What exactly is meant by Fourier techniques for edge detection?

[tjackson3]

What exactly is meant by "Fourier techniques" for edge detection?

I'm trying to work on an edge detection assignment for a Fourier Analysis class. I've nailed every bit of it up to this point, but now the assignment is wanting me to do edge detection via "Fourier techniques." No amount of looking around on the internet has given me any clue as to what this could be. I already had to do edge detection via convolution techniques, so my first instinct, which was to FFT the image, multiply it by the filter, and convert back is redundant (by the convolution theorem). Any thoughts on this?

Thanks so much for your help!

 

[hamster143]

FFT and a filter is exactly what I'd do if I had to do edge detection via Fourier transform. Fourier is not particularly useful for edge detection. In practical applications, people use high-pass filters, gradients, or wavelet methods. Perhaps there's some misunderstanding of the assignment?

 

[tjackson3]

FFT and a filter is exactly what I'd do if I had to do edge detection via Fourier transform. Fourier is not particularly useful for edge detection. In practical applications, people use high-pass filters, gradients, or wavelet methods. Perhaps there's some misunderstanding of the assignment?

How would the FFT/filter combo accomplish edge detection though? Sorry, I'm kinda new to the image processing game. Anyway, I don't think there's much room for misunderstanding. Here's the exact wording: "Use Fourier techniques to do the same as #1." #1 was "Use convolution techniques to separate the vertical, horizontal, and other edge components in the image." I wish it were some misunderstanding, as I feel that would make the whole thing much easier.

 

[tjackson3]

Alright, I went to the library today, and as far as I can tell, "Fourier techniques" refers to one of two things (mind you, none of these are explicitly labeled "Fourier techniques," but I at least see them as slightly different from "convolution techniques"):

1.) Using a wavelet transform with a highpass filter and a Fourier transform. It also looks like this works two-dimensionally, meaning I wouldn't have to do the horizontal and vertical components separately.

2.) Using a highpass filter with a Fourier transform. However, this means that we take the FFT of the image, multiply it by the highpass filter, and IFFT it, which is the same as convolving the filter with the image, so I think that's more of a convolution technique.

Any thoughts on that?

 

[hamster143]

I'm inclined to say #2. Presumably, if you're working on Fourier, wavelet techniques are not part of your toolbox yet.

Yes, things are often dual to one another in spatial & frequency space. One or the other approach may work better. FFT+highpass has complexity of O(N log N), where N is the number of pixels. Spatial-domain convolution would be O(N*M), where M is the area of high-pass filter. Depending on M and N, FFT could be much faster.

 

[tjackson3]

Oh wow. That's really really helpful and greatly simplifies things for me. Thank you so much!