I am looking for a kernel acting as a low pass filter that satisfies these conditions:
$$K(-\mathbf{u})=K(\mathbf{u}) \tag{1}$$ $$K(\mathbf{u}) \ge K(\mathbf{v}), \;\; \text{if} \;\; |\mathbf{u}|<|\mathbf{v}|, \;\; \text{and} \;\; \lim_{|\mathbf{u}|\rightarrow\infty} K(\mathbf{u})=0 \tag{2}$$ $$\int K(\mathbf{x})d\mathbf{x} = 1 \tag{3}$$
In the reference paper, the author suggested a Gaussian kernel that is:
$$K_\sigma(\mathbf{u})=\frac{1}{\left(\sqrt{2\pi}\sigma\right)^n}e^{-|\mathbf{u}|^2/2\sigma^2} $$
with a scale parameter $\sigma > 0$.
The Gaussian kernel is very good at approximating the required properties. But the kernel reduces edge information when the kernel size is large.
Could you suggest to me any kernel that can satisfy the three conditions above, and is more robust than Gaussian in noise reduction, while maintaining edge information? I found a modified kernel, but it was very difficult to implement. Thank you so much.
