OpenCV  4.1.1-pre
Open Source Computer Vision
Affine Transformations

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Goal

In this tutorial you will learn how to:

Theory

What is an Affine Transformation?

  1. A transformation that can be expressed in the form of a matrix multiplication (linear transformation) followed by a vector addition (translation).
  2. From the above, we can use an Affine Transformation to express:

    1. Rotations (linear transformation)
    2. Translations (vector addition)
    3. Scale operations (linear transformation)

    you can see that, in essence, an Affine Transformation represents a relation between two images.

  3. The usual way to represent an Affine Transformation is by using a \(2 \times 3\) matrix.

    \[ A = \begin{bmatrix} a_{00} & a_{01} \\ a_{10} & a_{11} \end{bmatrix}_{2 \times 2} B = \begin{bmatrix} b_{00} \\ b_{10} \end{bmatrix}_{2 \times 1} \]

    \[ M = \begin{bmatrix} A & B \end{bmatrix} = \begin{bmatrix} a_{00} & a_{01} & b_{00} \\ a_{10} & a_{11} & b_{10} \end{bmatrix}_{2 \times 3} \]

    Considering that we want to transform a 2D vector \(X = \begin{bmatrix}x \\ y\end{bmatrix}\) by using \(A\) and \(B\), we can do the same with:

    \(T = A \cdot \begin{bmatrix}x \\ y\end{bmatrix} + B\) or \(T = M \cdot [x, y, 1]^{T}\)

    \[T = \begin{bmatrix} a_{00}x + a_{01}y + b_{00} \\ a_{10}x + a_{11}y + b_{10} \end{bmatrix}\]

How do we get an Affine Transformation?

  1. We mentioned that an Affine Transformation is basically a relation between two images. The information about this relation can come, roughly, in two ways:
    1. We know both \(X\) and T and we also know that they are related. Then our task is to find \(M\)
    2. We know \(M\) and \(X\). To obtain \(T\) we only need to apply \(T = M \cdot X\). Our information for \(M\) may be explicit (i.e. have the 2-by-3 matrix) or it can come as a geometric relation between points.
  2. Let's explain this in a better way (b). Since \(M\) relates 2 images, we can analyze the simplest case in which it relates three points in both images. Look at the figure below:

    Warp_Affine_Tutorial_Theory_0.jpg

    the points 1, 2 and 3 (forming a triangle in image 1) are mapped into image 2, still forming a triangle, but now they have changed notoriously. If we find the Affine Transformation with these 3 points (you can choose them as you like), then we can apply this found relation to all the pixels in an image.

Code

Explanation

You may want to draw these points to get a better idea on how they change. Their locations are approximately the same as the ones depicted in the example figure (in the Theory section). You may note that the size and orientation of the triangle defined by the 3 points change.

We get a \(2 \times 3\) matrix as an output (in this case warp_mat)

with the following arguments:

We just got our first transformed image! We will display it in one bit. Before that, we also want to rotate it...

Result