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Image Rectification for Stereo Vision Charles Loop Zhengyou Zhang Microsoft Research Problem Statement Compute a pair of 2D projective transforms (homographies) Original images Rectified images Motivations To simplify stereo matching: Instead of comparing pixels on skew lines, we now only compare pixels on the same scan lines. Graphics applications: view morphing Problem: Rectifying homographies are not unique Goal: to develop a technique based on geometrically well-defined criteria minimizing image distortion due to rectification Epipolar Geometry M m m’ C C’ •Epipoles anywhere •Fundamental matrix F: a 3x3 rank-2 matrix •Epipole at i 1 0 •Fundamental matrix 0 0 0 F [i ] 0 0 1 0 1 0 0 T Stereo Image Rectification H and H’ such that Compute rectified image points: Compute Problem: H and H’ are not unique. Properties of H and H’ (I) Consider Recall: each row of H and H’ as a line: both e and e’ are sent to [1 0 0]T Observations (I): v and w must go through the epipole e v’ and w’ must go through the epipole e’ u and u’ are irrelevant to rectification Properties of H and H’ (II) Observation (II): Lines v and v’, and lines w and w’ must be corresponding epipolar lines. Observation (III): Lines w and w’ define the rectifying plane. Decomposition of H H HsHr H p Special projective transform: Similarity transform: Shearing transform: Special Projective Transform (I) Sends the epipole to infinity epipolar lines become parallel Captures all image distortion due to projective transformation Subgoal: Make Hp as affine as possible. Special Projective Transform (II) How to do it? Let original image point be the transformed point will be with weight Observation: If all weights are equal, then there is no distortion. Key idea: minimize the variation of wi over all pixels Similarity Transform Rotate and translate images such that the epipolar lines are horizontally aligned. Images are now rectified. Shearing Transform Free to scale and translate in the horizontal direction. Subgoal: Preserve original image resolution as close as possible. Example Original image pair Intermediate result After special projective transform: Intermediate result After similarity transform: Final result After shearing transform