multi-spectral image manipulation lecture 6 prepared by r. lathrop 10/99 revised 2/09

Multi-spectral image manipulation

Lecture 6

prepared by R. Lathrop 10/99

Revised 2/09

Where in the World?

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

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

Feature Space: 2 band scatterplot of image data

0 255

0

255

Band A

Band B

Histogram Band A

Histogram

B

and B

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.

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

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

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

//

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

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

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

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)

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

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

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

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 …

Feature Space: Image data ellipsoid

0 255

0

255

Band A

Band B

Histogram Band A

Histogram

B

and B

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

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

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

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

Eigenvalue: length of new PC axisEigenvector: angular orientation of new PC axis

0 255

0

255

Band A

Band B

PC2

PC1

PCA: Example for tm_oceanco_95sep04.img

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

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

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

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

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%

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

*),(

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

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

*),(

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

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

PCA: Example for tm_oceanco_95sep04.img

PCA: Example for tm_oceanco_94sep04.img

PC1

PC2

PC3

PCA: Example for tm_oceanco_94sep04.img

R-G-B

PC1-PC2-PC3

PCA: Homework PNR_110494.img

PCA: Example for PNR_110494.img

PC1

PC2

PC3

PCA: Example for PNR_110494.img

R-G-B

PC1-PC2-PC3

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

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

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

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.

Tasseled Cap Transform: example

brightness greeness

wetness haze

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)

MNF Transform: example 1

Original TM image using ENVI software

Plot of eigenvalue number vs. eigenvalue

MNF 6 = noise

MNF Transform: example 1

MNF 5

MNF 1 MNF 2

MNF 6MNF 4

MNF 3

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

MNF Transform: example 2

MNF 5

MNF 1 MNF 2

MNF 6MNF 4

MNF 3

MNF 7

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.

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).