the optil is of the mer is in the equtoril plne of the viewsphere, nother in whih the optil is is ligned with the poles of the viewsphere. An emple of the profile of mirror in eh lss is shown its performne is illustrted in r-tring environment. The pper onludes with rief disussion of the dvntges disdvntges of these mirrors. Clultion of mirror profile The theor underling the lultion of the mirror profiles is desried elow. C A O Cmer Fig. 1 Illustrtion of refletion geometr Fig. 1 shows r BO from the eternl environment impinging on the mirror, the refleted r OA entering the mer. The mpping properties of the required mirror speif the reltionship etween the diretions of the N B

2 inident refleted rs. Given this onstrint, the lol orienttion of the mirror surfe must e suh tht the inident r is refleted into the mer. For this to our, the refleted r must ( lie in the plne ontining the inident r the norml (ON to the surfe t the point of refletion; ( the surfe norml must iset the ngle etween the inident refleted rs. We denote the diretion osines (d.. s of the r entering the mer (,,. For n point (,, on the mirror surfe, these d.. s re given = ( = ( = (3 where the origin of the o-ordinte sstem is t the nodl point of the mer. We denote the d.. s of the reversed diretion of the r impinging on the mirror (,,. We represent the surfe profile the funtion f = + q(, =. It is well known [7] tht the d.. s of the norml to surfe defined f f re proportionl to, (where = 1. We denote these prtil derivtives ( p,,. p p In order to stisf onstrint ( ove, let us first lulte the d.. s of line OC tht is perpendiulr to OA s well s ON. Denoting the d.. s of OC (,,, we hve = (4. p. p +. p = (5 Setting + p = 1 solving for, we otin ( p. = (6 ( p.. p ( p. = (7 ( p.. p Constrint ( requires tht OB e perpendiulr to OC. This implies tht.. +. = (8 + Sustituting for from (6 (7 into (8, we otin p.(.. + p.(.. =.. (9 To stisf onstrint (, we require tht the ngle etween ON OA e equl to tht etween ON OB. This requires tht. p +. p +. p =. p +. p +. p (1 whih, fter setting p = 1, m e rewritten s p.( + p.( + ( = (11 Solving for p p from (9 (11, we otin (.(.. + (.(.. p = = (12 (.(.. + (.(.. (.(.. + (.(.. p = = (13 (.(.. + (.(.. p = = 1 (14 We now onsider two lsses of mirrors: (A Equtoril view mirrors (B Polr view mirrors. A. Equtoril view mirrors The onfigurtion of n equtoril view mirror is illustrted in Fig. 2. The figure shows the mirror surfe, the nodl point of the mer, the mer s imge plne (whih is drwn in front of the nodl point, for simpliit, rther thn ehind it. The mpping we desire is one in whih the optil is of the mer is in the equtoril plne of the viewsphere. Points of onstnt elevtion ( φ in the outside world mp to horiontl lines in the imge plne. These horiontl lines represent lines of onstnt φ in the mer (Fig. 3. Points of onstnt imuth ( in the outside world mp to vertil lines in the imge plne. These vertil lines represent lines of onstnt θ in the mer. This mpping leds to the flowing epressions for the diretion osines of r tht is inident on the mirror: = osφ. sinθ (15 = os( 9 φ = sinφ (16 θ

3 = osφ. osθ (17 We seek mirror profile tht ensures tht the imuthl elevtionl ngles in the eternl world re proportionl to the imuthl elevtionl ngles in the mer. Tht is, θ = αθ φ = βφ where α β re the imuthl elevtionl +9 deg Elevtion (φo deg -9 deg -18 deg φ θ A Aimuth (θo Cmer nodl point θο=α.θ C B 18 deg φο= β.φ Fig. 2 Configurtion of equtoril view mirror gins, respetivel. Inserting these reltionships into (15-17, we otin the following epressions for the diretion osines of the refleted r: = osβφ. sinαθ (18 = sin βφ (19 = osβφ. osαθ (2 where φ = tn 1 ( (21 θ = tn 1 ( (22 The solution to the required mirror surfe, f (,, is otined inorporting (21 (22 into (18 - (2, inserting the resulting epressions for,, s well s the epressions for, into (12 (13. Equtions (12 (13 re then integrted numerill, with respet to, respetivel, fter speifing the oundr onditions t prtiulr point. Tpill, the oundr onditions would speif the distne of point on the surfe from the nodl point of the mer, the orienttion of the surfe t tht point. For emple, one ould speif the distne etween the mer s nodl point the surfe long the mer s optil is, onstrin the surfe to e perpendiulr to the optil is t this point. Tht is, f (, = r p, p = p = 1t = ( =, =. Using these oundr onditions, the profiles of the mirror long the es n e omputed integrting equtions (12 (13 with respet to, respetivel. We then ompute the solution to the surfe t other points (,. This n e done in two ws: Integrte (12 long the is using eh previousl omputed vlue long the is, in turn, s oundr ondition; or ( integrte (13 long the is using eh previousl omputed vlue long the is, in turn, s oundr ondition. It turns out tht the surfes derived using these two different proedures re not the sme (detils re not shown here due to lk of spe. A r tring eerise revels tht neither surfe stisfies the mpping properties tht re required of the mirror. This finding indites tht there is no smooth surfe tht will hieve the desired mpping. The reson is tht the lol surfe orienttion tht is required to produe the orret mpping t eh point is, in generl, different from the orienttion of the glol surfe profile t tht point. The ove prolem n e overome, however, llowing the surfe to e disontinuous. Suh solution is illustrted the mirror portred in Fig. 3, whose shpe hs een derived using the following proedure: (i ompute the profiles long the es s desried ove; (ii ompute the profiles of ross setions perpendiulr to the is integrting equn (13 using the vlues of the profile long the is s the initil onditions for suessive profiles; (iii determine the lol slope long the is of the surfe etween suessive ross setions using equn (12. The resulting mirror profile is surfe tht onsists of numer of strips, or rions tht re ontinuous long, ut displ swtooth-like disontinuities long. The swtooth-like profiles re espeill prominent t the orners of the mirror, where θ φ re lrge. Tht this surfe does indeed produe the desired mpping is illustrted in Fig. 5. This figure shows the reltionship etween the diretions θ φ of the rs tht impinge upon the mirror, the diretions θ φ of the orresponding rs tht enter the mer, for 25 points on the mirror surfe. The r-tring ws performed using progrm written in Mtl (v It is ler tht θ is linerl relted toθ, tht the slope of this reltionship grees with the desired vlue of α = 5. Similrl, φ is r

4 linerl relted toφ, the slope of this reltionship orresponds to the desired vlue of β = 3. Thus, the mirror hieves the desired mpping. Fig. 5 Illustrtion of mpping properties of equtoril view mirror shown in Fig. 3 Fig. 3 Solution for equtoril view mirror The opertion of the mirror is nlogous, in some ws, to the opertion of Fresnel lens, whih lso possesses swtooth-like surfe profile. In the se of the lens, light rs re refrted rther thn refleted. But there, gin, the lol surfe orienttion tht is required to refrt the r in the orret diretion is different from the glol orienttion of the surfe profile, leding to the disontinuities. The mirror is pled with its pe t the entre of the viewsphere. A pinhole mer, with visul field of 7 deg (horiontl 5 deg (vertil is positioned fing the mirror with its nodl point 1 unit from the pe. The mer-mirror sstem is loted t the position indited the white sphere in the enter of the ring frmework. Fig. 6 Strd ojet used in r-tring environment to test imging properties of the mirrors. Fig. 4 Close-up view of mirror of Fig. 3 The performne of the mirror of Fig. 3 is shown in Figs. 6-8, using r-tring environment reted through POVRAY. Fig. 6 shows n ojet tht ws designed to test the mpping produed the mirror. It onsists of 12 vertill oriented rings, eh of rdius 3 units thikness 3 units, seprted in imuth 3 deg. The onstitute referene longitudes in viewsphere. There re lso 5 horiontl rings, seprted n elevtionl ngle of 3 deg. The provide referene ltitudes in the viewsphere. The imge registered the mer is shown in Fig. 7. It is ler tht the imge refleted the mirror hieves the desired mpping. The longitudinl rings mp into vertil olumns, representing lines of onstnt imuth ( θ. The ltitude rings mp into horiontl rils tht represent lines of onstnt elevtion (. The imges of the longitudinl φ rings eome wider with inresed elevtion. This is s it should e, euse, lthough the rings re of uniform thikness, their imuthl sutense inreses s the elevtion of view is inresed, pprohing vlue of 3 deg ner the top of the viewsphere where the rings interset.

5 α =.4 β =. 6. Interestingl, the solution surfe ehiits no disontinuities when the gins re oth lower thn unit. The right-h pnel shows the imge ptured this surfe. The result is mirror tht hs the sme qulittive mpping properties s the mirror in Fig. 3, ut whih mgnifies rther thn minifies. Fig. 7 Wide-ngle imge ptured equtoril view mirror of Fig. 3 This mirror ptures totl imuthl field of. 21 deg totl elevtionl field of. 15 deg. The drk dis in the entre of the imge represents the refletion of the mer. The wide-ngle imging pit of the mirror is illustrted omprison of Fig. 7 8, where Fig. 8 shows mer imge of the sme sene quired without the use of the mirror. Fig. 9 Profile of mirror in whih the imuthl elevtionl gins re less thn unit Fig. 1 Imge ptured mirror of Fig. 9 B. Polr view mirrors Fig. 8 Cmer imge of test ojet without the use of mirror Nrrow-ngle imging In the ove emple the mirror gins (α β were greter thn unit, thus enling wide-ngle imging (i.e. enling the mer to pture lrger segment of the world thn it normll does. The opposite effet nrrow ngle imging n e hieved seleting vlues of α β tht re lower thn 1. Fig. 9 shows n emple of mirror surfe with gins The onfigurtion of this lss of mirrors is illustrted in Fig. 11. Here, the desired mpping is one in whih the optil is of the mer oinides with the polr is of the viewsphere. With this onfigurtion, points of onstnt elevtion ( φ in the outside world mp to lines tht re prllel to the is in the imge plne. These lines represent lines of onstnt in the mer (Fig. 11. φ Points of onstnt imuth ( θ in the outside world mp to lines tht re prllel to the is in the imge plne. These horiontl lines represent lines of onstnt θ in the mer. This mpping leds to the following onstrints for the d.. s of r tht is inident on the mirror:

6 (13 s efore, using the epressions (29-31 for., B θο=α.θ A φο= β.φ C Aimuth (θo 27 deg 18 deg deg Aimuth (θo 27 deg Fig. 12 Profile of polr view mirror 9 deg 3 deg 3deg Elevtion (φo θ φ 9 deg Cmer nodl point Fig. 11 Configurtion of polr view mirror = sinφ. osθ (26 = osφ (27 = sinφ. sinθ (28 The profile for surfe withα = 5., β = 5. r = 1 is shown in Fig. 12. A lose-up view is shown in Fig. 13. Polr-view mirrors re lso serrted ner the orners. The imging properties of this mirror re illustrted in Fig. 14. With this prtiulr mirror, the left hlf of the imge ptures n imuthl setor of. 21 deg, the right hlf ptures smmetril setor on the other side (there is some overlp in the imuthl views t the upper lower ends of the imge. The mirror ptures n elevtion of. 1 deg on either side. The mimum elevtion n e inresed, if desired, inresing the vlue of β. We require tht θ = α. θ φ = β. φ where α β re the imuthl elevtionl gins, respetivel (see Fig. 11. Inserting these reltionships into (26-28 we otin the following epressions for the diretion osines of the refleted r: = sinβφ. osαθ (29 = os βφ (3 = sinβφ. sinαθ (31 Denoting the profile of the mirror surfe = g(,, we ppl the oundr onditions g(, = r, where r is the distne of the surfe from the nodl point of the mer t (,, p, p = p = 1 t ( =, =. = The surfe is omputed integrting equtions (12 Fig. 13 Close-up view of mirror of Fig. 12

7 Fig. 14 Wide-ngle imge ptured polr view mirror of Fig. 12 [1] Y. Ygi, S. Kwto S. Tsuji. Rel time omnidiretionl imge sensor (opis for visionguided nvigtion. IEEE Trnstions on Rootis Automtion, 1: 11-22, [2] J.S. Chl M.V. Srinivsn. Refletive surfes for pnormi imging. Applied Optis, 36 (31: , [3] M. Ollis, H. Hermn S. Singh. Anlsis design of pnormi stereo vision using equingulr piel mers. Teh. Report CMU-RI-TR-99-4, Crnegie Mellon Universit, Pittsurgh, Pennslvni, [4] S. Nr. Ctdioptri omindiretionl mer. Proeedings, IEEE Conferene on Computer Vision Pttern Reognition, , [5] R. A. Hiks R. Bjs. Ctdioptri sensors tht pproimte wide-ngle perspetive projetions. Proeedings, IEEE Workshop on Omnidiretionl Vision, Hilton Hed Isl, South Crolin 97-13, [6] R. Benosmn, E. DeFors J. Devrs. A new tdioptri sensor for the pnormi vision of moile roots. Proeedings, IEEE Workshop on Omnidiretionl Vision, Hilton Hed Isl, South Crolin , [7] L.M. Kells. Anlti Geometr Clulus. Prentie Hll, In., p , 963. Conlusions future work This pper hs presented the design of lss of mirrors tht ( perform wide-ngle imging ( produe n imge tht represents the world in Crtesin o-ordinte sstem in whih the siss represents imuth the ordinte represents elevtion. The disdvntges of these mirrors re (i their mnufture n e somewht omple, given the serrted nture of the profiles (eept for the minifing mirrors, whih possess smooth surfe, (ii the presene of the serrtions ould degrde imge qulit. The dvntges of these mirrors re (i diret optil unwrping, thus eliminting the need for omputer; (ii the Crtesin mpping hieves uniform resolution in imuth elevtion ( onstnt numer of piels per degree of imuth or elevtion, nwhere in the imge (iii the mpping mkes full use of the mer s imging plne (there re no unused orners, unlike the sitution with the irulr imges ptured onventionl onil imging mirrors. Future work will involve the onstrution of prototpes to ssess imge qulit under rel onditions, to mesure prmeters of performne suh s resolution depth of field. Referenes