136 CONSTANTINE KOTROPOULOS
et al.
The image processing applications of nonlinear mean filters, defined by the s p
mean filter, were first proposed by Kundu, Mitra, and Vaidyanathan [Kun84]. Fur-
ther generalizations of the
L p
mean filter were later advanced by Pitas and Venet-
sanopoulos [Pit86a, Pit86b, Pit86c]. The nonlinear mean filters remove the impulse
noise very effectively, especially when the impulses occur with a high probability.
They have a very simple structure and are suitable for real-time processing ap-
plications. The class of nonlinear mean filters can be treated as a generalization
of linear filters (e.g., the arithmetic mean filter). They are closely related to ho-
momorphic filters, perhaps one of the oldest classes of nonlinear filters. From a
statistical point of view, nonlinear mean filters rely on the nonlinear means that are
well-known location estimators [Ken73]. The latter approach is found to be more
fruitful because it produces a general filter structure that encompasses the filters
that are based on order statistics, the homomorphic filters, and the morphological
filters in addition to the nonlinear mean filters.
The nonlinear mean filters are first defined and their statistical as well as edge
preservation properties reviewed in Sec. 5.2. Their performance in the presence
of impulse noise is outlined as well. In Sec. 5.3,
L p
mean filters are proven to
be a general filter structure able to suppress signal-dependent noise. A class of
edge detectors obtained from the difference of nonlinear mean filters is studied in
Sec. 5.4. Next, soft grayscale morphological filters based on Lp mean filters are de-
scribed in Sec. 5.5. L2 mean filters are shown to be optimal for both multiplicative
Rayleigh speckle and signal-dependent Gaussian speckle noise. These properties
supported the application of the s mean filter, the signal-adaptive L2 mean filter,
and the L2 learning vector quantizer in ultrasonic image processing and analysis,
which are reviewed in Sec. 5.6. The use of
Lp
mean filters as approximators of
max and min operators for positive and negative p, respectively, is exploited in
Sec. 5.7 to design analog implementations of sorting networks. Edge preserving
filtering by combining nonlinear means and order statistics is studied in Sec. 5.8.
5.2 Nonlinear Mean Filters
The nonlinear mean of the N numbers xi, i = 1, 2,..., N, is defined by
y=f(x~,x2,...,XN)=g-~(y'~Nla~g(x~))
2i =
i ai
(5.1)
where g (x) is, in general, a single-valued analytic nonlinear function and a i are
weights. The properties of the nonlinear mean depend on the function g (x) and
the weights ai. Equation (5.2) indicates several choices of g(x) that result in filters
playing a special role in image processing:
x
1/x
g(x) = lnx
x p,
p ~ R- {-1,0, 1}
arithmetic mean ~,
harmonic mean YH,
geometric mean YG,
Lp
mean yLp,
(5.2)

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