geometric image manipulation: part 2cs510/yr2015sp/more...© bruce a. draper & j. ross...

© Bruce A. Draper & J. Ross Beveridge, January 23, 2015

Geometric Image Manipulation: Part 2

Lecture #2 January 26, 2015


Translation Applied to Images

Translate 20 in x Translate -20 in x

€

1 0 200 1 00 0 1

"

#

$ $ $

%

&

' ' '

€

1 0 −200 1 00 0 1

#

$

% % %

&

'

( ( (

Why Black?


Scale Applied to Images

Scale Uniformly by 2 Scale Uniformly by 0.5

€

2 0 00 2 00 0 1

"

#

$ $ $

%

&

' ' '

€

0.5 0 00 0.5 00 0 1

"

#

$ $ $

%

&

' ' '

Note the

origin


Rotation Applied to Images

Rotate by 15° Rotate by -15°

Note that a positive rotation rotates the positive X axis toward the positive Y axis


Remember Composition of Matrices To rotate by θ around a point (x,y):

1 0 x0 1 y0 0 1

!

"

###

$

%

&&&

cos θ( ) −sin θ( ) 0

sin θ( ) cos θ( ) 0

0 0 1

!

"

####

$

%

&&&&

1 0 −x0 1 −y0 0 1

!

"

###

$

%

&&&=

1 0 x0 1 y0 0 1

!

"

###

$

%

&&&

cos θ( ) −sin θ( ) ysin θ( )− xcos θ( )sin θ( ) cos θ( ) −xsin θ( )− ycos θ( )0 0 1

!

"

####

$

%

&&&&

=

cos θ( ) −sin θ( ) ysin θ( )− xcos θ( )+ xsin θ( ) cos θ( ) −xsin θ( )− ycos θ( )+ y0 0 1

!

"

####

$

%

&&&&


Similarity & Affine Transformations •  All the similarity transforms can be combined

into one generic matrix:

•  But! This matrix does more. What?

–  hint: 2 more transformations. –  hint: 6 degrees of freedom.

•  How can you specify this matrix?

!!!

"

#

$$$

%

&

!!!

"

#

$$$

%

&

=

!!!

"

#

$$$

%

&

11001yx

fedcba

vu Hint: diagonal

terms are not equal, and b ≠ -d.

Equivalent to adding 2 shear parameters, or unequal scaling & 1 shear parameter.


Affine Examples: Shear

€

1 0.5 00 1 00 0 1

"

#

$ $ $

%

&

' ' '

€

1 1 00 1 00 0 1

"

#

$ $ $

%

&

' ' '


Similarity vs. Affine Matrices •  Similarity : 4 DOF

•  Affine : 6 DOF

!!!

"

#

$$$

%

&

!!!

"

#

$$$

%

&

=

!!!

"

#

$$$

%

&

11001yx

fedcba

vu

uv1

!

"

###

$

%

&&&=

a b c−b a d0 0 1

!

"

###

$

%

&&&

xy1

!

"

###

$

%

&&&


Specifying Affine Transformations

•  There are six unknowns in the matrix (a through f) •  If you specify one point in the source image and a

corresponding point in the target image, that yields two equations:

•  So providing three point-to-point correspondences specifies an affine matrix

feydxvcbyaxu

iii

iii

++=

++=


Affine Specification: Example

There is one affine transformation that will map the green point on the right to the green point on the left, and align the red and blue points also.


Solving Affine Transformations These linear equations can be easily solved:

–  WLOG, assume x1=y1=0 –  then u1= c and v1 = f –  so:

( )

( )

( )( ) ( )

2323

123232

32

23

122

313

122

3133

2

23

3132

2123

2

212

1333

1222

xyyxuuxuux

yxyx

uuxxuu

b

uuxx

uubyxyx

byuux

byuuxxbyuu

a

ubyaxuubyaxu

−−

−−−=

−−

−−−=

−−−=""#

$%%&

'−

−

−−=−−

−−=

++=

++=

Calculation of a, b & c is independent of calculation of e, f & g.


Solving Affine (cont.)

•  This can be substituted in to solve for a •  The same process with y’s solves for d,e,f •  About the WLOG:

–  It was true because you can translate the original coordinate system by (-x1, -y1)

–  So what do you do to compensate?

•  Alternatively, set up a system of linear equations and solve... –  Will show this for a harder case shortly….


Perspective Transformations

•  We can go beyond affine transformations. •  We can do any perspective transformation of a

one 3D view of a plane to another view. •  Therefore, we can model an image as a plane

in space, and project it onto any other image. – How does this differ from the perspective

projection pipeline in CS410?


Perspective Matrix

•  Why does element [3,3] = 1? •  How many points are needed to specify this matrix?

€

u'v 'w

"

#

$ $ $

%

&

' ' '

=

a b cd e fg h 1

"

#

$ $ $

%

&

' ' '

xy1

"

#

$ $ $

%

&

' ' '

u = u'w , v = v'

w


Solving for Perspective

•  Four corresponding points produce eight equations, eight unknowns --- but we can’t observe w

1'

1'

++

++==

++

++==

ii

ii

i

ii

ii

ii

i

ii

hygxfeydx

wvv

hygxcbyax

wuu


Solving (cont.)

•  Multiply to get rid of the fraction…

•  Now, remember that the u’s, v’s, x’s & y’s are known; group the unknown terms

( )( ) feydxhygxv

cbyaxhygxu

iiiii

iiiii

++=++

++=++

11

iiiiiii

iiiiiii

vhyvgxfeydxvuhyugxcbyaxu

−−++=

−−++=


Solving (III)

•  And express the result as a system of linear equations

!!!!!!!!!!!

"

#

$$$$$$$$$$$

%

&

!!!!!!!!!!!

"

#

$$$$$$$$$$$

%

&

−−

−−

−−

−−

−−

−−

−−

−−

=

!!!!!!!!!!!

"

#

$$$$$$$$$$$

%

&

hgfedcba

vyvxyxuyuxyxvyvxyxuyuxyxvyvxyxuyuxyxvyvxyxuyuxyx

vuvuvuvu

444444

444444

333333

333333

222222

222222

111111

111111

4

4

3

3

2

2

1

1

10000001100000011000000110000001


Solving (IV)

•  Finally, invert the constant matrix and solve

!!!!!!!!!!!

"

#

$$$$$$$$$$$

%

&

=

!!!!!!!!!!!

"

#

$$$$$$$$$$$

%

&

!!!!!!!!!!!

"

#

$$$$$$$$$$$

%

&

−−

−−

−−

−−

−−

−−

−−

−−−

hgfedcba

vuvuvuvu


4

4

3

3

2

2

1

11

444444

444444

333333

333333

222222

222222

111111

111111

10000001100000011000000110000001


Solving (V) : Questions

•  Is there always a solution? •  Under what conditions is the matrix invertible? •  Is the solution always unique?


Perspective Image Transforms (Intuition)

•  What does the following matrix do?

!!!

"

#

$$$

%

& −

100022022


Matrix Decomposition

•  Note that such decompositions are: –  not unique (why?) –  difficult to intuit

!!!!!!

"

#

$$$$$$

%

& −

!!!

"

#

$$$

%

&

=!!!

"

#

$$$

%

& −

100

021

21

021

21

100020002

100022022

original Scale by 2 Rotation

by 45


More Intuitions

•  What will the following matrix do?

•  More specifically, what will it do to the giraffe image?

1 0 00 1 01 −1 1

"

#

$$$

%

&

'''


Check Your Intuitions

•  What’s going on here?


More Intuition Checking

•  Part of what you are seeing is a scale effect –  positive terms in the bottom row create larger w values, and

therefore smaller u, v values

•  Something much weirder is also going on: –  What happens when y = x+1? –  How do you interpret this geometrically? –  Isn’t the perspective transform linear?

•  So how do you select transform matrices?


Perspective Transform of 2D Planes Continued.

•  Recall the basic equation for the perspective transform

•  The only practical way to specify an image

transform is by providing four point correspondences

€

u'v 'w

"

#

$ $ $

%

&

' ' '

=

a b cd e fg h 1

"

#

$ $ $

%

&

' ' '

xy1

"

#

$ $ $

%

&

' ' '

u = u'w , v = v'

w


Computing Transformations

•  Remember how to build a transformation from four point correspondences….

!!!!!!!!!!!

"

#

$$$$$$$$$$$

%

&

!!!!!!!!!!!

"

#

$$$$$$$$$$$

%

&

−−

−−

−−

−−

−−

−−

−−

−−

=

!!!!!!!!!!!

"

#

$$$$$$$$$$$

%

&

hgfedcba


vuvuvuvu

444444

444444

333333

333333

222222

222222

111111

111111

4

4

3

3

2

2

1

1

10000001100000011000000110000001


Computing... •  So if we want the following mapping:

(0,0)→(0,0), (0,144)→(0,144), (152,0)→(152,50), (152,144)→(152,94)

!!!!!!!!!!!

"

#

$$$$$$$$$$$

%

&

−−

−−

−

−

−

13536142881144152000218882310400011441520760010152000023104000101522073601144000000000114400010000000000100


…More Computing...

!!!!!!!!!!!

"

#

$$$$$$$$$$$

%

&

=

!!!!!!!!!!!

"

#

$$$$$$$$$$$

%

&

=

!!!!!!!!!!!

"

#

$$$$$$$$$$$

%

&

!!!!!!!!!!!

"

#

$$$$$$$$$$$

%

&

−−

−−

−−−

−−−−

−

−−−−

001497.01077.100274.3

9415250152144000

0000.0000.0000.0000.000.000.000.000.000.000.000.000.000000100007.0007.007.007.007.007.007.002.014.002.007.002.014.002.0000000100000007.0007.022.007.022.014.023.007.023.014.

hgfedcba

M-1 u&v

vector

What does This say about x?

How does This alter it? Remember the earlier

WLOG? c = u1,…


…yields !!!

"

#

$$$

%

&

1001497.01077.100274.3


Geometric Transformation : Review

•  Goal: Ii(u,v,w) = Ij(G�[x,y,1]T) •  Transformation classes:

– Similarity (4 DoF : translation, rotation, scale) – Affine (6 DoF : adds shear & non-uniform scale) – Perspective (8 DoF : 2D plane to plane)

•  Specified via point correspondences


Still To Do….

•  Photometric registration •  Interpolation

– Transformations are not integer to integer

•  Then we can tackle matching under various transformation classes

geometric image manipulation: part 2cs510/yr2015sp/more...© bruce a. draper & j. ross...

Documents