Introduction

With image filtering, we were changing the range of image i.e. intensity values within the pixels $

With image transformations, we are changing the domain of images i.e. translation, stretch, squeeze etc $

We know the usual ones like stretch, squeeze, scale etc

But you can also change the perspective and affine.

Image scaling

Uniform Scaling

$\begin{vmatrix} x^{\prime} \\ y^{\prime} \\ \end{vmatrix} = M \begin{vmatrix} x \\ y \\ \end{vmatrix}$

Non-uniform Scaling

$\begin{vmatrix} x^{\prime} \\ y^{\prime} \\ \end{vmatrix} = \begin{vmatrix} a && 0 \\ 0 && b \\ \end{vmatrix} . \begin{vmatrix} x \\ y \\ \end{vmatrix}$

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Properties of Linear Transformations

• Origin maps to origin
• Lines map to lines
• Parallel lines remain parallel
• All ratios remain preserved

Image Translation

To make this happen, we can't use the above approach

So, we go with Homegeneous Coordinates

• Represent coordinates in 2 dimensions with a 3-djim vector
• Add a 3rd coordinate to every 2d point

$\begin{vmatrix} x \\ y \\ \end{vmatrix} => \begin{vmatrix} x \\ y \\ w \\ \end{vmatrix}$

$(x, y, w) => (x/w, y/w) \\ (x, y, 0) => \text{ infinity (so not allowed)} \\ (0, 0, 0) \text{ is not allowed} \\$

Degrees of Freedom

Are basically how many values you need to know within the transformation matrix to make the transformation happen. For example for a x-y translation, the degree of freedom is only 2

Homogeneous Transformations

Affine Transformations

Projective Transformations

Transformation Recovery Summary