Multi-View Geometry

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Multi-View Geometry
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Review Pinhole Camera
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Review Perspective Projection
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Review Perspective Projection

• Points go to Points
• Lines go to Lines
• Planes go to whole image or Half-planes
• Polygons go to Polygons

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Perspective cues
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Perspective cues
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Perspective cues
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Review Intrinsic Camera Parameters
Y
M
Image plane
C
Z
v
X
Focal plane
m
u
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Review Extrinsic Parameters
Y
M
Image plane
Y
C
Z
v
X
X
Z
Focal plane
m
u
By Rigid Body Transformation
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Recovering 3D from images

• What cues in the image provide 3D information?

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Visual cues

• Shading

Merle Norman Cosmetics, Los Angeles
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Visual cues

• Shading
• Texture

The Visual Cliff, by William Vandivert, 1960
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Visual cues

• Shading
• Texture
• Focus

From The Art of Photography, Canon
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Visual cues

• Shading
• Texture
• Focus
• Motion

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Visual cues

• Shading
• Texture
• Focus
• Motion

• Shape From X
• X shading, texture, focus, motion, ...

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Multi-View Geometry
Relates
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Multi-View Geometry
Relates

• Camera Orientations

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Stereo
scene point
image plane
optical center
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Stereo

• Basic Principle Triangulation
• Gives reconstruction as intersection of two rays

• Requires
• calibration
• point correspondence

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Stereo Constraints
p
?
p
Given p in left image, where can the
corresponding point pin right image be?
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Stereo Constraints
M
Image plane
Y1
p
O1
Z1
X1
Focal plane
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Epipolar Constraint
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From Geometry to Algebra
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From Geometry to Algebra
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Linear ConstraintShould be able to express as
matrix multiplication.
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Review Matrix Form of Cross Product
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Review Matrix Form of Cross Product
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Matrix Form
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The Essential Matrix