CSCI 497P/597P: Computer Vision

Lecture 13 Forward and Inverse Warping

Projective Transformations

Announcements• Please add a username.txt to the base directory

of your P1 repository, containing (only) your WWU username.

• Reminder: HW1 due Wednesday; P1 due Friday

Goals• Know how to resample images using forward and

inverse warping

• Know the definition of a projective (homography) transformation, and gain some geometric intuition for what it represents in 2D.

• Understand the intuition behind a "point at infinity"

• Know why a homography has 8 degrees of freedom, not 9

Last time: Affine Transformations• Affine transformations

úúû

ù

êêë

é

úúû

ù

êêë

é=

úúû

ù

êêë

é

wyx

fedcba

wyx

100''

Last time: Affine Transformations• Affine transformations

• Properties: • Lines map to lines • Parallel lines remain parallel • Ratios of lengths along lines are preserved • Closed under composition • Origin maps to origin

úúû

ù

êêë

é

úúû

ù

êêë

é=

úúû

ù

êêë

é

wyx

fedcba

wyx

100''

linear

Last time: Affine Transformations• Affine transformations

• Properties: • Lines map to lines • Parallel lines remain parallel • Ratios of lengths along lines are preserved • Closed under composition • Origin does not necessarily map to origin

úúû

ù

êêë

é

úúû

ù

êêë

é=

úúû

ù

êêë

é

wyx

fedcba

wyx

100''

linearaffine

Warping

x

y

x'

y'(480, 241) (265, 238)

We've found correspondence. What's the transformation? 2

4x0

y0

1

3

5 =

2

4

3

5

2

4xy1

3

5

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Warping

x

y

x'

y'

We've found the transformation. How do we warp the image?

(480, 241) (265, 238)

2

4x0

y0

1

3

5 =

2

41 0 �1150 1 �30 9 1

3

5

2

4xy1

3

5

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Warpingy

x'

y'x x'

y'

Warping

x'

y'

x'

y'

Forward Warping

x x’

T(x,y)y y’

Warping

x

y

x'

y'

We've found the transformation. How do we warp the image?

(480, 241) (265.2, 238.1)

2

4x0

y0

1

3

5 =

2

41 0 �114.80 1 �2.90 9 1

3

5

2

4xy1

3

5

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f(x,y)x

y

g(x’,y’)x’

T

Inverse Warping

T-1(x,y)y y’

Bilinear Interpolation

Other interpolation filters exist

Image by Cmglee - Own work, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=53064904

Are these related by an affine transformation?

Affine Transformations

Scott: No messing with the third row!

Affine Transformations

Scott: No messing with the third row!

mess with the third

row

Students:

Well... you asked.

Hang on a minute: what does it even mean if w != 1?

Well... you asked.

To get the 2D coordinates, we need to normalize.

What does this do?

https://iis.uibk.ac.at/public/piater/courses/demos/homography/homography.xhtml

Projective Transformation (also known as Homograpy)

Homogeneous Coordinates:Intuition for our math hack

A 3-vector belongs to a family of 3-vectors representing the same 2D point.

1D example, for intuition: A 2-vector belongs to a family of 2-vectors representing the same 1D point.

Homographies for image alignment