When should the sum of all elements of a gaussian kernel be zero?

The sum of all the elements in a kernel should be zero when you want to completely remove the "DC" or constant or offset term.

1. Why is it necessary for a differentiation kernel to have all its coefficients sum to zero?
2. Why the sum total of coefficients in a smoothing kernel is 1?
3. What is the standard deviation of a Gaussian kernel?
4. How does Gaussian kernel work?

Why is it necessary for a differentiation kernel to have all its coefficients sum to zero?

Differentiating kernels, in which all the elements of the kernel sum to zero, ∑i gi = 0, accentuate the places where the signal is changing rapidly in value and are therefore useful to extract information.

Why the sum total of coefficients in a smoothing kernel is 1?

This is why their kernels sum to 1. If you look at their frequency response, you'll see that the zero-frequency component (DC component) is 1. This component is the sum over the kernel. And it being 1 means that the DC component of the image is not modified when applying the convolution.

What is the standard deviation of a Gaussian kernel?

The standard deviation for a two-dimensional kernel is the radius in pixels containing 68% of the integrated magnitude of the coefficients.

How does Gaussian kernel work?

In other words, the Gaussian kernel transforms the dot product in the infinite dimensional space into the Gaussian function of the distance between points in the data space: If two points in the data space are nearby then the angle between the vectors that represent them in the kernel space will be small.