CS4495/6495 Introduction to Computer Vision 3C-L2 Intrinsic camera - PowerPoint PPT Presentation

CS4495/6495 Introduction to Computer Vision 3C-L2 Intrinsic camera calibration

Geometric Camera calibration Composed of 2 transformations: • From some (arbitrary) world coordinate system to the camera’s 3D coordinate system. Extrinisic parameters (or camera pose)

Camera Pose 𝑧 𝑦 𝑨 𝑨 Camera c w T Coordinates World Coordinates 𝑧 𝑦

From World to Camera          |         C C C W p R t p Homogeneous        W W coordinates          |       0 0 0 1       From world to camera is the extrinsic parameter matrix (4x4) (sometimes 3x4 if using for next step in projection – not worrying about inversion)

Geometric Camera calibration Composed of 2 transformations: • From some (arbitrary) world coordinate system to the camera’s 3D coordinate system. Extrinisic parameters (or camera pose) • From the 3D coordinates in the camera frame to the 2D image plane via projection. Intrinisic parameters

Ideal intrinsic parameters Ideal Perspective projection: x  u f z y  v f z

Real intrinsic parameters (1) But “pixels” are in some arbitrary spatial units x   u z y   v z

Real intrinsic parameters (2) Maybe pixels are not square x   u z y   v z

Real intrinsic parameters (3) We don’t know the origin of our camera pixel coordinates x    u u 0 z y    v v 0 z

Really ugly intrinsic parameters (4) May be skew between camera pixel axes v  v u   u    sin( ) v v         cos( ) cot( ) u u v u v

Really ugly intrinsic parameters (4) May be skew between camera pixel axes x y       cot( ) u u 0 z z v  v u    y u   v v    sin( ) v v  0         sin( ) z cos( ) cot( ) u u v u v

Intrinsic parameters, non-homogeneous coords Notice division by z x y       cot( ) u u 0 z z  y   v v  0 sin( ) z

Intrinsic parameters, homogeneous coords         x cot( ) 0 u   0 * z u        y      * 0 0 z v v       0  sin( ) z        z  0 0 1 0 1     C p  p' K In camera- In homogeneous Intrinsic based 3D pixels matrix coords

Kinder, gentler intrinsics • Can use simpler notation for intrinsics – remove last column which is zero: f – focal length   f s c s – skew x     a – aspect ratio 0 K a f c  y c x, c y - offset   0 0 1   (5 DOF)

Kinder, gentler intrinsics • If square pixels, no skew, and optical center is in the center (assume origin in the middle):   0 0 f In this case   only one DOF,   0 0 K f  focal length f   0 0 1  

Kinder, gentler intrinsics • Can use simpler notation for intrinsics – remove last column which is zero: f – focal length   f s c s – skew x     a – aspect ratio 0 K a f c  y c x, c y - offset   0 0 1   (5 DOF)

Quiz The intrinsics have the following: a focal length, a pixel x size, a pixel y size, two offsets and a skew. That’s 6. But we’ve said there are only 5 DOFS. What happened: Because f always multiplies the pixel sizes, those 3 a) numbers are really only 2 DOFs. In modern cameras, the skew is always zero so we b) don’t count it. In CCDs or CMOS cameras, the aspect is carefully c) controlled to be 1.0, so it is no longer modeled.

Combining extrinsic and intrinsic calibration parameters  C ' K p p Intrinsic Pixels World 3D          coordinates |       Camera 3D   C C C W Extrinsic p R t p       coordinates  W W          |          0 0 0 1   

Combining extrinsic and intrinsic calibration parameters    C C W ' p K R t p W W K 3x3 3x4  W ' p M p

Other ways to write the same equation pixel coordinates Conversion back world coordinates from W p p  homogeneous ' M coordinates   W p        T x   m P * . . . u s u m  1 1   u W     p     T y m P * . . . v s v m         2 3 W p        m P T   z 1 . . .    s  m      2 v 3  1   m P 3 projectively similar

Finally: Camera parameters • A camera (and its matrix) M (or Π ) is described by several parameters • Translation T of the optical center from the origin of world coordinates • Rotation R of the camera system • focal length and aspect (f, a) [or pixel size (s x , s y ) ] , principle point ( x’ c , y’ c ) , and skew (s) • blue parameters are called “ extrinsics ,” red are “ intrinsics ”

Finally: Camera parameters • Projection equation – the cumulative effect of all parameters:   X      * * * * sx       Y  x M X   * * * * sy       Z       * * * *  s     1 

Finally: Camera parameters • Projection equation – the cumulative effect of all parameters:      ' 1 0 0 0 f s x    c R I T     0 M      3 1 x 3 1 x 0 ' 0 1 0 0 af y 3 3 3 3 x x      c    1 1  0   0  (3x4)     0 0 1 0 0 1 0 1 3 x 1 3 x     intrinsics projection rotation translation DoFs: 5+0+3+3 = 11