Research on the Robustness of Focus Measure Operators Based on RRMSE

This paper establishes a quantitative relationship model between the relative root mean square error (RRMSE) and noise parameters under the additive white Gaussian noise model. According to the difference in operator operation structure, focus measure operators are divided into two categories: squared-type and absolute-value type. Theoretical derivation shows that the RRMSE of squared-type operators is proportional to the noise variance σ2, while for absolute-value type operators, when the noise variance is large, their RRMSE is approximately proportional to the noise standard deviation σ. On this basis, a new quantitative metric—noise response slope—is proposed to characterize the robustness of operators against noise perturbation. For squared-type operators, the value of the noise response slope can be accurately derived; for absolute-value type operators, the value of the noise response slope can only be approximated. Five squared-type operators and five absolute-value type operators are selected for experiments, and the experimental slopes are obtained via linear regression fitting. The experimental results show that for squared-type operators, the coefficient of determination exceeds 0.999; except for the sine image sequence, the relative error between the theoretical slope and the experimental slope is less than 2%, and for the sine image sequence the error is less than 10%, because the sine image has a large number of pixels whose grayscale values are close to 0 or 255. For absolute-value type operators, the coefficient of determination exceeds 0.98; there is a significant difference between the theoretical slope and the experimental slope, but the Spearman correlation coefficient between them is 1, with a two-tailed test significance level of 0.05. The proposed model can estimate the robustness of operators without adding noise to the image sequence, providing an effective analytical method for the robustness evaluation and design of focus measure operators.