Spatial operators are closely related to concepts from signal processing called correlation and convolution. They are similar but different and we discuss the important properties of convolution for Gaussian kernels.
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Using the properties of convolution we can combine a simple derivative kernel with Gaussian smoothing to create a derivative of Gaussian (DoG) kernel which is very useful for edge detection, or a Laplacian of Gaussian (LoG) kernel which is useful for detecting regions.
We introduce spatial operators by a simple example of taking the average value of all pixels in a box surrounding each input pixel. The result is a blurring or smoothing of the input image.