Least Squares Surface Fitting Program

Least squares surface fitting is a fundamental mathematical optimization technique used to determine optimal polynomial models that best approximate surface data. This method minimizes the sum of squared errors between data points and the fitted surface, deriving optimal polynomial expressions describing three-dimensional data z=f(x,y).

In MATLAB implementation, least squares surface fitting typically involves constructing design matrices and solving systems of linear equations. Key implementation steps include:

Data Preparation: Organize input 3D coordinate points (x, y, z) where z represents the dependent variable, while x and y serve as independent variables. MATLAB code typically stores these as column vectors or matrices for efficient computation.

Polynomial Model Selection: Choose appropriate polynomial degrees based on data characteristics. Common models include quadratic surfaces (z = a + bx + cy + dx² + exy + fy²) and cubic surfaces. The polyfitn function from MATLAB's File Exchange can automatically determine optimal degrees using statistical criteria.

Equation Formulation: Transform polynomial models into linear system formulations. Construct design matrix X containing various powers and cross-terms of x and y variables, converting the problem into solving the least squares solution for Xβ ≈ z. Implementation involves using MATLAB's array operations to generate Vandermonde-like matrices.

Coefficient Solving: Compute polynomial coefficients β using MATLAB's matrix operations or built-in functions. The backslash operator (mldivide) employs QR decomposition for numerically stable solutions, while specialized tools like polyfitn provide additional statistical metrics. The resulting coefficients define the analytical expression of the fitted surface.

This methodology finds applications in terrain modeling, image processing, and experimental data analysis. MATLAB's implementation advantages include robust matrix computation capabilities for handling large datasets efficiently, with flexibility to adjust polynomial degrees balancing fitting accuracy and overfitting risks through techniques like cross-validation.