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According to my question about separability of Gabor filters in this link, I want now to convolve my image with this separable filter by using the normalized cross correlation operation. Assume my Gabor filter is G, my image is I. My Gabor is separated into Low-Pass gaussian filter f(x) and Band-Pass gaussian filter g(y). Therefore the image is convolved with the Gabor using the following equation:

I(x,y)*G(x,y) = (I(x,y)*f(x))*g(y).

But I want to achieve this separable convolution using the normalized cross-correlation operation described below:

enter image description here

Where ^G is the zero mean, unit normal version of the filter and H(x,y) represents a filter with all ones and the same size of the Gabor filter.

1) I didn't understand what is ^G. What should be its value? what it differs from G ?

2) How the normalized cross-correlation is computed for the separable Gabor? I don't know if I use correctly the formula: I(x,y)*f(x)*g(y) / I^2(x,y)*H(x,y) . I don't think that it's true. because I didn't understand what should be the value of zero mean, unit normal version of the Gabor.

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1 Answer 1

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From what the paper described, I think $\hat G$ is achieved by standardized normalization on $u$ and $v$ before you implement the formula (1) on $G$:

x = -filtSizeL : filtSizeR; 
y = x;
u = x * cos(theta) + y * sin(theta);
v = -x * sin(theta) + y * cos(theta);
u = (u - mean(u))/std(u);
v = (v - mean(v))/std(v);

The filter is still separable after this normalization step, but I don't think you need to implement that, because it is just equivalent to select a different $\sigma$ value in the filter.

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  • $\begingroup$ Firstly, thank you for your answer. can you show please the pages 3 and 4 from this paper: "cbcl.mit.edu/publications/ai-publications/2011/…" $\endgroup$
    – Christina
    Commented Jan 20, 2014 at 22:40
  • $\begingroup$ Did you mean that I should only use I(x,y)*G(x,y)=I(x,y)*f(x)*g(y) ? without using the normalized cross correlation ? I didn't understand well your response :) $\endgroup$
    – Christina
    Commented Jan 20, 2014 at 22:42
  • $\begingroup$ @Christina. Yes you can directly use Ifg without normalization. It won't make a huge difference on your S1 results:) Yet I edited my answer trying to show how the normalization is obtained $\endgroup$
    – lennon310
    Commented Jan 20, 2014 at 22:46
  • $\begingroup$ Ohh thank you!! but what is the advantage of the normalization ? and why they were used the cross-correlation normalization in the paper ? $\endgroup$
    – Christina
    Commented Jan 20, 2014 at 22:50
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    $\begingroup$ @Christina I will try to read that part, and maybe later today I would answer your new question. Thanks. $\endgroup$
    – lennon310
    Commented Jan 20, 2014 at 22:59

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