1. Multi-spectral image manipulation Lecture 6 prepared by R. Lathrop 10/99 Revised 2/09

2. Where in the World?

3. Learning objectives Remote sensing science concepts –Rationale and theory behind Spectral ratioing and normalized difference ratioing PCA (Principal component analysis) Tasseled cap transformation; Minimum Noise Fraction (MNF) transformation Math Concepts –Matrices and PCA Skills –Visualizing in feature space –Undertaking and analyzing PCA

4. 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

5. Feature Space: 2 band scatterplot of image data 0 255 0 Band A Band B Histogram Band A Histogram Band B

6. Each dot represents a pixel; the warmer the colors, the higher the frequency of pixels in that portion of the feature space.

7. Spectral ratioing Enhancements resulting from the division of BV values in one spectral band by the corresponding values in another band BV i,j,r = BV i,j,k /BV i,j,l Useful for discriminating subtle spectral variations that are masked by the brightness variations in images; for examining the relationship between one band vs. another Useful for eliminating brightness variations due to topographic slope effects

8. Sunlight Terrain Shadowing Shadow Land coverBand ABandBRatio A/B Sunlit1401500.93 Shadow56610.92 Sunlit1021450.70 Shadow41580.71 Deciduous Conifer Adapted from Lillesand & Kiefer, 3 rd ed

9. Spectral ratioing Ratioing compensates for multiplicative rather than additive illumination effects. / /

10. Spectral Ratioing: for Absorption Enhancement Objective: enhance particular absorption features of materials of interest vs. background reflectance Numerator is a baseline of background absorption Denominator is an absorption peak for the material of interest (based on absorption spectra) As material concentration increases, denominator decreases, index increases

11. Spectral Ratioing: Geological Indices TM5/TM7 to enhance clay minerals –TM5: 1.55->1.75um provides background reflectance –TM7: 2080->2350um: specific absorption peak for clay minerals From ERDAS Field Guide 4 th ed. To more effectively discriminate between the various types of clay minerals can use hyperspectral ratios kaolinite: 2160/2190nm montmorillonite 2220/2250nm illite 2350/2488nm

12. Spectral Ratioing: for Reflectance Enhancement Objective: enhance particular reflectance features of materials of interest vs. background reflectance Numerator represents wavelengths where there is an increase in reflectance due to enhanced backscattering from the material of interest Denominator is a baseline of background reflectance As material concentration increases, numerator increases, denominator stays roughly the same (may go up or down slightly) index increases As long as the numerator increases faster than the denominator, the index increases

13. Normalized Difference Ratioing Objective: contrast bands where there is high absorption (low reflectance) vs. low absorption (high reflectance) Numerator is the difference between two bands where B1 has high reflectance and B2 has low reflectance for the feature of interest Denominator is the sum B1 + B2 Normalizes the difference with the overall scene brightness (B1 – B2) /(B1 + B2)

14. Normalized Difference Snow Index (NDSI) Snow reflectance high in the visible (0.5-0.7um) and low in the short-wave (mid-IR) infrared (1-4um) MODIS: B4 (0.555um) visible B6 (1.640 um) mid-IR NDSI = (B4 – B6) / (B4 + B6) Fore more info: Salomonson et al, 2004. RSE 89:351-360. MODIS 4-6-3 R-G-B

15. Working with Ratios Remember: – ratio outputs will be real floating point numbers and generally need to be rescaled for proper viewing –Can’t divide by zero, so need to exclude zeroes Generally good practice to transform the band BVs to their radiance or reflectance equivalent before ratioing – i.e., ratioing their true reflectance rather BV equivalent

16. Principal Components Analysis (PCA) Multispectral image data may have extensive inter-band correlation - i.e. two bands may be similar and convey essentially the same information PCA used to reduce the dimensionality of a data set - i.e. compress the information contained in an original n-channel data set into fewer than n “new” channels or components

17. Principal Components Analysis (PCA) N-dimensional ellipsoid in image feature space Goal of PCA is to translate the original axes to a new set of axes, with each axis orthogonal to the others 1st axis or PC is associated with the maximum amount of variance (the ellipsoid’s major axis) 2nd axis (orthogonal to the 1st) contains the next highest amount of variation and so on …

18. Feature Space: Image data ellipsoid 0 255 0 Band A Band B Histogram Band A Histogram Band B

19. Information Content = Image Variance major axis of data ellipsoid represents axis of greatest information content 0 255 0 Band A Band B Range of Band A Range of Band B Hypotenuse of triangle longer than any leg

20. PC axes: each orthogonal to the others, each explaining the next greatest amount of information variation 0 255 0 Band A Band B PC2 PC1

21. Principal Components Analysis (PCA) Matrix algebra used in PCA, computed from the covariance matrix Eigenvalue ( l ) provides the length of the new axes; one value for each PC Eigenvector provides the direction of the new axes; column of numbers with one coefficient for each of the original input bands

22. PCA: eigenvalue and eigenvector Definition: Let A be a square matrix. A non-zero vector C is called an eigenvector of A if and only if there exists a number (real or complex) λ such that AC=λC. If such a number λ exists, it called eigenvalue of A. The vector C is called eigenvector associated with the eigenvalue λ. AC 1 =-4C 1 λ 1 = -4 AC 2 =3C 2 λ 2 = 3

23. Eigenvalue: length of new PC axis Eigenvector: angular orientation of new PC axis 0 255 0 Band A Band B PC2 PC1

24. PCA: Example for tm_oceanco_95sep04.img

25. PCA: Example Covariance matrix for tm_oceanco_95Sep04.img 1 2 3 4 5 6 7 50.0331.8551.63-15.2671.5420.1755.08 31.8524.1137.94-3.5956.5713.5440.35 51.6337.9464.44-10.8394.8922.9768.12 -15.26-3.59-10.83167.4071.78-9.344.36 71.5456.5794.8971.78273.9038.61140.79 20.1713.5422.97-9.3438.6117.4227.59 55.0840.3568.124.36140.7927.5995.49

26. PCA: Example Sum of Variances = total information content of the image 1 2 3 4 5 6 7 ΣVariance = 692.77 50.0331.8551.63-15.2671.5420.1755.08 31.8524.1137.94-3.5956.5713.5440.35 51.6337.9464.44-10.8394.8922.9768.12 -15.26-3.59-10.83167.4071.78-9.344.36 71.5456.5794.8971.78273.9038.61140.79 20.1713.5422.97-9.3438.6117.4227.59 55.0840.3568.124.36140.7927.5995.49

27. Principal Components Analysis (PCA) The magnitude of the eigenvalue provides an index of the information content explained by that PC Sum of Variances = total information content = Σeigenvalueλ p To calculate proportion of the total information content explained by each PC.

28. PCA: Example The Eigenvalues for tm_oceanco_95sep04.img PC1 452.85 PC2 185.84 PC3 32.51 PC4 7.99 PC5 7.83 PC6 4.63 PC7 1.12 S eigenvalue l p = 692.77 To calculate proportion of the total information content explained by each PC. What percentage of the total information content is explained by the 1 st three PC’s?

29. PCA: Example The Eigenvalues for tm_oceanco_95sep04.img PC1 452.85 452.85/692.77* 100 = 65.4% PC2 185.84185.84 /692.77 * 100 = 26.8%96.9% PC3 32.5132.51/692.77 * 100 = 4.7% PC4 7.997.99/692.77 * 100 = 1.2% PC5 7.837.83/692.77 * 100 = 1.1% PC6 4.634.63/692.77 * 100 = 0.7% PC7 1.131.13/692.77 * 100 = 0.2%

30. Principal Components Analysis (PCA) Factor loading: the correlation of each original band with each PC, used to interpret the physical meaning of the PC axes PCA is heavily data dependent, unique for each image data set – not fixed like Tasseled Cap e kp = eigenvector for row (band) k and column (principal component) p λ p = eigenvalue for PC p (i.e., the pth eigenvalue) σ kk = variance for band k in the covariance matrix

31. PCA: Example Eigenvector Matrix for tm_oceanco_95sep04.img PC1PC2PC3 PC4 PC5PC6PC7 0.2488 -0.2403 ‑ 0.5026 0.1233 0.2167 ‑ 0.7385 -0.1405 0.1894 -0.1301 ‑ 0.3233 -0.068 0.1690 0.1977 0.8777 0.3150 -0.2390 ‑ 0.4440 ‑ 0.1426 0.2107 0.6111 -0.4563 0.1655 0.8982 -0.3906 0.0324 -0.1072 0.0079 -0.0251 0.7582 0.1321 0.5376 0.0688 0.3345-0.0442 0.0049 0.1250 -0.1196 -0.0600 0.9168 -0.3046 0.1809 0.0224 0.4302 ‑ 0.1723 ‑ 0.0216 ‑ 0.3369 -0.8146 ‑ 0.0851 0.0234

32. PCA: Example e kp = eigenvector for row (band) k and column (principal component) p λ p = eigenvalue for PC p (i.e., the pth eigenvalue) σ kk = variance for band k in the covariance matrix Corr (B1,PC1) = (0.2488 *sqrt(452.85) /sqrt(50.03) = (0.2488 * 21.28) / 7.07 = 0.75

33. What is the correlation between PC1 and Band 2? Corr (PC1,B2) = (e 21 *sqrt(λ 1 )) /sqrt(σ 22 ) e 21 = eigenvector for row (band) 2, col (PC) 1 λ 1 = eigenvalue for PC 1 σ 22 = variance for band 2 Corr (PC1,B2) = (0.1894 * sqrt(452.85)) /sqrt(24.11) = (0.1894* 21.28) / 4.91 = 0.82

34. PCA: Example for tm_oceanco_95sep04.img Correlation Matrix (Original TM Band vs. PC) PC1 PC2 PC3 PC4 PC5 PC6 PC7 1 2 3 4 5 6 7 0.75-0.46-0.410.050.09-0.22-0.02 0.82-0.36-0.38-0.040.100.090.19 0.84-0.41-0.32-0.050.070.16-0.06 0.270.95-0.170.01-0.020.00 0.970.110.190.010.06-0.010.00 0.64-0.39-0.080.62-0.200.090.01 0.94-0.24-0.01-0.10-0.23-0.020.00

35. PCA: Example for tm_oceanco_95sep04.img

36. PCA: Example for tm_oceanco_94sep04.img PC1 PC2 PC3

37. PCA: Example for tm_oceanco_94sep04.img R-G-B PC1-PC2-PC3

38. PCA: Homework PNR_110494.img

39. PCA: Example for PNR_110494.img PC1 PC2 PC3

40. PCA: Example for PNR_110494.img R-G-B PC1-PC2-PC3

41. PCA Spectral domain fusion Low and high resolution images are co-registered and resampled to same GRC PCA of multispectral image Substitution of PAN image for 1st principal component, often the “brightness component”, then backtransform to image space This technique can be used for any number of bands Generally a good compromise between limited spectral distortion and visually attractiveness

42. Tasseled Cap Transform Fixed feature space transformation designed specifically for agricultural monitoring, stable from scene to scene Red-NIR feature space shows a triangular distribution described as a “tasseled cap”. Over the growing season, crop pixels moved from the base “plane of soils” up the tasseled crop and then back down Linear transformation of original image data to new axes: brightness, greenness, wetness

43. Red Reflectance N I R R e f l e c t a n c e Spectral Feature Space Example Pixel X proportions: IS: 50% Grass: 30% Trees: 20% Sub-pixel Estimation Soil Line Increasing Vegetation

44. Tasseled Cap Transform Landsat Thematic Mapper 4 coefficients Brightness =.3037(TM1) +.2793(TM2) +.4743(TM3) +.5585(TM4) +.5082(TM5) +.1863(TM7) Greenness = -.2848(TM1) -.2435(TM2) -.5436(TM3) +.7243(TM4) +.0840(TM5) -.1800(TM7) Wetness =.1509(TM1) +.1973(TM2) +.3279(TM3) +.3406(TM4) -.7112(TM5) -.4572(TM7) Haze =.8832(TM1) -.0819(TM2) -.4580(TM3) -.0032(TM4) -.0563(TM5) +.0130(TM7) From ERDAS Field Guide 4 th Ed.

45. Tasseled Cap Transform: example brightness greeness wetnesshaze

46. Minimum Noise Fraction (MNF) Transform MNF: 2 cascaded PCA transformations to separate out the noise from image data for improved spectral processing; especially useful in hyperspectral image analysis 1st: is based on an estimated noise covariance matrix to de-correlate and rescale the noise in the data such that the noise has unit variance and no band-to-band correlation 2nd: create separate a) spatially coherent MNF eigenimage with large eigenvalues (high information content, λ >1) and b) noise-dominated eigenimages (λ close to = 1)

47. MNF Transform: example 1 Original TM image using ENVI software Plot of eigenvalue number vs. eigenvalue MNF 6 = noise

48. MNF Transform: example 1 MNF 5 MNF 1MNF 2 MNF 6MNF 4 MNF 3

49. MNF Transform: example 2 Tm_oceanco_95sep04.img Original TM image using ENVI software Plot of eigenvalue number vs. eigenvalue MNF 5,6 7 = noise

50. MNF Transform: example 2 MNF 5 MNF 1MNF 2 MNF 6MNF 4 MNF 3 MNF 7

51. Main points of the lecture Feature space; Spectral ratioing and Normalized difference ratioing (e.g., NDSI, NDVI) PCA (Principal component analysis); Tasseled Cap transformation; Minimum Noise Fraction (MNF) transformation.

52. Homework 1 Homework: Principal Component Analysis; 2 Reading Ch. 5:164-169, 296-301; Ch 11: 443-445 3 Reading ERDAS Ch. 6:162-183. 4 Take-home exam due March 4 (Wednesday in lab).