1. Chapter 7 Maximum likelihood classification of remotely sensed imagery 遥感影像分类的最大似然法 Chapter 7 Maximum likelihood classification of remotely sensed imagery 遥感影像分类的最大似然法 ZHANG Jingxiong ( 张景雄 ) School of Remote Sensing Information Engineering Wuhan University

2. Introduction n Thematic classification: an important technique in remote sensing for provision of spatial information n For example, land cover information key input to biophysical parameterization, environmental modeling, resources management, and other applications.

3. n Historically, visual interpretation ( 目视判 读 ) was applied to identify homogeneous areal-classes based on aerial photographs. n With digital images, it is possible to substantially automate this process by using two methods.

4. Unsupervised （非监督）

5. Supervised （监督）

6. The maximum likelihood classification ( 极大似然分类 ) - MLC The maximum likelihood classification ( 极大似然分类 ) - MLC n The most common method for supervised classification n Classes ( 类别 ): {  k, k=1,…,c} n Measurement vector （观测向量 / 数据） : z(x) n Likelihood （似然） : p(  k |z(x)) n Classification rule （分类规则） : x  i, if p(  i |z(x)) > p(  j |z(x)) for all j  i

7. Bayes’ theorem （贝叶斯定理）

8. n Candidate cover types {  k, k=1,…,c}, c being the total number of classes considered such as {urban, forest, agriculture} n Bayes‘ theorem then allows for evaluation of the posterior probability ( 验后概率 ) p(  k |z(x)) the posteriori probability of class  k conditional to data z(x)) Bayes’ theorem at work …

9. n Based on: 1) prior probability ( 先验概率 ) – expected occurrence of candidate classes  k p(  k ) or p k (before measurement) 2) class-conditional probability density – the occurrence of measurement vector z(x) conditional to class  k p(z(x) |  k ) or p k (z(x)), (derived from training data)

10. n Bayes‘ theorem then allows for evaluation of the posterior probability ( 验后概率 ) p(  k |z(x)) the posteriori probability of class  k conditional to data z(x)) n p(  k |z(x)) = p(  k, z(x)) / p(z(x)) = p k p k (z(x)) / p(z(x)) where p(z(x)) = sum( p(  k,z(x)) = sum(p k p k (z(x)) )

11.  To assign observation z(x) to class  k that has the highest posteriori probability p(  k |z(x)), or  that returns the highest product p k p k (z(x)), as p(z(x)) won’t change the relative magnitude of p(  k |z(x)) across k.

12. Computing p(z(x) |  k ) with normally distributed data n The occurrence of measurement vector z(x) conditional to class  k, assuming multivariate normal distribution: p k (z(x)) = (2  ) -b/2 det|cov Z|k | -1/2 exp(-dis 2 /2)

13. n The squared Mahalanobis distance between an observation z(x) and the means of class- specific Z variable m Z|k : n cov Z|k : variance and covariance matrix of variable Z conditional to class  k n m Z|k : the means of class-specific Z variable n b: the number of features (e.g., spectral bands)

14. n The objective/criterion ( 准则 ): to minimize the conditional average loss: Mathematical detail mis(i, j): the cost resulting from labeling a pixel actually belonging to class  j to class  i.

15. n “0-1 loss function” : 0 - no cost for correct classification 1 - unit cost for misclassification. n the expected loss incurred if pixel x with observation z(x) is classified as  i :

16. n a decision rule that will lead to minimized loss is: decide z(x)   i if and only if: n Bayesian classification rule works actually as a kind of maximum likelihood classification.

17. Examples n two classes  1 and  2 n means and dispersion matrices: m 1 = [4 2], m 2 = [3 3] COV 1 = |3 4| COV 2 = |4 5| |4 6| |5 7| COV 1 -1 = |3 -2| COV 2 -1 = |7/3 -5/3| |-2 1.5| |-5/3 4/3|

18. n To decide to which class the measure vector z = (4, 3) belongs. n squared Mahalanobis distance between this measurement vector and the two class means: dis 1 2 = 3/2 dis 2 2 = 4/3 n class-conditional probability densities: p 1 (z) = 1/(2pi*1.414) exp (-3/4) = 0.0532 p 2 (z) = 1/(2pi*1.732) exp (-2/3) = 0.0472

19. n Assume equal prior class probabilities: n p 1 = 1/2 and p 2 = 1/2 n the posterior probabilities are: p(  1 |z) = 0.0532 * (1/2) / (0.0266 + 0.0236) = 0.53 p(  2 |z) = 0.0472 * (1/2) / (0.0266 + 0.0236) = 0.47 n to classify the measurement z into class  1

20. n When p 1 = 1/3 and p 2 = 2/3, n the posterior probabilities are: p(  1 |z) = 0.0532 * (1/3) / (0.0177 + 0.0315) = 0.36 p(  2 |z) = 0.0472 * (2/3) / (0.0177 + 0.0315) = 0.64 n Then, class  2 is favored for z over class  1.

21. Looking back … n Pixel-based n Parcel-based methods adaptations to the MLC? object-oriented approaches?

22. Extraction of Impervious Surfaces Using Object-Oriented Image Segmentation Extraction of Impervious Surfaces Using Object-Oriented Image Segmentation × USGS NAPP 1 × 1 m DOQQ of an area in North Carolina × USGS NAPP 1 × 1 m DOQQ of an area in North Carolina Impervious surfaces

23. Methods for thematic mapping n Parametric MLC  n Non-parametric Artificial neural networks n Non-metric Expert systems Decision-tree classifiers Machine learning

24. References n Tucker, C. J. and Townshend, J. R. G. and Goff, T. E. (1985) African Land-Cover Classification Using Satellite Data. Science, 227(4685): 369-375. n Anyamba, A. and Eastman, J. R. (1996) Interannual Variability of NDVI over Africa and its relation to El Niño / Southern Oscillation. International Journal of Remote Sensing 17(13) : 2533-2548.

25. Questions 1. Discuss the importance of statistics in thematic classification of remotely sensed imagery.