Lecture 2 Image Formation¶

Camera and Lens¶

Pinhole Camera¶

A barrier to block off most of the rays. The opening is the aperture 光圈.

Flaws

Less light gets through.
Diffraction effects (衍射).

Lens 透镜¶

\[ \frac{1}{i} + \frac{1}{o} = \frac{1}{f}. \]

Focal length $f$ 焦距¶

If $o = \infty$, then $f = i$.

Magnification $m$ 放大率¶

\[ m = \frac{h_i}{h_o}. \]

Field of View (FOV) 视野¶

Factor

Focal Length.
- Longer focal length, Narrower angle of view. Vice versa.

Note

50mm / 46° (and full frame) is the most similar FOV with human eyes. Thus 50mm lens is called standard lens (标准镜头).

telephoto lens (长焦镜头，望远镜头): 视野小, 放大率大.
short focal lens (短焦镜头，广角镜头); 视野大, 放大率小.

Sensor Size
- Bigger sensor size, Wider angle of view. Vice versa.

Aperture 光圈¶

The representation of aperture is its Diameter $D$.

F-Number

\[ N = \frac{f}{D}\ \ \text{ (mostly greater than $1$, around $1.8 \sim 22$)}. \]

Lens Defocus¶

Blur Circle Diameter (光斑半径)

\[ b = \frac{D}{i'}|i' -i|,\ \ b \propto D \propto \frac{1}{N}. \]

Focusing 对焦¶

Depth of Field (DoF) 景深¶

\[ {\tt DoF} = o_2 - o_1 = \frac{2of^2cN(o-f)}{f^4-c^2N^2(o-f)^2}. \]

From the equation above, we can find that DoF is almost proportional to $N$, and thus the Larger aperture, the Smaller F-Number and the Smaller DoF.

How to blur the background¶

Large aperture.
Long focal length.
Near foreground.
Far background.

Geometric Image Formation（定位）¶

Camera model

\[ \begin{bmatrix} u \\ v \end{bmatrix} = \begin{bmatrix} f \cfrac{x}{z} \\ f \cfrac{y}{z} \end{bmatrix}. \]

Homogeneous Coordinates / Projective Coordinates¶

Suppose that $\begin{bmatrix} x \\ y \\ w \end{bmatrix}$ is the same as $\begin{bmatrix} x/w \\ y/w \\ 1 \end{bmatrix}$, then we get

\[ \begin{bmatrix} f & 0 & 0 & 0 \\ 0 & f & 0 & 0 \\ 0 & 0 & 1 & 0 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \\ 1 \end{bmatrix} = \begin{bmatrix} fx \\ fy \\ z \\ \end{bmatrix} \cong \begin{bmatrix} f\cfrac{x}{z} \\ f\cfrac{y}{z} \\ 1 \end{bmatrix}. \]

For convenience of dicussion, we can also put the image plane in front of the camera (opposite to the previous picture).

Perspective Projection¶

What are Preserevd: Straight lines are still straight.
What are Lost: Length and Angle.

Vanishing Points¶

Properties

Any two parallel lines have the same vanishing point $\bf v$.
- $\bf v$ can be outside the image frame or at infinity.
Line from $\bf C$ to $\bf v$ is parallel to the lines.

Vanishing Lines¶

Multiple vanishing points compose the vanishing lines.

The direction of the vanishing line tells us the orientation of the plane.

Distortion¶

Converging verticals.

lec2_13 — Problem and Solution (View Camera 移轴相机)

Exterior columns appear bigger. (Due to lens flaws.)
Radial distortion. (Due to imperfect lens.)

\[ \begin{aligned} r^2 &= {x'}_n^{2} + {y'}_n^{2}, \\ {x'}_d &= {x'}_n(1 + \kappa_1r^2 + \kappa_2r^4), \\ {y'}_d &= {y'}_n(1 + \kappa_1r^2 + \kappa_2r^4). \\ \end{aligned} \]

Solution

Take a photo of a grid at the same point and then use the mathematics to calculate and correct radial distortion.

Orthographic Projection¶

\[ \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \\ 1 \end{bmatrix} = \begin{bmatrix} x \\ y \\ 1 \\ \end{bmatrix} \Rightarrow (x, y). \]

Photometric Image Formation（定颜色/亮度）¶

Image Sensor 传感器¶

CMOS
CCD (Charge Coupled Device)

Shutter 快门¶

Shutter speed controls exposure time.

Color Sensing¶

Color Spaces
- RGB
- HSV
Practical Color Sensing: Bayer Filter

Shading 着色¶

BRDF (Bidirectional Reflectance Distribution Function)¶

\[ L_r(\hat{\textbf{v}_r};\lambda) = \int L_i(\hat{\textbf{i}_r};\lambda)f_r(\hat{\textbf{v}_r}, \hat{\textbf{v}_i}, \hat{\textbf{n}}; \lambda)\cos^+\theta_i\ d\hat{\textbf{v}_i}. \]

Diffuse (Lambertian) Reflection¶

Shading independent of view direction.

Specular Term¶

Intensity depends on view direction.

Blinn-Phong Reflection Model¶

\[ L = L_a + L_d + L_s = k_aI_a + k_d(I/r^2)\max(0, \textbf{n}\cdot\textbf{l}) + k_s(I/r^2)\max(0, \textbf{n}\cdot\textbf{h})^p. \]

最后更新: 2023.11.21 17:00:13 CST
创建日期: 2022.11.08 20:35:08 CST