Existing algorithms for camera calibration and metric reconstruction are not appropriate for image sets containing geometrically transformed images for which we cannot apply the camera constraints such as square or zero-skewed pixels. In this paper, we propose a framework to use scene constraints in the form of camera constraints. Our approach is based on image warping using images of parallelograms. We show that the warped image using parallelograms constrains the camera both intrinsically and extrinsically. Image warping converts the calibration problems of transformed images into the calibration problem with highly constrained cameras. In addition, it is possible to determine affine projection matrices from the images without explicit projective reconstruction. We introduce camera motion constraints of the warped image and a new parameterization of an infinite homography using the warping matrix. Combining the calibration and the affine reconstruction results in the fully metric reconstruction of scenes with geometrically transformed images. The feasibility of the proposed algorithm is tested with synthetic and real data. Finally, examples of metric reconstructions are shown from the geometrically transformed images obtained from the Internet.