Can gaussian low pass filter remove ringing effect from the image?

Question

An image is convolved with an ideal low pass filter. The resultant image is convolved with a Gaussian low pass filter to obtain a final image. Will you observe ringing effect in the final image. WPlease give reason.

Answer 1

The ringing from convolving with an ideal low pass filter is Gibbs oscillations. If you convolve with a smooth function, you won't get ringing, but of course you will get some smoothing and attenuation. You could try multiplying by a function like $\exp(-x^4)$ in the Fourier domain or its ilk. These functions are flat around $0$, unlike the Gaussian, so they are more ideal low pass filter-like.