Projective transformations

Basic definitionsTransformations of lines and conicsTransformations of linesTransformations of conics Projective transformations)

一个读者的感悟。 所有的知识点都是来源于书Multiple View of Geometry in Computer Vision 。 这一篇中很多概念都在上一篇2D投影平面中介绍。注：在没有上一篇基础的前提下这一篇难以理解。 这一篇涉及到投影变换的基本概念。 依然，数学真是很奇妙的东西。

Basic definitions

Definition 2.9. A projectivity (also called a collineation) is an invertible mapping $h$ from ${\mathbb{P}}^2$ (projective plane) to itself such that three points $x_1$, $x_2$, $x_3$ lines on the same line if and only if $h(x_1)$, $h(x_2)$, $h(x_3)$ do.

Theorem 2.10. A mapping $h$: ${\mathbb{P}}^2\to {\mathbb{P}}^2$ is a projecticity if and only if there exists a non-singular 3 $\times$ 3 matrix $H$ such that for any point in ${\mathbb{P}}^2$ relrese ted h a vector $x$ it is true that $h(x)=Hx$

Note that the points mentioned above is homogeous coordinates. The theorem indicates that the projectivity is a linear transformation and the linear mapping is projectivity

Proof: $x_1$, $x_2$, $x_3$ lie on the line $l$. Thus $l^T{x}_i=0$ for $i=1,2,3$. Given $H$ is a non-singular 3 $\times$ 3 matrix. We can get $l^T{H}^{-1}H{x}_i=0$. Thus $H{x}_i$ all lie on the line $l{}^{\prime }=H^{-T}l$.

Definition 2.11. Projective transformation A planar projective transformation is a linear transformation on homogeneous 3-vectors represented by a non-sigular 3 $\times$ 3 matrix: $$\left(\begin{array}{c}{x}_1{}^{\prime }\\ x_2{}^{\prime }\\ x_3{}^{\prime }\end{array}\right)=\left(\begin{array}{ccc}{h}_{11}& h_{12}& h_{13}\\ h_{21}& h_{22}& h_{23}\\ h_{31}& h_{32}& h_{33}\end{array}\right)=\left(\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right)$$, more briefly, $x^{{}^{\prime }}=Hx$.

Note that the equation won’t change when multiply any non-zero scala (非零常数) to $H$. The $H$ has 8 ratios so has 8 degrees of freedom. In the ray model in Figure 2.1 (2D投影平面), any projective transformation is linear transformation of ${\mathbb{R}}^3$

Example 2.12. Removing the projective distortion from a perspective image of a plane.

(需补充图片2.4)

Let original image point be $(x,y)$ and the point at image plane is $(x{}^{\prime },y{}^{\prime })$ in inhomogeous system. Thus we can get: $x{}^{\prime }=\frac{x_1{}^{\prime }}{x_3{}^{\prime }}=\frac{h_{11}x+h_{12}y+h_{13}}{h_{31}x+h_{32}y+h_{33}}$, $y{}^{\prime }=\frac{x_2{}^{\prime }}{x_3{}^{\prime }}=\frac{h_{21}x+h_{22}y+h_{23}}{h_{31}x+h_{32}y+h_{33}}$.

Thus we can get two equations at each point for the elements of $H$, which can be written as:

$x{}^{\prime }(h_{31}x+h_{32}y+h_{33})=h_{11}x+h_{12}y+h_{13}$, $y{}^{\prime }(h_{31}x+h_{32}y+h_{33})=h_{21}x+h_{22}y+h_{23}$,

where the equations are linear for the element of H. For 4 points of image, we can get 8 equations, which sufficient to solve $H$. Note that the 4 points can not be collinear. The example is shown as Figure 2.4.

Properties：

It doesn’t need knowledge of camera (parameters or position).4 points are not necessary (describedin section 2.7(之后的博客中会提及到)).

(需要补充图片2.5)

A variety of projective transformations are shown in the Figure 2.5.

Transformations of lines and conics

Transformations of lines

From the Theorem 2.10 and 2.11, after projective transformation, the points $x_i$ belongs to line $l$ can be written as $x_i{}^{\prime }=H{x}_i$ line on the line: $$l{}^{\prime }=H^{-T}l$$. Or $l^{\prime T}=l^T{H}^{-1}$. Thus we can say that points transformation according to $H$ while lines transformation according to $H^{-1}$. We can say points transform contravariantly and lines transform covariantly.

Transformations of conics

Given $x{}^{\prime }=Hx$, $$x^{T}Cx=x^{\prime T}[H^{-1}{]}^{T}C{H}^{-1}x{}^{\prime }=x^{{}^{\prime }T}{H}^{-T}C{H}^{-1}{x}^{{}^{\prime }}$$, which is quadratic form $x^{\prime T}{C}^{\prime }x{}^{\prime }$, where $C^{\prime }=H^{-T}C{H}^{-1}$.

Result 2.13. Under a point transformation $x{}^{\prime }=Hx$, a conic $C$ transforms to $C^{\prime }=H^{-T}C{H}^{-1}$.