In Color We Live C IP E I. In Color We Live. t n. In Color We Live

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1 I r We Live Eviro me I r We Live IP E I 0 12 I 2 T E vir ome 0 12 I 2 T IPEI Evirome IPEI 012 I 2 T I r We Live

2 Iesiy Idepede RGB-o-XYZ ur amera alibraio Bria FUNT, ad Pouya BSTNI School of ompuig Sciece, Simo Fraser Uiversiy Vacouver, aada BSTRT I order for a digial cur camera o represe he curs i he evirome accuraely, i is ecessary o calibrae he camera RGB oupus i erms of a crimeric space such as IE XYZ or srgb (Wyszecki 1976). ssumig ha he camera respose is a liear fucio of scee lumiace, he mai sep i he calibraio is o deermie a 3 3 liear rasformaio marix M mappig daa from liear camera RGB o XYZ. Deermiig M is usually doe by phoographig a calibraed arge, ofe a cur checker, ad he performig a leas-squares regressio o he differece bewee he camera s RGB digial cous from each cur checker pach ad heir correspodig rue XYZ values. I order o measure accuraely he XYZ coordiaes for each pach, eiher a compleely uiform lighig field is required, which ca be hard o accomplish, or he illumia power specrum icide o each pach mus be measured. I his paper we prese a opimizaio mehod for camera cur calibraio ha does o ecessiae cosa radiace across he scee, ye sill deermies a accurae cur correcio marix. 1. INTRODUTION Due o he fac ha he specral sesiiviy fucios of mos cameras do o correspod o ha of he huma eye, cameras ofe eed o be calibraed i order o produce oupus ha are cosise wih sadard cur spaces, such as srgb or IE XYZ. ssumig for he mome ha he camera respose is a liear fucio of scee lumiace, he mai sep i he calibraio is o deermie a 3 3 liear rasformaio marix M mappig daa from (liear) ucalibraed camera RGB o XYZ. No-liear gamma correcio is he added as a secod sep. Deermiig M is ofe doe by phoographig a calibraed arge such as he X-Rie Digial rhecker SG ad he performig a leas-squares regressio o he differece bewee he camera's RGB digial cous from each cur checker pach ad heir correspodig rue XYZ values. Oe difficuly wih his mehod is ha i is hard o creae a evirome i which he lighig is compleely uiform. If here is a sigle ligh source, he we ca assume ha he relaive specral power disribuio of he ligh is cosa; however, is irradiace is likely o vary across he cur checker ad his will affec he RGB digial cous. If he amou of variaio i he irradiace is ukow, he he rue XYZ values will o correcly model he scee ad so he cur correcio marix (M) M compued from hem will also be icorrec. differece i irradiace o a pach chages is lumiace ad hece resuls i a scalig of he associaed RGB iesiies (i.e., digial cous). I oher words, reaig RGB as a vecor, is legh chages, bu o is direcio. I radiioal leas-squares, he differece bewee rasformed RGBs ad XYZs is wha is miimized. 128

3 Oe way o accou for illumiaio variaio is o measure he irradiace a each pach, bu his exra sep ca be quie ime cosumig. I would ceraily be preferable o measure he illumia specrum a oly oe locaio ad have a mehod for derivig marix M ha works eve whe he irradiace is o-uiform. Jus such a illumia-idepede mehod is preseed here. The proposed mehod deals wih illumiaio variaio by miimizig he agle bewee rasformed RGBs ad XYZs. Ulike he radiioal leas-squares mehod, which akes io accou boh he direcio ad magiude of RGB vecors, ad hus is affeced by ay irradiace-iduced scalig; he proposed mehod seeks he 3 3 liear rasform ha bes aligs he vecors from RGB space wih hose i XYZ space, wihou regard o heir magiude. s a resul, he effec of irradiace is elimiaed from he miimizaio. 2. BKGROUND The camera cur calibraio process ivolves imagig a arge wih a camera. Le represe he 3 N marix of camera resposes for N paches. Similarly, le B represe he 3 Nmarix of he correspodig risimulus values compued usig he measured SPD of he illumia, he specral reflecace fucios of each pach, ad he IE cur machig fucios. oveioal leas-squares regressio miimizes E M = M B!. I is well kow ha he bes mappig is give by he Moore-Perose pseudo-iverse M = M!!!!. s is clear from he above equaio, boh he direcio ad magiude of RGB vecors represeed i affec he regressio resuls. We ex propose a magiude-idepede regressio echique ha is especially useful for camera calibraio uder circumsaces where he irradiace cao be guaraeed o be uiform across he arge. 3. INTENSITY-INDEPENDENT REGRESSION osider a se of camera RGBs, {a}!!!!, capured from N paches ad heir correspodig XYZ values, {b! }!!!!. The proposed echique fids a 3 3 liear rasform M mappig RGB o XYZ by miimizig! Ma E M = cos!!! b! M a! b!!!! This miimizaio akes io accou oly he agles bewee pairs of vecors. We used he Nelder-Mead (Nelder 1965) oliear opimizaio mehod o solve for he M. However, sice he miimizaio is performed wihou regard o he overall vecor magiudes, he resulig rasformaio marix M ca be arbirarily large or small i magiude. To correc for he overall magiude of M, he marix is rescaled so ha he sum of all is elemes is 3 i srgb space. This cosrai esures ha he rasformaio does o aler he overall image brighess. We ex show he resuls of performig camera calibraio usig syhesized daa by applyig leas-squares ad our proposed iesiy-idepede regressio. 129

4 4. EXPERIMENTS We performed wo ses of experimes o sudy he effeciveess of our proposed mehod i camera calibraio. I he firs experime, we syhesized a image of he ierior paches of he X-Rie Digial rhecker SG uder a o-uiform illumia (Figure 1a) wih irradiace-iduced image iesiy variaio as show i Figure 1b. The XYZ coordiaes of each pach are calculaed assumig cosa radiace across he scee i boh cases. a b c d Figure 1: a) syhesized image of he ierior paches of he cr checker uder o-uiform lighig b) iesiy variaio i he syhesized image c) capured RGB values of he cr checker paches i he ligh booh d) iesiy variaio i he capured image. The images here have bee modified for visualizaio. The acual image daa is liear. If we compue he M mappig camera RGB o XYZ space usig coveioal leassquares miimizaio, we obai he followig wo marices M! ad M! for he images syhesized uder uiform ad o-uiform lighig, respecively: M! = M! = The relaive differece bewee he wo marices, measured usig he Frobeius orm, is 2!!!!!! = 0.893, idicaig ha he calibraio resuls are impaced sigificaly by he!!!!!! irradiace variaio. However, usig our proposed echique, he camera calibraio resuls are uaffeced by he variaio i he scee irradiace, ad he wo Ms obaied for scees 1 ad 2 are ideical: M! = M! = Of course, i is o sufficie for he rasform simply o be iesiy idepede; we are also ieresed i seeig how well he rasform maps curs from camera RGB o XYZ. 130

5 For his purpose, we compue he M based o he shaded image ad measure he error o he uiformly li image usig IEDE2000. We repea his process for boh mehods. The resuls i Table 1 uder syhesized daa colum show ha our proposed calibraio echique has beer cur correcio properies ha he leas-squares mehod uder ouiform lighig due o he fac ha i is o affeced by he irradiace variaio across he scee. Syhesized Image Daa apured Image Daa mea ΔE!" max ΔE!" mea ΔE!" max ΔE!" Leas-Squares Iesiy Idepede Table 1: ΔE!" saisics based o calibraig he camera usig a image of he X-Rie Digial rhecker SG uder o-uiform lighig ad esig i o a uiformly li image. I addiio o usig syhesized images, we used a camera o acquire images ad he compued he M based o he capured image daa. To reduce he effecs of oise, he camera RGB values for each pach were deermied by averagig he values over he whole pach. The resulig values are displayed as he image show i Figure 1c. To measure he irradiace variaio across he cur checker, we replaced i wih a grey surface i he same locaio ad capured a secod image. The average RGBs from each pach locaio are show i Figure 1d. s before we calibraed he camera usig he origial (o-uiformly li) image ad also a image i which he iesiy was adjused o be uiform usig he backgroud irradiace variaio map. s expeced, he resuls showed ha while he wo Ms compued by he leas-squares mehod based o he origial versus iesiy-correced images differed sigificaly (Frobeius orm of 0.10), our iesiy-idepede regressio resuled i a ideical M i boh cases. The capured daa colum of Table 1 also shows ha he proposed mehod has beer correcio error ha he sadard leas-squares mehod. 5. ONLUSION We proposed a cr calibraio echique ha does o require uiform irradiace across he calibraio arge. While coveioal mehods, such as leas-squares regressio, ake io accou boh he magiude ad direcio of cr vecors i mappig ucalibraed camera RGB oupu o XYZ coordiaes, our mehod elimiaes he depedece o he scee irradiace ad resulig image iesiies by cosiderig oly vecor direcios. The mehod s effeciveess was demosraed o a o-uiformly li calibraio arge. KNOWLEDGEMENT The fiacial suppor of he Naural Scieces ad Egieerig Research oucil of aada is graefully ackowledged. REFERENES Wyszecki, G., ad W. S. Siles r sciece, coceps ad mehods, quaiaive daa ad formulas. New York: Joh Wiley ad Sos. J.. Nelder, R. Mead, simplex mehod for fucio miimizaio. omp. Joural 7, pp , 1965 ddress: Bria Fu, School of ompuig Sciece, Faculy of pplied Scieces Simo Fraser Uiversiy, 8888 Uiversiy Drive, Buraby, B, aada fu@sfu.ca, pbasai@cs.sfu.ca 131