[Practical Exercise] MATLAB Camera Calibration Images

# 1. Introduction to MATLAB Camera Calibration

Camera calibration is a crucial technique in computer vision that determines the internal and external parameters of a camera, providing accurate geometric information for image analysis and processing. MATLAB, as a powerful scientific computing platform, offers a rich set of tools and functions for efficient and precise camera calibration. This chapter will provide an overview of the basic concepts of camera calibration, the methods and applications of camera calibration in MATLAB.

# 2. MATLAB Camera Calibration Theory

**2.1 Camera Model and Projection Matrix**

The essence of camera calibration is to establish a camera model that describes the imaging process. The pinhole camera model is the most commonly used, assuming the camera is a small hole through which light enters and forms an image on the imaging plane.

In the pinhole camera model, the origin of the camera coordinate system is located at the aperture, with the x and y axes parallel to the imaging plane and the z axis perpendicular to it. The origin of the world coordinate system is located outside the camera to be calibrated, with the x, y, and z axes pointing to the right, up, and forward, respectively.

The camera projection matrix projects three-dimensional points in the world coordinate system onto two-dimensional points on the imaging plane. The projection matrix P is a 3x4 matrix, where the first three columns represent the camera intrinsic matrix K, and the last column represents the extrinsic matrix T.

P = [K | T]

Where:

* K =

[fx 0 cx]
[0 fy cy]
[0 0 1]

* T =

[tx ty tz]
[rx ry rz]

fx, fy are the focal lengths, cx, cy are the coordinates of the principal point, tx, ty, tz are the translation vector, and rx, ry, rz are the rotation vector.

**2.2 Calibration Parameters and Optimization Algorithms**

Camera calibration requires the estimation of 11 parameters in the projection matrix P. These parameters can be divided into intrinsic and extrinsic parameters. Intrinsic parameters are related to the camera itself, including focal length, principal point coordinates, and distortion coefficients. Extrinsic parameters are related to the position and orientation of the camera in the world coordinate system, including translation and rotation vectors.

Camera calibration typically uses optimization algorithms to estimate the projection matrix. The most commonly used optimization algorithm is the Levenberg-Marquardt algorithm. This algorithm minimizes projection errors through iterative methods, gradually approaching the optimal solution.

Optimization algorithms require an initial value as a starting point. The initial value can be manually specified or estimated through other methods. For example, the focal length can be estimated by the pixel size in the image and the size of a known object.

**Optimization Algorithm Process**

1. Initialize the parameters of the projection matrix P.
2. Calculate the projection error, which is the distance between the three-dimensional points in the world coordinate system projected onto the two-dimensional points on the imaging plane and the actual observed two-dimensional points.
3. Calculate the gradient of the projection error.
4. Update the parameters of the projection matrix P using the gradient and Hessian matrix.
5. Repeat steps 2-4 until the projection error reaches a minimum.

# 3. MATLAB Camera Calibration Practice

### 3.1 Calibration Board Fabrication and Image Acquisition

**Calibration Board Fabrication**

The calibration board is a key tool for camera calibration, providing precise locations of feature points. When fabricating a calibration board, consider the following factors:

***Size and Pattern:** The calibration board should be large enough to include a sufficient number of feature points. The pattern usually consists of alternating black and white squares or circles, which should be easy to detect and extract.
***Material:** The calibration board should be made of a material that is not prone to deformation or damage, such as cardb