Lecture 13: Spectral Mixture Analysis

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Lecture 13 Spectral Mixture Analysis
Tuesday 16 February 2010
Reading Ch 7.7 7.12 Smith et al. Vegetation
in deserts (class website)
Last lecture framework for viewing image
processing and details about some standard
algorithms
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19.1 Trees 43.0 Road 24.7 Grass/GV 13.2
Shade
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Spectral images measure mixed or integrated
spectra over a pixel
19.1 Trees 43.0 Road 24.7 Grass/GV 13.2
Shade
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Each pixel contains different materials, many
with distinctive spectra.
19.1 Trees 43.0 Road 24.7 Grass/GV 13.2
Shade
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Some materials are commonly found together.
These are mixed.
19.1 Trees 43.0 Road 24.7 Grass/GV 13.2
Shade
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Others are not. They may be rare, or may be pure
at multi-pixel scales
19.1 Trees 43.0 Road 24.7 Grass/GV 13.2
Shade
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Spectral Mixtures
100
Reflectance
0
Wavelength
100
Reflectance
0
Wavelength
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Linear vs. Non-Linear Mixing

• Linear Mixing
• (additive)
• Non-Linear Mixing
• Intimate mixtures,
• Beers Law

r fgrg rs (1- fg)
r rg rs(1- rg)exp(-kgd)
(1-rg) exp(-kgd) .
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Spectral Mixture Analysis works with spectra that
mix together to estimate mixing fractions for
each pixel in a scene.
The extreme spectra that mix and that correspond
to scene components are called spectral
endmembers.
0 1 2
Wavelength, µm
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Spectral Mixtures
25 Green Vegetation (GV) 75 Soil
60
100 GV
100 Soil
40
75 GV
25 GV
TM Band 4
50 GV
20
0
0
0
20
40
60
TM Band 3
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Spectral Mixtures
25 Green Vegetation 70 Soil 5 Shade
60
100 GV
100 Soil
40
TM Band 4
20
100 Shade
0
0
20
40
60
TM Band 3
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Linear Spectral Mixtures
There can be at most mn1 endmembers or else
you cannot solve for the fractions f uniquely
Reflectance of observed (mixed) image spectrum
at each band b Fraction of pixel filled by
endmember em Reflectance of each endmember at
each band Reflectance in band b that could not
be modeled number of image bands, endmembers
eb
n,m
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In order to analyze an image in terms of
mixtures, you must somehow estimate the
endmember spectra and the number of endmembers
you need to use
Endmember spectra can be pulled from the image
itself, or from a reference library (requires
calib- ration to reflectance). To get the right
number and identity of endmembers,
trial-and-error usually works. Almost always,
shade will be an endmember shade a spectral
endmember (often the null vector)
used to model darkening due to terrain slopes
and unresolved shadows
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Inverse SMA (unmixing)
The point of spectral mixture analysis (SMA) is
usually to solve the inverse problem to find the
spectral endmember fractions that are
proportional to the amount of the physical
endmember component in the pixel. Since the
mixing equation (two slides ago) should be
underdetermined more bands than endmembers
this is a least-squares problem solved by
singular value decomposition in
ENVI. http//en.wikipedia.org/wiki/Singular_value
_decomposition
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Landsat TM image of part of the Gifford Pinchot
National Forest
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Old growth
Burned
Mature regrowth
Shadow
Immature regrowth
Broadleaf Deciduous
Clearcut
Grasses
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Spectral mixture analysis from the Gifford
Pinchot National Forest
In fraction images, light tones indicate high
abundance
Green vegetation
NPV
R NPV G green veg. B shade
Shade
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Spectral Mixture Analysis - North Seattle
Blue concrete/asphalt Green - green
vegetation Red - dry grass
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As a rule of thumb, the number of useful
endmembers in a cohort is 4-5 for Landsat TM
data. It rises to about 8-10 for imaging
spectroscopy. There are many more spectrally
distinctive components in many scenes, but they
are rare or dont mix, so they are not useful
endmembers. A beginners mistake is to try to
use too many endmembers.
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Foreground / Background Analysis (FBA)

• Objective Search for known material against a
  complex background
• Mixture Tuned Matched Filter in ENVI is a
  special case of FBA in which the background is
  the entire image (including the foreground)
• Geometrically, FBA may be visualized
• as the projection of a DN data space
• onto a line passing through the centroids
• of the background and foreground clusters
• The closer mystery spectrum X plots to
• F, the greater the confidence that the pixel
  
• IS F. Mixed pixels plot on the line between
• B F.

DNk
X
?
?
F
?
?
DNj
B
?
?
?
?
?
?
?
?
?
?
DNi
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Foreground Background
Vector w is defined as a projection in hyperspace
of all foreground DNs (DNF) as 1 and all
background DNs as (DNB) 0. n is the number of
bands and c is a constant. The vector w and
constant c are simultaneously calculated from the
above equations using singular-value
decomposition.
http//en.wikipedia.org/wiki/Singular_value_decomp
osition
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• Mixing analysis is useful because
• It makes fraction pictures that are closer to
  what you want to know about abundance of
  physically meaningful scene components
• It helps reduce dimensionality of data sets to
  manageable levels without throwing away much data
• 3) By isolating topographic shading, it provides
  a more stable basis for classification and a
  useful starting point for GIS analysis

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