Degraded image: Wiener Filter vs Wiener deconvolution

Question

I am studying methods for recovering images (that has been blurred and noised).

Can someone explain the difference between:

• http://en.wikipedia.org/wiki/Wiener_deconvolution
• http://en.wikipedia.org/wiki/Wiener_filter

I can understand that wiener deconvolution is used for Deblurring while wiener filter is used for Denoising.

Is that right? Are there any difference between the two?

Answer 1

Wiener deconvolution is an approach to solve the deconvolution problem that relies on the filter proposed by Wiener. The equation is the same in denoising and deblurring, except that the filter $G$ (to stick with Wikipedia's notations) that you should use is different.

To make things clear:

• denoising consists in the case where the degradation kernel $H$ is the identity,
• in deconvolution this kernel is equal to the blur kernel that destroyed your image (motion blur, lens PSF...).

Hence, in denoising, the $H$ part that appears in the expression of the Wiener filter $G$ disappears (it is equal to 1), while in deconvolution it is equal to the Fourier transform of your blur kernel.

More generally, when using an inverse problem approach for these tasks, i.e. solving $\| A x - y \|^2 + \lambda g(x)$ (with $x$ your solution and $y$ the noisy observation, $g(x)$ some prior on $x$), you can always switch between denoising and deconvolution by choosing $A = Id$.