A Fast Method for Fitting a Multidimensional Gaussian Function

Abstract

This paper estimates the multidimensional Gaussian profile parameters from the noisy measurements in the exponential function’s argument domain. The proposed method minimizes the weighted squared error between the natural logarithm of the model and the logarithm of the normalized input data with the weights set to the input data values or model values. The proposed method is an iterative method where the parameters of the covariance matrix and the profile’s peak position are alternatively estimated. The main advantage of the proposed method is a one-step analytical solution for the parameters of the covariance matrix and the linear profile scale for the given initial centroid position for arbitrary dimensions. The profile’s peak position is then updated given the estimated parameters by solving a system of nonlinear coupled equations using an iterative optimization procedure. Finally, the proposed method in the log domain is compared with the LS method in the domain of Gaussian profile values, where all profile parameters are simultaneously estimated using an iterative procedure for solving a system of nonlinear equations using numerical optimization. The proposed log domain estimation method yields similar results as the numerical LS method in the value domain for sufficiently high signal-to-noise ratios (SNRs) and narrow regions-of-interest (ROIs) concerning their precision. However, it converges much faster due to the analytic solution.

Keywords

• Multidimensional Gaussian profile fitting
• weighted least-squares method
• estimation in the log domain