transformations-in-2d

Reference

• Stanford CS231A
• Jinsol Kim’s blog

2차원 공간에서의 변환 종류

이미지에 있는 정보를 알아내기 위해서는 2D 공간에서의 다양한 변환을 알아야 한다.

Isometric transformations

• preserve distances
• described as a rotation $R$ and translation $t$ $\left[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\\ 1\end{array}\right]=\left[\begin{array}{cc}R& t\\ 0& 1\end{array}\right]\left[\begin{array}{c}x\\ y\\ 1\end{array}\right]$

Similarity transformations

• preserve shape (= isometric transformations + scaling)
• preserve ratio of lengths and angles
• every isometric transformations = similarity transformations when $s=1$ $\left[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\\ 1\end{array}\right]=\left[\begin{array}{cc}SR& t\\ 0& 1\end{array}\right]\left[\begin{array}{c}x\\ y\\ 1\end{array}\right],S=\left[\begin{array}{cc}s& 0\\ 0& s\end{array}\right]$

Affine transformations

Figure 1. Affine transformation

• preserve points, straight lines, and parallelism For some vector $v$, an affine transformation $T$ is, (where $A$ is a linear transformation) $T(v)=Av+t$

In homogeneous coordinate, affine transformation is $\left[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\\ 1\end{array}\right]=\left[\begin{array}{cc}A& t\\ 0& 1\end{array}\right]\left[\begin{array}{c}x\\ y\\ 1\end{array}\right]$

Projective transformations (or homographies)

Figure 2. Perspective transformation

• any transformations that maps lines to lines
• does note preserve parallelism
• does preserve collinearity of points, as it maps lines to lines
• extra degrees of freedom are added with the addition of $v$ $\left[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\\ 1\end{array}\right]=\left[\begin{array}{cc}A& t\\ v& 1\end{array}\right]\left[\begin{array}{c}x\\ y\\ 1\end{array}\right]$ Cross ratio of four collinear points remains invariant under projective transformations $\text{cross ratio}=\frac{\Vert P_3-P_1\Vert \Vert P_4-P_2\Vert }{\Vert P_3-P_2\Vert \Vert P_4-P_1\Vert }$ 한 직선 위에 점 4개를 가지고 계산하면, 그 값을 사진 찍기 전 (3D) 이나 찍은 후 (2D) 나 똑같이 유지한다.