MATLAB Implementation of Least Squares Method for Ellipse Detection

The least squares method is a classic data fitting technique that can be efficiently implemented in MATLAB for ellipse detection. This approach estimates optimal fitting parameters by minimizing the sum of squared errors. For ellipse detection problems, the least squares implementation typically involves the following steps: In the image preprocessing phase, edge detection is performed on the input image using common operators like Canny or Sobel. This step aims to extract potential ellipse edge points from the image. In MATLAB, this can be implemented using functions like edge() with appropriate method specifications. Ellipse equation parameterization typically uses the general quadratic curve equation representation. This transforms the ellipse fitting problem into a parameter estimation problem. The general conic equation Ax² + Bxy + Cy² + Dx + Ey + F = 0 is commonly employed. Constructing the least squares system involves substituting edge point coordinates into the ellipse equation to form an overdetermined system. MATLAB's powerful matrix operations are particularly suitable for handling such problems. The implementation typically creates a design matrix where each row corresponds to an edge point's coordinates. Utilizing MATLAB's linear algebra capabilities, such as the backslash operator () or pinv() function for pseudo-inverse calculation, to solve the least squares solution. This effectively solves for the ellipse equation parameters. The backslash operator automatically computes the least squares solution when dealing with overdetermined systems. The post-processing phase may include validation and optimization of fitting results, such as removing outliers or employing robust methods like RANSAC to improve fitting accuracy. MATLAB's statistical toolbox provides functions for outlier detection and robust fitting implementations. This method finds extensive applications in industrial inspection, medical image analysis, and other fields where automatic identification of elliptical objects and precise estimation of geometric parameters are required. MATLAB's strong matrix computation capabilities and visualization tools make algorithm implementation and result verification intuitive and efficient. The implementation benefits from MATLAB's built-in functions for matrix operations, linear algebra, and graphical output for result validation.