《图像处理与计算机视觉 Image Processing and Computer Vision》课程教学资源（PPT课件讲稿）Chapter 02 Image processing and computer vision（Camera models and parameters）

Camera model) Intrinsic parameters I extrinsic parameters I projection Relate world 3d to camera 3D coordinates Pw=a point in World 3D coordinates=[XwYw Zu]' P= the same point in Camera 3d coordinates = C)C)C R=Rotate of the camera in world coord Ton=t=translation of the camera in world coord p=Rm(P-tam)=r(p-t We will show later wh R 1722 Tcam=t2=T R is more convenient to be used r3 2 Ch2. Cameras v0.b2 11

Camera model | Intrinsic parameters | extrinsic parameters | projection Relate world 3D to camera 3D coordinates • Pw=a point in World 3D coordinates=[Xw,Yw,Zw] T • Pc= the same point in Camera 3D coordinates =[Xc ,Yc ,Zc ] T • Rcam=Rotate of the camera in world coord. • Tcam =Tc=Translation of the camera in world coord. ( ) c cam c c cam w cam c w c T t t t T r r r r r r r r r R P R P T R P T =       =       = = − = − − 3 2 1 31 32 33 21 22 23 11 12 13 1 , ( ) ( ) We will show later why Rc is more convenient to be used Ch2. Cameras v.0.b2 11

Camera model) Intrinsic parameters I extrinsic parameters I projection Step1: Motion of camera from world to camera coordinates Camera motion(rotation=Rcam=R, translation=T) will cause change of pixel position/x, y, See p156(1/ camera center c-aXIs cam/ TC Y =RllYI-T---(,where C-axIs W-axIs c-axis w-axIs To the camera r31732733 t3 it is p X o the world W-ax World center p=y Lp=Y Z it is p Cameras v, 3d Ch2. Cameras v0.b2 12

Camera model | Intrinsic parameters | extrinsic parameters | projection Step1:Motion of camera from world to camera coordinates • Camera motion (rotation=Rcam=Rc -1 , translation=Tc ) will cause change of pixel position (x,y), See p156[1] Cameras v.3d Yc-axis Zc-axis Xc-axis Rcam,Tc World center Camera center           =           =           −           =           3 2 1 3 1 3 2 3 3 2 1 2 2 2 3 1 1 1 2 1 3 , ---(i), where t t t T r r r r r r r r r R T Z Y X R Z Y X c c c w w w c c c c Xw-axis Zw-axis Yw-axis y x z Ch2. Cameras v.0.b2 12 Pc it is To the camera, Pw it is To the world       =       = c c c c w w w w Z Y X P Z Y X P

Camera model Intrinsic parameters extrinsic parameters projection To learn more about rotation and translation The advantage of Z 313×l using the homogenous where R33=h21123,T=|2 coordinates 73173273 is to make the X X transform =RY R c3×3 c3×x3c3×l from world to camera 3×x1」3×l coordinates a X X inear one Raa*Tess )by, add l at the bottom for both sides/ Note: now Z the 3D X coordinates R T Yy of a point is ∥ this is the same as(i) 0 4x1, instead l×l 4×4 of 3x1 The 4xl vector[x Y z I] is in 3-D homogeneous Coordinates 13

Camera model | Intrinsic parameters | extrinsic parameters | projection To learn more about : Rotation and Translation • ( ) ( ) The 4x1 vector X Y Z 1 isin 3- D homogeneous coordinates. , // thisis the same as 1 0 1 1 ,add1at the bottom for both sides where , ( ) T 4 1 1 3 1 1 4 4 3 1 3 3 3 3 4 1 3 3 3 1 3 1 3 1 3 1 3 3 3 1 3 3 3 1 3 1 3 3 3 1 3 1 3 1 3 3 3 1 3 3 1 2 1 31 32 33 3 3 21 22 23 11 12 13 3 3 3 1 3 1 3 1 3 3 3 1 3 1 3 1 3 1 (i) Z YX R R T Z YX R T Z YX R Z YX R T Z YX T R Z YX R Z YX t t t T r r r r r r r r r R T i Z YX R Z YX w w w c c c c c c c c w w w c c c c c c w w w c c w w w c c c c c c c w w w c c c c                                             −  =       −              =       −       =             −       =             =       = − − −             −       =          The advantage of using the homogenous coordinates is to make the transform from world to camera coordinates a linear one. Note: now the 3D coordinates of a point is 4x1, instead of 3x1 Ch2. Cameras v.0.b2 13

Camera model) Intrinsic parameters I extrinsic parameters I projection Step 2 Projection of camera(F) Result> image(x,y) Ch2. Cameras v0.b2 14

Camera model | Intrinsic parameters | extrinsic parameters | projection Step2 Projection of camera (F) Result → image (x,y) Ch2. Cameras v.0.b2 14

Camera model) Intrinsic parameters I extrinsic parameters I projection Step2: Camera coordinates to image plane Perspective projection of camera in meters) X=F*X/Z---( yF“/z-(2) 1024x768 Focal lengthe F F=focal length in meters A Point in 3D space(camera reference space)is [X in meters The 2D image point is / x, y]in meters Ch2. Cameras v0.b2 15

Camera model | Intrinsic parameters | extrinsic parameters | projection Step2: Camera coordinates to image plane (Perspective projection of camera in meters) • x=F*Xc /Zc ---(1) • y=F*Yc /Zc ---(2) • F=focal length in meters • A Point in 3D space (camera reference space) is [Xc ,Yc ,Zc ] T in meters • The 2D image point is [x,y]T in meters Ch2. Cameras v.0.b2 15