and
yields the same result. From the definition of the tensor (5.1) we immediately see that this requirement is met.
It is also necessary that the tensor representing orientation be invariant to the type of signal actually encountered, i.e. obtaining the same angle independent of am-plitude, or whether it is an edge, line etc. As an example, the orientation obtained from the two signals in figure 5.3 should be the same. Again, from the definition of the tensor, which has no reference to the contents of the signal, we see that this requirement is met. The actual orientation estimation is shown in section 5.3.
The orientation tensor should also locally preserve the angle metric, i.e., the tensor, T, should change proportionally to the change in local orientation. The proof that this holds is omitted here, but can be found in [27].
5.3 Orientation estimation
The general idea of energy-based direction estimation is to probe the Fourier space in several directions with filters each picking up energy in a particular angular sector, and then combining the filter outputs yielding the direction of most signal variance. This section shows how a combination of outputs from specific filters can be used to estimate the orientation tensor presented in section 5.2.1 on the facing page.
5.3.1 The filter sets
Due to the angular invariance required by the orientation tensor, it is immediately recognized that the filters must be distributed equally over the entire orientation space. Note that the phase independent filter outputs will be symmetric so that the
Figure 5.4: Relative contribution from two filters
output of and gives the same result. In the 2D case it may seem at first that the obvious number of filters is two, but as indicated in figure 5.4 there will be no way of differentiating contributions from the filters (bold lines) along opposing directions of the two thin lines. Two particular areas in the Fourier plane giving identical contributions from the two filters are indicated by circles.
Thus, the minimum number of filters required for the 2D orientation space is three.
Distributing them equally gives the orienting vectors:
+
+
+
(5.2)
where and , as also illustrated in figure 5.5 on the next page.
By following the same reasoning one can show that the minimum number of filters in three-dimensions is six, and their orienting vectors are:
+
+
+
+
+
+
where , and !
. 5.3.2 Filter outputs
In the following the analysis is restricted to real valued simple neighborhoods, i.e., neighborhoods that can be expressed as
@
(5.3)
where and are real functions. Two-dimensional examples of such signals are given in figure 5.3
Figure 5.5: Two-dimensional filter orientations
As shown in section 5.1 on page 37 the filters used in the phase-independent energy estimation are zero over one half of the Fourier space, also known as quadrature filters[26]. Given the filter direction vector,
The output % of the corresponding filter will, as mentioned above, be complex.
The magnitude
% %
will be independent of whether the local signal is an edge or line. If is a sinusoidal signal, the magnitude will be completely phase invariant, as demonstrated in section 5.1.
The Fourier transform of the simple signal
is an impulse-line in the direction of
. Let the filter functions be spherically separable, i.e.,
The filtering of by results in:
and denotes complex conjugate. The complex conjugate comes from the Her-mitian property of the Fourier transform,
, of the real signal
. Since the filter is zero over one half of the Fourier domain, either
or
will be zero. This makes it possible to write the filter output as
can be seen as the local signal amplitude.
As first suggested by Knutsson[26] the following directional function will be shown to have quite desirable properties:
% @
is the filter orienting vector.
Combining equation (5.4) and (5.5) gives the filter output magnitude
% @
+ %
(5.6) where is independent of the filter orientation and depends only on the radial dis-tribution of the signal spectrum, , and the filter function, .
The radial function,
, can be chosen arbitrarily, and is typically some kind of band-pass function. When actually implementing the filters the usual limita-tions of filter design must be considered. The function suggested by Knutsson[14]
is given by
where is the relative band-width and is the center frequency. Figure 5.2 shows the ideal filter reference function with
and
G @ !
. 5.3.3 Tensor construction
Here it is shown that, given a simple neighborhood, the orientation tensor can be estimated exactly by linearly combining the result of the quadrature filters. In non-simple neighborhoods the resulting tensor still conveys interesting properties, as explained in section 5.4 on page 46. Due to notational simplicity, we focus on the two-dimensional construction, but the three-dimensional case is done in exactly the same manner.
Define the tensors
%
as being the outer product of the orienting vectors
Letting the signal orienting vector be:
gives, by using the equations (5.2) and (5.6), the following filter output magnitudes:
!
, by their corresponding output magni-tudes, % , we obtain the tensor
2 % %
%
which has the following components:
!
!
!
If we subtract from the diagonals we end up with a tensor of the desired form (5.1):
making a linear combination of the
% Thus, the orientation tensor can be estimated by filtering the signal with the appro-priate set of quadrature filters, and using the filter output magnitudes as weights in equation (5.7). Note that the
%
tensors do not depend on the signal, and can thus be precalculated.
As mentioned above, the method of obtaining the three-dimensional results is iden-tical, and the
%
tensors in equation (5.7) will for the three-dimensional case be
[14]:
% %