RESEARCH ON CLOSE-RANGE PHOTOGRAMMETRY WITH BIG ROTATION ANGLE

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1 RESEARCH ON CLOSE-RANGE PHOOGRAMMERY WIH BIG ROAION ANGLE Lu Jue a a he Departent of Surveyng and Geo-nforatcs Engneerng, ongj Unversty, Shangha, 9. - lujue985@6.co KEY WORDS: Bg Rotaton Angle; Colnearty Equaton; Coplanarty Equaton; Exteror Orentaton ABSRAC: hs paper studes the nfluence of bg rotaton angles on the classcal soluton of analytcal process. Close-range photograetry dffers fro tradtonal aeral photograetry, for the forer usually adopts oblque photography or convergent photography, whle the latter s slar to take vertcal photos wheren the orentaton eleents are very sall. In ths paper, an exploraton of the algorth n every step of the analytcal process fro dependent relatve orentaton to absolute orentaton s descrbed. In order to ephasze the dfferences, contrasts wll be ade between the soluton of the classcal aeral photography and the algorth of the close-range photograetry. A practcal close-range photograetry experent s presented to deonstrate ths new algorth. hrough the exaple, t s ponted that wth ths new algorth, the coordnates of the ground ponts and the exteror orentaton can be aurately calculated. he results ndcate that the algorth deduced n ths paper s applcable n the close-range photograetry wth bg rotaton angle.. INRODUCION Close-range photograetry deternes the shape, sze, and geoetrc poston of close targets wth photograetrc technology []. Close-range photograetry dffers fro tradtonal aeral photograetry n several ways. For exaple, the forer usually takes oblque or convergent photographs, so the rotaton angles n the exteror orentaton eleents are often very bg and the dscrepances between the adjacent caera statons three-densonal coordnates wll also be qute large, whle the latter s slar to takng vertcal photos wheren the orentaton eleents are very sall. So n the feld of closerange photograetry, perforance of every step of the analytcal process, such as dependent relatve orentaton, space ntersecton, strp foraton process, absolute orentaton and space resecton, etc, a splfed odel wth sall angles or a sple atheatcal odel can not be contnued to use to calculate the paraeters or coordnates that we need. In ths paper, an exploraton of the algorth n every step of the analytcal process n close-range photograetry s descrbed. Specal ephass s placed on the contrasts between the soluton of the classc aeral photography and the closerange photograetry. And correspondng exaples wll be presented to explan and verfy the algorth.. BASIC PRINCIPLE OF SPACE RESECION AND LS Collnearty condton equatons are essental n analytcal photograetry. he fundaental theory of the collnearty equatons s that the perspectve center, the age pont, and the correspondng object pont all le on a lne. Based on these equatons, age and object coordnate systes wll be related by three poston paraeters and three orentaton paraeters, called exteror orentatons []. No atter what forat of the three orentaton paraeters are present, they wll be plctly expressed n the nne eleents of a orthogonal rotaton atrx R. So the collnearty equatons can be expressed as follow: a ( X X b ( Y Y c ( Z Z a ( X X b ( Y Y c ( Z Z s s s x x = f a ( X X s b ( Y Y s c ( Z Z s s s s y y = f a ( X X s b ( Y Y s c ( Z Z s a R = b c a b c a b c Where ( x, y, f are the nteror orentatons. ( XYZ,, and ( X s, Ys, Zs are the object space coordnates of the ground ponts and the perspectve centres. a a a b b b c c c are the 9 eleents of the rotaton atrx. In resecton, once the nteror paraeters are known, the collnearty equatons can be eployed to deterne 6 exteror orentaton paraeters. Wth ore than control ponts, to estate better results, we lnearze the collnearty equatons and ntroduce the Least Square ethod []. However, good ntal values are requred for the LS estaton process to converge to approprate values []. In tradtonal aeral photograetry, the ntal values of three orentaton paraeters can be set at zeros, for ts vertcal anner [4]. But n close-range photograetry wth bg rotaton angles, f they are stll zeros, the process ay not converge to proper results. So we need to deterne the exteror orentatons wthout the knowledge of ther approxate values. In ths paper, the collnearty soluton based on the orthogonal atrx s appled. In ths ethod, the nne eleents of the atrx are used nstead of the three angles, wth three poston paraeters to take the roles of the unknown. he lnear observaton equatons can then be ade along wth sx ( ( 57

2 he Internatonal Archves of the Photograetry, Reote Sensng and Spatal Inforaton Scences. Vol. XXXVII. Part B6b. Bejng 8 addtonal condton equatons based on the orthogonal condtons of the rotaton atrx (RR =R R=I [5], the 6 orthogonal condtons are : a a a = b b b = c c c = a a b b c c= a a b b c c= a a b b c c= ( C Cx W a a a b b b c c c = a a b b c c a a b b c c a a b b c c a a a - b b b - c c c - W = aa bb aa bb aa bb (6 hen the adjustent wth condtons s appled. he expressons of the error equatons are: v= Ax-l vx lx x-( x v=, l v = y l = y y-( y x = [ dx s dys dzs da da da db db db dc dc dc] a a a a4 a5 a6 a7 a8 a9 a a a A = b b b b4 b5 b6 b7 b8 b9 b b b x x x x x x x x x x x x Xs Ys Zs a a a b b b c c c = y y y y y y y y y y y y Xs Ys Zs a a a b b b c c c (4 If Z = ( a ( X - XS b ( Y - YS c ( Z - ZS,then the eleents n the coeffcent atrx A are: a = ( a f a ( x- x /Z, b = ( a f a ( y- y /Z a = ( b f b ( x- x / Z, b = ( b f b ( y- y / Z a = ( c f c ( x- x /Z, b = ( c f c ( y- y/z a4 = - f ( X - XS / Z, b4 a5, b5 = - f ( X - XS / Z a6 = -( x- x ( X - XS / Z, b6 = -( y- y ( X - XS / Z a7 = - f ( Y - Y S / Z, b7 a8, b8 = - f ( Y - YS /Z a9 = -( x- x ( Y - YS / Z, b9 = -( y- y ( Y - YS / Z a = - f ( Z - ZS / Z, b a, b = - f ( Z- ZS / Z a = -( x- x ( Z - Z / Z, b = -( y- y ( Z - Z S / Z S (5 hs process has elnated the need for dfferentatng the nonlnear observaton equatons and t can be used for any angle regardless of the forat of the three ratos and what sze they are.. ALGORIHM OF HE ANALYICAL PROCESS WIH BIG ROAION ANGLES. Dependent Relatve Orentaton he odel coordnate syste n dependent relatve orentaton s parallel to the frst age s coordnate syste, leavng two poston paraeters and the three orentaton paraeters of the second age wth respect to the odel coordnate syste to be deterned. In one age par, f the left and the rght ages space coordnates are S -X Y Z and S -X Y Z, the coordnates of the age ponts A, A n these two coordnate systes are ( X, Y, Z and ( X, Y, Z, and the coordnates of the perspectve centre S n coordnate syste S -X Y Z are ( Bx, By, B z, then the coplanarty condton equatons can be expressed as follow: Bx By Bz Bx By Bz F = X Y Z = X Y Z X B Y B Z B X Y Z x y z In aeral photograety, snce the perspectve centres le n nearly a straght lne, the usual procedure s to extract the coponent of the base lne B x, and then B y, B z can be expressed by nute angles, lke μ and ν. Wth the other three rotaton angles, the fve relatve orentaton paraeters μ, ν, ϕ, ω,κ are fored. hen the coplanarty condton equatons are splfed as: (7 And the condton equatons can be wrtten as: tgν tgμ cos μ μ ν F = B X Y Z B X Y Z x x X Y Z X Y Z (8 Where ( X, and R s the rotaton atrx, Y, Z = R ( X, Y, Z fro rght age to left one, lke equaton (. hs ethod s 58

3 he Internatonal Archves of the Photograetry, Reote Sensng and Spatal Inforaton Scences. Vol. XXXVII. Part B6b. Bejng 8 avalable n the classc aeral photography, but n wde-angle photograetry, these fve angles (fve relatve orentaton paraeters are not sall any ore, so two probles ay be arsed. Frst, snce the values of the relatve orentaton paraeters ϕ, ω,κ are very bg, f the ntal values are not approprate enough, the LS estaton process of the nonlnear observaton equatons ay not converge to rght values. So the relatve orentaton soluton based on the orthogonal atrx s eployed. Second, n the aeral photography, μ ν are adopted nstead of tanμ tanν/cosμ, for the values of the two angles are sall. hs step can avod the need for dfferentatng the nonlnear functon. But n close-range photograetry, μ ν ay be bg angles, so the splfed forats are not stll sutable. he developent of the soluton s to calculate the three coponents of the base lne B x, B y and B z drectly, not usng the two angles μ ν. So we use twelve unknowns, three of whch are coponents of the base lne, whle the others are the nne eleents of the rotaton atrx. Snce relatve orentaton nvolves the deternaton of fve degrees of freedo, seven condton equatons wll be added after the observaton equatons: one for the ltaton of base lne coponents and the others based on the orthogonal condtons of the rotaton atrx. he error equatons are: v= Ax-l x = dbx dby dbz da da da db db db dc dc dc l= F = XYB z XZB y YZB x YZB x XYB z XZB y A= a a a a a a a a a a a a [ ] = Bx By Bz a a a b b b c c c (9 And the coeffcents of the data atrx A are: a = = YZ Y Z, a = = X Z X Z Bx By a = = X Y X Y, a = = ( x x ( B Z BY 4 y z Bz a a = = ( y y ( B Z BY, a = = ( f ( B Z BY 5 y z 6 a a a = = ( x x ( B X B Z, a = = ( y y ( B X B Z 7 z x 8 z x b b a = = ( f ( B X B Z, a = = ( x x ( B Y B X 9 z x x y b c a = = ( y y ( BY B X, a x y c y z = = ( f ( B Y B X x y c ( Bx By Bz = B a a a = b b b = c c c = a a b b c c a a b b c c a a b b c c ( Where B s the arbtrary constant, so the condton equatons can be expressed as: C Cx W Bx By Bz a a a b b b = c c c a a b b c c a a b b c W c a a b b c c Bx By Bz - B a a a - b b b - = c c c - aa bb aa bb aa bb ( By ths eans, the nonlnear functon can be changed nto lnear one as to elnate the process of dfferentatng and t can converge to the correct result quckly wthout the knowledge of the approxate values of the paraeters. he process of the deducton s not coplcated, and the constructon of the coeffcent of the adjustent equatons s clear and concse.. Intersecton Intersecton refers to the deternaton of a pont s poston n object space by ntersectng the age rays fro two or ore ages. Here the standard ethod wll be used that s the applcaton of the collnearty equatons, wth two equatons for each age of the pont. Wth ths ethod, ore than two ages can be perfored to calculate the coordnates of the odel ponts, akng t a strct soluton and not bounded by the nuber of photos [6]. x x x v x = dx dy X Y Z dz ( x ( x y y y v y = dx dy X Y Z dz ( y ( y ( he condtons are:. Brdgng of Models he essence of odel lnk s to calculate the scale between the two adjonng odles [6]. In aeral photograetry, the connectve condton s based on the equaton of the depths of the coon ponts [7]. 59

4 he Internatonal Archves of the Photograetry, Reote Sensng and Spatal Inforaton Scences. Vol. XXXVII. Part B6b. Bejng 8 In contrast to the tradtonal ethod, n close-range photograetry, parallax s not only appeared along the X axs, but also exsted along the Z axs. So the two adjonng stereo pars of photographs are ncongruous both n scale and orentaton. If the connectve condton s only the depths of the coon ponts, t s obvously not strct enough. So the threedensonal coordnates of the coon ponts wll be eployed to brdge the suessve odel, for they should have the sae coordnates n the sae syste. Wth followng equatons, the scale paraeter k can be calculated:.5 Absolute Orentaton he fnal step s the absolute orentaton. Analytcal absolute orentaton s based on the three-densonal slarty transfor. Seven paraeters are nvolved: a unfor scale λ, three translatons (X,Y,Z, and three rotatons (Φ, Ω, Κ. In classc aeral photography, the splfed odel wth sall angles s usually perfored. But n wde-angle photograety, t can not be appled for the bg rotaton angles any longer. X Bx X Y (,, Z By = k R ϕ ω κ Y Bz Z (4 he soluton here s also based on the orthogonal rotaton atrx. Instead of the calculaton of the three rotatons Φ, Ω,Κ drectly, the nne eleents of the atrx are used to establsh the lnear observaton equatons. So the adjustent wth condtons wll be appled [8]. Here (X are the coordnates of the odel pont n,y,z the stereo par of photographs. hese values sut for the age coordnate syste of the left age. ( Bx,,, By Bz ( ϕ,, ω κ are the relatve orentaton paraeters between the left age of the ( stereo par and the left age of the stereo par. So the unknown paraeter s only the scale k..4 Buldng the Integrated Suessve Model In close-angle photograetry, there are two nconsstences between the two adjacent odels. One s the scale, the other s the orentaton. If R and k s the rotaton atrx and the scale of the rght age to the left one, the R and k are the rotaton atrx and the scale to the frst age: In the process of calculaton, the frst step s to centralze the coordnates. he ntal values of X, Y, Z are the dfferences of the correspondng center of gravty coordnates n the two coordnate systes, λ =,R =I. he error equatons are: v= Ax l x = [ dx dy dz dλ da da da db db db dc dc dc ] ax ( ay ( az ( x ( y ( z ( A bx ( by ( bz ( x ( y ( z ( cx ( cy ( cz ( λ x( λ y ( λ z( λ λ λ λ λ λ X a a a x ( X( l= Y λ b b b y ( - Y( Z c c c z( Z( (8 R ( = R( = a( a( a( R ( = R( R( = R( = b( b( b( c( c( c( R ( j = R( j R( j R( R(, ( j =,4..., n k ( = k( = k ( = k( k( = k( k ( j = k( j k( j k( k(, ( j =,4,..., n On ths bass, the coordnates of every perspectve center n odel coordnate syste ( XXs(j,YYs(j,ZZs(j can be deduced as: XXs(j XXs(j Bx (j YYs(j YYs(j k (j R = (j By (j ZZs(j ZZs(j Bz (j (j =,,...,n (5 (6 (7 ( X (, Y (, Z ( denotes the object space coordnates of the control ponts, ( x(, y(, z( the relatve odel space coordnates. hs odel of three-densonal datu transforaton can sut any angle. 4. CASE SUDY In ths secton, the precedng approaches n close-range photograetry wth bg rotaton angles wll be studed. he target s a sple house, and 4 photos are taken, as expressed n Fgure, where the red ponts are perspectve centers, the blue ponts are known ground ponts, the green lnes represent the an lnes of the house, and the yellow lnes are for the wndows. he span of the house s fro (,, to (,,, the coordnate of the top s (5,5,. Randoly select No., No. 4 and No. 5 ponts as ground control ponts, and the resdual ones perfor as check ponts. Interor orentatons are ( x, y, f = (,,5, and the exteror orentatons are descrbed n able. he errors n age coordnates are about.. After ths process, every odel coordnate syste n the strp s parallel to the frst age s coordnate syste. So the odel coordnates of all unknown ponts can be calculated wth the applcaton of the standard ethod of ntersecton usng the collnearty equatons. 6

5 he Internatonal Archves of the Photograetry, Reote Sensng and Spatal Inforaton Scences. Vol. XXXVII. Part B6b. Bejng 8 Iage No. dxs ( dys ( dzs ( Dφ ( Dω ( Dκ ( Fgure. Iage for experental data Iage Xs( Ys( Zs( φ( ω( κ( No able. Gven exteror orentatons of every age Wth the algorths n the precedng secton, the object space coordnates of the unknown ponts can be calculated. wo ethods wll be presented to evaluate the result. Frstly, the auracy s evaluated usng check ponts whose world coordnates are known but not used as control n the soluton. he root ean square of the dfferences between the coputed coordnates and the known values provdes a easure of the soluton auracy. he dstrbute of the errors of the check ponts s lke Fgure : able. Dfferences of the exteror orentatons between the coputed result and the known values Fro able, we can see the result s approprate. More data have been analyzed outsde ths paper, and the outcoe s also correct. So they verfy the valdty and robustness of the algorths n close-range photograetry wth bg rotaton angles n ths paper. 5. CONCLUSION he algorths for close-range photograetry wth bg rotaton angles have been developed n ths paper to perfor the analytcal process, ncludng dependent relatve orentaton, space ntersecton, strp foraton process, absolute orentaton and space resecton, etc. Usng these algorths, we have obtaned hgh auracy and robust results. So they verfy the valdty and robustness of the algorths deduced n ths research paper. Copared wth the approaches of the analytcal process n classc aeral photograetry, the algorths n ths paper have such advantages: ( hey are applcable n close-range photograetry wth bg rotaton angles and they dffer fro the ethods used n classc photograetery because the algorths n ths paper are strcter, but not eploy the splfed odels. Fgure. Plot of check pont error Fgure ndcates that the orentatons of the errors are rando and the absolute values are sall, and the axal one s No., the values are ax( dx, dy, dz = (.89,.8,-.66. If the nuber of check ponts s n check, then the ean square errors are: dx dy dz ( σ X, σy, σ Z = (,, = (.46,.,.79 n n n check check check Secondly, usng the soluton of space resecton wth bg rotaton angles (descrbed n secton to calculate the exteror orentatons of every age, and ake a coparson between the coputed values and the gven ones. he result s expressed n able. ( hey contnue to be sutable when the absolute values of the dfference between the adjacent caera statons threedensonal coordnates are qute large. Wth these algorths, the ages can be ted nto a suessve odel lke a strp, so fewer control ponts (ore than are needed to perfor the absolute orentaton. It reduces the aount of the feld work. 6. ACKNOWLEDGEMENS he paper s substantally supported by Natonal Hgh echnology Research and Developent Progra of Chna (86 Progra Coplcated Features Extracton and Analyss fro Central Dstrcts n Metropols, ChangJang Scholars Progra, Mnstry of Educaton, PRC and ongj Unversty 985 Project Sub-syste Integrated Montorng for Cty Spatal Inforaton. he author would lke to thank ChangJang Scholars Dr. Ron., L and Dr. Hongxng Sun for ther substantal help. REFERENCE Aero Photograetry. Bejng: People s Lberaton Ary Press, pp

6 he Internatonal Archves of the Photograetry, Reote Sensng and Spatal Inforaton Scences. Vol. XXXVII. Part B6b. Bejng 8 Chen Y, Shen Yun-zhong, Lu Da-je. A Splfed Model of hree Densonal-Datu ransforaton Adapted to Bg Rotaton Angle, 4. Geoatcs and Inforaton Scence of Wuhan Unversty, 9: pp. -5. Chen Y, 4. New Coputaton Method of Collnearty Equaton Sutng Dgtal Photograetry. Journal of ongj Unversty, (5: pp Edward M. Mkhal, Jaes S. Bethel, J. Chrs McGlone,. Introducton to Modern Photograetry [M]. Wley, Har/Cdr edton, pp Feng Wen-hao,. Close Range Photograetry-he Rules for Photography wth the Shape and Moton State of Objects. Wu Han: Wuhan Unversty Press, pp. -4. L De-ren, Zheng Zhao-bao, 99. Analytcal Photograetry. Bejng: Surveyng Press, pp Qan Ceng-bo, Lu Jng-z, Xao Guo-chao, 99 L De-ren, Zhou Yue-qn, Jn We-xan,. he Outlne of Photograetry and Reote Sensng. Bejng: Surveyng Press, pp Wang Zh-Zhuo, 984. he Prncple of Photograetry. Wu Han: Surveyng Press, pp