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Multi-View Geometry (Cont.) - PowerPoint PPT Presentation

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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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

p

p’

O’

O

From Geometry to Algebra (Review)

p

p’

O’

O

From Geometry to Algebra (Review)

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

• 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

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

Similarly,

Essential Matrix is singular with rank 2

e’

Translational Component of Velocity

Angular Velocity

Motion Models (Review)

3D Rigid Motion

Velocity Vector

Translational Component of Velocity

Angular Velocity

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

Y

M

Image plane

C

Z

v

X

Focal plane

m

u

P

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

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

The 8-point algorithm

Each point correspondence can be expressed as a linear equation

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

Define the disparity: