Spectral Transformation

1
Spectral Transformation

• ????? ??????
• ??? ??

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 -gt 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
NDVI0.7
PVI0.5
NIR
NDVI0.5
NIR
PVI0.3
Soil Line
Soil Line
NDVI0.0
PVI0.0
Red
Red
Values are not in scale.
10
Principle Component Transformation
PG Steamer Image Processing -gt 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
4 7 5 4 1
3 4 4 5 3
4 2 3 4 3
2 4 5 5 6
2 6 7 4 3

• Scattergram
• Covariance Matrix
• Eigenvalue
• Eigenvector
• Transformation Matrix
• Diagonalize C
• PCT
• Scattergram

4 4 1 4 2
3 3 4 3 3
1 2 3 4 1
2 4 7 3 3
1 3 5 4 1
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

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 -gt PCT -gt stretch (equalize variances) -gt
  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