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I'm studying image processing and I know that there are different filters you can apply. I don't understand the difference between the low-pass/bandpass/notch filter (or high-pass/band-reject/notch). I tried to draw them but I don't know if I did it correctly:

enter image description here

From what I have realized it is that the difference between a low-pass filter and a band-pass is that the first is symmetrical with respect to the y axis while the second is not. Therefore, a low-pass filter passes the lower frequencies and resets those higher than a certain threshold (cutoff frequency). Instead, a band-pass filter passes the frequencies included in a certain range and cancels the others. It's correct?

But I have not really understood the purpose of the notch filters. I realized that there are notch band-pass and band-reject notch. In addition the book I'm studying (Digital Image Processing - Rafael C. Gonzalez) says that a notch filter is composed of the product of two high-pass filters. Just I don't understand how they work, how they are made and what they do.

The book also talks then about optimal notch filters that is used where there are several interference components in the image in the Fourier domain. Again I haven't understood how it works.

Could someone help me? Thank you very much!

So the difference between a notch filter and a band-pass is that I use the notch when I don't know what are the frequencies to pass (or delete) but I get it visually analyzing the image of the Fourier transform, right?

However, this is what there are written in my book: enter image description here

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  • $\begingroup$ What does (-k) index imply? $\endgroup$
    – MimSaad
    Commented Apr 11, 2017 at 6:44
  • $\begingroup$ Thanks for your patience. H_{k}(u,v) and H_{-k}(u,v) are highpass filters whose centers are at (u_k, v_k) and (-u_k, -v_k). However, the main message contains the entire paragraph. $\endgroup$
    – sleepwalking
    Commented Apr 11, 2017 at 10:35

2 Answers 2

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First,note that your drawn filter responses are one dimensional filters.

Notch filter, selectively suppresses some frequency bands that are not of interest. One of the well known applications of notch filter in image processing is image denoising. As an example, consider the following image, which is corrupted by periodic noise (the tiled parallel bars).

Periodic noise.

If we take Fourier transform of the image, it will be something like this: enter image description here

Based on our prior observations and knowledge we know this image should not have larges peaks other that central section, so those other peaks are the noise and are responsible for the parallel bars in the image. To remove those, we can apply a notch filter which allows all frequencies to pass, except for frequency components of the periodic noise.

a notch filter is composed of the product of two high-pass filters

I would say it is product of a low-pass and a high-pass filter.

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  • $\begingroup$ Thank you for the reply. I edit my main message. Can you help me again? $\endgroup$
    – sleepwalking
    Commented Apr 10, 2017 at 17:14
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    $\begingroup$ A notch filter usually has a complex pole pair and a complex zero pair. Not really a product between a high pass and a low pass. $\endgroup$
    – Ben
    Commented Feb 8, 2019 at 20:00
  • $\begingroup$ @MimSaad the product of a low-pass and a high-pass filter is Band-Pass Filter. $\endgroup$
    – Bilal
    Commented Dec 28, 2020 at 20:08
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In simple terms, the x axis measures the frequency. Beginnign with lower frequency to higher frequency. Low pass filter means you allow lower frequency to pass through. Higher frequency is "blocked" or attenuated. High pass filter, you allow Higher frequency but "block" low Frequency.

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