The Meaning of Camera Calibration (Addressing What Camera Calibration Is)

In image measurement processes and machine vision applications, establishing a geometric model of camera imaging is essential to determine the relationship between the three-dimensional geometric position of a point on a spatial object's surface and its corresponding point in the image. The parameters of these geometric models constitute the camera parameters.

Under most conditions, these parameters must be determined through experimentation and calculation; this process of parameter determination is referred to as camera calibration.

The Significance of Camera Calibration (Explaining Why Camera Calibration Is Necessary)

One of the fundamental tasks of computer vision is to extract geometric information of objects in three-dimensional space from camera-acquired images, enabling object reconstruction and recognition. The spatial relationship between a point's three-dimensional geometric position on an object's surface and its corresponding point in the image is determined by the geometric model formed by the camera's imaging process; these model parameters constitute the camera parameters.

Under most conditions, these parameters must be determined through experimentation and computation. Whether in image measurement or machine vision applications, camera parameter calibration is a critical step; the accuracy of the calibration results and the stability of the algorithm directly influence the precision of the camera's operational outputs.

Therefore, proper camera calibration is a prerequisite for subsequent tasks, and enhancing calibration accuracy is the focal point of scientific research.

Common methods for camera calibration

Camera calibration methods include: traditional camera calibration method, active vision-based camera calibration method, and self-calibration method.

The orientation of the camera relative to the world coordinate system. The calibration accuracy directly determines the precision of computer vision (machine vision).

To date, numerous methods have been proposed for camera calibration, and the theoretical aspects of camera calibration have been adequately addressed. For current research in this field, the focus should be on developing specific, simple, practical, rapid, and accurate calibration methods tailored to particular real-world application challenges.

The traditional calibration method will not be elaborated here; the camera coordinate system transformation is as follows:

The camera coordinate system consists of three axes—X, Y, and Z—with its origin at point C.

A Cartesian coordinate system consists of two axes, x and y, with the origin located at point p.

The image coordinate system generally refers to the relative coordinate system of an image, which can be considered to lie on the same plane as the image plane coordinate system.

The diffraction-based DOE device introduces a novel dimension for camera calibration. Leveraging its translation-invariant property, calibration can be achieved using just a single image.

For implementing camera-based measurements in machine vision, high-precision geometric camera calibration is absolutely essential. The objective is to determine the internal camera parameters required to map 3D world coordinates onto 2D image coordinates.

The common approach involves using a predefined calibration grid for photogrammetric calibration (d and r), followed by estimating camera parameters by minimizing the nonlinear error function, which requires observations from multiple different directions.

Due to the limited grid size, this technique is more or less restricted to close-range camera calibration. Another method suitable for far-field camera calibration employs a collimator-compass setup to illuminate a set of individual pixels (n × m). By knowing the direction of the collimated light, the camera parameters d and the projection d' can be estimated, providing a more comprehensive overview of the key developments in camera calibration.

The calibration procedure described here combines the unique advantages of calibration grid arrangement and single-pixel illumination. By employing diffractive optical elements as beam splitters, only an image with n × m diffractive points is required to estimate the internal camera parameters.

Diffractive optical elements (DOE) can be used to split an incident laser beam with wavelength λ into multiple beams each possessing a well-defined propagation direction. Since the image on the sensor constitutes a Fraunhofer diffraction pattern, each projection image point represents a point at infinity, expressed in the 3D projection space P³ by homogeneous coordinates d = [X, Y, Z, 0]T.

f = (fx, fy) represents the spatial frequency encoded in the DOE.

The above formula is only valid when the incident light wave is a plane wave with a uniform intensity distribution, completely perpendicular to the DOE surface. In practical setups, the beam's extension is finite and typically exhibits an inhomogeneous intensity distribution, usually a Gaussian distribution. Moreover, it is difficult to avoid a slight tilt of the DOE relative to the incident beam.

Include the incident angle in the calculation

rotated the x and y axes of the DOE coordinate system using angles α and β in the collimator coordinate system. The direction of the diffracted beam is now as follows:

In the DOE coordinate system, the direction of the diffracted beam can be directly calculated through simple matrix operations; therefore, we will omit some of the lengthy expressions derived from this process.

To convert the beam direction into the camera coordinate system, the camera's external orientation in the DOE coordinate system must be considered:

Here, R is a 3×3 rotation matrix defining the camera orientation, and t is the translation vector representing the camera position. The formula demonstrates that the mapping of an ideal point at infinity remains invariant under translation—a fundamental requirement for the computational process. Compared to traditional calibration grids, this approach offers a significant advantage: a single image suffices for calibration, thereby requiring fewer parameters to be estimated.

This paper describes a novel method for geometric sensor calibration that employs customized diffraction optical elements as beam splitters with precisely known diffraction angles. Since the virtual source of the diffracted beam is an infinitely distant point, the imaged object resembles a starry sky, endowing the image with translation invariance.

This specialized feature enables complete camera calibration using a single image, eliminating the need for complex beam adjustments and resulting in a highly rapid and reliable calibration process.

The obtained results are consistent with those from classical camera calibration using the pinhole camera model and radial distortion model. Our analysis also incorporated eccentric distortion, yet no improvement was observed.

The results demonstrate that a reliable solution can be achieved, which separates the parameters of the internal orientation from the rotation of the DOE and the external orientation of the camera. Consequently, complex alignment of calibration setup components is unnecessary, simplifying the calibration process and enabling on-site calibration in principle.