1. 1.Image Error and Quality 2.Sampling Theory 3.Univariate Descriptive Image Statistics 4.Multivariate Statistics 5.Geostatistics for RS Next Remote Sensing1 Image Quality Assessment & Statistical Evaluation Geography KHU Jinmu Choi

2. 1. Error in Remote Sensor Cause of Error (or noise) into the remote sensor data by: the environment (e.g., atmospheric scattering), random or systematic malfunction of the remote sensing system (e.g., an uncalibrated detector creates striping), or improper airborne or ground processing of the remote sensor data prior to actual data analysis (e.g., inaccurate analog-to-digital conversion). Remote Sensing2

3. Image Quality Assessment Looking at the frequency of occurrence of individual brightness values in a histogram Viewing on individual pixel brightness values at specific locations or within a geographic area, Computing univariate descriptive statistics to determine if unusual anomalies in the image, Computing multivariate statistics to determine the amount of between-band correlation (e.g., to identify redundancy). Remote Sensing3

4. RS Raster (Matrix) Data Format i = a row (or line) in the imagery j = a column (or sample) in the imagery k = a band of imagery l = another band of imagery n = total number of picture elements (pixels) in an array BV ijk = brightness value in a row i, column j, of band k BV ik = ith brightness value in band k (source: Jensen, 2011)

5. Image Processing Mathematical Notation BV il = ith brightness value in band l min k = minimum value of band k max k = maximum value of band k range k = range of actual brightness values in band k quant k = quantization level of band k (e.g., 2 8 = 0 to 255; 2 12 = 0 to 4095) µ k = mean of band k var k = variance of band k s k = standard deviation of band k skewness k = skewness of a band k distribution kurtosis k = kurtosis of a band k distribution cov kl = covariance between pixel values in two bands, k and l r kl = correlation between pixel values in two bands, k and l X c = measurement vector for class c composed of brightness values (BV ijk ) from row i, column j, and band k M c = mean vector for class c M d = mean vector for class d µ ck = mean value of the data in class c, band k s ck = standard deviation of the data in class c, band k v ckl = covariance matrix of class c for bands k through l; shown as V c v dkl = covariance matrix of class d for bands k through l; shown as V d

6. 2. RS Sampling Theory Population: an infinite or finite set of elements. Sample: a subset of the elements taken from a population If selecting images obtained only in the summer, it is a biased sample. Sampling error: the difference between a population and a sample Remote Sensing6

7. RS Sampling Theory Large samples produce a symmetrical frequency distribution. Bell-shaped graph of the distribution: a normal distribution. Statistical tests in the analysis of RS data assume that the brightness values are normally distributed. Remote Sensing7 (source: Jensen, 2011)

8. RS Sampling Theory Histogram: useful graphic representation of the information content of a remotely sensed image Remote Sensing8 (source: Jensen, 2011)

9. 3. Display of Brightness Values Viewing individual pixel brightness values is for assessing the quality and information content of the data. Remote Sensing9 (source: Jensen, 2011)

10. Univariate Descriptive Statistics Central Tendency The mode : most frequent value in a distribution Median: midway value in the frequency distribution Mean: arithmetic average Remote Sensing10 (source: Jensen, 2011)

11. Hypothetical Dataset of Brightness Values Remote Sensing11 PixelBand 1 (green) Band 2 (red) Band 3 (near- infrared) Band 4 (near- infrared) (1,1)13057180205 (1,2)16535215255 (1,3)10025135195 (1,4)13550200220 (1,5)14565205235 (source: Jensen, 2011)

12. Band 1 (green) Band 2 (red) Band 3 (near- infrared) Band 4 (near- infrared) Mean (m k )13546.40187222 Variance (var k ) 562.50264.801007570 Standard deviation (s k ) 23.7116.2731.423.87 Minimum (min k ) 10025135195 Maximum (max k ) 16565215255 Range (BV r )65408060 Univariate Statistics Remote Sensing12 (source: Jensen, 2011)

13. RS Univariate Statistics Range: difference between the maximum (max k ) and minimum (min k ) values Variance: average squared deviation (SS) of all possible observations from the sample mean Standard deviation: positive square root of the variance. 13

14. Standard Deviation (source: Jensen, 2011)

15. Asymmetry and Peak Sharpness Skewness : a measure of the asymmetry of a histogram: Kurtosis: very sharp peak or not compared with a perfectly normal distribution Remote Sensing15

16. 4. RS Multivariate Statistics Measurement of radiant flux reflected or emitted from an object in more than one band Multivariate statistical measures (covariance, correlation) among the several bands to determine how the measurements covary. Usage Principal components analysis (PCA) Feature selection Classification and accuracy assessment. Remote Sensing16

17. Covariance Covariance : the joint variation of two variables about their common mean For covariance, first compute the corrected sum of products (SP) defined by the equation: Covariance between brightness values in bands k and l, cov kl, is equal to: Remote Sensing17

18. Variance-Covariance Matrix Remote Sensing18 Band 1 (green) Band 2 (red) Band 3 (near- infrared) Band 4 (near- infrared) Band 1SS 1 cov 1,2 cov 1,3 cov 1,4 Band 2cov 2,1 SS 2 cov 2,3 cov 2,4 Band 3cov 3,1 cov 3,2 SS 3 cov 3,4 Band 4cov 4,1 cov 4,2 cov 4,3 SS 4

19. Computation of Var.-Cov. Remote Sensing19 Band 1(Band 1 x Band 2)Band 2 1307,41057 1655,77535 1002,50025 1356,75050 1459,42565 67531,860232

20. Result Var.-Cov. Matrix Remote Sensing20 Band 1 (green) Band 2 (red) Band 3 (near- infrared) Band 4 (near- infrared) Band 1 562.25 --- Band 2 135 264.80 -- Band 3718.75275.25 1007.50 - Band 4537.5064663.75 570

21. Correlation Correlation coefficient, (r) : the degree of interrelation between two bands of RS data, r kl, Ratio of their covariance (cov kl ) to the product of their standard deviations (s k and s l ) Coefficient of determination (r 2 ) : proportion of total variation in the values of “band l” that can be explained by a linear relationship with the values of “band k.” Remote Sensing21

22. Using Univariate Statistics Remote Sensing22 Band 1 (green) Band 2 (red) Band 3 (near- infrared) Band 4 (near- infrared) Mean (m k )13546.40187222 Variance (var k ) 562.50264.801007570 Standard deviation (s k ) 23.7116.27 31.423.87 Minimum (min k ) 10025135195 Maximum (max k ) 16565215255 Range (BV r )65408060

23. Resulting Correlation Matrix Remote Sensing23 Band 1 (green) Band 2 (red) Band 3 (near- infrared) Band 4 (near- infrared) Band 1 - --- Band 2 0.35 - -- Band 30.950.53 - - Band 40.940.160.87 -

24. Band Min Max Mean Standard Deviation 1 51 242 65.163137 10.231356 2 17 115 25.797593 5.956048 3 14 131 23.958016 8.469890 4 5 105 26.550666 15.690054 5 0 193 32.014001 24.296417 6 0 128 15.103553 12.738188 7 102 124 110.734372 4.305065 Covariance Matrix Band Band 1 Band 2 Band 3 Band 4 Band 5 Band 6 Band 7 1 104.680654 58.797907 82.602381 69.603136 142.947000 94.488082 24.464596 2 58.797907 35.474507 48.644220 45.539546 90.661412 57.877406 14.812886 3 82.602381 48.644220 71.739034 76.954037 149.566052 91.234270 23.827418 4 69.603136 45.539546 76.954037 246.177785 342.523400 157.655947 46.815767 5 142.947000 90.661412 149.566052 342.523400 590.315858 294.019002 82.994241 6 94.488082 57.877406 91.234270 157.655947 294.019002 162.261439 44.674247 7 24.464596 14.812886 23.827418 46.815767 82.994241 44.674247 18.533586 Correlation Matrix Band Band 1 Band 2 Band 3 Band 4 Band 5 Band 6 Band 7 1 1.000000 0.964874 0.953195 0.433582 0.575042 0.724997 0.555425 2 0.964874 1.000000 0.964263 0.487311 0.626501 0.762857 0.577699 3 0.953195 0.964263 1.000000 0.579068 0.726797 0.845615 0.653461 4 0.433582 0.487311 0.579068 1.000000 0.898511 0.788821 0.693087 5 0.575042 0.626501 0.726797 0.898511 1.000000 0.950004 0.793462 6 0.724997 0.762857 0.845615 0.788821 0.950004 1.000000 0.814648 7 0.555425 0.577699 0.653461 0.693087 0.793462 0.814648 1.000000 Univariate and Multivariate Statistics of Landsat TM Data of, Charleston SC (source: Jensen, 2011)

25. Feature Space Plots Remote Sensing25 Visual inspection How two bands features in a remote sensing dataset The greater the frequency of occurrence of unique pairs of values, the brighter the feature space pixel.

26. Geostatistical Analysis of RS Data How measure autocorrelation in images? Geostatistical analysis incorporates spatial autocorrelation information in the kriging interpolation process Remote Sensing26 Image values record spatial properties of the Earth’s surface Autocorrelation: Things that are close to one another are more alike than those farther away (source: Jensen, 2011)

27. 5. Geostatistical Analysis - Kriging Kriging : a family of least-squares linear regression algorithms for interpolation Weights in Kriging based not only on the distance between the measured points and the point to be predicted, but also on the overall spatial arrangement among the measured points (i.e., their autocorrelation) Methods of kriging : simple, ordinary, universal, probability, indicator, disjunctive, and multiple variable co-kriging Remote Sensing27

28. Variogram Two tasks of kriging process: quantifying the spatial structure of the surrounding data points, (Variography) producing a prediction at a new location Semivariogram: measurements of the spatial structure (the amount of spatial separation between samples) Here, (BV) z of pixels x, h intervals (lag distance), m possible pairs, semivariogram g(h). Remote Sensing28

29. Semivariance calculation with lag distance (h) along a transect of pixels extracted from an image. Semivariance (source: Jensen, 2011)

30. Characteristics of Semivariogram Important characteristics of the semivariogram include: lag distance (h) on the x-axis, sill (s), range (a), nugget variance (C o ), and spatially dependent structural variance partial sill (C). Remote Sensing30 (source: Jensen, 2011)

31. Image Error and Quality Sampling Theory Univariate Descriptive Image Statistics Multivariate Statistics Geostatistics for RS Remote Sensing31 Summary

32. Next Exercise: Image Assessment with RS software Lecture: Initial Display Alternatives and Scientific Visualization Source: Jensen and Jensen, 2011, Introductory Digital Image Processing, 4 th ed, Prentice Hall.