# What is Local Mean Filter?

The research paper "Multidirectional Scratch Detection and Restoration in Digitized Old Images" says that,

4.1. Preprocessing. The preprocessing step aims to enhance image features along a set of chosen directions. First, image is grey-scaled and filtered with a sharpening filter (we subtract from the image its local-mean filtered version), thus eliminating the DC component.

Now, I am Googling the term "Local Mean Filter" but there is nothing available like that.

Can anyone please provide me any reference of "Local-Mean Filter"?

What filter are they using for Sharpening?

## 1 Answer

They probably just wanted to say that the image was blurred (by some local method, e.g. convolution with a Gaussian kernel) in a more scientific way.

On the sharpening: They don't sharpen the image directly. What they do is to blur the image and the subtract the blurred version from the original. The result is the same as sharpening directly via an appropriate filter.

Why this works

Assume the image $I$ and an blurring kernel $g$ (which could be a Gaussian kernel). In Fourier-Space, blurring the image by convolution becomes a multiplication:

$\mathscr{F}(g\star I) = \mathscr{F}(g)\mathscr{F}(I)$

with $\mathscr{F}(\cdot)$ being the Fourier-Transformation.

Then $I - g \star I$ becomes: $\mathscr{F}(I) - \mathscr{F}(g)\mathscr{F}(I) = \mathscr{F}(I)(1-\mathscr{F}(g))$.

Since $\mathscr{F}(g)$ enhances the low-frequency components in in Fourier-Space, $1-\mathscr{F}(g)$ enhances the higher frequencies (I assume here that $\mathscr{F}(g)$ is normalized appropriately in magnitude, which can be achieved via scaling factors). So, in image-space, the resulting image is a sharpened version of $I$.

• So, what filter are they using for Sharpening? – user18425 Jun 1 '16 at 15:19
• They don't sharpen the image directly. What they do is to blur the image and the subtract the blurred version from the original. The result is the same as sharpening directly via an appropriate filter. – M529 Jun 1 '16 at 15:22
• But, why are they doing so instead of sharping directly? – user18425 Jun 1 '16 at 15:28
• Sorry, I can't read their minds. What difference does it make for you? – M529 Jun 1 '16 at 15:30
• In this case, I would just go for it and check the results. They describe the step as a "sharpening", without any specification about it. Those specifics are certainly not the key-point of the paper. Hence, go for it, try several sharpening filters, or use the one you have at hand, and check the results and their dependence on the details of this step. If you are still in doubt then, you could contact the authors and ask them directly. Play with the problem! :) – M529 Jun 1 '16 at 17:35