Methods of Plane Fitting

Plane fitting is a common problem in 3D geometry, typically used to find an optimal fitting plane from scattered point sets. The least squares method is a classical approach for solving such problems, which determines the optimal plane parameters by minimizing the sum of squared perpendicular distances from points to the plane.

Fundamental Principles of Least Squares The general form of a plane equation is (Ax + By + Cz + D = 0), where ((A, B, C)) represents the plane's normal vector and (D) is the constant term. To ensure a unique solution, constraints such as (A^2 + B^2 + C^2 = 1) are typically applied. The objective of least squares is to optimize these parameters so that the sum of perpendicular distances from all observed points to the plane is minimized. Implementation Approach: This can be achieved by constructing a covariance matrix from point coordinates and solving through Singular Value Decomposition (SVD).

Implementation Steps Data Preprocessing: Adjust point set coordinates to center them around the origin, reducing numerical errors. Code implementation typically involves calculating centroid coordinates and subtracting them from all points. Matrix Equation Construction: Use point coordinates to build a covariance matrix or directly formulate the least squares optimization problem. In MATLAB, this can be implemented using functions like cov() or by assembling the design matrix. Parameter Solving: Obtain optimal plane parameters through Singular Value Decomposition (SVD) or eigenvalue decomposition. Key functions: svd() in MATLAB for stable numerical solutions.

Calculation of Flatness and Uncertainty Flatness describes how closely the fitted plane approximates the point set, typically evaluated by calculating the standard deviation or maximum deviation of perpendicular distances from all points to the plane. Implementation involves computing residuals after fitting. Uncertainty reflects the reliability of fitted parameters, which can be estimated using covariance matrix analysis or Monte Carlo simulation methods. Code implementation may include error propagation calculations or statistical resampling techniques.

Application Scenarios Plane fitting is widely used in industrial inspection (e.g., workpiece flatness analysis), 3D reconstruction (point cloud data processing), and computer vision (depth perception). Selecting appropriate fitting methods and error evaluation strategies can enhance algorithm robustness and accuracy. Implementation considerations include handling outliers through RANSAC or weighted least squares approaches.