1. Modeling Spatial Context in Classification/Prediction Using MRF and SAR techniques.
Shashi Shekhar
Weili Wu
Sanjay Chawla
Ranga Raju Vatsavai
Department of Computer Science.
Army HPC Research Center.
University of Minnesota.

2. Outline Introduction Problem Definition Supervised Classification
Spatial Context
Markov Random Fields (MRF)
Spatial Autoregression (SAR)
Results

3. Introduction Spatial Databases Objectives Techniques
Maps, Ground Observations, Multi-spectral/Multi-temporal remote sensing images
e.g. Ecology (Wetlands), Forest Inventory, Aerial Photographs and Satellite Remote Sensing images
Objectives
Predict spatial distribution of marsh-breeding birds
Thematic Classification – identification of objects in a imagery
Techniques
Supervised, Unsupervised
Statistical, Neural
Knowledge Based

4. Example Datasets

5. Example Datasets

6. Problem Definition Given Find Objective Constraints
A spatial framework S consisting of {s1, …,sn} for an underlying geographic space G
A collection of explanatory functions fXk : S -> Rk, k = 1, …,K. Rk is the range
A dependent class variable fL : S -> L = {l1, …, lM}
A value for parameter , relative importance of spatial accuracy
Find
Classification model: f^L : R1 x …RK –> L.
Objective
Maximize similarity ( = (1- )classification_acc(f^L , fL ) + ()spatial_acc(f^L , fL ))
Constraints
S is multi-dimensional Euclidean space
Explanatory functions and response function may not be independent, i.e. spatial autocorrelation exists.

7. Accuracy Measure
ADNP

8. Problem Definition Given Find
Multi-spectral image X, composed of N-dimensional Pixels
X(i,j) = [bv1, …bvn]’
Find
Appropriate Label

9. Classical Techniques Logistic Regression Assumptions
y = X  + 
Assumptions
independent, identical, zero-mean, normal distribution
i.e. i = N ( 0, 2 ).
Bayesian Classification
Pr(li | X) = Pr(X | li) Pr(li) / Pr(X).

10. Classification Schemes

11. Pixel Classification
Class to which a pixel at location x belongs it is strictly the conditional probability
Decision Rule

12. Classification

13. Comparison Criteria Logistic Regression Bayesian Input fx1, …,fxk, fl
Intermediate Results 
Pr(li), Pr(X|li)
Output
Pr(li | X) based on 
Pr(li | X) based on Pr(li), Pr(X|li)
Decision
Select most likely class
For a given feature value
Assumptions
- Pr(X|li)
- Class bndry
- Autocorrelation
Exponential family
Linearly separable
None -

14. Spatial Context What is spatial context? Why?
Observations at location i depend on observations at location j i.
Formally yi = f(yj), j = 1,2,…..n j i.
Why?
Natural Objects (Species) occur together
Higher sensor resolution (i.e. object is bigger than pixel).

15. Prior Distribution Model:
For Markov random field , the conditional distribution of a point in the field given all other points is only dependent on its neighbors. x s x x x x x x x x s x x x s x x x x x x x x x

16. MRF Continued
A Clique is defined as a subset of points in S such that if s and r two points contained in clique C, then s and r are neighbors. s

17. Gibbs Distribution
For a given neighborhood system, a Gibbs distribution is defined as any distribution expressed as:

18. Hammersley-Clifford Theorem
H-C theorem states that is an MRF with strictly positive distribution and a neighborhood system
the distribution of can be written as a Gibbs distribution with cliques induced by the neighborhood system.
Therefore, if p( ) is formulated as Gibbs distribution, would have
properties of MRF.

19. Gibbs Distribution

20. ICM 1. Compute 2. For all pixels (i,j), update 3. Repeat 2.
Assuming multivariate normal distribution:

21. SAR y =  W y + X  +  W is the contiguity matrix
 strength of spatial dependencies

22. Experimental Results
Experimental Setup

23. Results

24. Results

25. Results

26. Results

27. Results

28. Results

29. Conclusion Comparison on a common probabilistic framework.
Modeling spatial dependencies
SAR makes more restrictive assumptions than MRF
MRF allows flexible modeling of spatial context
Relationship between SAR and MRF is analogues to logistic regression and Bayesian classifiers.
Efficient Solution procedures
Graph-cut
Extend the graph-cut to SAR.