Performance of bilateral filtering on Gaussian noise

Bilateral filtering is a nonlinear technique that reduces noise from images while preserving strong image edges. Due to the nonlinear nature of bilateral filtering, it is difficult to analyze the performance of the filter. We derive a closed-form equation of bilateral filtering for flat regions which shows the relationship between noise reduction and filtering parameters. This work explicitly shows that noise reduction depends on the ratio of the range parameter to the noise standard deviation, which confirms reported empirical observations. The derived result is a significant contribution for the analysis of bilateral filters toward estimating the optimal param- eters for minimum mean square error. We demonstrate that the theoretical analysis presented is consistent with simulations. © 2014 SPIE and IS&T (DOI: 10.1117/1.JEI.23.4.043024) explicitly shows that noise reduction depends on the ratio of the range parameter to the noise standard deviation. We observe that most of the noise is removed when the value of the range filter parameter is over two times the standard deviation of the noise from our analysis, consistent with the work of Zhang and Gunturk. 8,9 We also present experimental results with test images. If we can develop accurate signal modeling in the future, then the optimal parameters for the range and domain filter can be estimated. The remainder of this paper is structured as follows: Sec. 2 describes the bilateral filter. We derive the equation of the bilateral-filtered Gaussian noise using the ergodic theorem in Sec. 3. Simulations with test images and analysis are shown in Sec. 4 to validate our proposed method. A dis- cussion and conclusions are presented in Sec. 5.