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I'm reading this paper about about an algorithm to measure image sharpness, and am confused by these sentences in the 6th paragraph:

It is well known that the attenuation of high-frequency content can lead to an image that appears blurred. One way to measure this effect is to examine the image’s magnitude spectrum $M(f)$, which is known to fall inversely with frequency, i.e. $M(f)\propto f^{-\alpha}$, where $f$ is the frequency and $-\alpha$ is the slope of the line $\log{M}\propto-\alpha \log{f}$.

I don't know what frequency this is referring to as the images this paper and algorithm are concerned with are grayscale, does anyone have a clue?

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  • $\begingroup$ Removed the judgemental description of that paragraph from your title. $\endgroup$ – Marcus Müller Jun 19 at 16:03
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Detail in images require higher frequency basis functions. The frequency in this case is measuring fluctuations in intensity as a distance is traversed. With a lot of detail, a lot of fluctuations, thus higher frequency.

Tamp down the higher frequency and you lose detail, i.e. the image blurs.

The dampening is measured (described) best on a log scale.

Consider this answer Honey Mustard.

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  • $\begingroup$ But isn't this measure of fluctuations spatial rather than spectral? I would assume a spectrum of magnitude is just an ordered sequence of the magnitudes of all the pixels in an image? $\endgroup$ – Ivan Zabrodin Jun 19 at 16:15
  • $\begingroup$ @IvanZabrodin You are measuring a range of intensities in the spatial domain. Doing a fourier transforms (Magnitude spectrum) gives you a representation in which you are measuring a range of tonal component strength for a set of frequencies in the domain. Your last sentence doesn't make sense to me. Consider an image consisting of vertical stripes. In one direction there is no change in intensity and thus no high frequencies. Traverse across the stripes and you have a repeating pattern with a frequency. If you are not familiar with the 1D DFT, you should start with that. $\endgroup$ – Cedron Dawg Jun 19 at 16:30
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I find that pretty well-written; just as you can have time signal that has frequency properties, you can have a spatial signal that has frequencies. That's a pretty important concept in image descriptions and processing!

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Frequencies in this context just means how fast the intensities change in the greyscale images as we move in the plane of the 2-D image.

As you might be knowing the edges or boundaries or outlines of objects in the images are composed of high frequency elements. Why? Because in order to show the outline of an object in the image, the intensities at the boundary of the object needs to suddenly change to a higher value or a lower value.

For example, if the intensity changes from 0 to 255 within 1 or 2 pixels, it will seem like an edge of an object in the image.

Now if you attenuate this sudden change in intensity by filtering out high frequency components. Then the intensity change for example can become 0 to 10 in that region and now the edge of the object will be fudged and it won't be as sharp as it was originally.

Bottomline: by filtering out high frequency components in the image spectrum, we are softening the rapid changes in intensities which often define the edges of patterns and objects in an image. And, by softening the edges, we are gonna get blurred images because we can't figure out the boundaries anymore.

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