Lec.02: Image Formation¶

Camera and lens¶

Pinhole camera
- Add a barrier to block off most of the rays, The opening known as the aperture
- 孔太小一方面亮度不够，另一方面会发生衍射现象diffraction

Lens: \(\dfrac{1}{i}+\dfrac{1}{o}=\dfrac{1}{f}\)

Magnification: \(m=\dfrac{h_i}{h_o}\)

Field of View: depend on focal length and sensor size
- 长焦会放大远处的物体。焦距越长，视角越窄，适合远摄；焦距越短，视角越宽，适合广角拍摄。
- FOV also depends on sensor size

F-number: More convenient to represent aperture as a fraction of focal length
- \(N=\dfrac{f}{D}\)
Lens Defocus
- From similar triangles: \(\dfrac bD=\dfrac{|i^{\prime}-i|}{i^{\prime}}\)
- Blur circle diameter: \(b=\dfrac D{i^{\prime}}|i^{\prime}-i|,b\propto D\propto\dfrac1N\)
- 如何认为image是well focused的，只要b的大小小于像素大小即可

Depth of Field：图像足够聚焦的物体距离的范围

景深和哪些因素有关

o 相机到物体的距离

f 镜头焦距

c 图像中可接受的最大模糊圈直径

N 光圈的F-number

\[ \begin{aligned}&\bullet c=\frac{f^2(o-o_1)}{No_1(o-f)}, c=\frac{f^2(o_2-o)}{No_2(o-f)}\newline &\bullet\text{ Depth of Field: }o_2-o_1=\frac{2of^2cN(o-f)}{f^4-c^2N^2(o-f)^2}\end{aligned} \]

How to blur the background
- Large aperture
- Long focal length
- Near foreground
- Far Background

Geometric Image Formation¶

Pin-hole camera model: Perspective Projection

Perspective Projection in homogeneous coordinates

\[ \begin{bmatrix}f & 0 & 0 & 0\newline 0 & f & 0 & 0\newline 0 & 0 & 1 & 0\end{bmatrix}\begin{bmatrix}x\newline y\newline z\newline 1\end{bmatrix}=\begin{bmatrix}fx\newline fy\newline z\end{bmatrix}\cong\begin{bmatrix}f\frac xz\newline f\frac yz\newline 1\end{bmatrix} \]

Each point has an infinite set of homogeneous coordinates

Vanishing points: 三维空间中平行线在照片中会收敛于同一点

The ray from C through v is parallel to the lines
v tells us the direction of the lines
v may be outside the image frame or at infinity

Vanishing lines
- Any set of parallel lines on the plane define a vanishing point
- The union of all of these vanishing points is the vanishing line
- Note that different planes define different vanishing lines
  - The direction of the vanishing line tells us the orientation of the plane

造成畸变的原因有哪些

Perspective distortion
- 向上倾斜照相机会导致垂直方向收敛
- 用一个普通的镜头来保持相机的水平，只能捕捉到建筑的底部
- 外部柱看起来更大
- 变形不是由于镜头的缺陷
Radial distortion
- Caused by imperfect lenses(透镜)
- More noticeable for rays that pass through the edge of the lens
  
  对于通过透镜边缘的光线来说更为明显
- 广角容易导致桶形畸变
- 长焦容易导致枕形畸变

\[ \begin{aligned}&r^{2} = {x_{n}^{\prime}}^{2}+{y_{n}^{\prime}}^{2}\newline &x_{d}^{\prime} = x_{n}^{\prime}(1+\kappa_{1}r^{2}+\kappa_{2}r^{4})\newline &y_{d}^{\prime} = y_{n}^{\prime}(1+\kappa_{1}r^{2}+\kappa_{2}r^{4})\end{aligned} \]

Photometric image formation¶

Shutter speed controls exposure time
The pixel value is equal to the integral of the light intensity within the exposure time

像素值等于光强\(\times\)曝光时间

Rolliing shutter effect：卷帘快门，逐行曝光，而非global shutter，即理想中的整张照片同时曝光，拍摄结果就可能出现"倾斜"、"摇摆不定"或"部分曝光"等情况，即果冻效应。
Color spaces: RGB, HSV(Hue色调, Value亮度, Saturation饱和度)
HSV比RGB更接近人类对颜色感知，在图像处理中使用比较多
两者之间可以互相转换