Classification of Remotely Sensed Imagery
Using Markov Random Fields
Brandt TSO (Taiwan), Paul M. Mather(UK)1
Associated Professor, Research Centre, Operation R&D Division
National Defence Management College, National Defence University
Jong-Ho P.O. Box 90046, Taipei
Tel: +886-2-2254-8131 or +886-2-2222-2137 ext 8452, Fax +886-2-2254-8131
E-mail : brandttso@kimo.com.tw
1Professor, Remote Sensing Society Vice President, School of Geography
University of Nottingham, Nottingham NG7 2RD England
Email: paul.mather@nottingham.ac.uk
Key Words
Markov Random Field, Genetic Algorithm, Multisource Classification
Abstract
The use of contextual information for modelling the prior probability mass function (generally called the smoothness prior) in the traditional statistical Bayesian classification formula has been widely adopted in recent years. Random field models, especially Markov random fields (MRF), provide a theoretical robust yet mathematical tractable way of coding multisource information and modelling such contextual behaviour. In dealing with remotely-sensed multisource data, the determination of source-weighting and MRF model-related parameters is a difficult task. We've used the genetic algorithm GA to address the parameter estimation issue in a case study over Red Sea Hill, Sudan for lithological type identification (total eight categories were involved in our classification analyses). GA has been proved in many studies to be a powerful, however cost-effective tool for searching optimal solution. The data set used for the experiment includes LANDSAT Thematic Mapper (TM) six-band and Shuttle Image Radar (SIR) L band, C band, and total power multisource data. Three kinds of MRF classification mechanisms known as Iterated Conditional Mode-ICM, Maximiser of Posterior Marginals-MPM, and Line-Process were investigated. It was shown that the incorporation of contextual information leads to impressively improved results (up to 80% of average producer's accuracy was achieved) in comparison with the output derived from traditional non-contextual maximum likelihood classifier (only around 67% of average producer's accuracy was obtained). The resulted classified imagery using context were also found to reveal more patch-like, meaningful patterns. We therefore conclude that incorporating contextual relationship in terms of MRF, with well assignments of model-related parameters and suitable classification algorithms being used, can be a powerful tool for real world, remotely-sensed imagery classification.
Introduction
In recent years, there has been an increasing interest in use of contextual information for modeling the prior probability density function (p.d.f.) (Derin and Elliott, 1987, Dubes and Jain, 1989, Jhung and Swain, 1996, Schistad et al., 1996, Tso and Mather, 1999). Using context to model prior probability to help the interpretation of remotely-sensed imagery is considered as a reasonable procedure, since a pixel classified as "ocean" is likely to be surrounded by the same class and more unlikely to have neighbors from categories such as "pasture" or "forest". In other words, using the concept of context, each pixel is not treated in isolation but as part of a spatial pattern. The relationship between the pixel of interest and its neighbor is therefore not considered to be statistically independent. If context information can be suitably modeled and incorporated with class-conditional p.d.f., then improved classification results can be expected(Li, 1995a, Li, 1995b)..
Discrete random field models, especially the Gibbs Random Fields (GRF) and Markov Random Fields (MRF), have been found to be useful tools for characterizing contextual information and are widely used in image segmentation and restoration(German and German, 1984, German and Gidas, 1991, Tso and Mather, 1999). This study presents some basics ideas of MRF as a framework to model prior probability (and so as to achieve MAP estimate) for remotely-sensed imagery classification.
Theoretical Background
In the interpretation of remotely-sensed imagery, there will be a set of observed feature vectors (e.g. pixel gray value in different bands), d. Traditionally, each pixel r is labeled based on d
r alone without considering contextual information. Once the context is included as a prior information and modeled in terms of MRF, current practice is to use a Bayesian formulation to construct the posterior energy and to search the MAP-MRF labeling in terms of energy minimization.
Multisource Posterior Energy
Based on Bayesian formula, the posterior distribution P(w½d) can be expressed as
Recall from (1) that under a pair-site MLL model, the prior probability for pixel r is
where the U(wr) is the prior energy, and is defined as
where b £ 0, and define d(wr), wr')) as a step function:
The conditional distribution dr) given the true label wr) is often assumed to be normal. For wr) = label j, the conditional probability which can be formulated as
which is the class-conditional energy. By combining (10) and (13), one obtains the posterior energy as
The MAP estimate is equivalent to minimizing the posterior energy.
For multisource classification, in terms of simplicity, we have made a class-conditional independence assumption, i.e.
where dsi is the observation from data source i. Eq. (7) means that we consider the observations from different sources to be conditionally independent. Such an assumption may not be universally valid. However, by adopting this assumption, the mathematical analysis and computation become treatable.
If we take the data-associated reliability factor into account, one simple method is to assign weighting parameters through exponential form, then Eq.(7) can be further refined as
where the 
are the data-associated reliability factor for the source k. The larger the value of lk the greater the influence of data source k on the classification process. Eq. (8) indicates that multisource posterior energy can be expressed as
Therefore, based on (9), an alternative expression of the MAP-MRF estimation of multisource data is:
where
