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Multi-View Geometry (Cont.)PowerPoint Presentation

Multi-View Geometry (Cont.)

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Stereo Constraints (Review)

p’

?

p

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

p’

Y2

X2

Z2

O2

Epipole

Stereo Constraints (Review)M

Image plane

Y1

p

O1

Z1

X1

Focal plane

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

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 Matrix

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

Translational Component of Velocity

Angular Velocity

Motion Models (Review)3D Rigid Motion

Translating Camera

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

Fundamental Matrix

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

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

The 8-point algorithm

Each point correspondence can be expressed as a linear equation

Basic Stereo Derivations

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

Basic Stereo Derivations

Define the disparity:

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