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represented by histograms.
In the case of non-parametric representation of the texture and by
assuming independent probabilities for the observed pixels (
order Markovian model). For a first order Markov process, the 0th
order statistics of the samples must be an Eigenvector of
with Eigenvalue
. Unfortunately, this means that a uniform
prior
for
over
is
inconsistent with the uniform prior used in
the 0th order case. To re-establish the consistency, it is necessary
to choose a
order prior such that the expected
value of a
column of the transition matrix
is
obtained by adding
rather than 1 to the number of observations in that column of the
transition matrix before normalizing the column to sum to
. This
means that the transition matrix,
![\begin{displaymath}
E(p_{i\vert j}\vert S_{1}^{c})=\frac{C_{ij}+1/I}{1+\sum
_{i}C_{ij}}=\frac{C_{ij}+1/I}{1+o_{j}}\end{displaymath}](img31.png) |
(6) |
where
is
the number of times that intensity
follows intensity
in the
sequence
.
And hence the expected 0th order
distribution
(which is the vector
) has the
desired
properties since:
![\begin{displaymath}\sum
_{j}E(p_{i\vert j}\vert S_{1}^{c})\frac{(o_{j}+1)}{(c+I)}=\frac{\sum
_{j}C_{ij}+1/I}{c+I}=\frac{o_{i}+1}{c+I}\end{displaymath}](img37.png) |
(7) |
Which is the
same as Equation 5. This
modification is equivalent to
imposing a prior over
that favors structure
in the Markov
process and is proportional to
). This
gives Algorithm 2.
Next: Texture Boundary
Detection by
Up: Texture Estimation
Previous: texture
represented by histograms.
Ali Shahrokni
2004-06-21