Normalized Cross Correlation Operation

Question

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:

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.

Answer 1

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.