Wednesday 16 February 2011
Lecture 13: Spectral Mixture
Analysis
Reading
Ch 7.7 – 7.12
Smith et al. Vegetation in deserts (class website)
What was covered in the previous lecture
•
•
LECTURES
Jan 05 1. Intro
Jan 07 2. Images
Jan 12 3. Photointerpretation
Jan 14 4. Color theory
Jan 19 5. Radiative transfer
Jan 21 6. Atmospheric scattering
Jan 26 7. Lambert’s Law
Jan 28 8. Volume interactions
Feb 02 9. Spectroscopy
Feb 04 10. Satellites & Review
Feb 09 11. Midterm
Feb 11 12. Image processing
previous
Feb 16 13. Spectral mixing
today
Feb 18 14. Classification
Feb 23 15. Radar & Lidar
Feb 25 16. Thermal infrared
Mar 02 17. Mars spectroscopy (Matt Smith)
Mar 04 18. Forest remote sensing (Van Kane)
Mar 09 19. Thermal modeling (Iryna Danilina)
Mar 11 20. Review
Mar 16 21. Final Exam
Friday’s lecture
framework for viewing image processing
and details about some standard
algorithms
Today’s lecture:
•Spectral mixture analysis
2
19.1%
43.0%
24.7%
13.2%
Trees
Road
Grass/GV
Shade
Spectral images measure mixed or
integrated spectra over a pixel
19.1%
43.0%
24.7%
13.2%
Trees
Road
Grass/GV
Shade
Each pixel contains different materials,
many with distinctive spectra.
19.1%
43.0%
24.7%
13.2%
Trees
Road
Grass/GV
Shade
Some materials are commonly
found together. These are mixed.
19.1%
43.0%
24.7%
13.2%
Trees
Road
Grass/GV
Shade
Others are not. They may be rare, or
may be pure at multi-pixel scales
19.1%
43.0%
24.7%
13.2%
Trees
Road
Grass/GV
Shade
Spectral Mixtures
Reflectance
100
0
Wavelength
Reflectance
100
0
Wavelength
Linear vs. Non-Linear Mixing
• Linear Mixing
(additive)
r = fg·rg+ rs ·(1- fg)
• Non-Linear Mixing
– Intimate mixtures,
Beer’s Law
r = fg·rg+ rs·(1- fg)·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
Reflectivity, %
100
0% leaves
80
25% leaves
60
50% leaves
40
75% leaves
20
100% leaves
0
00
11
22
Wavelength,
μm
Wavelength,
micrometers
3
The extreme spectra
that mix and that
correspond to scene
components are
called spectral
endmembers.
Spectral Mixtures
25% Green Vegetation (GV)
75% Soil
60 100% GV
TM Band 4
100% Soil
40
75% GV
25% GV
100
50% GV
20
80
60
0
0
20
40
TM Band 3
60
40
20
0
350
850
1350
1850
2350
Spectral Mixtures
25% Green Vegetation
70% Soil
5% Shade
TM Band 4
60
100% GV
100%
Soil
40
100
20
80
60
0
100% Shade
0
20
40
TM Band 3
60
40
20
0
350
850
1350
1850
2350
Linear Spectral Mixtures
m
rmix ,b 
( f
m
f
emrem,b )  e b
em1
em1
There can be at most m=n+1 endmembers
or else you cannot solve for the fractions f
uniquely
r
em 1
rms 
1
n
n

e b2
b 1
mix,b = Reflectance of observed (mixed) image spectrum at each band b
fem
= Fraction of pixel filled by endmember em
r
= Reflectance of each endmember at each band
em,b
eb
= Reflectance in band b that could not be modeled
n,m
= number of image bands, endmembers
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 calibration 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 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
% leaves
% wood
Green
% soil
leaves
Red
wood
shadow
soil
0
Blue
Landsat TM image
of part of the
Gifford Pinchot
National Forest
Mature
regrowth
Old growth
Burned
Shadow
Immature
regrowth
Broadleaf
Deciduous
Grasses
Clearcut
Spectral mixture analysis from the
Gifford Pinchot National Forest
NPV
In fraction images, light tones
indicate high abundance
Green
vegetation
R = NPV
G = green veg.
B = shade
Shade
Spectral Mixture Analysis - North Seattle
Blue – concrete/asphalt
Green - green vegetation
Red - dry grass
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.
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)
DNk
• Geometrically, FBA may be visualized
as the projection of a DN data space
▫X
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.
B▪▪
▪▪▪▪▪ ▪
▪
▪▪▪ F
DNj
DNi
n
Foreground:
 w DN
b
F ,b
 c 1
B ,b
c0
b 1
n
Background:
 w DN
b
b 1
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_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
Next lecture –
Image classification