(the log scale in the frequency domain) is a parabola. Variance reduction How much does a Gaussian filter with standard deviation 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 , then after applying the filter the reduced standard deviation can be approximated as Sample Gaussian matrix This sample matrix is produced by sampling the Gaussian filter kernel (with σ = 0.84089642) at the midpoints of each pixel and then normalizing. The center element (at [0, 0]) has the largest value, decreasing symmetrically as distance from the center increases. Since the filter kernel's origin is at the center, the matrix starts at and ends at where R equals the kernel radius. The element 0.22508352 (the central one) is 1177 times larger than 0.00019117 which is just outside 3σ. Implementation A Gaussian blur effect is typically generated by convolving an image with an FIR kernel of Gaussian values. In practice, it is best to take advantage of the Gaussian blur’s separable property by dividing the process into two passes. In the first pass, a one-dimensional kernel is used to blur the image in only the horizontal or vertical direction. In the second pass, the same one-dimensional kernel is used to blur in the remaining direction. The resulting effect is the same as convolving with a two-dimensional kernel in a single pass, but requires fewer calculations. Discretization is typically achieved by sampling the Gaussian filter kernel at discrete points, normally at positions corresponding to the midpoints of each pixel. This reduces the computational cost but, for very small filter kernels, point sampling the Gaussian function with very few samples leads to a large error. In these cases, accuracy is maintained (at a slight computational cost) by integration of the Gaussian function over each pixel's area. When converting the Gaussian’s continuous values into the discrete values needed for a kernel, the sum of the values will be different from 1. This will cause a darkening or brightening of the image. To remedy this, the values can be normalized by dividing each term in the kernel by the sum of all terms in the kernel. A much better and theoretically more well-founded approach is to instead perform the smoothing with the discrete analogue of the Gaussian kernel, which possesses similar properties over a discrete