Maximizing accuracy of 2D Gaussian profile estimation using differential entropy

The goal of this paper is to find an optimal width of circular region-of-interest (ROI) for the precise estimation of 2D Gaussian profile parameters in the presence of additive noise. The radius of circular ROI for the rotationally symmetrical profile can be represented as a product of the profile’s STD and the factor of Mahalanobis distance k. The centre of ROI coincides with the centre of the profile being estimated. It was shown that in the case of a random sampling within such circular ROI, the estimation accuracy of the least-squares method is highly affected by the chosen factor k for the constant number of random input samples and given SNR. The differences in estimation accuracy are the results of variations of profile data informativity for different ROI widths. If sample positions are random variables uniformly distributed within the circular ROI, it was derived that the 2D Gaussian profile values as a function of random variables follow the log-uniform distribution. Therefore, in the paper we derive the differential entropy of log-uniform distribution which is maximized with respect to the factor of Mahalanobis distance k, thus yielding the optimal ROI width. The theoretical results are verified using Monte-Carlo simulation and we show that the loss of estimation accuracy for other non-optimal widths is proportional to the reduction of the profile’s differential entropy. Such a solution is valid under a fixed number of samples as an estimation constraint. However, for the case of sample density constraint, the solution is different, as we will demonstrate in the paper.