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In 1D signals f(t) (e.g. a sound signal - independent variable t-->time) when we want to use informations from the Fourier transform F(ω) in order to reconstruct the signal, using the magnitude |F(ω)| yields better reults than using <F(ω). In 2D signals f(x,y) (e.g. an image - independent variables x,y-->space coordinates) using <F(ω1,ω2) (usually) yields better results than using |F(ω1,ω2)|. Why is this happening? Is there an intuitive explanation for this result?

Edit: A simple example of what I am referring to is the below, where for image reconstruction phase seems to be more critical than magintude. enter image description here

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  • $\begingroup$ Phase is often very important in the 1D Fourier Transform as well. An example is the Phase Vocoder for audio pitch change. $\endgroup$ Mar 13 at 1:03
  • $\begingroup$ But still most of the times in 1D signals magnitude is more useful than phase, but for reason in 2D signals the opposite seems to happen. $\endgroup$ Mar 13 at 1:12
  • $\begingroup$ I agree that magnitude only is useful when we use the FFT for observing a power spectrum but phase is also very important in many other 1D applications where the FFT is used for actual signal processing. Fast correlation and beam steering are other examples. I’d answer below but I’m less versed in 2D processing. $\endgroup$ Mar 13 at 1:57
  • $\begingroup$ It's not clear what you are specifically referring to in terms of "better". Better for what exactly? Signal reconstruction, parameter estimation, etc. Generally speaking, images tend to be sensitive to phase issues when doing processing, which is why linear phase FIRs are typically preferred, though not always. But that doesn't mean there aren't 1D applications that aren't sensitive to phase noise. $\endgroup$
    – Baddioes
    Mar 13 at 5:55
  • $\begingroup$ I am referring to signal reconstruction, for a better explanation see the edited part in my question above. $\endgroup$ Mar 13 at 15:08

2 Answers 2

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This will not be a very mathematically rigorous answer, but an answer nonetheless. For spatial data (as opposed to multi-channel data), the phase of the FFT encodes the spatial relationship, and therefore the structure, in both dimensions. The magnitude on the other hand encodes the intensity distribution. If you look at the Einstein magnitude with raccoon phase image, clearly a raccoon is visible, but the shading of the image is primarily vertical, ie whole columns seem to have roughly the same intensity value. This correlates more with the shading distribution of the Einstein image. On the other hand, for the Einstein phase raccoon magnitude image, the shading is more horizontal, corresponding to the more horizontal shading in the raccoon image, while Einstein is still clearly visible. The spatial relationship between pixels is likely more important in images than the overall intensity distribution in terms of interpretability.

In 1-D, where the x-axis is time based, the phase encodes the relative temporal relationship of the different components of the signal. For certain applications, this may not be as important for signal reconstruction as the intensity distributions, though I'm sure in some areas like comms, speech recognition, etc, it is quite important.

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    $\begingroup$ I wasn't looking for anything mathematically rigorous anyway, your answer gives some intuition! $\endgroup$ Mar 13 at 20:45
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using the magnitude |F(ω)| yields better reults than using <F(ω)

I disagree with the basic premise. For example a simple delay in the time domain ONLY affects the phase in the frequency domain whereas the magnitude remains unaffected

The notion that one is "more important" than the other is IMO misguided. Both are essential parts of the signal representation. Sometimes a certain physical property maps more to one than the other, but in most cases you will need both and so you might as well get used to it.

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  • $\begingroup$ Maybe I didn't explained clearly before, I am bascically referring to cases like the one mentioned in the edited part of my question. $\endgroup$ Mar 13 at 15:10
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    $\begingroup$ I think you could make an (weak) argument that the "what" of a feature is mostly magnitude but the "where:" is phase. Recall that translation in the spatial domain is a pure phase shift in the spatial frequency domain. If you want to explore this a bit more, start with something simple: a blob or a line. Move it around and see what changes. $\endgroup$
    – Hilmar
    Mar 13 at 18:53
  • $\begingroup$ I thinks it kinda gives an sensation of why that works, thanks! $\endgroup$ Mar 13 at 20:42

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