Camera Parameters INEL 6088 Computer Vision Camera Parameters - - PowerPoint PPT Presentation

Camera Parameters INEL 6088 Computer Vision

Camera Parameters • Extrinsic parameters: define the location and orientation of the camera with respect to the world reference frame • Intrinsic parameters: link the pixel coordinates of an image point to the corresponding coordinates in the camera reference frame • Main reference: chapter 2 of Introductory Techniques for 3D Computer Vision by Emanuele Trucco and Alessandro Verri

EXTRINSIC PARAMETERS • Identify the transformation between the unknown camera and known world reference frames. - 3-D translation vector, T , that relate the origins of the two frames - 3 × 3 rotation matrix, R orthogonal matrix (R T R=RR T =I) brings axes of the two frames into each other P c = R ( P w − T ) P c = R P w − T or

      1 0 0 c ϕ 0 s ϕ c θ − s θ 0 r x = 0 c ψ − s ψ r y = 0 1 0 r z = s θ c θ 0       s ψ c ψ − s ϕ c ϕ 0 0 0 0 1     r 11 r 12 r 13 c θ c ϕ − s θ c ϕ s ϕ  = r x r y r z = r 21 r 22 r 23 c θ s ψ s ϕ + c ψ s θ − s ψ s ϕ s θ + c ψ c θ − s ψ c ϕ R =    r 31 r 32 r 33 − c ψ s ϕ c θ + s ψ s θ c ψ s ϕ s θ + s ψ c θ c ψ c ϕ Rotation matrix in Wikipedia

Basic Properties 1. Any ray entering the lens parallel to the THIN LENS axis goes through the focus on the other side; 2. any ray entering the lens from the focus in one side emerges parallel to the axis on the other side 1 + 1 = 1 Thin lens equation: d o d i f

Perspective Projection

INTRINSIC PARAMETERS • o x , o y = Position of image centre (Principal Point) • Focal length f • s x , s y = pixel dimensions • k 1 = radial distortion coefficient

Perspective Camera Model ( x ′ � , y ′ � ) ( x , y , z ) World coordinates ( x , y , z ) Image plane coordinates ( x ′ � , y ′ � ) x ′ � = f x x ′ � f = x z ⇒ x ′ � = f x z ¯ y ′ � = f y z z ¯ y ′ � f = y z ⇒ y ′ � = f y Weak-perspective camera model z (difference in distance to scene Perspective camera model points is small compared to average distance)

Transformation between Camera and Image frame coordinates Ignoring the lenses’ geometric distortions and assuming that the sensor array is made of a rectangular grid of photosensitive elements, x ′ � = − ( x im − o x ) s x y ′ � = − ( y im − o y ) s y where • (x im , y im ) : coordinates of an image point, in pixels • (o x , o y ) : coordinates of the image center (the principal point), in pixels • (s x , s y ): effective pixel size (in millimetres) in the horizontal and vertical directions, respectively • (x',y') : coordinates if the same point (x im , y im ) in the camera reference plane Transformation between World and sensor coordinates P c = R ( P w − T )

P c = R ( P w − T ) T ( P w − T ) T ( P w − T ) x ′ � = − ( x im − o x ) s x = − f R 1 y ′ � = − ( y im − o y ) s y = − f R 2 T ( P w − T ) T ( P w − T ) R 3 R 3 [ r 11 r 21 r 31 ] [ t x x z ] − t y y t z T ( P w − T ) x ′ � = − ( x im − o x ) s x = − f R 1 = − f [ r 13 r 23 r 33 ] [ T ( P w − T ) R 3 t x x z ] − y t y t z [ r 12 r 22 r 32 ] [ t x x z ] − t y y t z T ( P w − T ) y ′ � = − ( y im − o y ) s y = − f R 2 = − f [ r 13 r 23 r 33 ] [ T ( P w − T ) R 3 t x x z ] − y t y t z

Define: − f / s x 0 o x Intrinsic parameter matrix: Transformation between camera M int = 0 − f / s y o x and image reference frame 0 0 1 T T r 11 r 12 r 13 − R 1 Extrinsic parameter matrix: Transformation between world T T M ext = r 21 r 22 r 23 − R 2 and camera reference frame T T r 31 r 32 r 33 − R 3 Linear Matrix Equation of Perspective Projections: X w (x im , y im ) : pixel x 1 x im = x 1 / x 3 coordinates Y w x 2 = M int M ext that Z w y im = x 2 / x 3 x 3 we measure 1

For simplicity, asume • (o x , o y ) = (0,0) • s x = s y = 1 Projection matrix M = M int M ext : T T − fr 11 − fr 12 − fr 13 f R 1 T T M = − fr 21 − fr 22 − fr 23 f R 2 T T r 31 r 32 r 33 − R 3 M describes the full perspective camera mode Weak-perspective camera model difference in distance to scene points is small compared to average distance T T − fr 11 − fr 12 − fr 13 f R 1 T T M = − fr 21 − fr 22 − fr 23 f R 2 T (¯ 0 0 0 R 3 P − T )

Distortion No distortion Pin cushion Barrel Radial distortion of the image • Caused by imperfect lenses • Deviations are most noticeable for rays that pass through the edge of the lens

Correcting radial distortion from Helmut Dersch

Modeling distortion Project 
 to “normalized” 
 image coordinates Apply radial distortion Apply focal length 
 translate image center To model lens distortion • Use above projection operation instead of standard projection matrix multiplication