IPMV L4 Camera Calibration

04. Camera Calibration

WCS. CCS. IPCS. PCS

坐标系变换关系汇总

WCS: World Coordinate System$(X^W,Y^W,Z^W)$

CCS: Camera Coordinate System$(X^C,Y^C,Z^C)$

IPCS/FCS: Image Plane Coordinate System/Film Coordinate System $(x,y)$

PCS: Pixel Coordinate System$(u,v)$

$$WCS\leftrightarrow CCS\leftrightarrow IPCS\leftrightarrow PCS$$

补充1：不同旋转平移方式

补充2：偏斜系数(skew coefficient)

Affine transformation

Definition

通俗解释及动画演示参考：https://www.matongxue.com/madocs/244/

Affine transformation = Linear transformation + Translation

Linear transformation:

• 变换前是直线的，变换后依然是直线
• 直线比例保持不变
• 变换前是原点的，变换后依然是原点

Affine translation:

• 变换前是直线的，变换后依然是直线
• 直线比例保持不变

Examples

Homogeneous Coordinates进一步理解

Description

Homogeneous coordinates(Projective coordinates)$\leftrightarrow$Projective Geometry

(类比:Cartesian coordinates$\leftrightarrow$Euclidean Geometry)

齐次坐标空间与普通坐标空间不同！

Advantages: coordinates of points(including$\infty$) can be represented using finite coordinates

Example 1

Determine the line passing through a(2,0) and b(1,3).

转化为齐次坐标，作外积得到的向量三个分量即为直线方程的系数A,B,C.

Example 2

Vanishing Point(消失点)

Definition

2D perspective projections of mutually parallel in 3D space appear to converge.

Example1

Example 2

How many vanishing points are there in the picture?

Ans: $\infty$

Lens distortion

Introduction

Deform the physically straight lines and makes them appear as curves in the images.

Category

1. Radial distortion(径向)

Reason: Lens’ geometric shape affects the straight line transformation.

2. Tangential distortion(横向)

Reason: Lens installed in front of the camera is not perfectly parallel to the image plane.

In practical experiments, the image geometry is affected to a much higher extent with radial distortion than with tangential distortion(sometimes neglected).

Radial distortion

Classification

Calculation

$$x_{undist}=x_{dist}(1+k_1r^2+k_2r^4+k_3r^6) \\ y_{undist}=y_{dist}(1+k_1r^2+k_2r^4+k_3r^6) \\ where \\ x_{dist}=\frac{x_c}{z_c}=\frac{u-u_o}{f_x} \\ y_{dist}=\frac{y_c}{z_c}=\frac{v-v_o}{f_y} \\ r^2=x_{dist}^2+y_{dist}^2 \\$$

​ $k_1,k_2,k_3$ are 3 intrinsic parameters used for radial distortion correction.

$$P_{dist}=[x_{dist},y_{dist}]^T\rightarrow P_{undist}=[x_{undist},y_{undist}]^T$$

Tangential Distortion

Camera Calibration

Introduction

Process of estimating camera parameters by using images that contain a calibration pattern.

Camera parameters includes:

• Intrinsic matrix
• Distortion coefficients
• Extrinsic matrix

Applications:

• Remove lens distortion
• Measure planar objects
• Reconstruct 3D scenes from multiple cameras

Process

Step 1. Define real world coordinates with checkerboard pattern.

Step 2. Capture multiple images of the checkerboard from different viewpoints.

Step 3. Find 2D coordinates of the checkerboard.

Step 4. Calibrate the camera.

Left: Camera-centric Right: Pattern-centric

Step 5. Compute the re-projection error and refine(精制，提炼) the calibration.

Re-projection error