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I have trouble understanding in an intuitive way (not by writing complicated math formulas) what is value of $x$ & $y$ in the Gabor functions.

Here is the formula,

$$g(x,y) = \frac{1}{2\pi\sigma_x\sigma_y}\exp\left[-\frac{1}{2}\left(\frac{x'^2}{\sigma_x^2}+\frac{y'^2}{\sigma_y^2}\right)\right]\exp\left[2\pi\cdot jWx'\right]$$

Where

$$x' = x \cos \theta + y \sin \theta \qquad(1)$$

and

$$y' = -x \sin \theta + y \cos \theta\qquad(2)$$

Actually what will be the value of $x$ & $y$ in $(1)$ and $(2)$

Thanks

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1 Answer

Gabors are two-dimensional filters and $(x,y)$ specifies the coordinates. They are equal to $(0,0)$ at the center of the Gabor (as are $(x^\prime,y^\prime)$). The $(x,y)$ are the "original" coordinates, corresponding to $(\sigma_x,\sigma_y)$, while $(x^\prime,y^\prime)$ are the rotated coordinates.

Note that in the $(x,y)$ coordinates, the Gabor can only be stretched along the x-axis or the y-axis, but not diagonally. The equations (1) and (2) allows the Gabor to be stretched in any orientation (defined by $\theta$) in the $(x^\prime,y^\prime)$ coordinates.

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