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3. UNIFIED REGION BASED MULTISCALE MRF 
MODEL 
This section discusses the proposed unified multi-scale region 
based MRF model. It is an extension of the multi-scale random 
field model(MSRF) proposed by Bouman(C. Bouman, M. 
Shapiro, 1994). We first briefly present the MSRF model, and 
then discuss the proposed unified multi-scale region based MRF 
model. 
3.1 MSRF Model and Statistical Inference 
3.1.1 MSRF model Defined on Quadtree: Multi-scale 
random model composed of a series of random field at different 
scale, which aims at to capture the interaction between 
neighbored scales of random field. The random field which 
represents the spectral properties at each site at given scale is 
the observations field that needs to be classified into different 
groups of distinct statistical behavior. At given scale, we can 
define restriction of label random field X to level S*: 
X'S(X,.se5"), S? is the set of sites at scale ». Similar 
notations holds for occurrence of X: x'$(x,,se S"). In the 
same way, observation field Y and its occurrence y can be 
partitioned as Y" (Y, se S") and y' $(y,,se 5"j. 
Two assumptions are made about the label field X. First, the 
element in X" will be conditionally independent, given 
elements in X"" . Second, each element y" will be only 
dependent on a local neighbourhood of elements at the next 
coarse scale. Based on these two assumptions, the transition 
probabilities must have the form. 
Píz^ [a ye TT PG, |, Vn ed. N) (1) 
ses” 
P(x,|x,) is the probability density for X, given its parent 
X,, at the next coarse scale. 
For the observation field, the conditional distribution of 
observation Y at the finest scale has the following form: 
P(y|x")=] | PO" |x") (2) 
ses" 
P(y;|x?)is the conditional density for an individual element 
Y, given the class label. 
The MSRF model of observation and label are define on 
quadtree. Each node at coarse scale corresponds to four node at 
next fine scale. Given the label of parent node, the probability 
of the child node take the value of given label is given as 
follows: 
  
i 0°, i=j 
P(x, = 7x, =D #11 _0° (3) 
ij 
M -1 
Parameter 0" is the probability that the labeling remain 
unchanged from scale n-1 to n. This model favors identity 
between the parent and children, all other transitions being 
equally likely. 
3.1.2 Statistical Inference on Quadtree: In most case, the 
MAP criterion is too conservative to get meaningful 
Segmentation since it does not consider how much difference 
these configurations are. Bouman[3] introduced a new estimator 
to take into account the location of estimation errors in the 
hierarchical structure, which has been named "sequential MAP". 
The higher a node on the tree, the more error estimation cost is 
set by this estimator. 
By using Bayesian rule, the Markov properties of X, and 
assuming that labeling field at the coarsest scale is uniformly 
distributed, the inference of X is performed with following top- 
down recursion: 
£, -argmaxlog P(y | x^) i) 
X =argmax {log P(y|x")+log p(x" | x" )} 
The conditional distribution of Y can be computed in 
preliminary bottom-up sweep: 
I (k) 2 log p(y, | k) 
2 M 5 
= Y log le exp(/"(k))+ A Sex’) (5) 
red ' (s) Su mm 
  
I? (k) is the site-wise likelihood function at the finest scale, 9, is 
the transition probability from scale n--/ to n when the labeling 
remain unchanged. 0? is estimated by using EM algorithm as 
discussed in[3].The conditional distribution of Y is assumed to 
be characterized by a Gaussian model, defined by a mean vector 
and a covariance matrix. Once the likelihood function is 
computed, the SMAP classified result can be obtained by using: 
&, c argmax Y^ (I G7) log pG7 1827] (e 
x ses” 
3.2 Unified Region Based MSRF Model and Inference 
MRF defined on pixel is difficult to model large range 
interaction, while MRF defined on region level is influenced by 
the segmented results. If we can consider the information 
contained in the pixel and region data at the same time, it would 
generate better classified results. Based on this idea, a unified 
MSRF (UMSRF) model is proposed. Namely, the UMSRF is 
defined on the quadtree structure with the purpose of regular 
spatial neighborhood context. Moreover, it assumes that the 
observed image Y is composed of two independent observations. 
One is the pixel level observations Y" - (Y^ |se 5") and the 
other is the region level observations Y" say" ises, ie. 
Y - (Y^, Y^). Therefore, both the multi-scale pixel and the region 
information could be considered in the UMSRF model. In this 
case, the best realization x given by (4) is reformulated as: 
& —argmax(In P()^ | x.) -In P^ | x^) 
& -argmax[In PO | à) dn P^ |") In po? |") 
Compared with equation (4), the likelihood P( y? | x^) at region 
U 
| 
level is considered besides the pixel level likelihood P(y^ | x"). 
The pixel level likelihood and transition probability is estimated 
by using the method proposed in[]. Region level likelihood 
P(Y^|X) is assumed to be Gaussian distribution and is 
determined based on region feature extraction results. 
3.3 Image Segmentation and Region Feature extraction 
We need to first segment the input image in multi-scale space to 
get initial segment results. After multi-scale segmentation is 
performed, region features are extracted to calculate region 
likelihood P(Y | X,) at each scale.