Calibration Programming Exercises (40 Points): Implement The ✓ Solved

Calibration Programming Exercises (40 points): Implement the

Implement the direct parameter calibration method in order to (1) learn how to use SVD to solve systems of linear equations; (2) understand the physical constraints of the camera parameters; and (3) understand important issues related to calibration, such as calibration pattern design, point localization accuracy and robustness of the algorithms. Since calibrating a real camera involves lots of work in calibration pattern design, image processing and error controls as well as solving the equations, we will mainly use simulated data to understand the algorithms. As a by-product we will also learn how to generate 2D images from 3D models using a "virtual" pinhole camera.

Calibration pattern "design". Generate data of a "virtual" 3D cube similar to the one shown in Fig. 1 of the lecture notes in camera calibration. For example, you can hypothesize a 1x1x1 m³ cube and pick up coordinates of 3-D points on one corner of each black square in your world coordinate system. Make sure that your data is sufficient for the following calibration procedures. In order to show the correctness of your data, draw your cube (with the control points marked) using Matlab (or whatever tools you are selecting).

"Virtual" camera and images. Design a "virtual" camera with known intrinsic parameters including focal length f, image center (o_x, o_y) and pixel size (s_x, s_y). As an example, you can assume that the focal length is f = 16 mm, the image frame size is 512x512 (pixels) with (o_x, o_y) = (256, 256), and the size of the image sensor inside your camera is 8.8 mm x 6.6 mm (so the pixel size is (s_x, s_y) = (8.8/512, 6.6/512)). Capture an image of your "virtual" calibration cube with your virtual camera in a given pose (R and T). For example, you can take the picture of the cube 4 meters away and with a tilt angle of 30 degrees. Use three rotation angles alpha, beta, gamma to generate the rotation matrix R (refer to the lecture notes in camera model). You may need to try different poses in order to have a suitable image of your calibration target.

Direction calibration method: Estimate the intrinsic (f_x, f_y, aspect ratio µ, image center (o_x, o_y)) and extrinsic (R, T and further alpha, beta, gamma) parameters. Use SVD to solve the homogeneous linear system and the least square problem, and to enforce the orthogonality constraint on the estimate of R. i. Use the accurately simulated data (both 3D world coordinates and 2D image coordinates) to the algorithms, and compare the results with the "ground truth" data (which are given in step (a) and step (b)). Remember you are practicing a camera calibration, so you should pretend you know nothing about the camera parameters (i.e. you cannot use the ground truth data in your calibration process). However, in the direct calibration method, you could use the knowledge of the image center (in the homogeneous system to find extrinsic parameters) and the aspect ratio (in the Orthocenter theorem method to find image center).

ii. Study whether the unknown aspect ratio matters in estimating the image center, and how the initial estimation of image center affects the estimating of the remaining parameters. Give a solution to solve the problems if any.

iii. Accuracy Issues. Add in some random noises to the simulated data and run the calibration algorithms again. See how the "design tolerance" of the calibration target and the localization errors of 2D image points affect the calibration accuracy. For example, you can add 0.1 mm random error to 3D points and 0.5 pixel random error to 2D points. Also analyze how sensitive the Orthocenter method is to the extrinsic parameters in imaging the three sets of the orthogonal parallel lines. (* extra points:10) In all of the steps, you should give your results using either tables or graphs, or both of them. Figure. A 2D image of the "3D cube" with control 16+16 points.

Paper For Above Instructions

The calibration of camera parameters is a crucial aspect of computer vision and image processing, enabling effective mapping between 3D real-world coordinates and 2D image coordinates. This paper explores the implementation of the direct parameter calibration method, utilizing a simulated "virtual" calibration cube in conjunction with a "virtual" camera. The focus is on applying SVD (Singular Value Decomposition) for solving systems of linear equations during the calibration process, while taking various factors such as calibration pattern design and noise into account.

Calibration Pattern Design

The calibration begins with the design of a virtual 3D cube, a fundamental geometric shape that aids in defining the calibration pattern. For computational modeling, we assume a 1x1x1 m³ cube. The coordinates of 3D points are designated as vertices at the corners of the black squares on the cube's surface, such as (0, 0, 0), (1, 0, 0), (1, 1, 0), etc. These defined points are crucial as they serve as control points, ensuring accuracy during the calibration.

To represent the cube graphically, Matlab or similar tools can be employed. The control points will be marked clearly, enabling verification of data sufficiency. The generated image serves as a visual representation for subsequent calibration steps, validating that the simulation reflects the physical reality on which it is based.

Virtual Camera Setup

Next, a virtual camera is constructed with intrinsic parameters such as focal length, image center, and pixel size. For illustration, we set the focal length (f) at 16 mm, defining the image frame size at 512x512 pixels, with an image center at (o_x, o_y) = (256, 256). The size of the camera sensor is 8.8 mm x 6.6 mm, resulting in a pixel size defined by (s_x, s_y) = (8.8/512, 6.6/512). This configuration simulates a standard camera model, laying the groundwork for image capture.

According to the instructions, an image of the 3D calibration cube is captured from a specific pose, positioned 4 meters away with a tilt angle of 30 degrees. The rotation matrix R is generated using rotation angles alpha, beta, and gamma, which must be accurately calculated to ensure a perspective that captures the calibration target appropriately. Various poses might be tested to achieve optimal representation in the resulting image.

Direction Calibration Method

The ultimate goal is to estimate both intrinsic and extrinsic camera parameters through the direct calibration method. Intrinsic parameters include focal lengths (f_x, f_y), aspect ratio µ, and the image center (o_x, o_y). Extrinsic parameters comprise the rotation matrix (R) and translation (T) vectors, along with rotation angles.

To solve the system, SVD will be applied to analyze both homogeneous linear systems and least square problems, ensuring the orthogonality of the rotation matrix R. Utilizing the simulated data (3D world coordinates and corresponding 2D image coordinates), results will be compared against "ground truth" data established in the pattern design and camera setup phases. Nonetheless, it is essential to approach this calibration with the premise of ignorance towards actual camera parameters, except where the calibration method allows it.

Influence of Aspect Ratio on Calibration

One critical exploration in this project is the impact of the unknown aspect ratio on the estimation of the image center. Understanding how initial estimations affect subsequent results will be an essential inquiry, as these parameters directly impact calibration accuracy. A corrective methodology will be devised should complications arise.

Accuracy Issues: Noise Impact Analysis

Calibration accuracy is further scrutinized by incorporating random noises into the simulated data. For instance, adding 0.1 mm random errors to the 3D points and 0.5 pixel random errors to the 2D points will enable evaluation of how the design tolerance of the calibration target influences overall calibration accuracy. Additionally, the sensitivity of the Orthocenter method to extrinsic parameters will be analyzed. This will help in assessing how imaging errors affect the determination of the orthogonal parallel lines' sets.

Results Presentation

Throughout the steps of the calibration exercises, results will be documented systematically, utilizing tables and graphs to provide clear and understandable visual representations. This comprehensive analysis will ultimately present ways to improve calibration processes and highlight areas for further discussion and investigation.

Conclusion

In conclusion, the calibration programming exercises will enhance understanding of camera calibration through practical implementation. These exercises underscore the importance of both theoretical and practical aspects of calibration, leading to successful integration of simulated data and model outcomes.

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