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If the standard form of a Gabor function is as follows,
$$ g_{\lambda, \theta,\varphi, \sigma,\gamma}(x, y)=\exp\left(-\frac{x'^2+\gamma^2y'^2}{2\sigma^2}\right)\cos\left(2\pi\frac{x'}{\lambda}+\varphi\right)$$
where $$ x'=x\cos\theta+y\sin\theta\\ y'=-x\sin\theta+y\cos\theta$$ How can I find the Center Frequency from this equation?
The center frequency of a Gabor function is given by the reciprocal of its wavelength, which in turn is determined by the value of the parameter $\lambda$ in the equation.
To find the value of $\lambda$, we can rewrite the cosine term as follows:
$$\cos\left(2\pi\frac{x'}{\lambda}+\varphi\right) = \cos\left(\frac{2\pi x'}{\lambda} \right)\cos(\varphi) - \sin\left(\frac{2\pi x'}{\lambda} \right)\sin(\varphi)$$
Substituting the expressions for $x'$ and $y'$ in terms of $x$ and $y$ yields:
$$\cos\left(2\pi\frac{x\cos\theta+y\sin\theta}{\lambda}+\varphi\right) = \cos\left(\frac{2\pi x\cos\theta+2\pi y\sin\theta}{\lambda} \right)\cos(\varphi) - \sin\left(\frac{2\pi x\cos\theta+2\pi y\sin\theta}{\lambda} \right)\sin(\varphi)$$
Then, we can use the trigonometric identity $\cos(a+b) = \cos(a)\cos(b) - \sin(a)\sin(b)$ to write:
$$\cos\left(2\pi\frac{x\cos\theta+y\sin\theta}{\lambda}+\varphi\right) = \cos\left(\frac{2\pi x}{\lambda}\cos\theta+\frac{2\pi y}{\lambda}\sin\theta\right)\cos(\varphi) - \sin\left(\frac{2\pi x}{\lambda}\cos\theta+\frac{2\pi y}{\lambda}\sin\theta\right)\sin(\varphi)$$
We can now recognize the expression inside the cosine and sine functions as the dot product of the vector $(x,y)$ with the vector $(\cos\theta,\sin\theta)$, and write:
$$\cos\left(2\pi\frac{x\cos\theta+y\sin\theta}{\lambda}+\varphi\right) = \cos\left(\frac{2\pi}{\lambda}(x\cos\theta+y\sin\theta)\right)\cos(\varphi) - \sin\left(\frac{2\pi}{\lambda}(x\cos\theta+y\sin\theta)\right)\sin(\varphi)$$
Finally, we can use the fact that the wavelength $\lambda$ corresponds to a full period of the cosine term, which occurs when the argument of the cosine function increases by $2\pi$, to obtain:
$$\frac{2\pi}{\lambda} = 1$$
Therefore, the center frequency is given by:
$$f_c = \frac{1}{\lambda} = \frac{1}{2\pi}$$
Note that the center frequency is independent of the other parameters of the Gabor function, such as the orientation $\theta$, the phase $\varphi$, and the width parameters $\sigma$ and $\gamma$.
