This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Assignment1 of CMSC 733 By Assignment1 of CMSC 733 1. (Camera models) For a camera, the image x of a point X in space is given by: x = PX , with x and X homogeneous 3 and 4vectors respectively. (a) If PC = 0 , where C is a homogeneous 4vector, show that C is the camera center. ======================================================== Consider the point C and another point A in 3D space, any point on the line passing through A and C can be expressed as, X ( λ ) = λA + (1 λ ) C Give a mapping that maps X to x by matrix P, x can be represented as, x = PX ( λ ) = P ( λA + (1 λ ) C ) = λPA + (1 λ ) PC If PC = 0 , then we have x = λPA which means all points on the line AC are mapped to the same image point λPA . Thus this ray must be a ray through the camera center, and C is the homogeneous representation of the camera center. ======================================================== (b) The camera projection matrix P can be written as : P = KR [ I   ˜ C ] , where K is the calibration matrix, R is the rotation between the camera and world coordinate frames, and the camera center C is expressed as C = ( ˜ C, 1) . Show that the calibration matrix can be obtained from a RQ decomposition of the rst 3 × 3 sub matrix of the camera matrix P. ======================================================== On one hand, we know that the camera projection matrix P = KR [ I   ˜ C ] . K is the calibration matrix, which is also an upper triangular: K = α x s x β x y 1 And R is the orthogonal rotation matrix between the camera and world coordinate frames. On the other hand, given a 3 × 4 camera projection matrix P, then, P = [ M   M ˜ C ] = M [ I   ˜ C ] where M is the rst 3 × 3 sub matrix of P, and ( ˜ C, 1) is the camera center. By RQ decomposition of M, we obtain, M = AB , where A is an upper triangular with the last element of the last row is 1, and B is orthogonal matrix. Thus, P = M [ I   ˜ C ] = KR [ I   ˜ C ] As a result, we can see that the two matrix A and B of RQ decomposition of M match with the calibration matrix K and rotation matrix R respectively. Thus the calibration matrix K can be obtained from a RQ decomposition of the rst 3 × 3 sub matrix of the camera matrix P. ======================================================== 1 Assignment1 of CMSC 733 By (c) In some application we obtained the following camera matrix P. Find the camera center and the calibration parameters. P = 707 679 . 2 555 . 4 2898920 207 46 . 6 919 . 2 6325250 1 . 4 . 7 1 . 22 1837 ======================================================== Given the camera matrix P and the equation P = [ M   M ˜ C ] , we have, M = 707 679 . 2 555 . 4 207 46 . 6 919 . 2 1 . 4 . 7 1 . 22  M ˜ C =  2898920 6325250 1837 Then we can compute ˜ C by, 707 679 . 2 555 . 4 207 46 . 6 919 . 2 1 . 4 . 7 1 . 22 ˜ C =  2898920 6325250 1837 By solving the linear equations, we get,...
View
Full
Document
This note was uploaded on 01/12/2012 for the course CMSC 733 taught by Professor Staff during the Spring '08 term at Maryland.
 Spring '08
 staff

Click to edit the document details