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Unformatted text preview: Pictorial Information Homework 2 1 Question 1: Calibrating cam1 The calibration process is separated into two steps: initialization and non-linear optimization. During the initialization step user manually selects four extreme corners of the grid. The coordinates of selected points are refined by a corner detection algorithm which searches for a corner in the vicinity of the point selected by user. Then given the number of squares in the grid the rest of the corner points of the grid are estimated. The closed form solution for the calibration parameters (intrinsic and extrinsic) is computed from these image points. Projection matrix P is obtained. The optimization step involves minimization of the reprojection error, which is defined as follows. Metric configuration of the grid is known: number of squares in the grid and length of each square along X and Y directions. Hence, placing the grid so that it lies in the Z = 0 plane we know 3D coordinates of the corners of the grid X i (note that the location of the grid with respect to the world coordinate frame needs to be fixed during the initialization step). We can reproject these points to the image plane using x i = P X i . The reprojection error of point X i is d ( x i , x i ), where d is the Euclidean distance and x i is the 2D coordinate of the corner point X i in the image obtained in the initialization step. The optimized projection matrix is the one that minimizes the sum of the reprojection errors for all grid corners P = argmin P X i d ( x i , x i ) Once P is obtained the coordinates of grid corners in the image are updated to x i = P X i . 1. Calibration parameters estimated after extracting grid corners with corner finder window size set to 5. Note that the estimated camera centre is way off from the expected [320 240] for a 640 480 image. Focal Length: fc = [ 569.89119 561.69978 ] +- [ 66.90937 73.86583 ] Principal point: cc = [ 356.36231 299.73559 ] +- [ 58.51278 104.49382 ] Skew: alpha_c = [ 0.00000 ] +- [ 0.00000 ] => angle of pixel axes = 90.00000 +- 0.00000 degrees Distortion: kc = [ -0.71192 0.72151-0.05281-0.00828 0.00000 ] +- [ 0.51963 2.40495 0.07955 0.03679 0.00000 ] Pixel error: err = [ 1.13242 1.04376 ] The numbers following +- (numerical errors/standard deviation??) of the corresponding parameters after the non-linear minimization of the reprojection error. The corresponding calibration matrix K is given by K = x s x y y 1 where x and y are the focal length of the camera expressed in units of horizontal and vertical pixels (the values are different if pixels are not perfect squares), s is the skew factor, x and y is the principal point expressed in pixels. In our case camera matrix is K = 569 . 89119 356 . 36231 561 . 69978 299 . 73559 1 1 Homework 2 2 2. Points that are further away from the camera are imaged at a lower resolution. If in addition to that the grid is imaged as a skewed rectangle this might lead to a situation where the corners between two...
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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