Multi-level B-spline Surface Fitting

Multi-level B-spline surface fitting is an efficient non-uniform parameterized surface modeling method that achieves high-precision smooth surface fitting through progressive optimization. Compared to traditional single-level B-spline fitting, this approach refines the control point grid hierarchically and adjusts surface shapes at different scales, making it better suited for capturing complex geometric features. The algorithm implementation typically involves initializing a coarse control grid and recursively subdividing it based on residual errors.

The core concept begins with a sparse initial control grid, computes fitting errors, and then densifies control points layer by layer to enhance local detail representation. Each optimization level only adjusts newly added control points while preserving the overall topology from the previous layer, ultimately achieving progressive optimization from global to local scales. This method ensures surface continuity while effectively avoiding overfitting issues. In code implementation, this involves maintaining a hierarchy of control point matrices and applying localized optimization using basis function calculations.

Typical applications include point cloud reconstruction in reverse engineering, surface deformation in computer animation, and geometric modeling in CAD/CAM systems. Due to its uniform parameterization strategy, the algorithm maintains numerical stability while enabling acceleration of large-scale data processing through parallel computing. For noisy measurement data, regularization methods can be integrated to enhance fitting robustness. Key computational steps involve B-spline basis evaluation, error thresholding, and hierarchical grid refinement operations.