Camera model) Intrinsic parameters I extrinsic parameters I projection Inspection exercise Explain what are these variables F=( focal length) C=ox, oy Zc= principal axis Ow=Camera center (u, v) image 2-D axes Xc,Yc, Zc)3-D axes Focal length is the length between the lens and the image screen Camera center is the origin of the camera coordinate system Principal axis is the vector perpendicular to the image and intersects with the camera center Image Center is the center the 2d image or the point that the principal axis intersects with the image Ch2. Cameras v.O. b2 6
Camera model | Intrinsic parameters | extrinsic parameters | projection Inspection exercise • Explain what are these variables. – F =(focal length) – C=(ox,oy) – Zc = principal axis – Ow =Camera center – (u,v) image 2-D axes – (Xc,Yc,Zc) 3-D axes • Focal length is the length between the lens and the image (screen) • Camera center is the origin of the camera coordinate system • Principal axis is the vector perpendicular to the image and intersects with the camera center • Image Center is the center the 2D image , or the point that the principal axis intersects with the image Ch2. Cameras v.0.b2 6
Camera model) Intrinsic parameters I extrinsic parameters I projection The most important concept for a camera is the Image formation process 3D Object points Xw YwZw world coordinates Step1 Motion of camera(Rcam=R, Tcam) Camera Coordinates result→Xc,YcZ Step2 Projection of camera(F Result> image(x,y) Ch2. Cameras v0.b2
Camera model | Intrinsic parameters | extrinsic parameters | projection The most important concept for a camera is the image formation process • 3D Object points (Xw,Yw,Zw): world coordinates Step2: Projection of camera (F) Result → image (x,y) Step1: Motion of camera (Rcam=Rc -1 ,Tcam) Camera Coordinates result → Xc ,Yc ,Zc Ch2. Cameras v.0.b2 7
Camera model) Intrinsic parameters I extrinsic parameters I projection Step1 Motion of camera (Rcam Tcam)Camera Coordinates result→XCYC,ZC Positive rotation angle about an axis: The thumb is pointing to the axis direction, a positive Camera coordinates rotation angle is the same direction as the other fingers (R) Thumb cam pointing and t cam Axis C direction Yw eh Right hand coordinates world coordinates) W Ch2 Came
Camera model | Intrinsic parameters | extrinsic parameters | projection Step1 Motion of camera (Rcam,Tcam) Camera Coordinates result → Xc,Yc,Zc Yc Zc Xc Xw Zw Yw Rcam=(Rc) -1 , and Tcam=Tc Right-hand coordinates (world coordinates) Positive rotation angle about an axis: The thumb is pointing to the axis direction, a positive rotation angle is the same direction as the other fingers. Thumb pointing Axis direction y x Camera coordinates z Ch2. Cameras v.0.b2 8
Camera model) Intrinsic parameters I extrinsic parameters I projection Rotation notations: Roll pitch yaw in aviation systems They are the same systems- you will see it if you turn the plane up side down). It is called the right hand system http://www.grcnasagov/www/k-12/aIrplanE/rotaTions.html Xw(Pitch angle Center of Pitch Axis pitch Roll Axis/V Yw +Yaw Yaw Axis Zw(Roll angle)V Right-hand coordinates Yw (Yaw angle)
Camera model | Intrinsic parameters | extrinsic parameters | projection Rotation notations: Roll pitch yaw in aviation systems They are the same systems– (you will see it if you turn the plane up side down). It is called the Right hand system. • Xw Right-hand coordinates Yw (Yaw angle) Zw (Roll angle) Xw (Pitch angle) http://www.grc.nasa.gov/WWW/k-12/airplane/rotations.html Zw Yw y x z Ch2. Cameras v.0.b2 9
Camera model Intrinsic parameters extrinsic parameters projection A revision of basic matrix operations in case%matlab you forgot them A=[138 62-9 1075] a=d e f=6 9 LB B=[210 1075 p q 346 7-2-1 g 346 A'=A=transpose of A=b e h=327 A AB note: transposition means rows become columns and columns become rows A+B A*B=d*j+e*m+f*pd*k+e*n+∫*qd*1+e·0+f*r Answer:>> gj+h'm+ip g'k+h'n+ig 81+h*0+i*r a= 138T210 1*2+3*7+8*31*1+3°(-2)+8*41*0+3*(-1)+8*6 1610 62-97-2-1=6*2+2*7+(-9)*36*1+2*(-2)+(-9)*46*0+2*(-1)+(-9)*6 327 10751346」L10*2+7*7+5*310+1+7*(-2)+5*410°*0+7*(-1)+5*6 8-95 472745 A*B= 472745 1-34-56 1+23+18+01348 841623 A+B=6+72 A+B= 10+37+45+6131111 348 130-10 If r a 3 x 3 rotation matrix, RT=R- andR*R=R*R=1=0 1 0 131111 Ch2. cameras v0.b2 10
Camera model | Intrinsic parameters | extrinsic parameters | projection A revision of basic matrix operations in case you forgot them = = = = = − + + + + − − − + + + + = = − + + + − + + − + + + − + − + − + − + − + + + − + + − + = − − = − + + + + + + + + + + + + + + + + + + = − = = = = = − − = = − = − − 0 0 1 0 1 0 1 0 0 If is a 3 3 rotation matrix, , and * * 13 11 11 13 0 10 3 4 8 10 3 7 4 5 6 6 7 2 2 9 1 1 2 3 1 8 0 84 16 23 1 - 34 - 56 47 27 45 10*2 7*7 5*3 10*1 7*( 2) 5*4 10*0 7*( 1) 5*6 6*2 2*7 ( 9) *3 6*1 2*( 2) ( 9) *4 6*0 2*( 1) ( 9) *6 1*2 3*7 8*3 1*1 3*( 2) 8*4 1*0 3*( 1) 8*6 3 4 6 7 2 1 2 1 0 10 7 5 6 2 9 1 3 8 * * * * * * * * * * * * * * * * * * * * * * * * * * * * note : transposition means rows become columns and columns become rows , 8 9 5 3 2 7 1 6 10 transpose of A 3 4 6 7 2 1 2 1 0 , 10 7 5 6 2 9 1 3 8 1 1 ' R R R R R R R I A B g j h m i p g k h n i q g l h o i r d j e m f p d k e n f q d l e o f r a j b m c p a k b n c q a l b o c r A B c f i b e h a d g A A p q r m n o j k l B g h i d e f a b c A T T T %matlab A=[1 3 8 6 2 -9 10 7 5] B=[2 1 0 7 -2 -1 3 4 6] A’ A*B A+B Answer:>> A’= 1 6 10 3 2 7 8 -9 5 A*B= 47 27 45 -1 -34 -56 84 16 23 A+B= 3 4 8 13 0 -10 13 11 11 Ch2. Cameras v.0.b2 10