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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,...
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 Spring '08
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 Linear Algebra, Matrices, Orthogonal matrix

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