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The Gabor wavelet is a kind of the Gaussian modulated sinusoidal wave (source)

 

Gabor wavelets are formed from two components, a complex sinusoidal carrier and a Gaussian envelope. (source)

Figure 3: 1D Gabor Wavelet

and

In fact, the wavelet shown in Figure 2a (called the Morlet wavelet) is nothing more than a Sine wave (green curve in Figure 2b) multiplied by a Gaussian envelope (red curve). (source)

Morlet wavelet

Are these just different names for the same thing?

Update:

Not to be confused with the "Gabor transform", which seems to be just another name for "STFT with a Gaussian window". There's also the Gabor atom, which I guess is the same as the Gabor wavelet?

Since asking this on math.SE, I've also found terms like "Gabor/Morlet wavelet" and "Gabor-Morlet transform", implying that they are the same thing.

Also this has been asked before: Gabor transform/wavelet vs. Morlet wavelet but the answers are not clear to me.

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  • $\begingroup$ They have different formulas. dsprelated.com/showmessage/122107/1.php $\endgroup$ – pharmine Oct 4 '11 at 10:19
  • $\begingroup$ @pharmine: Those are just different ways to write the same thing, though, no? $\endgroup$ – endolith Oct 4 '11 at 15:14
  • $\begingroup$ @endolith Is it possible to put up the plots they are referring to? $\endgroup$ – Spacey Apr 19 '12 at 19:02
  • $\begingroup$ @endolith Based on this evidence they certainly appear to be the exact same... $\endgroup$ – Spacey Apr 19 '12 at 19:31
  • $\begingroup$ @Mohammad: Added $\endgroup$ – endolith Apr 19 '12 at 19:31
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The Gabor wavelet is basically the same thing. It's apparently another name for the Modified Morlet wavelet. Quoting from Wavelets and Signal Processing:

[The Modified Morlet wavelet] does not satisfy the admissibility condition but is nonetheless commonly used. Sometimes this wavelet is called the "Gabor wavelet," but that term is improper because Gabor had nothing to do with wavelets. He was ... one of the founders of time-frequency analysis.

That book is a collection of papers, and that paper ("The Wavelet Transform and Time-Frequency Analysis") is by Leon Cohen (of time-frequency distribution "Cohen class" fame), so I think it's reasonably authoritative.

At the very least, it sounds like the confusion is just a naming disagreement. According to A Friendly Guide to Wavelets (pg. 114), Gabor was the first person to propose the use of Gaussian windows for time-frequency localizations, so his name tends to get attached anytime they're involved.

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    $\begingroup$ Aha! "We distinguish between the Gabor function (non­zero-mean function) and the Gabor kernel (zero-mean function). The Gabor kernel satisfies the admissibility condition for wavelets, thus being suited for multi-resolution analysis. Apart from a scale factor, it is also known as the Morlet Wavelet." Pattern Recognition and Image Analysis: Proceedings $\endgroup$ – endolith Apr 20 '12 at 0:48
  • $\begingroup$ Better link: A Real-Time Gabor Primal Sketch for Visual Attention $\endgroup$ – endolith Apr 20 '12 at 0:56
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    $\begingroup$ Is there a short answer for what the wavelet 'admissibility condition' is? Simply the conservation of energy during transformations? $\endgroup$ – Spacey Apr 20 '12 at 3:47
  • $\begingroup$ @Mohammad: "It is important to note that, for this to be a wavelet, the positive and negative areas `under' the curve must cancel out. This is known as the admissibility condition." "The admissibility condition ensures that the inverse transform and Parseval formula are applicable." $\endgroup$ – endolith May 7 '12 at 14:40

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