Self-Adjoint Filter Doesn't Work

Question

My understanding is that a symmetrical kernel is its own self-adjoint. For example, if we had the following kernel:

kernel = np.array([[-1, -1, -1],
                   [-1,  8, -1],
                   [-1, -1, -1]])

And applied a convolution with that filter:

from skimage import data
image = data.camera()

conv_result_1 = signal.fftconvolve(image, kernel, mode='same')

Then applied that same kernel again, it would undo the operation:

conv_result_2 = signal.fftconvolve(grad, kernel, mode='same')

But it doesn't undo the operation! What am I missing here?

Answer 1

In what ring of convolutions the given kernel is idempotent?

Certainly not for the regular convolution

kernel = np.array([[-1, -1, -1],
                   [-1,  8, -1],
                   [-1, -1, -1]])
scipy.signal.fftconvolve(kernel, kernel, mode='full')

array([[  1.,   2.,   3.,   2.,   1.],
       [  2., -14., -12., -14.,   2.],
       [  3., -12.,  72., -12.,   3.],
       [  2., -14., -12., -14.,   2.],
       [  1.,   2.,   3.,   2.,   1.]])

Actually, there is no finite kernel that is idempotent for the regular convolution. Assume that a kernel is a polynomial $h(z_x, z_y) \ne 1$, and you want $h(z_x, z_y)^2=1$, this simply has no solutions, unless you are in a special ring of polynomials (e.g. the $h(z_x, z_y)^2 \equiv 1 \mod x^N - 1$, for the cyclic convolution).