# Understanding this Gabor Filter equation

I am working on a project where I need to implement road edge detection. This research paper suits my requirement in the best way, so I am studying it: http://imagine.enpc.fr/publications/papers/TIP10a.pdf But I cannot understand the Gabor Filter equation they have used and also the set of parameter values they have used. Almost everywhere the Gabor Filter equation is different then theirs. If someone can explain, it would be great.

• You may have to track this down by reading the references. In , equation 1 is similar to what you have, and they call it a "generalized Gabor function", and they point to two further references. I'd suspect this is a form of the Gabor filter that is especially suited to a very specific application. – MBaz Sep 16 '15 at 18:34
• In , they have referred to "J.G. Daugman, Two-Dimensional Spectral Analysis of Cortical Receptive Field Profile". I cannot find this research paper for free anywhere, its paid! Also my college has no provision to access those, can anyone tell me where do I find it, or if anyone has access to those, please post the equation here? – nipunmaster Sep 17 '15 at 4:42
• If you email the authors, it's very likely they'll send you the article (or at least a pre-print). – MBaz Sep 17 '15 at 12:47
• Your question has beeen answered. Do not hesitate to vote for the useful ones and accept the most suitable – Laurent Duval Feb 9 '17 at 17:31

## 2 Answers

I will describe the rationale behind the "non-standard" terms, with respect to more classical expressions.

Essentialy, the $$\omega e^{-\omega^2 }$$ term indicates a derivative of a Gaussian (with 0 average), similar to an antisymmetric wavelet, instead of the modulated Gaussian window in standard Gabor filters: a wavelet filter bank, versus a modulated filter banks. This wavelet "smoothed derivative" aspect might help detect edges.

Then, the $$4a^2+b^2$$ induces an elliptic shape constraining anisotropy in the 2D wavelet, with a $$2$$-fold factor ratio.

Last, the $$e^{-c^2/2}$$ might serve an admissibility condition for this Morlet-Gabor like wavelet, to ensure easier inversion/reconstruction.

It is truly multiscale, as they specify they have 5 dyadic scales.

Indeed, the formula looks scary at first sight but is very intuitive. The basic idea is to design a filter that has the following properties:

• it is localized, meaning that it's envelope has a specific size (or scale in the terminology of wavelets) and responds optimally to certain orientations - while controlling the sensitivity of the measure.

Remember from the Heisenberg principle that you cannot be infinitely be good at the same time to localization and to frequency estimation...

Dually, you can think of it as :

• it is localized in the 2D Fourier plane, meaning that it has a preferred orientation and spatial frequency and a corresponding bandwidth along these axis

The formula above does this by multiplying the different components independently. Note that it's spitting out complex number such that you can consider symmetric and anti-symmetric features.

Shameless plug: I have contributed to a library which implements such filters but with a few differences to better fit what is supposed to happen in the visual system: https://github.com/bicv/LogGabor/blob/master/LogGabor.ipynb The page gives several examples of filters and filtered images.