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Published byEvan Forbes Modified over 3 years ago

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Image Rectification for Stereo Vision Charles Loop Zhengyou Zhang Microsoft Research

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Problem Statement u Compute a pair of 2D projective transforms (homographies) Original images Rectified images

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Motivations u To simplify stereo matching: Instead of comparing pixels on skew lines, we now only compare pixels on the same scan lines. u Graphics applications: view morphing u Problem: Rectifying homographies are not unique u Goal: to develop a technique based on geometrically well-defined criteria minimizing image distortion due to rectification

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Epipolar Geometry M C C m m Epipoles anywhere Fundamental matrix F: a 3x3 rank-2 matrix Epipole at Fundamental matrix

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Stereo Image Rectification u Compute H and H such that u Compute rectified image points: u Problem: H and H are not unique.

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Properties of H and H (I) u Consider each row of H and H as a line: u Recall: both e and e are sent to [1 0 0] T u Observations (I): Ý v and w must go through the epipole e Ý u and u are irrelevant to rectification

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Properties of H and H (II) u Observation (II): Lines v and v, and lines w and w must be corresponding epipolar lines. u Observation (III): Lines w and w define the rectifying plane.

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Decomposition of H u Special projective transform: u Similarity transform: u Shearing transform:

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Special Projective Transform (I) u Sends the epipole to infinity þ epipolar lines become parallel u Captures all image distortion due to projective transformation u Subgoal: Make H p as affine as possible.

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Special Projective Transform (II) How to do it? u Let original image point be u the transformed point will be u Observation: If all weights are equal, then there is no distortion. u Key idea: minimize the variation of w i over all pixels with weight

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Similarity Transform u Rotate and translate images such that the epipolar lines are horizontally aligned. u Images are now rectified.

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Shearing Transform u Free to scale and translate in the horizontal direction. u Subgoal: Preserve original image resolution as close as possible.

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Example u Original image pair

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Intermediate result u After special projective transform:

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Intermediate result u After similarity transform:

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Final result u After shearing transform

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