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.
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.