For the 2D discrete Gabor transform, why is it that we cannot use a set of orthonormal basis for its representation, instead we have to use frames for representing it?

  • $\begingroup$ Are you talking about this Gabor transform, which is a form of STFT, or a Gabor wavelet transform, or something else? $\endgroup$ – endolith Sep 10 '13 at 0:11
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    $\begingroup$ It's on page 4 of this pdf (Spatial (2-D) Gabor Filter) mplab.ucsd.edu/tutorials/gabor.pdf $\endgroup$ – freak_warrior Sep 10 '13 at 3:06
  • $\begingroup$ There was a doctoral dissertation at UMass Lowell that developed an orthagonal version of the Gabor transform. Unfortunately I do not recall the author's name nor that of the dissertation. I believe the dissertation was completed somewhere between 1990 and 1992. You might want to contact the University. $\endgroup$ – user20514 Apr 12 '16 at 15:44
  • $\begingroup$ @freak_warrior what do you call a 2D discrete Gabor transform, precisely? $\endgroup$ – Laurent Duval Apr 12 '16 at 19:54

The frames end up giving you an orthogonal representation, just not exactly a basis representation that you're talking about. Simply by definition that's not what a Gabor filter does. It's a time-frequency representation of your signal, while "orthogonal basis vectors" wouldn't have a time-specific property -- they stretch for as long as your signal stretches.

If you want to use orthogonal basis transforms, use Fourier, cosine, empirical mode decomposition and similar transforms.

If you want time-frequency representation, your can use short-time Fourier transform, wavelet transforms (of which Gabor can be considered a special case) and other similar transforms.

  • $\begingroup$ Hi, why is the Gabor transform a special case of the wavelet transform? $\endgroup$ – freak_warrior Jan 14 '14 at 23:58
  • $\begingroup$ Some discrete wavelets are actually orthogonal bases. For example the Daubechies and Coiflet families. At the most basic level, the Haar (db2) wavelets are extremely time localized, but only weakly frequency localized. $\endgroup$ – LutzL Feb 24 '14 at 15:53
  • $\begingroup$ The Gabor transform is a continuous wavelet transform, if you only use one specific function in its scaled and shifted variants. If an affine system of shifts and dilations is selected, it becomes a discretized wavelet transform that is invertible if it is a frame. A frame is a generating set that is a non-basis but nevertheless contains no superfluous elements. $\endgroup$ – LutzL Feb 24 '14 at 15:58

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