Spectral Transformation

1. Spectral Transformation 강원대학교 지구물리학과 이훈열 교수 Reference: Chapter 5 of Schowengerdt, 1997, Remote Sensing Models and Methods for Image Processing, 2nd Ed., Academic Press.

2. Image Space (Spatial + Spectral) • Multi-spectral Image Space • 2D spatial + multi-D spectral information DN(x,y) = (DN1(x,y), DN2(x,y), … DNn(x,y)) • Here, we only concern “spectral space” • Multi-dimensional, Multi-spectral vectors • N-dimensions = N-bands • Spectral Transformation (Ratios, NDVI, PCT, Contrast enhancement…) • Spatial Transformation (filters, FFT… will be discussed later) • Various feature space , which is useful for thematic classification, can be derived from the spectral space.

3. Feature Space • A general, linear matrix transformation of the spectral vector W: a transformation matrix (rotation, scaling, …) • A general, nonlinear transformation

4. Band Math PG Steamer: Image Processing -> Band Math Dataset: class_ex.idm

5. DN values • DN values in band k is approximately a linear function of earth surface reflectance (Lambertian Surface): : radiometric gain (sensor) : radiometric offset (sensor) : incidence angle (topography) : surface reflectance (target)

6. Multispectral Ratios • A non-linear transformation. One of the earliest feature extraction technique. • Bias-corrected: topographic effect (cosine term) disappear. • Fully-calibrated: • Modulation ratio: normalized to [-1, 1]

7. Vegetation Indices • Ratio Vegetation Index(RVI) • Normalized Difference Vegetation Index (NDVI): poor when the ground cover is low, as in arid and semi-arid region. • Soil-Adjusted Vegetation Index (SAVI): good for low land-cover environment • Transformed Vegetation Index (TVI) • Perpendicular Vegetation Index (PVI): distance to the soil line. • Tasseled Cap Transform

8. NDVI Result RGB321 NDVI

9. Vegetation Index Examples: NDVI and PVI NDVI=0.7 PVI=0.5 NDVI=0.5 NIR NIR PVI=0.3 Soil Line Soil Line NDVI=0.0 PVI=0.0 Red Red Values are not in scale.

10. Principle Component Transformation PG Steamer: Image Processing -> Image Transformation Dataset: class_ex.idm

11. Spectral Redundancy • Spectral redundancy (waste of money) occurs if the correlation between spectral bands are high: • Material spectral correlation • Topographic shading • Sensor bands overlap • Principle Component Transformation (PCT) is a feature space transformation designed to remove the spectral redundancy

12. Scattergram (Landsat TM) B1 B2 B3 B3 B3 B3 B4 B5 B6 B3 B3 B3

13. Spectral Rotation B4 B2 PC1 PC2 PC2 PC1 B3 B3

14. Principle Component Transformation • PCT is a rotation in K-D of the original coordinate axes to coincide with the major axes of the data. • PC axes are orthogonal, but it may not look orthogonal in 2-D scattergram. • It optimally redistributes the total image variance in the transformed data. The first PC image contains the maximum possible variance for any linear combination of the original bans. • PC images are uncorrelated with each other. • The total image variance is preserved. • You can diagonalize the covariance matrix since it is symmetric. • You need to find eigenvalues and eigenvectors for the covariance matrix. • Each eigenvalue is equal to the variance of the respective PC image along the new coordinate axes. • The sum of all the eigenvalues must equal the sum of all the band variances of the original image, thus preserving the total variance in the data. • PC compress most of the total image variance into fewer dimensions.

15. PCT Procedure

16. PCT – DIY • Scattergram • Covariance Matrix • Eigenvalue • Eigenvector • Transformation Matrix • Diagonalize C • PCT • Scattergram

17. PCT Images PC1 PC2 PC3 PC4 PC5 PC7

18. PCT Scattergram – low covariances PC1 PC2 PC3 PC1 PC1 PC1 PC5 PC6 PC4 PC1 PC1 PC1

19. PCT - Statistics [Band Statistics] Band 1 2 3 4 5 6 7 Min. 34.31 15.35 -22.74 -12.45 -9.58 -8.70 -0.33 Max. 411.56 234.96 163.63 119.69 84.39 46.47 0.23 Mean 115.90 71.41 70.76 30.56 39.72 13.44 -0.02 S.D. 50.20 28.74 6.28 3.15 2.57 1.49 0.05 [Covariance Matrix] Band 1 2 3 4 5 6 7 1 2519.92 -2.40 -0.20 0.05 -0.01 0.01 0.00 2 -2.40 826.02 -0.05 0.19 0.15 -0.08 -0.00 3 -0.20 -0.05 39.42 0.14 0.06 -0.05 0.00 4 0.05 0.19 0.14 9.91 -0.04 0.02 0.00 5 -0.01 0.15 0.06 -0.04 6.60 0.03 0.00 6 0.01 -0.08 -0.05 0.02 0.03 2.23 0.00 7 0.00 -0.00 0.00 0.00 0.00 0.00 0.00 [Correlation Matrix] Band 1 2 3 4 5 6 7 1 1.00 -0.00 -0.00 0.00 -0.00 0.00 0.00 2 -0.00 1.00 -0.00 0.00 0.00 -0.00 -0.00 3 -0.00 -0.00 1.00 0.01 0.00 -0.01 0.00 4 0.00 0.00 0.01 1.00 -0.00 0.00 0.00 5 -0.00 0.00 0.00 -0.00 1.00 0.01 0.00 6 0.00 -0.00 -0.01 0.00 0.01 1.00 0.00 7 0.00 -0.00 0.00 0.00 0.00 0.00 1.00

20. Color Decorrelation Stretch using PCT • If spectral correlation of the three bands are high, then the color lies along a line in the color cube, and very little of the available color space is utilized. • RGB -> PCT -> stretch (equalize variances) -> Inverse PCT to RGB

21. Tasseled Cap Transformation • PCT is data dependent. • TCT has a fixed WTC and is independent of the scene. • Designed for agricultural monitoring • Axes: soil brightness, greenness, yellow stuff, non-such B4 Greenness Soil Brightness B3