Image Processing Reference
In-Depth Information
At this point we note that if the complex moments I 40 and I 60 of a local power
spectrum with an energy concentration having only 2-folded symmetry are com-
puted, the magnitude responses will be high. However, these magnitude responses
will be lower relative to the cases in which the spectrum shows 4- and 6-folded sym-
metries because 2 is a factor of 4 and 6; hence, such a concentration “leaks” to 4-
and 6-folded symmetries. The converse is not true, which allows complex moments
to discriminate between these symmetries. In the ideal case, a 6-folded symmetric en-
ergy concentration gives a zero response for complex moments with p
q =4and
2. Likewise, a 4-folded symmetric concentration yields a zero response for p
q =2
and 6.
One might think that the arguments of the complex moments which represent
k 2 ,k 4 ,... according to theorem 14.1 need to be divided by 2, 4, etc., for a straight-
forward representation of the directions of the component lines or edges. Likewise,
the moduli of these moments do not provide for the minimum error directly. Al-
though it is simple to compute this by solving the minimum error in Eqs. (14.6)
and (14.7), we will argue that for pattern recognition purposes, the complex number
representation of I pq has advantages that allow us to circumvent the following three
problems.
First, the continuity of the I n, 0 w.r.t. group directions is possible to achieve in the
sense that two such directions that differ a small amount also differ a small amount in
the numerical representations afforded by the real and imaginary parts of I n, 0 . That
is, a number arbitrarily close to 0 is not arbitrarily close to 2 π , whereas the corre-
sponding physical angles (as well as I n, 0 ) are [90]. In the Cartesian representation,
the complex numbers are continuous with respect to changes in their arguments, ex-
cept at the origin. Second, the factor n makes the representation of the direction of
the symmetry unique. This is because this factor makes the argument of I n, 0 a group
direction , as discussed in Sect. 14.1. Eliminating the factor n in the argument would
thus discard the equivalence of certain rotation angles which are the same because
they have exactly the same physical effect on n -folded symmetric images. Third,
knowing the value of e ( k n
min
) alone is not sufficient to judge the quality of the esti-
mate; one must know whether this error is large or small (the range problem). The
comparison with the worst case, i.e., e ( k n
max
e ( k n
min
), provides a means to assess
the quality. The complex quantity I 20 already represents the difference of the er-
rors through its magnitude, which is real. Alternative quality measures can be found
easily, e.g.,
)
) =
e ( k n
max
e ( k n
min
)
)
I n, 0
n
2
(14.9)
)+ e ( k n
min
, n
2
e ( k n
max
which is always in the interval [0,1] and attains the end points 0 and 1 if the quality
of the fit is totally uncertain and totally certain, respectively.
The continuity, group direction representation, and range problems might easily
imperil the performance of e.g., a clustering method that could follow the extraction
of orientation and certainty features, if an adequate representation of these features
is lacking. By using complex moments in the Cartesian representation one can avoid
these three problems.
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