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

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

Wavelength

100

0

Ref

lect

ance

Spectral Mixtures

Wavelength

0

Ref

lect

ance

100

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

Landsat TM image of part of the

Gifford Pinchot National Forest

BurnedMatureregrowth

Old growth

Immatureregrowth

BroadleafDeciduous

ClearcutGrasses

Shadow

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

Next lecture –

Image classification