Wednesday, November 15. 2006
The exponential in $R^d$, II
(Q15) I’m gonna show you some cool Gabor atoms on .
, as described previously, indeed looks like this:
CODE:
[xx yy] = meshgrid(linspace(-2,2,100));
v=[1 1.5];
e1 = exp(pi*i*( v(1)*xx + v(2)*yy ));
imagesc(real(e1))
w=[-1 4];
e2 = exp(pi*i*( w(1)*xx + w(2)*yy ));
imagesc(real(e2))
See how the value of the frequency changes with the length of , and the direction with the orientation of
, just as described previously.
In the STFT, these frequencies get reduced locally by an “envelope function”. One could take the Gaussian window to achieve this:
CODE:
g1 = exp(-(xx.^2+yy.^2));
imagesc(g1)
g2 = exp(-4*(xx.^2+yy.^2));
imagesc(g2)
And now these are the modulated Gaussians, whose set of translates across forms the building blocks for Gabor analysis on
:
CODE:
imagesc(real(g1.*e1))
imagesc(real(g2.*e2))
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