MultiView Geometry Cont Stereo Constraints Review p p

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Multi-View Geometry (Cont. )

Multi-View Geometry (Cont. )

Stereo Constraints (Review) p p’ ? Given p in left image, where can the

Stereo Constraints (Review) p p’ ? Given p in left image, where can the corresponding point p’ in right image be?

Stereo Constraints (Review) M Image plane Y 1 Epipolar Line p p’ Y 2

Stereo Constraints (Review) M Image plane Y 1 Epipolar Line p p’ Y 2 Z 1 O 1 X 2 X 1 Focal plane O 2 Epipole Z 2

Epipolar Constraint (Review)

Epipolar Constraint (Review)

From Geometry to Algebra (Review) P p O p’ O’

From Geometry to Algebra (Review) P p O p’ O’

From Geometry to Algebra (Review) P p O p’ O’

From Geometry to Algebra (Review) P p O p’ O’

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

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

The Essential Matrix (Review)

The Essential Matrix (Review)

The Essential Matrix • Based on the Relative Geometry of the Cameras • Assumes

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 the

The Essential Matrix Similarly p is the epipolar line corresponding to p in the right camera

The Essential Matrix Similarly, Essential Matrix is singular with rank 2 e’

The Essential Matrix Similarly, Essential Matrix is singular with rank 2 e’

Small Motions and Epipolar Constraint

Small Motions and Epipolar Constraint

Motion Models (Review) 3 D Rigid Motion Velocity Vector Translational Component of Velocity Angular

Motion Models (Review) 3 D Rigid Motion Velocity Vector Translational Component of Velocity Angular Velocity

Small Motions Velocity Vector Translational Component of Velocity Angular Velocity

Small Motions Velocity Vector Translational Component of Velocity Angular Velocity

Translating Camera Focus of expansion (FOE): Under pure translation, the motion field at every

Translating Camera Focus of expansion (FOE): Under pure translation, the motion field at every point in the image points toward the focus of expansion

FOE for Translating Camera

FOE for Translating Camera

FOE from Basic Equations of Motion q O p v

FOE from Basic Equations of Motion q O p v

What if Camera Calibration is not known

What if Camera Calibration is not known

Review: Intrinsic Camera Parameters Y Image plane C Z v M X Focal plane

Review: Intrinsic Camera Parameters Y Image plane C Z v M X Focal plane u m P

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

Fundamental Matrix 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

Fundamental Matrix is singular with rank 2 In principal F has 7 parameters up to scale and can be estimated from 7 point correspondences Direct Simpler Method requires 8 correspondences

Estimating Fundamental Matrix The 8 -point algorithm Each point correspondence can be expressed as

Estimating Fundamental Matrix The 8 -point algorithm Each point correspondence can be expressed as a linear equation

The 8 -point Algorithm

The 8 -point Algorithm

Shape from Stereo

Shape from Stereo

Pinhole Camera Model

Pinhole Camera Model

Basic Stereo Derivations Derive expression for Z as a function of x 1, x

Basic Stereo Derivations Derive expression for Z as a function of x 1, x 2, f and B

Basic Stereo Derivations

Basic Stereo Derivations

Basic Stereo Derivations Define the disparity:

Basic Stereo Derivations Define the disparity:

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