Image Rectification for Stereo Vision

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# Image Rectification for Stereo Vision - PowerPoint PPT Presentation

Image Rectification for Stereo Vision. Charles Loop Zhengyou Zhang Microsoft Research. Problem Statement. Compute a pair of 2D projective transforms ( homographies ). rectification. Original images. Rectified images. Motivations. To simplify stereo matching:

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## Image Rectification for Stereo Vision

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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)

rectification

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’

Epipole at

Fundamental matrix

• Epipoles anywhere
• Fundamental matrix

F: a 3x3 rank-2 matrix

Stereo Image Rectification
• Compute H and H’ such that
• Compute rectified image points:
• Problem:

H and H’ are not unique.

Properties of H and H’ (I)
• Consider each row of H and H’ as a line:
• Recall: 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
• 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
• Observation:

If all weights are equal, then there is no distortion.

• Key idea:

minimize the variation of wi over all pixels

with weight

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