Projective Geometry Based on slides by Peter Corke Homogeneous - - PowerPoint PPT Presentation

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Nov 02, 2022 492 likes •697 views

Projective Geometry Based on slides by Peter Corke Homogeneous coordinates Cartesian homogeneous homogeneous Cartesian lines and points are duals Pin-hole model in homogeneous form Perspective transformation, with the pesky

SLIDE 1

Projective Geometry

Based on slides by Peter Corke

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Homogeneous coordinates

homogeneous → Cartesian
Cartesian → homogeneous

lines and points are duals

SLIDE 5

Pin-hole model in homogeneous form

Perspective transformation, with the pesky divide by

Z, is linear in homogeneous coordinate form.

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3D to 2D scaling/zoomi ng

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Central projection model

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Change of coordinates

scale from metres to pixels
shift the origin to top left

corner

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Complete camera model

intrinsic parameters extrinsic parameters camera matrix

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Camera matrix

Mapping points from the world to an image (pixel)

coordinate is simply a matrix multiplication using homogeneous coordinates

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Scale invariance

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Normalized camera matrix

Since scale factor is arbitrary we can fix the value of
ne element, typically C(3,4) to one.

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Points on a plane

all points on the plane have Z=0

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Planar homography

Once again the scale factor is arbitrary
8 unique numbers in the homography matrix
Can be estimated from 4 world points and their

corresponding image points

homography matrix

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Perspective rectification

>> H = homography(p1, p2) H= 1.4003 0.3827 -136.5900 - 0.0785 1.8049 -83.1054 - 0.0003 0.0016 1.0000

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>> homwarp(H, im, 'full')

Perspective rectification

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Warping

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Name

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