# Image Transformation

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## Transformation in 2D
Let us start from 2D space, and then extend the concept to 3D space later. Noted that we use the homogeneous coordinates in this post. So matrix $M_{3 \times 3}$ is actually a 2D -> 2D transformation.

| Name | Matrix | #DoF | Preserves | Shape |
| -------- | -------- | -------- |--- |--- |
| translation | $\begin{bmatrix} I & t \\ 0^{T} & 1 \end{bmatrix}_{3 \times 3}$ | 2 | orientation + ... | <img src="https://hackmd.io/_uploads/B1eQG25qT.png" width="50"> |
| rigid |$\begin{bmatrix} R & t \\ 0^{T} & 1 \end{bmatrix}_{3 \times 3}$ | 3 | lengths + ... | <img src="https://hackmd.io/_uploads/SJXvG3q5p.png" width="50">
| similarity | $\begin{bmatrix} sR & t \\ 0^{T} & 1 \end{bmatrix}_{3 \times 3}$ | 4 | angles + ... | <img src="https://hackmd.io/_uploads/SJXvG3q5p.png" width="60"> |
| affine | $\begin{bmatrix} A & t \\ 0^{T} & 1 \end{bmatrix}_{3 \times 3}$ | 6 | parallelism + ... | <img src="https://hackmd.io/_uploads/S1yiz35c6.png" width="60"> |
| perspective | $\begin{bmatrix} A & t \\ v^{T} & 1 \end{bmatrix}_{3 \times 3}$ | 8 | straight lines + ... | <img src="https://hackmd.io/_uploads/BkX2fhqc6.png" width="60"> |

The matrix representation can be specifically written as follow:
1. Translation transformation:
$$
\begin{bmatrix}
x' \\
y' \\
1
\end{bmatrix} =
\begin{bmatrix}
1 & 0 & t_{x} \\
0 & 1 & t_{y} \\
0 & 0 & 1
\end{bmatrix}
\begin{bmatrix}
x \\
y \\
1
\end{bmatrix}=
\begin{bmatrix}
I & t \\
0^{T} & 1
\end{bmatrix}
\begin{bmatrix}
x \\
y \\
1
\end{bmatrix}
$$
==DoF is 2: $t_{x}, t_{y}$==

1. Euclidean / Rigid / Isometric transformation:
$$
\begin{bmatrix}
x' \\
y' \\
1
\end{bmatrix} =
\begin{bmatrix}
cos \theta & -sin \theta & t_{x} \\
sin \theta & cos \theta & t_{y} \\
0 & 0 & 1
\end{bmatrix}
\begin{bmatrix}
x \\
y \\
1
\end{bmatrix} =
\begin{bmatrix}
R & t \\
0^{T} & 1
\end{bmatrix}
\begin{bmatrix}
x \\
y \\
1
\end{bmatrix}
$$
==DoF is 3: $\theta, t_{x}, t_{y}$==

<figure>
<img src="https://hackmd.io/_uploads/H1vvD2bcp.gif" width="200">
</figure>

2. Similarity transformation:
$$
\begin{bmatrix}
x' \\
y' \\
1
\end{bmatrix} =
\begin{bmatrix}
s ~ cos \theta & -s ~ sin \theta & t_{x} \\
s ~ sin \theta & s ~ cos \theta & t_{y} \\
0 & 0 & 1
\end{bmatrix}
\begin{bmatrix}
x \\
y \\
1
\end{bmatrix} =
\begin{bmatrix}
sR & t \\
0^{T} & 1
\end{bmatrix}
\begin{bmatrix}
x \\
y \\
1
\end{bmatrix}
$$
==DoF is 4: $s, \theta, t_{x}, t_{y}$==

<figure>
<img src="https://hackmd.io/_uploads/S1SdD3-cp.gif" width="200">
</figure>

3. Affine transformation:
$$
\begin{bmatrix}
x' \\
y' \\
1
\end{bmatrix} =
\begin{bmatrix}
a & b & t_{x} \\
c & d & t_{y} \\
0 & 0 & 1
\end{bmatrix}
\begin{bmatrix}
x \\
y \\
1
\end{bmatrix} =
\begin{bmatrix}
A & t \\
0^{T} & 1
\end{bmatrix}
\begin{bmatrix}
x \\
y \\
1
\end{bmatrix}
$$
==DoF is 6: $a, b, c, d, t_{x}, t_{y}$==

<figure>
<img src="https://hackmd.io/_uploads/r12OwnW5T.gif" width="200">
</figure>

4. Perspective / Projective / Homography transformation:
$$
\begin{bmatrix}
x' \\
y' \\
1
\end{bmatrix} =
\begin{bmatrix}
a & b & t_{x} \\
c & d & t_{y} \\
v_{1} & v_{2} & 1
\end{bmatrix}
\begin{bmatrix}
x \\
y \\
1
\end{bmatrix} =
\begin{bmatrix}
A & t \\
v^{T} & 1
\end{bmatrix}
\begin{bmatrix}
x \\
y \\
1
\end{bmatrix}
$$
==DoF is 8: $a, b, c, d, v_{1}, v_{2}, t_{x}, t_{y}$==

<figure>
<img src="https://hackmd.io/_uploads/H1wtwhbca.gif" width="200">
</figure>

## Transformation in 3D

| Name | Matrix | #DoF | Preserves | Shape |
| -------- | -------- | -------- |--- |--- |
| translation | $\begin{bmatrix} I & t \\ 0^{T} & 1 \end{bmatrix}_{3 \times 4}$ | 3 | orientation + ... | <img src="https://hackmd.io/_uploads/SyFMEh55a.png" width="50"> |
| rigid |$\begin{bmatrix} R & t \\ 0^{T} & 1 \end{bmatrix}_{3 \times 4}$ | 6 | lengths + ... | <img src="https://hackmd.io/_uploads/B1cQEh556.png" width="50">
| similarity | $\begin{bmatrix} sR & t \\ 0^{T} & 1 \end{bmatrix}_{3 \times 4}$ | 7 | angles + ... | <img src="https://hackmd.io/_uploads/B1cQEh556.png" width="60"> |
| affine | $\begin{bmatrix} A & t \\ 0^{T} & 1 \end{bmatrix}_{3 \times 4}$ | 12 | parallelism + ... | <img src="https://hackmd.io/_uploads/HJbSV2956.png" width="60"> |
| perspective | $\begin{bmatrix} A & t \\ v^{T} & 1 \end{bmatrix}_{3 \times 4}$ | 15 | straight lines + ... | <img src="https://hackmd.io/_uploads/ByurE3ccp.png" width="60"> |

1. Euclidean / Rigid / Isometric transformation:
$$
\begin{bmatrix}
x' \\
y' \\
1
\end{bmatrix} =
\begin{bmatrix}
R & t \\
0^{T} & 1
\end{bmatrix}
\begin{bmatrix}
x \\
y \\
z \\
1
\end{bmatrix}
$$
$$
R = \begin{bmatrix}
1 & 0 & 0\\
0 & cos ~\alpha & -sin ~\alpha \\
0 & sin ~\alpha & cos ~\alpha
\end{bmatrix}
\begin{bmatrix}
cos ~\beta & 0 & sin ~\beta\\
0 & 1 & 0 \\
-sin ~\beta & 0 & cos ~\beta
\end{bmatrix}
\begin{bmatrix}
cos ~\gamma & -sin ~\gamma & 0\\
sin ~\gamma & cos ~\gamma & 0 \\
0 & 0 & 1
\end{bmatrix}
$$
$$
t = \begin{bmatrix}
t_{x} \\
t_{y} \\
t_{z}
\end{bmatrix}
$$
==DoF is 6: $\alpha, \beta, \gamma, t_{x}, t_{y}, t_{z}$==

## Reference
- https://medium.com/@insight-in-plain-sight/deconstructing-the-homography-matrix-35989ecc0b2