multi-spectral image manipulation lecture 6 prepared by r. lathrop 10/99 revised 2/09
TRANSCRIPT
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Multi-spectral image manipulation
Lecture 6
prepared by R. Lathrop 10/99
Revised 2/09
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Where in the World?
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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
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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
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Feature Space: 2 band scatterplot of image data
0 255
0
255
Band A
Band B
Histogram Band A
Histogram
B
and B
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Red Reflectance
NIRReflectance
Spectral Feature Space
Each dot represents a pixel; the warmer the colors, the higher the frequency of pixels in that portion of the feature space.
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Spectral ratioing• Enhancements resulting from the division of BV values
in one spectral band by the corresponding values in another band
• BVi,j,r = BVi,j,k/BVi,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
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Sunlight Terrain Shadowing
Shadow
Land cover Band A BandB Ratio A/B Sunlit 140 150 0.93 Shadow 56 61 0.92
Sunlit 102 145 0.70 Shadow 41 58 0.71
Deciduous
Conifer
Adapted from Lillesand & Kiefer, 3rd ed
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Spectral ratioing
• Ratioing compensates for multiplicative rather than additive illumination effects.
ijl
ijk
ijl
ijk
BV
BV
BV
BV
*2
*2
ijl
ijk
ijl
ijk
BV
BV
BV
BV
2
2
//
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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
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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 4th 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
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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
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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)
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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
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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
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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
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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 …
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Feature Space: Image data ellipsoid
0 255
0
255
Band A
Band B
Histogram Band A
Histogram
B
and B
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Information Content = Image Variance major axis of data ellipsoid represents axis of
greatest information content
0 255
0
255
Band A
Band B
Range of Band A
Range of Band B
Hypotenuse of triangle longer than any leg
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PC axes: each orthogonal to the others, each explaining the next greatest amount of information variation
0 255
0
255
Band A
Band B
PC2
PC1
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Principal Components Analysis (PCA)
• Matrix algebra used in PCA, computed from the covariance matrix
• Eigenvalue () 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
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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.
121
016
121
A
1
2
1
1C
2
3
2
2C
AC1=-4C1 λ1 = -4AC2=3C2 λ2 = 3
4
8
4
1AC
6
9
6
2AC
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Eigenvalue: length of new PC axisEigenvector: angular orientation of new PC axis
0 255
0
255
Band A
Band B
PC2
PC1
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PCA: Example for tm_oceanco_95sep04.img
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PCA: Example
Covariance matrix for tm_oceanco_95Sep04.img
1 2 3 4 5 6 7
50.03 31.85 51.63 -15.26 71.54 20.17 55.08
31.85 24.11 37.94 -3.59 56.57 13.54 40.35
51.63 37.94 64.44 -10.83 94.89 22.97 68.12
-15.26 -3.59 -10.83 167.40 71.78 -9.34 4.36
71.54 56.57 94.89 71.78 273.90 38.61 140.79
20.17 13.54 22.97 -9.34 38.61 17.42 27.59
55.08 40.35 68.12 4.36 140.79 27.59 95.49
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PCA: Example
Sum of Variances = total information content of the image
1 2 3 4 5 6 7
ΣVariance = 692.77
50.03 31.85 51.63 -15.26 71.54 20.17 55.08
31.85 24.11 37.94 -3.59 56.57 13.54 40.35
51.63 37.94 64.44 -10.83 94.89 22.97 68.12
-15.26 -3.59 -10.83 167.40 71.78 -9.34 4.36
71.54 56.57 94.89 71.78 273.90 38.61 140.79
20.17 13.54 22.97 -9.34 38.61 17.42 27.59
55.08 40.35 68.12 4.36 140.79 27.59 95.49
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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 = Σeigenvalueλp
• To calculate proportion of the total information content explained by each PC.
100% p
pp eigenvalue
eigenvalue
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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
eigenvalue 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 1st three PC’s?
100% p
pp eigenvalue
eigenvalue
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PCA: ExampleThe Eigenvalues for tm_oceanco_95sep04.img
PC1 452.85 452.85/692.77* 100 = 65.4%
PC2 185.84 185.84 /692.77 * 100 = 26.8% 96.9%
PC3 32.51 32.51/692.77 * 100 = 4.7%
PC4 7.99 7.99/692.77 * 100 = 1.2%
PC5 7.83 7.83/692.77 * 100 = 1.1%
PC6 4.63 4.63/692.77 * 100 = 0.7%
PC7 1.13 1.13/692.77 * 100 = 0.2%
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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
ekp = eigenvector for row (band) k and column (principal component) pλp = eigenvalue for PC p (i.e., the pth eigenvalue)σkk = variance for band k in the covariance matrix
kk
pkp
pk
ePCBCorr
*),(
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PCA: ExampleEigenvector Matrix for tm_oceanco_95sep04.img
PC1 PC2 PC3 PC4 PC5 PC6 PC7
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
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PCA: Example
ekp = eigenvector for row (band) k and column (principal component) pλp = eigenvalue for PC p (i.e., the pth eigenvalue)σkk = 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
kk
pkp
pk
ePCBCorr
*),(
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What is the correlation between PC1 and Band 2?
Corr (PC1,B2) = (e21 *sqrt(λ1)) /sqrt(σ22)
e21 = eigenvector for row (band) 2, col (PC) 1
λ1 = eigenvalue for PC 1
σ22 = variance for band 2
Corr (PC1,B2) = (0.1894 * sqrt(452.85)) /sqrt(24.11) = (0.1894* 21.28) / 4.91 = 0.82
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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 -0.41 0.05 0.09 -0.22 -0.02
0.82 -0.36 -0.38 -0.04 0.10 0.09 0.19
0.84 -0.41 -0.32 -0.05 0.07 0.16 -0.06
0.27 0.95 -0.17 0.01 -0.02 0.00 0.00
0.97 0.11 0.19 0.01 0.06 -0.01 0.00
0.64 -0.39 -0.08 0.62 -0.20 0.09 0.01
0.94 -0.24 -0.01 -0.10 -0.23 -0.02 0.00
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PCA: Example for tm_oceanco_95sep04.img
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PCA: Example for tm_oceanco_94sep04.img
PC1
PC2
PC3
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PCA: Example for tm_oceanco_94sep04.img
R-G-B
PC1-PC2-PC3
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PCA: Homework PNR_110494.img
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PCA: Example for PNR_110494.img
PC1
PC2
PC3
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PCA: Example for PNR_110494.img
R-G-B
PC1-PC2-PC3
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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
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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
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Red Reflectance
N I R Re f l e c tance
Spectral Feature Space
Example
Pixel X proportions:
IS: 50%
Grass: 30%
Trees: 20%
Sub-pixel Estimation
Soil Line
Increasing Vegetation
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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 4th Ed.
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Tasseled Cap Transform: example
brightness greeness
wetness haze
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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)
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MNF Transform: example 1
Original TM image using ENVI software
Plot of eigenvalue number vs. eigenvalue
MNF 6 = noise
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MNF Transform: example 1
MNF 5
MNF 1 MNF 2
MNF 6MNF 4
MNF 3
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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
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MNF Transform: example 2
MNF 5
MNF 1 MNF 2
MNF 6MNF 4
MNF 3
MNF 7
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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.
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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).