Digital Imaging and Remote Sensing Laboratory Mixed Pixels and Spectral Unmixing 1 Spectral Mixtures areal aggregate intimate Linear mixing models (areal

Mixed Pixels and Spectral Unmixing 1Digital Imaging and Remote Sensing Laboratory

Mixed Pixels and Spectral UnmixingMixed Pixels and Spectral Unmixing

Spectral Mixtures

• areal

• aggregate

• intimate

Linear mixing models (areal and aggegrate)


Mixed Pixels and Spectral Unmixing (cont’d)Mixed Pixels and Spectral Unmixing (cont’d)

where R is the effective reflectance of the mixed

pixel, Ri is the reflectance of the ith material (end

member), fi is the spatial fraction covered by the ith

material and N is the number of materials in the pixel.

N

iiiRfR

1



Since L=mR+b

where Li is the radiance from a pure pixel of

material/end member i.

bmRfbRfmbmRfLfL i

N

i

N

iiiii

N

iii

1 11)(



Given an M spectral band sensor with the bands

designated with a j subscript, we can write M

equations of the form

space ereflectanc in 1

N

iijij RfR

space radiance in or1

N

iijij LfL



If we can claim to know the spectral reflectance or

radiance for the materials potentially in each pixel

(i.e., the end member) we can write M simultaneous

linear equations in N unknowns (i.e., the fractions are

the only unknowns).



In matrix/vector form, this looks like

NNMMM

M

M

M f

f

f

LLL

LLL

LLL

L

L

L

2

1

21

22212

12111

2

1

FeeeL N

31

FEL



where L is the vector of radiance values for a pixel,

is the vector of radiance values for the ith end

member, E is the matrix made up of the N end

member radiance column vectors and F is the

vector of unknown end member fractions.

If the end members are independent, then we can

solve for the end member fractions using least

squares methods as long as M is greater than or

equal to N.

ie



Take a simple 3-band 3-end member case with end

member reflectance vectors.

4242

8202

4164

E so

4

8

4

,

24

20

16

,

2

2

4

veglandwater eee

32i

32i

32i

4f24f2f8

8f20f2f8

4f16f4f7

yielding

8

8

7

R



Solving for f1, f2, and f3 yields

When we add error into the system, we must

recognize that without constraining the system, a

least squares solution may yield a best estimate for

the fractions that are physically unrealizable.

25.0

25.0

5.0

F



We can constrain the fractions to sum to 1 by adding

another linear equation.

This is referred to as the partially constrained case.

We can further constrain all the fractions to be

between 0 and 1. This is the fully constrained case.

N

iif

11



Both these cases can be solved using

extensions to the least squares theory

(Robinson, 1997 and Gross, 1996).


Spectral Mixture AnalysisSpectral Mixture Analysis

Consider linear mixing of 3 end members in 2 bands. All combinations of these lie along lines connecting the end members in spectral space (no matter how many bands). All combinations lie within the area defined by the outermost pair-wise combinations of end members.


Spectral Mixture Analysis (cont’d)Spectral Mixture Analysis (cont’d)

Ternary diagram maps space to a linear combination of end

members fractions in a geometric representation with 100% of end

members at the extremes. Mixtures of 2 along the solid lines and

mixtures of more than 2 in the interior. A geometric mixture model

with 4 end members would be a pyramid with 3 sides and a base

having end members at each apex.



• Graphical illustration of mixing models

• The nature of mixing suggests that all

combinations of end members must lie inside of

a convex hull made up of the end members



• end member concepts

• The end members are assumed to be spectral

extrema representing spectrally idealized

examples of a land cover type.

• End members greater that 1 and less that zero



requires more than 100%of E3

requires a negative amount of E2

E2

E3

E1Band 2

Band 1



In order to account for brightness variations due to

solar illumination effects and mixed pixels

containing shadows, a shadow end member may

be introduced. The shadow end member typically

is assigned the spectral reflectance expected from

a dark shadow element.



In general, to avoid over-fitting the end member

model, a smaller number of end members is

preferred. Adding more end members to the

model will always reduce the residual error, but

often we are just trying to fit information to the

noise.



Because we may not have “pure” end members

(i.e., true extremes), it is often physically

reasonable to have fractions slightly less than zero

or greater than one. For this reason, the partially

constrained model is often most appropriate.



In cases where shade fractions are needed, but not

of interest, the shade fraction can be redistributed

to the other fractions. i.e., Each fraction is

increased according to

1Nf

ff shadeii



• Image.

• Fractions maps.

• End member selection and model generation.


Steps in Practical Application of Steps in Practical Application of Mixture AnalysisMixture Analysis

(1) Select scene derived end members subject to the following criteria:

Use the fewest end members that reduce the image wide residual below some user-defined threshold (4 counts corresponding to ±2% reflectance was used in this study).


Steps in Practical Application of Steps in Practical Application of Mixture Analysis (cont’d)Mixture Analysis (cont’d)

Residual defined as

EEkp

( fkjrij ) DNkij1

N

i1

M

/Mk

p

/ p



where Ek is the per pixel residual averaged over the

spectral bands, fkj is the fraction of the jth end

member in the kth pixel, rij is the end member digital

count of the jth end member in the ith band, DNki is the

kth pixels digital count in band i, M is the number of

bands, N is the number of end members, and p is the

number of pixels.



In practice, the user smartly picks image derived

end members, including shade, in order of

decreasing spectral contrast. (i.e., High contrast

end members with respect to shade added first.)

Adding end members until E is below the

threshold value.



End members can be picked by taking pixels and

sequentially testing to see if they are linear

combinations of each other (thresholded to some

criteria). As each new pixel is tested, it is either a

combination or a new end member. New end

members can cause previous ones to be replaced.

Usually this is not done image wide, but rather the

user introduces candidate end members. Brightly lit

(low shade) samples should be selected.



End members may also be simply defined by

context if the user is familiar with the site. Note

image derived end members may be (and usually

are) mixtures themselves. The shade end member

accounts for illumination effects (cos), shape

factor (F), shadowing, and any other source of

mean level brightness change.



The shade end member can be selected from

pixels in shadow if resolvable. Sometimes a dark

object can be selected as an estimate, or

atmosphere and sensor calibration can be used

to estimate the signal expected from a shadow (

0% reflective).



Next the image end members can be modeled as mixtures of laboratory spectra and the calibration coefficients determined.

We can test end members by (1) image wide residual error (Equation 4 or an RMS equivalent) testing both magnitude and spatial pattern, by (2) fraction continuity - context (i.e., do the fraction maps make intuitive sense?). Included in this is a mapping of [1-

Fshade] scaled as brightness which should mimic to

topography.



See next page for Figure 7.1 description

Combinedrms-errorimage

Shade fractionimage

Vegetation fractionimage

Substrate fractionimage



(3) If fraction maps are scaled for display as

DC=(fi+1)·100, then the normal range will be 100-200 and

out of range values (negative and greater than one) can

be identified. Note these values can regularly occur in

the partially constrained case and either indicate the lack

of purity of an end member or failure to include an end

member. These out of range values can be color coded

to flag out of range values in the fraction maps giving us

a trend criteria for the quality of the end member

selection. See Figure 7.2.



Fig. 7.2. Cartoon illustrating two-dimensional spectral data space.

Visible and near-infrared reflectance spectra of vegetation, soil, and shade sampled by two wavelength channels, 1 and 2.

Vegetation, soil, and shade plot at corners of triangle in two-channel plot (open circles). Image spectra (Xs) that plot within the triangle can be described as realistic mixtures of the end members.

Image spectra (Xs) all cluster near the center of the triangle. Image end members selected from this set are themselves mixtures of the spectra of vegetation, soil and shade

Cartoon illustrating that image spectra measured in DN’s (light Xs outside triangle) must be corrected by a gain and an offset for each channel to plot the equivalent position in reflectance (Xs inside triangle).


Linked ModelsLinked Models

Rather than try to unmix a complex image simultaneous with many possible end members, it is often desirable to unmix on a smaller number of end members in a localized region and then link the solutions together. We can use masks to avoid unmixing the same area in two ways.


Residual ErrorResidual Error

• Error vectors as a means to analyze fraction maps

• The error vectors for each pixel is the vector

comprised of the difference between the image

radiance (reflectance) vector and the vector

predicted by the fraction model.

FELe


Residual Error (cont’d)Residual Error (cont’d)

• We can can make a map of the magnitude of the

errors as an indication of locations where the

model is inadequate

• We can also look at the spectral shape of the error

and compare it to spectral features of interest.



Fig. 9. Effect of transmission and scattering on residuals.



• If we have a target material(s) representing a small

number of pixels, it may be more effective to leave

it out of the end member analysis and just look

closely at the error vectors.

• We may map an error spectrally localized to

characteristic features in the target material.



• What is an end member and what use are end

member maps.

–Fraction maps

–Associations as end members

–Combining fractions to form associations

–Class maps with transition classes


Truth Fraction MapsTruth Fraction Maps

Lab

els

Fra

ctio

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Lin

ear

Tru

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Linear Unmixing

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Digital Imaging and Remote Sensing Laboratory Mixed Pixels and Spectral Unmixing 1 Spectral Mixtures areal aggregate intimate Linear mixing models (areal