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I am attempting to implement the Gabor Descriptor based on the following paper Image Copy Detection Using a Robust Gabor Texture Descriptor. It is the best paper that I have found on the subject. It is not perfect, as it has some ordering issues with some equations and paragraphs that only make sense after later sections. No biggie, it is only a matter of time consumed to decipher the intent.

What is stopping me right now, is that one of the parameters is never defined in the paper. It first appears in $(1)$ of Section 2. There we have the following factor: $e^{2\pi jWx}$

$W$ is later assumed to be 1 and $x$ is the parameter of the function, but $j$ is never defined (though it is used again in several subsequent equations, especially in section 3.1, of most interest is $(11)$).

So I am seeking either a definition of $j$. A nice bonus would be a better paper (or book) defining the Gabor Descriptor.

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In this context, $j$ is the imaginary unit: $j = \sqrt{-1}$. The complex exponential function represents what they call an "oriented sinusoidal grating." The fact that $j$ is the imaginary unit in this case can be verified by noticing the relationship between $(1)$ and $(2)$ in Section 2; this is an example of the frequency-shifting property of the Fourier transform. That is, multiplying the Gaussian function $h(x,y)$ by the complex exponential function results in a shift in the Fourier transform of the result along the axis that the sinusoid runs across (the $x$ axis in your example).

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  • $\begingroup$ I have never seen $j$ used for the imaginary unit, always been $i$. Didn't even occur to me. Though I'm now somewhat confused. What is the point of that expansion? Doesn't Euler formula tell us that $e^{2 \pi i} = 1$? So $e^{2\pi i W x} = (e^{2\pi i})^{Wx}=1$? $\endgroup$ – Alex K Feb 18 '12 at 21:16
  • $\begingroup$ $j$ is used as the imaginary unit in many engineering contexts, especially in electrical engineering (as Hilmar pointed out below, since $i$ is used as a variable for electrical current). In your example, if $Wx$ is not an integer, then $(e^{j 2\pi})^{Wx}$ is a root of $1$, which is not necessarily equal to $1$. For example, if $Wx = 0.5$, then $(e^{j 2\pi})^{Wx} = -1$. $\endgroup$ – Jason R Feb 18 '12 at 21:27
  • $\begingroup$ I don't think I follow. $e^{2\pi i} = 1$ by Euler. So, it's $(1)^{Wx}$. 1 to any real power is 1, is it not? $\endgroup$ – Alex K Feb 18 '12 at 21:39
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    $\begingroup$ "So, it's $(1)^{Wx}$. $1$ to any real power is $1$, is it not?" Have you ever considered that $1^{1/2}$ might equal $-1$? $\endgroup$ – Dilip Sarwate Feb 20 '12 at 4:32
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Electrical Engineers typically use $j$ for the imaginary unit since $i$ is reserved for current.

The whole terms is a complex exponential, i.e. it's magnitude is always one and the period is W*x. It's only unity if W*x is an integer.

The identity you use only works if W and x are positive integers, see http://en.wikipedia.org/wiki/Exponentiation#Failure_of_power_and_logarithm_identities

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