Derivation of Tranformation Parameter Computation for a 2D/3D Coordinates System from Laser Scanner Coordinates to Camera Coordinates

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1 Dervatn Tranratn Paraeter Cputatn r a D/3D Crdnate te r Laer anner Crdnate t Caera Crdnate Cnventn: In a 3D rdnate te wll be ued r bjet rdnate n the anner rdnate te. Th the rdnate te r whh the tranratn ade. wll be ued r the bjet rdnate epreed n the aera rdnate te at the latn the age plane n the aera CCD enr. Th the rdnate te t whh the anner rdnate are tranred. wll be ued r the bjet rdnate tranred t a rdnate te whh ha a t rgn the aera rdnate te rgn. are unaeted b the aera len pt. wll be ued r the tranlatn needed t plae the rgn the anner rdnate te at the rgn the aera rdnate te. In a 3D rdnate te Oega wll derbe rtatn abut the -a Ph Ф wll derbe rtatn abut the -a and Kappa wll derbe rtatn abut the - a. I Dervatn the 3D tranratn atr A rght-hand rdnate te wth rght hand rtatn wll be ued. M Rtatn abut the aera -a M Rtatn abut the aera -a M Rtatn abut the aera -a A gle rtatn abut the -a wuld have the r hwn belw. M Rtatn abut the three aera ae an be dne n an rder but ut then be dne ntentl thrughut the putatn n the ae rder. Generall atr ultplatn nt trantve that [M] [N] de nt equal [N] [M]. Rtatn abut the -a llwed b rtatn abut the and -ae the n equene Lnel hte 5/3/7

2 that ued. Thu a tranratn r the anner rdnate te t the aera rdnate te aplhed wth the llwng atr peratn. The ull tranratn wuld have the r hwn belw. M M M r epreed n plete r belw The bned tranratn atr M an be epreed a M M M M. And the tranratn r anner t aera rdnate an be epreed a M Multplng ut the rtatn atre M M and M hwn belw M Eah eleent the atr naed ardng t t' rw/lun ptn Lnel hte 5/3/7

3 Lnel hte 5/3/7 3 The aera rdnate te ha the rgn lated at the al pnt the len. The - a the aera parallel t the a the len te and perpendular t the CCD enr n the aera. The ptve dretn the -a tward the bjet beng phtgraphed. ne the CCD enr lated behnd the al pnt the CCD enr lated at a -rdnate [: al length the len]. The CCD enr repreent the - plane the aera rdnate te and rthgnal t the -a. Nte that wth the CCD enr - plane lated behnd the al pnt nvern n the and denn ur. Th rreted b vrtuall vng the CCD enr a dtane n rnt the al pnt. The abve tranratn atr tranr the anner rdnate t aera rdnate but de nt ale the denn t the ale the CCD enr. That the dennal redutn that ur due t the aera len nt releted n the tranratn atr at th pnt. An age denn related t the bjet denn b the rat the al length the len and the dtane t the bjet. g the relatnhp lar trangle a hwn belw / where the al length the aera the dtane r the al pnt the aera t the bjet and the denn the bjet. The and value wll be aled b the rat /. Thu the atr equatn abve when tranlated t the plane the CCD enr an be epreed a The abve plete tranratn atr epreed n equatn r a 3 3 3

4 hat are alled the llneart equatn are red b ubttutng r n the equatn r and. The llneart equatn are It wll be neear t negate the value abve t ae the aera and the anner rdnate te patble. Th wll be dne at the end the dervatn. The llwng agnent are ade n rder t pl the ntatn later reer t the nueratr and dennatr the abve equatn and are the rdnate a pnt n the aera CCD enr wth the rgn the rdnate te at the enter the CCD enr. Thee are nt rdnate. rdnate have the rgn at the upper let rner a phtgraph and have value between and.. and have denn whh relate t the e the CCD enr and range r -/ the enr wdth t / the enr wdth and r -/ the enr heght t / the enr heght. In lvng r the tranratn paraeter Φ and ur pnt are dented n between the phtgraph and the 3D del. Thu we have ur et value r and. e d nt have value r n the aera rdnate te. ubttutng r n the equatn r and the need t lve r elnated. Crtal athatal untn t lve r the tranratn paraeter A Talr' ere epann ued t lneare the llneart equatn that an nteratve leat quare ethd an be ued t lve r the paraeter. Onl the rt tw eber the Talr' ere are ued. a a ' a Thu a b... a a a a a a d b b b b b b d d d d d d d d d d In general r the dervatve an untn where u/v gven a d u du u dv d v v d v^ d d Lnel hte 5/3/7 4

5 Lnel hte 5/3/7 5 The abve epren r and are lnear equatn and an be ued n a leat quare ethd etatng the tranratn paraeter. An eaple the partal dervatve n the lutn prvded belw. a ^ There wll be a lar epren r the dervatve wth repet t eah the tranratn paraeter Φ. The Talr' ere epann r the llneart equatn are epreed n part belw.... ^ ^ d d a... ^ ^ d d b here a And b The llwng epren wll be ued t pl the Talr' ere: ^ 6 ^ 5 ^ 4 ^ 3 ^ ^ ^ 6 ^ 5 ^ 4 ^ ^ ^

6 Lnel hte 5/3/7 6 The Talr' ere epreed n pled r: a b And then epreed n atr r b a Epreed n the r neear r an teratve leat quare putatn b a Eah the partal dervatve -> 6 wll be derved.

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10 Lnel hte 5/3/7 The llwng are the plete epren r the eent thrugh 6. Nte that thrugh 6 are evaluated wth the paraeter und durng the prevu teratn The nal dered r F F

11 Lnel hte 5/3/ The r requred r the teratve lutn the tranratn paraeter F g atr algebra the unnwn ntaned n the atr an be lved r wth the llwng equatn. F At the pletn eah teratn the unnwn Φ are puted a hwn belw. Then the net teratn ue the new value r Φ and the arr the ubrpt " ". The abve equatn are repreented n the MatLab prgra naed d3.. Thee equatn wll nt prdue the tranratn paraeter neear t drape the phtgraph ver the TIN del the Regel anner pnt lud. It neear t negate the value n rder t aheve that tranratn.

12 Lnel hte 5/3/7 The equatn neear r th tranratn llw F F Nte that the value n the - atr here have been ultpled b -. MatLab prgra d3n. prvde the neear tranratn paraeter.