[Feat] Discrete Gaussian kernel (gaussian) should be adapted for discrete filtering

🐛 Bug

The current Gaussian kernel simply samples the continuous Gaussian function:

def gaussian(window_size, sigma):
    x = torch.arange(window_size).float() - window_size // 2
    if window_size % 2 == 0:
        x = x + 0.5
    gauss = torch.exp((-x.pow(2.0) / float(2 * sigma ** 2)))
    return gauss / gauss.sum()

Expected behavior

Discrete Gaussian filtering should be compatible with scale-space theory and rely on modified Bessel function for its kernel weights. See e.g.: https://en.wikipedia.org/wiki/Scale_space_implementation#The_discrete_Gaussian_kernel

Additional context

Note that this is a similar issue to that raised in MONAI: https://github.com/Project-MONAI/MONAI/issues/363

I came here while looking for a PyTorch implementation of a Discrete Gaussian filter but realised this one was also sampling the continuous Gaussian function.

ITK implements such a behaviour in itk::GaussianOperator: https://github.com/InsightSoftwareConsortium/ITK/blob/master/Modules/Core/Common/include/itkGaussianOperator.hxx

Wikipedia also provides a basic python implementation (albeit relying on scipy and without normalisation):

from scipy.special import iv
import numpy as np

def discrete_gaussian_kernel(n, t):
    T = np.exp(-t) * iv(n, t)
    return T

[ducha-aiki]

I will check the actual difference. In case of we are doing the change, we also would need to modify our spatial_gradient kernels. And it should help with local features localization precision as well.

[edgarriba]

@wyli that sounds good we could try to target to include this feature for v0.5