2D Transformations

• 2D Transformations

 

A method used to model the matching relationship between two images directly on a 2D plane.

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• Rigid Transformation(Euclidean transformation)

 

Only change its position and orientation while maintaining its shape and size.

In other words, it is a transformation that only allows rotation and translation.

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• Translation

 

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$$t_x=\frac{1}{n}\sum _i(x_i^{\prime }-x_i)$$ $$t_y=\frac{1}{n}\sum _i(y_i^{\prime }-y_i)$$

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- Degree of Freedom(DOF) : t1, t2

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• Shearing

 

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If you want to scale the figure image on the left side by a multiple of the horizontal and vertical b times as shown on the right side, you can apply x'=ax and y' = bx.

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• Rotation

 

- A transformation matrix that rotates (x, y) counterclockwise by θ radian. The object is spinning based on (0,0). 

- Degree of Freedom(DOF) : 1, therefore, only one matching pair can determine the rotational transformation.

 

$$[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\end{array}]=\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ sin\theta & \cos \theta \end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]$$

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- considering scale change

 

$$[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\end{array}]=[\begin{array}{cc}a& -b\\ b& a\end{array}][\begin{array}{c}x\\ y\end{array}]$$

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• Euclidean

 

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- Degree of Freedom(DOF) : $\theta ,r_3,r_6$

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• Rigid

 

Translation + Rotation 

Rotate by θ around the origin of the image and then move in parallel to the original position.

The degree of freedom of the rotational transformation is 3. It requires at least 2 matching pairs.

 

$$\left[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\end{array}\right]=\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]+\left[\begin{array}{c}c\\ d\end{array}\right]$$
c : tx + rotation
d : ty + rotation
[tx, ty] : translation

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• Similarity

 

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- Degree of Freedom(DOF) : $s,\theta ,r_3,r_6$

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$$[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\end{array}]=[\begin{array}{cc}a& -b\\ b& a\end{array}][\begin{array}{c}x\\ y\end{array}]+[\begin{array}{c}c\\ d\end{array}]$$

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- Convert to homogeneous coordinate system

In order to be a single matrix from Rotation + Translation + Scaling

 

$$[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\\ 1\end{array}]=\left[\begin{array}{ccc}a& -b& c\\ b& a& d\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}x\\ y\\ 1\end{array}\right]$$

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• Affine Transformation

 

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- A transformation that preserves linearity, length (distance) ratio, and parallelism.

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- Degree of Freedom(DOF) : 6, $\left[\begin{array}{c}{a}_1{a}_2\\ a_4{a}_5\end{array}\right]=\left[\begin{array}{c}s{r}_{1}s{r}_2\\ s{r}_{3}s{r}_4\end{array}\right]=\left[\begin{array}{c}scos\theta -ssin\theta \\ ssin\theta scos\theta \end{array}\right]$ + $h_1,h_2$

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$$[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\end{array}]=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]+\left[\begin{array}{c}e\\ f\end{array}\right]$$

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- Convert to homogeneous coordinate system

 

$$\left[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\\ 1\end{array}\right]=\left[\begin{array}{ccc}a& b& e\\ c& d& f\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}x\\ y\\ 1\end{array}\right]$$

 

 

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• Homograpy (Projective Transformation)

 

If a planar surface is projected as an image A and an image B for different camera positions, the relationship between the image A and the image B may be expressed as a homography.

 

$$w[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\\ 1\end{array}]=\left[\begin{array}{c}{h}_{11}{h}_{12}{h}_{13}\\ h_{21}{h}_{22}{h}_{23}\\ h_{31}{h}_{32}{h}_{33}\end{array}\right]\left[\begin{array}{c}x\\ y\\ 1\end{array}\right]$$

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https://velog.io/@richpin/Computer-Vision-07-2D-Transformations

https://darkpgmr.tistory.com/80

https://darkpgmr.tistory.com/79