How much does a Gaussian filter with standard deviation σ f {\displaystyle \sigma _{f}} smooth the picture? In other words, how much does it reduce the standard deviation of pixel values in the picture? Assume the grayscale pixel values have a standard deviation σ X {\displaystyle \sigma _{X}} , then after applying the filter the reduced standard deviation σ r {\displaystyle \sigma _{r}} can be approximated as