What is the difference between an impulse response and a kernel?

Question

It seems to me they are two terms used to refer to the same concept, but, according to the Wikipedia page on Gaussian filter

a Gaussian filter is a filter whose impulse response is a Gaussian function

So, what is the difference between an impulse response and a kernel? I am particularly interested in knowing the difference in the context of computer vision and image processing.

Answer 1

The impulse response of an operation is the same as the kernel of that operation, but only if the operation can be represented by a kernel (the operation is linear and translation invariant).

For example, consider the following input (the impulse):

  [[0., 0., 0., 0., 0.],
   [0., 0., 0., 0., 0.],
   [0., 0., 1., 0., 0.],
   [0., 0., 0., 0., 0.],
   [0., 0., 0., 0., 0.]]

If you give it as input to a laplace filter, you get:

  [[ 0.,  0.,  0.,  0.,  0.],
   [ 0.,  0.,  1.,  0.,  0.],
   [ 0.,  1., -4.,  1.,  0.],
   [ 0.,  0.,  1.,  0.,  0.],
   [ 0.,  0.,  0.,  0.,  0.]]

This corresponds to the kernel of the laplace filter.

But if you give the same input to a 90 degrees rotation around the centre, you get:

  [[0., 0., 0., 0., 0.],
   [0., 0., 0., 0., 0.],
   [0., 0., 1., 0., 0.],
   [0., 0., 0., 0., 0.],
   [0., 0., 0., 0., 0.]]

(the same as the input)

Obviously, this is not a 'rotation kernel', it is just the identity. This happens because there is no such thing as a rotation kernel, because rotation operations are not translation independent (besides the trivial case of rotating by 0 degrees).

If the operation was not linear, the kernel would not match the impulse response either (for example, for a thresholding operation).