Real-Time Nonlinear Parameter Estimation Using the Levenberg–Marquardt Algorithm on Field Programmable Gate Arrays

The Levenberg-Marquardt (LM) algorithm is a nonlinear parameter learning algorithm that converges accurately and quickly. This paper demonstrates for the first time to our knowledge, a real-time implementation of the LM algorithm on field programmable gate arrays (FPGAs). It was used to train neural networks to solve the eXclusive Or function (XOR), and for 3D-to-2D camera calibration parameter estimation. A Xilinx Virtex-5 ML506 was used to implement the LMA as a hardware-in-the-loop system. The XOR function was approximated in only 13 iterations from zero initial conditions, usually the same function is approximated in thousands of iterations using the error backpropagation algorithm. Also, this type of training not only reduced the number of iterations but also achieved a speed up in excess of 3 ×106 when compared to the software implementation. A real-time camera calibration and parameter estimation was performed successfully on FPGAs. Compared to the software implementation the FPGA implementation led to an increase in the mean squared error and standard deviation by only 17.94% and 8.04% respectively. The FPGA increased the calibration speed by a factor of 1.41 × 106. There are a wide range of systems problems solved via nonlinear parameter optimization, this study demonstrated that a hardware solution for systems such as automated optical inspection systems or systems dealing with projective geometry estimation and motion compensation systems in robotic vision systems is possible in real time.