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BMT

What does it mean that Transformation T is global?

- Is the same for any point p
- Can be described by just a few numbers (parameters)

BMT

2D Linear Transformations:

- Identity
- Scaling
- Rotation
- Mirror
- Shear

BMT

Transformations can be combined by ...?

What is important?

Transformations can be combined by matrix multiplication (matrix composition).

The ordering here is IMPORTANT!

BMT

Forward Warping

Send each pixel f(x, y) to its corresponding location

(x', y')= T(x, y) in the second image.

And use Splatting to color the pixels

BMT

Problem Forward Warping?

Destination picture might have holes

BMT

Inverse (Backward) Warping

Get each pixel g(x', y') from its corresponding location

(x, y)= T^-1(x', y') in the ﬁrst image.

Use Interpolation to color the pixel.

BMT

Pro and Con Inverse Warping

+ ensures that no holes occur

- requires and invertible warp function

BMT

Non-Parametric (Local) Image Warping

Image Warping using Vector Fields

BMT

Image Warping using Vector Fields

• Let (vx, vy)= F(x, y) be an arbitrary vector ﬁeld and I an image.

• Question: How can we compute the value at I(x + vx, y + vy)?

• Answer: Use forward warping to propagate the pixels to a new

location.

• Problem: Same as before, resulting image will contain holes.

-> Solution:

- Answer 2: Use inverse warping with bilinear interpolation.
- Need to invert vector ﬁeld F(x, y)
- Look up source pixels using F(x', y')^-1
- Interpolate using one of the presented models, e.g. bilinear

-> Vector Fields transform each pixel separately

-> Inverse warping is better, but __requires__ invertible vector field

BMT

Good interpolation techniques attempt to find an optimal balance between three undesirable artifacts:

edge halos

blurring

aliasing

BMT

Bilinear Interpolation: Pros and Cons

No jagged artifacts as in Nearest Neighbor

BUT blurry edges

BMT

Transformation T is a coordinate-changing machine:

p' = T(p) (p = 2D vector)

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