Image Processing Reference
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Fig. 10.22. Gabor filter responses when an input image is linearly symmetric with a gradient
direction aligned with filter tuning orientations. The sizes of the circles represent the magni-
tudes of the complex filters at the respective site
I 11 =
k,l
F {k,l} | 2 =
k,l
z kl ( z kl ) 1 |
z kl | 2 |
F {k,l} | 2
|
(10.85)
Using theorem 10.4, we conclude that these complex numbers are the elements of
the direction tensor constructed from the discrete local spectrum. Accordingly, the
direction tensor encodes the axis having the least square error when fit to the discrete
power spectrum. In case the image is linearly symmetric, the error of the fit, en-
coded by I 11 −|
will vanish and the arg( I 20 ) will deliver the direction in double-
angle representation as 2 ϕ , with ϕ being the angle of the image gradient. Suggested
by [90,139], the double angle representation (to represent direction) maps an angle ϕ
and its mirror angle ϕ + π to the same angle. Previously, we arrived at this represen-
tation by a total least squares minimization. According to the above equations, the
direction 2 ϕ can be interpreted as an angle interpolation between fixed filter direc-
tions using the magnitude responses of a bank of filters. Because of the interpolation,
the result should not suffer from the direction quantization as much when compared
to using the maximum power direction. In Fig. 10.25, encoded as images, we show
the direction estimations as I 20 and I 11 , for three tune-on directions in a log-polar
Gabor decomposition , the details of which are discussed further below. However, it
should be emphasized that the structure tensor estimation via the formulas given by
Eqs. (10.84)-(10.85) are equally valid for a Cartesian Gabor decomposition .
I 20 |
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