Lourakis Implementation of the Levenberg-Marquardt Method

Lourakis' implementation of the Levenberg-Marquardt (LM) method is a classical optimization algorithm for solving nonlinear least squares problems. This algorithm combines the advantages of gradient descent and Gauss-Newton methods by dynamically adjusting the damping parameter to balance convergence speed and stability, making it particularly suitable for applications such as curve fitting and Bundle Adjustment in computer vision. Lourakis' code implementation is renowned for its efficiency, being written in C with support for sparse matrix processing. The core algorithmic principle involves iteratively adjusting parameters to minimize the objective function (typically the sum of squared errors). The implementation uses a trust region approach where the damping factor is automatically reduced when approaching the optimal solution, exhibiting quadratic convergence similar to the Gauss-Newton method, while maintaining higher damping during initial iterations to ensure global convergence. This implementation is frequently integrated into open-source libraries like OpenCV and features memory usage optimization, support for user-defined Jacobian matrix calculation, and interfaces to multiple linear algebra solvers. For developers working with large-scale nonlinear optimization problems, understanding Lourakis' LM implementation can significantly improve algorithm deployment efficiency through its careful handling of matrix operations and convergence control mechanisms.