Computer Vision: Image Alignment

125 Pages·2013·7.05 MB·English

Computer Vision: Image Alignment
Raquel Urtasun
TTIChicago
Jan 24, 2013
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Readings
Chapter 2.1, 3.6, 4.3 and 6.1 of Szeliski’s book
Chapter 1 of Forsyth & Ponce
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What did we see in class last week?
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What is the geometric relationship between these images?
!"
[Source: N. Snavely]
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What is the geometric relationship between these images?
Very important for creating mosaics!
[Source: N. Snavely]
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Image Warping
Image filtering: change range of image
g(x)=h(f(x))
!" %"
$"
#" #"
Image warping: change domain of image
g(x)=f(h(x))
!" %"
$"
#" #"
[Source: R. Szeliski]
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Parametric (global) warping
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!!"!#$%&'! !"!"!#$(%&('!
Transformation T is a coordinate-changing machine:
p(cid:48) =T(p)
What does it mean that T is global?
Is the same for any point p
Can be described by just a few numbers (parameters)
[Source: N. Snavely]
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Forward and Inverse Warping
Forward Warping: Send each pixel f(x) to its corresponding location
(x(cid:48),y(cid:48))=T(x,y) in g(x(cid:48),y(cid:48))
Inverse Warping: Each pixel at the destination is sampled from the original
image
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All 2D Linear Transformations
Linear transformations are combinations of
Scale,
Rotation
Shear
Mirror
(cid:20)x(cid:48)(cid:21) (cid:20)a b(cid:21)(cid:20)x(cid:21)
=
y(cid:48) c d y
[Source: N. Snavely]
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Parallel lines remain parallel
Ratios are preserved
Closed under composition
(cid:20)x(cid:48)(cid:21) (cid:20)a b(cid:21)(cid:20)e f(cid:21)(cid:20)i j(cid:21)(cid:20)x(cid:21)
=
y(cid:48) c d g h k l y
What about the translation?
[Source: N. Snavely]
All 2D Linear Transformations
Properties of linear transformations:
Origin maps to origin
Lines map to lines
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