MATLAB Code Implementation for Ellipse Fitting

Implementing ellipse fitting in MATLAB is a common geometric fitting problem, particularly useful for extracting ellipse parameters from discrete point data. The least squares method serves as the classical approach for solving such problems, achieving optimal fitting results by minimizing the sum of squared errors.

Implementation Approach: Data Preparation: A minimum of five 2D coordinate points is required (since the general ellipse equation has five degrees of freedom). Store these points in matrix format for efficient computation. In MATLAB code, this typically involves organizing data as N×2 matrices where each row represents an (x,y) coordinate pair.

Equation Construction: The general ellipse equation is expressed as Ax² + Bxy + Cy² + Dx + Ey + F = 0. By substituting each data point into this equation, we construct a linear system. MATLAB implementation involves creating design matrices using operations like x.^2, x.*y, and y.^2 to form the coefficient matrix.

Least Squares Solution: The problem transforms into finding the least squares solution for an overdetermined system. Use matrix operations such as SVD (svd function) or QR decomposition (qr function) to optimize parameters A, B, C, D, E, F while applying constraints (e.g., B² - 4AC < 0) to ensure the result represents an ellipse rather than other conic sections. The constraint can be implemented using Lagrange multipliers or eigenvalue methods.

Parameter Conversion: Convert the general equation parameters to standard ellipse parameters (center coordinates, major/minor axis lengths, rotation angle) for intuitive understanding and practical application. This involves solving eigenvalue problems and performing coordinate transformations using MATLAB's matrix computation capabilities.

Visualization Verification: Plot original data points and the fitted ellipse curve to validate fitting quality. Use MATLAB's plot and ellipse plotting functions with parametric equations derived from the fitted parameters.

Extension Considerations: For noisy data or outliers, incorporate robust algorithms like RANSAC to improve fitting stability. The MATLAB Computer Vision Toolbox provides ransac function implementation for this purpose. For special cases like axis-aligned ellipses, simplify the equation and reduce parameters by setting B=0, which decreases computational complexity. Ellipse fitting finds wide applications in computer vision, engineering measurement, and astronomical orbit analysis, such as detecting circular object projections or analyzing celestial trajectories.

Through these steps, MATLAB can efficiently complete ellipse fitting tasks and provide geometric parameters for subsequent analysis. Key MATLAB functions involved may include lsqnonlin for nonlinear least squares, eig for eigenvalue decomposition, and various plotting functions for visualization.