Image Rectification for Stereo Vision Charles Loop Zhengyou Zhang Microsoft Research.
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Transcript of Image Rectification for Stereo Vision Charles Loop Zhengyou Zhang Microsoft Research.
Image Rectification Image Rectification for Stereo Visionfor Stereo Vision
Charles LoopCharles Loop
Zhengyou ZhangZhengyou Zhang
Microsoft ResearchMicrosoft Research
Problem Statement
Compute a pair of 2D projective transforms Compute a pair of 2D projective transforms ((homographieshomographies))
Original images Rectified images
Motivations To simplify stereo matching:To simplify stereo matching:
Instead of comparing pixels on Instead of comparing pixels on skew linesskew lines, we now only compare , we now only compare pixels on pixels on the same scan linesthe same scan lines..
Graphics applications: view morphingGraphics applications: view morphing Problem:Problem:
Rectifying homographies are not uniqueRectifying homographies are not unique Goal: Goal: to develop a technique based onto develop a technique based on
geometrically well-defined criteriageometrically well-defined criteria minimizing minimizing image distortion due to rectificationimage distortion due to rectification
Epipolar Geometry
M
C
C’
m
m’
•Epipoles anywhere•Fundamental matrix F: a 3x3 rank-2 matrix
•Epipole at•Fundamental matrix
T001i
010
100
000
][iF
Stereo Image Rectification
Compute Compute H H and and H’ H’ such thatsuch that Compute rectified image points:Compute rectified image points:
Problem:Problem:
H H and and H’ are not unique.H’ are not unique.
Properties of H and H’ (I) Consider each row of Consider each row of H H and and H’ as a line:H’ as a line:
Recall: Recall: bothboth e e andand e’ e’ are sent toare sent to [1 0 0] [1 0 0]TT
Observations (I):Observations (I): v v andand w w must go through the epipolemust go through the epipole e e v’ v’ andand w’ w’ must go through the epipolemust go through the epipole e’ e’ u u and and u’u’ are irrelevant to rectification are irrelevant to rectification
Properties of H and H’ (II) Observation (II):Observation (II):
Lines Lines vv and and v’v’, and lines , and lines ww and and w’w’ must be corresponding epipolar lines. must be corresponding epipolar lines.
Observation (III):Observation (III):
Lines Lines ww and and w’w’ define the rectifying plane. define the rectifying plane.
Decomposition of H
Special projective transform:Special projective transform:
Similarity transform:Similarity transform:
Shearing transform:Shearing transform:
prs HHHH
Special Projective Transform (I)
Sends the epipole to infinitySends the epipole to infinity epipolar lines become parallel epipolar lines become parallel Captures all image distortion due to Captures all image distortion due to
projective transformationprojective transformation Subgoal: Subgoal: Make Make HHp p as affine as possible.as affine as possible.
Special Projective Transform (II)
How to do it?How to do it? Let original image point beLet original image point be the transformed point will bethe transformed point will be
Observation: Observation:
If all weights are equal, then there is no distortion.If all weights are equal, then there is no distortion.
Key ideaKey idea::
minimize the variation of minimize the variation of wwii over all pixels over all pixels
with weight
Similarity Transform
Rotate and translate images such that the Rotate and translate images such that the epipolar lines are horizontally aligned.epipolar lines are horizontally aligned.
Images are now rectified.Images are now rectified.
Shearing Transform
Free to scale and translate in the horizontal Free to scale and translate in the horizontal direction.direction.
Subgoal:Subgoal:
Preserve original image resolution as close Preserve original image resolution as close as possible.as possible.
Example
Original image pairOriginal image pair
Intermediate result After special projective transform:After special projective transform:
Intermediate result After similarity transform:After similarity transform:
Final result After shearing transformAfter shearing transform