Camera Geometry & Calibration

Transcription

1 1 Camera Geometry & Calibration CS 554 Computer Vision Pinar Duygulu Bilkent University

2 Coordinate systems 2 Zw X Xw P Yw O Y Z x f p y We will use WORLD, CAMERA and Image Coordinate Systems. Adapted from Octavia Camps

3 Geometric Camera Models 3 Issue camera may not be at the origin, looking down the z-axis extrinsic parameters one unit in camera coordinates may not be the same as one unit in world coordinates intrinsic parameters Intrinsic parameters Do not depend on the camera location Focal length, CCD dimensions, lens distortion Extrinsic parameters Depend on the camera location Translation, and Rotation parameters

4 Notions of Geometry 4 Homogeneous coordinates Matrix representation of geometric transformations Extrinsic and intrinsic parameters that relate the world and the camera coordinate frames

5 Reminder 5 Dot product When u has unit norm u.v is sign length of projection of v onto u Cross product u x v is orthogonal to these two If u and v have same direction u x v = 0 Forsyth & Ponce

6 Homogeneous coordinates 6 Adapted from David Forsyth, UC Berkeley

7 Homogeneous coordinates 7 Adapted from Trevor Darrell, MIT

8 Homogeneous coordinates 8 Adapted from Trevor Darrell, MIT

9 Pinhole Camera Model 9 X π Image plane Optical axis O f Z P=(x,f) x P=(X,Z) X x = f X Z Adapted from Octavia Camps x = f X Z

10 Pinhole Camera Model 10 X Y O x Z p y f P x y = = f f X Z Y Z Adapted from Octavia Camps

11 11 Perspective Matrix Equation Z Y f y Z X f x = = Using homogeneous coordinates: Using homogeneous coordinates: ' ' ' ' z y y z x x = = ' ' ' = Z Y X f f z y x Adapted from Octavia Camps

12 Perspective Matrix Equation CS554 Computer Vision Pinar Duygulu 12 Adapted from Gregory Hager, JHU

13 Weak Perspective Model CS554 Computer Vision Pinar Duygulu 13 P y Y x p f Z=Zo + δ O x = f X / Zo y = f Y / Zo Object Object depth δ << Camera distance Zo Linear Linear equations!! Adapted from Octavia Camps, PennState

14 Model for Weak Perspective Projection 14

15 Orthographic Projection 15 Forsyth & Ponce

16 The projection matrix for orthographic projection 16 Adapted from David Forsyth, UC Berkeley

17 Weak Perspective vs Ortographic Projection 17 Weak perspective = Orthographic projection + Isotropic Scaling Adapted from Octavia Camps, PennState

18 Camera parameters 18 Intrinsic parameters Focal length, principal point, aspect ratio, angle between axes Extrinsic parameters Translation, and Rotation parameters X U Transformation Transformation V = representing representing Y Z W intrinsic parameters extrinsic parameters T Adapted from David Forsyth, UC Berkeley

19 Intrinsic parameters 19 Adapted from Trevor Darrell, MIT

20 Intrinsic parameters focal length 20 p = M int. P Adapted from Octavia Camps, PennState

21 Intrinsic parameters aspect ratio The CCD sensor is made of a rectangular grid nxm of photosensors. Each photosensor generates an analog signal that is digitized by a frame grabber into an array of NxM pixels. 21 Pixels may not be square vs [ α ] M int = [ 0 β 0 0 ] [ 0 0 1/f 0 ] Adapted from Octavia Camps, PennState

22 Intrinsic parameters - origin 22 [ α 0 uo 0 ] M int = [ 0 β vo 0 ] [ 0 0 1/f 0 ]

23 Intrinsic parameters angle between axes 23

24 Intrinsic parameters 24 Adapted from Trevor Darrell, MIT

25 Extrinsic parameters Translation and rotation of camera frame 25 Adapted from Trevor Darrell, MIT

26 3D Rotation of Coordinates Systems 26 Rotation around the coordinate axes, clock-clockwise: X γ X Z,Z Y Y R R R x y z ( α ) = ( β ) = ( γ ) = cosα sin α 0 sin α cosα cos β 0 sin β sin β 0 cos β cosγ sin γ 0 sin γ cosγ Adapted from Octavia Camps

27 3D Translation of Coordinate Systems 27 Translate by a vector t=(t x,t y,t x ) T : z t z y T Y = t t t 1 x y z x x Adapted from Octavia Camps CS554 Computer Vision Pinar Duygulu

28 Combining Extrinsic and Intrinsic Parameters 28 y P Zw Y x p X R O Z f Xw T Yw P p = R T P = M P w ext w = M P = M M int int ext P w Adapted from Octavia Camps

29 Combining Extrinsic and Intrinsic parameters 29 Forsyth & Ponce

30 Combining Extrinsic and Intrinsic parameters 30 Forsyth & Ponce

31 Camera Calibration 31 Compute the camera intrinsic and extrinsic parameters using only observed camera data Place a known object in the scene identify correspondence between image and scene compute mapping from scene to image

32 Camera Calibration 32 Adapted from Trevor Darrell

33 Camera Calibration 33 Adapted from Trevor Darrell

34 Camera Calibration 34 M has 12 entries each image point provides 2 equations Can solve m ij 's by Least Square Solution Adapted from Trevor Darrell