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Multi-View Geometry (Cont.). Stereo Constraints (Review). p’. ?. p. Given p in left image, where can the corresponding point p’ in right image be?. Epipolar Line. p’. Y 2. X 2. Z 2. O 2. Epipole. Stereo Constraints (Review). M. Image plane. Y 1. p. O 1. Z 1. X 1. Focal plane.

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Stereo constraints review
Stereo Constraints (Review)

p’

?

p

Given p in left image, where can the corresponding point p’in right image be?


Stereo constraints review1

Epipolar Line

p’

Y2

X2

Z2

O2

Epipole

Stereo Constraints (Review)

M

Image plane

Y1

p

O1

Z1

X1

Focal plane



From geometry to algebra review

P

p

p’

O’

O

From Geometry to Algebra (Review)


From geometry to algebra review1

P

p

p’

O’

O

From Geometry to Algebra (Review)


Linear Constraint:Should be able to express as matrix multiplication.



The essential matrix
The Essential Matrix

  • Based on the Relative Geometry of the Cameras

  • Assumes Cameras are calibrated (i.e., intrinsic parameters are known)

  • Relates image of point in one camera to a second camera (points in camera coordinate system).

  • Is defined up to scale

  • 5 independent parameters


The essential matrix1
The Essential Matrix

Similarly p is the epipolar line corresponding to p in theright camera


The essential matrix2
The Essential Matrix

Similarly,

Essential Matrix is singular with rank 2

e’



Motion models review

Velocity Vector

Translational Component of Velocity

Angular Velocity

Motion Models (Review)

3D Rigid Motion


Small motions
Small Motions

Velocity Vector

Translational Component of Velocity

Angular Velocity


Translating camera
Translating Camera

Focus of expansion (FOE): Under pure translation, the motionfield at every point in the image points toward the focus ofexpansion





Review intrinsic camera parameters
Review: Intrinsic Camera Parameters

Y

M

Image plane

C

Z

v

X

Focal plane

m

u

P


Fundamental matrix
Fundamental Matrix

If u and u’ are corresponding image coordinates then we have


Fundamental matrix1
Fundamental Matrix

Fundamental Matrix is singular with rank 2

In principal F has 7 parameters up to scale and can be estimatedfrom 7 point correspondences

Direct Simpler Method requires 8 correspondences


Estimating fundamental matrix
Estimating Fundamental Matrix

The 8-point algorithm

Each point correspondence can be expressed as a linear equation





Basic stereo derivations
Basic Stereo Derivations

Derive expression for Z as a function of x1, x2, f and B



Basic stereo derivations2
Basic Stereo Derivations

Define the disparity:


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