Lecture 13: Spectral Mixture Analysis Tuesday 16 February 20 Last lecture: framework for viewing image processing and details about some standard algorithms Reading Ch 7.7 – 7.12 Smith et al. Vegetation in deserts (class website)
Dec 20, 2015
Lecture 13: Spectral Mixture Analysis
Tuesday 16 February 2010
Last lecture: framework for viewing image processing and details about some standard algorithms
Reading
Ch 7.7 – 7.12Smith et al. Vegetation in deserts (class website)
19.1% Trees19.1% Trees
43.0% Road43.0% Road
24.7% Grass/GV24.7% Grass/GV
13.2% Shade13.2% Shade
Spectral images measure mixed or integrated spectra over a pixel
19.1% Trees19.1% Trees
43.0% Road43.0% Road
24.7% Grass/GV24.7% Grass/GV
13.2% Shade13.2% Shade
Each pixel contains different materials, many with distinctive spectra.
Some materials are commonly found together. These are mixed.
19.1% Trees19.1% Trees
43.0% Road43.0% Road
24.7% Grass/GV24.7% Grass/GV
13.2% Shade13.2% Shade
19.1% Trees19.1% Trees
43.0% Road43.0% Road
24.7% Grass/GV24.7% Grass/GV
13.2% Shade13.2% Shade
Others are not. They may be rare, or may be pure at multi-pixel scales
Linear vs. Non-Linear Mixing
• Linear Mixing
(additive)
• Non-Linear Mixing– Intimate mixtures,
Beer’s Law
r = fg·rg+ rs ·(1- fg)
r = rg+ rs·(1- rg)·exp(-kg·d) · (1-rg)· exp(-kg·d) +…….
d
Spectral Mixture Analysis works with spectra that mix together to estimate mixing fractions for each pixel in a scene.
Spectral Mixtures, green leaves and soil
0
20
40
60
80
100
0 1 2 3
Wavelength, micrometers
Ref
lect
ivity
, %
0% leaves
25% leaves
50% leaves
75% leaves
100% leaves
The extreme spectra that mix and that correspond to scene components are called spectral endmembers.
0 1 2 Wavelength, μm
Spectral Mixtures
25% Green Vegetation (GV)75% Soil
TM
Ban
d 4
TM Band 3
0
60
40
20
0 40
75% GV
50% GV
25% GV
100% GV
100% Soil
0 20 60
0
20
40
60
80
100
350 850 1350 1850 2350
Spectral Mixtures25% Green Vegetation70% Soil 5% Shade
TM
Ban
d 4
TM Band 3
0
60
40
20
0 20 40 60
100% GV
100% Shade
100%Soil
0
20
40
60
80
100
350 850 1350 1850 2350
Linear Spectral Mixtures
r mix,b
fem
r em,b
= 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
b
bbem
m
em
embmix rfr
)( ,
1
, 11
m
em
emf
There can be at most m=n+1 endmembers or else you cannot solve for the fractions f uniquely
n
b
bnrms
1
21
n,m
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
Inverse SMA (“unmixing”)
The point of spectral mixture analysis (SMA) is usuallyto 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
Green vegetation
NPV
Shade
Spectral mixture analysis from the Gifford Pinchot National Forest
R = NPVG = green veg.B = shade
In fraction images, light tones indicate high abundance
Blue – concrete/asphaltGreen - green vegetationRed - dry grass
Spectral Mixture Analysis - North Seattle
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 don’t mix, so they are not useful endmembers.
A beginner’s mistake is to try to use too many endmembers.
• 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.
Foreground / Background Analysis (FBA)
▪▪▪▪▪▪▪▪
▪▪
▪
▪
▪B
F
DNi
DNj
DNk
▫X
Foreground: Foreground:
BackgroundBackground: :
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.
0
1
,
1
,
1
cDNw
cDNw
bB
n
b
b
bF
n
b
b
.
Clearcut(1990)
Clearcut(1994)Clearcut
(1984)
Uncut Forest
87 88 89 90 91 92 93 94 95
120
100
80
60
40
20
0
Year
MSS TM AVIRISMSS TM
FB
A G
reen
Veg
etat
ion
Inde
x
http://en.wikipedia.org/wiki/Singular_value_decomposition
Mixing analysis is useful because –
1) It makes fraction pictures that are closer to what you want to know about abundance of physically meaningful scene components
2) 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