This document summarizes a lecture on image transformations and warping. It discusses:
1) Different types of geometric transformations between images like similarity transformations involving translation, rotation and scaling.
2) Image warping which changes the domain of an image by applying a transformation to its coordinates, as opposed to filtering which changes the range/values.
3) Parametric or global warping which applies the same transformation to all points using parameters, and linear transformations represented by 2x2 matrices including translations, rotations, scaling and shears.
4) Homographies or projective transformations which include affine transformations and projective warps, mapping lines to lines but not preserving parallelism or ratios.
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Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
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3. Announcements
• Project 2 out, due Monday, March 4, by
11:59pm on CMS
– Please form teams of 2, and create your team on
CMS
• Project 1 artifact voting
5. What is the geometric relationship
between these two images?
?
Answer: Similarity transformation (translation, rotation, uniform scale)
6. What is the geometric relationship
between these two images?
?
7. What is the geometric relationship
between these two images?
Very important for creating mosaics!
First, we need to know what this transformation is.
Second, we need to figure out how to compute it using feature matches.
8. Richard Szeliski Image Stitching 8
Image Warping
• image filtering: change range of image
• g(x) = h(f(x))
• image warping: change domain of image
• g(x) = f(h(x))
f
x
h
g
x
f
x
h
g
x
9. Richard Szeliski Image Stitching 9
Image Warping
• image filtering: change range of image
• g(x) = h(f(x))
• image warping: change domain of image
• g(x) = f(h(x))
h
h
f
f g
g
10. Richard Szeliski Image Stitching 11
Parametric (global) warping
• Examples of parametric warps:
translation rotation aspect
11. Parametric (global) warping
• Transformation T is a coordinate-changing machine:
p’ = 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)
• Let’s consider linear xforms (can be represented by a 2x2 matrix):
T
p = (x,y) p’ = (x’,y’)
14. 2x2 Matrices
• What types of transformations can be
represented with a 2x2 matrix?
2D mirror about Y axis?
2D mirror across line y = x?
15. • What types of transformations can be
represented with a 2x2 matrix?
2D Translation?
Translation is not a linear operation on 2D coordinates
NO!
2x2 Matrices
16. All 2D Linear Transformations
• Linear transformations are combinations of …
– Scale,
– Rotation,
– Shear, and
– Mirror
• Properties of linear transformations:
– Origin maps to origin
– Lines map to lines
– Parallel lines remain parallel
– Ratios are preserved
– Closed under composition
y
x
d
c
b
a
y
x
'
'
y
x
l
k
j
i
h
g
f
e
d
c
b
a
y
x
'
'
17. Homogeneous coordinates
Trick: add one more coordinate:
homogeneous image
coordinates
Converting from homogeneous coordinates
x
y
w
(x, y, w)
w = 1
(x/w, y/w, 1)
homogeneous plane
20. Basic affine transformations
1
1
0
0
0
cos
sin
0
sin
cos
1
'
'
y
x
y
x
1
1
0
0
1
0
0
1
1
'
'
y
x
t
t
y
x
y
x
1
1
0
0
0
1
0
1
1
'
'
y
x
sh
sh
y
x
y
x
Translate
2D in-plane rotation Shear
1
1
0
0
0
0
0
0
1
'
'
y
x
s
s
y
x
y
x
Scale
21. Affine Transformations
• Affine transformations are combinations of …
– Linear transformations, and
– Translations
• Properties of affine transformations:
– Origin does not necessarily map to origin
– Lines map to lines
– Parallel lines remain parallel
– Ratios are preserved
– Closed under composition
w
y
x
f
e
d
c
b
a
w
y
x
1
0
0
'
'
29. Homographies
• Homographies …
– Affine transformations, and
– Projective warps
• Properties of projective transformations:
– Origin does not necessarily map to origin
– Lines map to lines
– Parallel lines do not necessarily remain parallel
– Ratios are not preserved
– Closed under composition
31. 2D image transformations
These transformations are a nested set of groups
• Closed under composition and inverse is a member
32. Implementing image warping
• Given a coordinate xform (x’,y’) = T(x,y) and a
source image f(x,y), how do we compute an
xformed image g(x’,y’) = f(T(x,y))?
f(x,y) g(x’,y’)
x x’
T(x,y)
y y’
33. Forward Warping
• Send each pixel f(x) to its corresponding
location (x’,y’) = T(x,y) in g(x’,y’)
f(x,y) g(x’,y’)
x x’
T(x,y)
• What if pixel lands “between” two pixels?
y y’
34. Forward Warping
• Send each pixel f(x,y) to its corresponding
location x’ = h(x,y) in g(x’,y’)
f(x,y) g(x’,y’)
x x’
T(x,y)
• What if pixel lands “between” two pixels?
• Answer: add “contribution” to several pixels,
normalize later (splatting)
• Can still result in holes
y y’
35. Inverse Warping
• Get each pixel g(x’,y’) from its corresponding
location (x,y) = T-1(x,y) in f(x,y)
f(x,y) g(x’,y’)
x x’
T-1(x,y)
• Requires taking the inverse of the transform
• What if pixel comes from “between” two pixels?
y y’
36. Inverse Warping
• Get each pixel g(x’) from its corresponding
location x’ = h(x) in f(x)
• What if pixel comes from “between” two pixels?
• Answer: resample color value from interpolated
(prefiltered) source image
f(x,y) g(x’,y’)
x x’
y y’
T-1(x,y)
37. Interpolation
• Possible interpolation filters:
– nearest neighbor
– bilinear
– bicubic
– sinc
• Needed to prevent “jaggies”
and “texture crawl”
(with prefiltering)
