Description
You may choose to write solutions by hand; in that case, please submit a scanned copy.
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Suppose that our image coordinate system has its origin at the bottom left corner, the x-axis is along the bottom-row (pointing to the right) and the y-axis to be pointing upwards at an angle of 85 degrees to the x-axis. Assume that the focal length is 25 millimeters and that pixel spacing along the x-axis is .04 millimeters and along the y-axis is .05 millimeters. Let the image be 1000 x 1000 pixels and the principal ray intersect the image plane in its center. For these conditions, derive the intrinsic matrix, K, which helps map a point, P, specified in the camera coordinate frame to the image coordinates (x, y, 1)T expressed in pixel units (ignore the issue of rounding off pixel coordinates to integers). Choose a convenient alignment of the axes of the camera coordinate frame.
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a) Show that a set of parallel lines converge to a common point, called a vanishing point, and that the location of this point is determined solely by the directions of the lines (expressed in the camera coordinate system).
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Now, consider sets of parallel lines lying in a plane. Show that the vanishing points corresponding to the lines of different orientation in the plane all lie on a common line, called the vanishing line. Derive an expression for the image of this vanishing line (note that the line will be a function of the orientation of the plane and the intrinsic parameters of the camera).
You are encouraged to use projective geometry formulations to help simplify the derivation.
