Remote Sensing Image Statistics

1. Remote Sensing Image Statistics Supplement to Lecture 2 prepared by R. Lathrop 9/99 updated 8/01 Readings: ERDAS Field Guide 5th Ed App A:Math Topics

2. Digital Images • Digital Number (DN) or Brightness Value (BV) - the tonal gray scale expressed as a number, typically 8-bit number (0-255) • Dimensionality - determined by the number of data layers (bands) • Measurement Vector of a pixel - is the set of data file values for one pixel in all n bands

3. Digital Image Storage Formats • Band sequential (BSQ) - each band contained in a separate file • Band interleaved by line (BIL) - each record in the file contains a scan line (row) of data for one band, with successive bands recorded as successive lines • Band Interleaved by Pixel (BIP)

4. Calculating disk space [ ( (x * y * b) * n) ] x 1.4 = output file size where: y = rows x = columns b = number of bytes per pixel n = number of bands 1.4 adds 30% for pyramid layers and 10% for other info

5. Summarizing data distributions • Frequency distributions - method of describing or summarizing large volumes of data by grouping them into a limited number of classes or categories • Histograms - graphical representation of a frequency distribution in the form of a bar chart

6. Measures of Central Location • Mean - simple arithmetic average, the sum of all observations divided by the number of observations • Median - the middle number in a data set, midway in the frequency distribution • Mode - the value that occurs with the greatest frequency, the peak in a histogram

7. Measures of Central Location example BV band 1 130 Mean = 675 / 5 = 135 165 100 Median = 135 135 145 sum = 675

8. Statistical Notation • uk = mean of band k • sk = standard deviation of band k • Vark = variance of band k • COVkl = covariance of bands k & l • rkl = correlation between bands k & l

9. Measures of Dispersion • Range - the difference between the largest and smallest value • Variance - the average of the squared deviations between the data values and the mean • Standard Deviation - the square root of the variance, in the units of data measurement

10. Measures of Dispersion example BVik Bvik-uk ( Bvik-uk)2 Var = SUM(( Bvik-uk)2 130 -5 25 n - 1 165 30 900 Vark = 2250 / 4 100 -35 1225 = 562.5 BV2 135 0 0 Sk = SQRT(562.5) 145 10 100 = 23.71 BV 675 0 2250

11. Feature Space Image • Visualization of 2 bands of image data simultaneously through a 2 band scatterplot - the graph of the data file values of one band of data against the values of another band • Feature space - abstract space that is defined by spectral units

12. Spectral distance • Spectral distance - the Euclidean distance in n-dimensional spectral space • D = SQRT[(sum (dk - ek)2] where dk = BV of pixel d in band k where ek = BV of pixel e in band k • the equation is summed across k = 1 to n bands

13. Spectral Distance example Y 92, 153 180, 85 X

14. Spectral Distance example Distance between [x1,y1] & [x2, y2] [180, 85] & [92, 153] D = SQRT[(sum (dk - ek)2] D = SQRT[(180-92)2 + (85-153)2] = SQRT[(88)2 + (-68)2] = SQRT[7744 + 4624] = SQRT[12,368] = 111.2

15. Spectral Distance example Y 92, 153 Yd = 85-153 180, 85 Xd = 180 -92 X

16. Measures of inter-relation between bands • Independence - no relationship • Dependence - inter-relationship • Statistical measures of covariance and correlation

17. Covariance - measure of the joint variation of 2 variables about their common mean • COVk,l = Sum [(BVik - uk)(BVil - ul)] (n-1) • COVk,l = 0, statistical independence • COVk,l > 0, positive relationship, as k increases, l increases • COVk,l < 0, negative relationship as k increases, l decreases

18. Correlation - the degree of inter-relation in a manner not influenced by the measurement units (unitless) • rkl = COVkl Sk * Sl • -1 < rkl < 1 • rkl = 0, statistically independent • rkl = -1, perfect negative relationship • rkl = 1, perfect positive relationship

19. Covariance & Correlation example BV1 BV1-u1 BV2BV2-u2(D1)(D2) 130 -5 57 10.6-53 COV12 = 540 / 4 165 30 35 -11.4 -342 = 135 100 -35 25 -21.4 749 r12 = _ 135 135 0 50 3.6 0 (23.7)(16.3) 145 10 6518.6 186 = 135 / 386.3 675 0 232 0 540 = 0.35 uk = 46.4

20. Covariance & Correlation Matrices • Provide a useful summary of data relationships • High variance suggests a higher information content for that band • High correlation suggests a substantial amount of redundancy • Low correlation suggests that each band provides information not found in the other

21. Covariance Matrix Diagonals represent band variances. Example, variance for Band 3 = 273.1Off-diagonals represent co-variances. Example, covariance of Band 1 and 4 = -35.3; same as covariance of Band 4 and 1. Negative covariance: as one band increases, the other decreases. Covariance matrix 1 2 3 4 5 6 7 1 232.3 139.1 237.2 -35.3 191.9 42.0 182.6 2 139.1 89.9 153.4 -4.6 142.1 24.1 122.0 3 237.2 153.4 273.1 -26.9 249.1 46.4 219.4 4 -35.3 -4.6 -26.9 341.1 216.1 -38.1 25.3 5 191.9 142.1 249.1 216.0 555.2 33.5 305.3 6 42.0 24.1 46.4 -38.1 33.5 31.22 40.6 7 182.6 122.0 219.4 25.3 305.3 40.6 227.6

22. Homework 2(Optional): Image Statistics NBTM88.img is a 7 band TM image with 1000 rows by 1500 columns. Image statistics for NBTM88.img: B1 B2 B3 B4 B5 B6 B7 Mean 102.8 41.7 47.0 74.9 77.4 155.8 32.8 Covariance matrix 1 2 3 4 5 6 7 1 232.3 139.1 237.2 -35.3 191.9 42.0 182.6 2 139.1 89.9 153.4 -4.6 142.1 24.1 122.0 3 237.2 153.4 273.1 -26.9 249.1 46.4 219.4 4 -35.3 -4.6 -26.9 341.1 216.1 -38.1 25.3 5 191.9 142.1 249.1 216.0 555.2 33.5 305.3 6 42.0 24.1 46.4 -38.1 33.5 31.22 40.6 7 182.6 122.0 219.4 25.3 305.3 40.6 227.6

23. Homework 2: Image Statistics 1. What is the computer disk space storage size for the NBTM88.img in ERDAS Imagine format? ______________ 2. What is the variance for Band 4? ____________________ 3. Do Band 3 and Band 4 positively covary? ______________ 4. List the bands in order of decreasing "information" content. High a. ______ b. ______ c. ______ d. ______ e. ______ f. ______ Low g. ______