Colour in a textured region is by definition not homogenous and
presents a very variable behaviour through different image
regions. Hence, methods which implicitly assume the same shape for
all the clusters in the space, are not able to handle the
complexity of the real feature space [19].
Therefore, we focus our attention on density estimation from a
non-parametric approach since these methods do not have embedded
assumptions, and specifically we adopt the kernel estimation
technique. Considering colour pixels inside seeds as a set of data
points assumed to be a sample of region colour, density estimation
techniques allow the construction of an estimate of the
probability density function which describes the behaviour of
colour in a region. Given data points
,
in the d-dimensional space
, the multivariate kernel density
estimator with kernel
and a bandwidth parameter
,
becomes the expression
which gives us the probability of a pixel to belong to a region
considering colour properties, , on the three-dimensional
colour space. Note that in order to use only one bandwidth
parameter
the metric of the feature space has to be
Euclidean.
On the other hand, texture of each region is modeled by a
multivariate Gaussian distribution considering the set of
texture features extracted from the luminance image. Thus, the
mean vector
and the covariance matrix
, which are initialised from the seeds, describe the
texture homogeneity region behaviour. Therefore, the probability
of a pixel of belonging to a region taking textural properties
into account,
, is given by the probability density
function of a multivariate Gaussian distribution.
Considering both properties together, colour and texture, the
probability of a pixel of belonging to a region
will be
obtained considering the similarity of the colour pixel with the
colour of the region, and the similarity of the texture around the
pixel with the texture of the region. The combination of both
terms gives the equation
where weights the relative importance of colour and
texture terms to evaluate the region information.