1. A Feature-Space-Based Indicator Kriging Approach for Remote Sensing Image Classification
鄭克聲
台灣大學生物環境系統工程學系

2. Image Classification
Statistical pattern recognition techniques are widely used for landuse/land cover classification.
Some supervised classification algorithms
Parametric Approach
Maximum likelihood classifier
Bayes classifier
Non-parametric Approach
Nearest Neighbour classifier
Artificial Neural network classifier

3. ANN Classifiers
ANN classifiers do not consider classification features as having probability distributions, and therefore, classification is not explicitly probability-based.
In a loosely defined sense, ANN classification is a process of searching optimal solution of weight vector that minimizes the sum of squared errors between network and desired output responses.

4. ANN Classifiers
It has been shown that the output of a backpropagation network can approximate the posterior density function, if its activation function is capable of representing the a posteriori probability function and the number of training samples is sufficiently large (Lee et al., 1991).
Manry et al. (1996) also showed that a neural network can approximate the minimum mean square estimator arbitrarily well, provided that it is of adequate size and is well-trained.

5. ANN Classifiers
Egmont-Petersen et al. (2002) point out that ANNs suffer from what is known as the black-box problem: given any input a corresponding output is produced, but it cannot be elucidated why this decision was reached, how reliable it is, etc.

6. Image Classification
The work of image classification can be considered as partitioning a hyperspace using discriminant rules established by samples.
Each sample point in feature space is labeled a class index.

7. Difficulties Encountered In Application of Parametric Approaches
Application of parametric approaches require knowledge of probability distribution of classification features.
Classification features often have finite mixture distributions (multi-modal class densities).
The class distribution may be non-Gaussian.

12. Geostatistical Approach of Spatial Estimation
Geostatistics is a set of techniques, often referred to as kriging methods, which utilize the spatial covariance function or the semivariogram for spatial data analysis.
Ordinary kriging yields best linear unbiased estimator (BLUE).
Indicator kriging yields estimate of the probability distribution at specified locations.

13. Since probability density and correlation structure between classification features are insightful, probability-based classification methods are appealing to many researchers and practitioners.
The work of probability-based classification can be conceived as a spatial estimation problem for which the estimates are probabilities that a pixel with certain feature-vector belongs to different classes.

15. Ordinary Kriging
Ordinary kriging assumes second-order stationary properties for the random field {Z(x), x}

16. Properties of OK Estimates
Unbiased
i.e.
Minimum variance of estimation error
Conditional minimization
Minimizing

17. Ordinary Kriging System
Semi-variogram

18. Typical Form of A Variogram
Variogram characterizes the spatial variation of a random field.

19. Matrix Form of OK System

20. Indicator Kriging
Indicator kriging is a method of spatial estimation that yields an estimate of probability distribution function of the random variable of interest.
Consider a random field of k classes where Ω represents the spatial domain of the random field. A total number of n features are used for classification of the k classes. For convenience of illustration, let’s assume k = 3 and n = 2. From a set of training pixels, we first establish the k-class scatter plot in feature space.

21. Scatter Plot in Feature Space

22. Indicator Variable
For a continuous random field, the indicator variable can be used to estimate the distribution of the random variable by using a set of cutoff values.
The indicator variable at location x is defined as
where is a selected cutoff value.

23. The weighted average of indicator variables is an estimate of the cumulative probability, i.e.,
If Z(xj), j = 1, 2, …, N, are mutually independent, then j = 1/N .
For a random field with spatial auto-correlation characteristics, indicator variogram must be established and used to estimate the cumulative probability of the random variable at unobserved locations.

24. Indicator Variable for Categorical Random Field
Similar to the case of continuous random field, the indicator variable can be used to estimate the probability that a pixel belongs to a certain class for categorical random field. Let the indicator variable be defined as
wherere presents the j-th class and represents the pixel at location x is the value of the indicator variable related to the j-th class.

25. The weighted average of values of indicator variables is an estimate of the probability that a pixel belongs to the j-th class, i.e.

26. Class-specific Indicator Variable Scatter Plot in Feature Space
Three-class scatter plot of indicator
variables in two dimensional
feature space.
Class-specific Scatter plot of indicator variables
(Binary Scatter Plots)
Class Class Class-3

27. For each binary scatter plot, we consider the variation of indicator variables as a random field associated with that particular class.
By conducting ordinary kriging, for each class, of indicator variables in feature space, we obtain the probability that the pixel of interest belongs to each individual class.
Class assignment of the pixel of interest is done based on the following criterion
If max , then assign
to class

28. Study Area and Data
An area of approximately 70 km2 in Central Taiwan is selected as our study site.
The study area includes a small township and nearby mountainous area. A major river flows westward along the northern edge of the area.
Five landcover classes (water, built-up, forest, crop, and bare land) are identified in this study.
SPOT satellite image acquired on September 21, 2001 was used for landuse classification.

29. SPOT Image of the Study Area

30. Three classification features (green, red and near infrared bands) were used.
A total of 1886 training pixels and 732 verification pixels were selected.

31. Confusion Matrix – ML (Training)

32. Confusion Matrix – IK (Training)

33. Confusion Matrix – IK (Verification)

34. A Further Test Case

43. Further Considerations
Replicates in feature space.
Anisotropic variation in feature space.

44. Conclusions
Indicator kriging approach is distribution-free; therefore, it does not require the knowledge of distribution types.
IK algorithm achieves high classification accuracies.

45. References
Bierkens, M. F. and P. A. Burrough , 1993.The indicator approach to categorical soil data.Ι.Theory .J. of Soil Science, 44, pp
Bierkens, M. F. and P.A. Burrough , 1993.The indicator approach to categorical soil data. II. Application to mapping and land use suitability analysis. J. of Soil Science, 44, pp
Meer, F. V. D., Classification of remotely-sensed imagery using an indicator kriging approach: application to the problem of calcite dolomite mineral. Int. J. of remote sensing. Vol. 17, no. 6, pp

46. References
Journel, A. G., non-parametric estimation of spatial distributions, math. Geol
Lillesand, T. M. , Johnson, W. L. , Deueil, R. L.,O.M. Lixdstrom and D.E. Meisner,1983.Use of Landsat data to predict the trophic state of Minnesota lakes. Photogrammetric Engineering & Remote Sensing, Vol. 49,No.2, pp. 219~229.
Lillesand, Thomas M. and Kiefer, Ralph W. ,1994. Remote sensing and image interpretation.