Multi view geometry
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Multi-View Geometry. Review: Pinhole Camera. Review: Perspective Projection. Review: Perspective Projection. Points go to Points Lines go to Lines Planes go to whole image or Half-planes Polygons go to Polygons. Perspective cues. Perspective cues. Perspective cues.

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Multi-View Geometry

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Multi view geometry

Multi-View Geometry


Review pinhole camera

Review: Pinhole Camera


Review perspective projection

Review: Perspective Projection


Review perspective projection1

Review: Perspective Projection

  • Points go to Points

  • Lines go to Lines

  • Planes go to whole image or Half-planes

  • Polygons go to Polygons


Perspective cues

Perspective cues


Perspective cues1

Perspective cues


Perspective cues2

Perspective cues


Review intrinsic camera parameters

Review: Intrinsic Camera Parameters

Y

M

Image plane

C

Z

v

X

Focal plane

m

u


Review extrinsic parameters

Review: Extrinsic Parameters

Y

M

Image plane

Y

C

Z

v

X

X

Z

Focal plane

m

u

By Rigid Body Transformation:


Recovering 3d from images

Recovering 3D from images

  • What cues in the image provide 3D information?


Visual cues

Visual cues

  • Shading

Merle Norman Cosmetics, Los Angeles


Visual cues1

Visual cues

  • Shading

  • Texture

The Visual Cliff, by William Vandivert, 1960


Visual cues2

Visual cues

  • Shading

  • Texture

  • Focus

From The Art of Photography, Canon


Visual cues3

Visual cues

  • Shading

  • Texture

  • Focus

  • Motion


Visual cues4

Visual cues

  • Shading

  • Texture

  • Focus

  • Motion

  • Shape From X

    • X = shading, texture, focus, motion, ...


Multi view geometry1

  • 3D World Points

  • Camera Centers

  • Camera Orientations

Multi-View Geometry

Relates


Multi view geometry2

  • 3D World Points

  • Camera Centers

  • Camera Intrinsic Parameters

  • Image Points

Multi-View Geometry

Relates

  • Camera Orientations


Stereo

Stereo

scene point

image plane

optical center


Stereo1

Stereo

  • Basic Principle: Triangulation

    • Gives reconstruction as intersection of two rays

  • Requires

    • calibration

    • point correspondence


Stereo constraints

Stereo Constraints

p’

?

p

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


Stereo constraints1

Epipolar Line

p’

Y2

X2

Z2

O2

Epipole

Stereo Constraints

M

Image plane

Y1

p

O1

Z1

X1

Focal plane


Epipolar constraint

Epipolar Constraint


From geometry to algebra

P

p

p’

O’

O

From Geometry to Algebra


From geometry to algebra1

P

p

p’

O’

O

From Geometry to Algebra


Multi view geometry

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


Review matrix form of cross product

Review: Matrix Form of Cross Product


Review matrix form of cross product1

Review: Matrix Form of Cross Product


Matrix form

Matrix Form


The essential matrix

The Essential Matrix


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