1. Lecture 13: Spectral Mixture Analysis
Tuesday February 17
Lecture 13: Spectral Mixture Analysis
Last lecture: framework for viewing image processing and details about some standard algorithms

2. 19.1% Trees 43.0% Road 24.7% Grass/GV 13.2% Shade
Others are not. They may be rare, or may be pure at multi-pixel scales
Some materials are commonly found together. These are mixed.
Each pixel contains different materials, many with distinctive spectra.
Spectral images measure mixed or integrated spectra over a pixel
19.1% Trees
43.0% Road
24.7% Grass/GV
13.2% Shade

3. Spectral Mixtures Reflectance Wavelength Reflectance Wavelength 100
Wavelength
100
Reflectance
Wavelength

4. Linear vs. Non-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

5. 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.
Wavelength, μm

6. Spectral Mixtures 25% Green Vegetation (GV) 75% Soil 60 100% GV40
75% GV
25% GV
TM Band 4
50% GV 20 20 40 60
TM Band 3

7. Spectral Mixtures 25% Green Vegetation 70% Soil 5% Shade 60 100% GV40
TM Band 4 20
100% Shade 20 40 60
TM Band 3

8. Linear Spectral Mixtures
There can be at most m=n+1 endmembers
or else you cannot solve for the fractions f
uniquely r
mix,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 f em r
em,b eb
n,m

9. 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

10. Landsat TM image of part of the Gifford Pinchot National Forest

11. Old growth Burned Immature regrowth Broadleaf Deciduous Grasses
Shadow
Immature
regrowth
Broadleaf
Deciduous
Grasses
Clearcut

12. Spectral mixture analysis from the Gifford Pinchot National Forest
In fraction images, light tones
indicate high abundance
NPV
Green vegetation
R = NPV
G = green veg.
B = shade
Shade

13. Spectral Mixture Analysis - North Seattle
Blue – concrete/asphalt
Green - green vegetation
Red - dry grass

14. 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.

15. 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

16. 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.

17. 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

18. Next lecture –
Estimating roughness from stereo images