EOS 740 Hyperspectral Imaging Systems

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EOS 740 Hyperspectral Imaging Systems
March 11, 2005 Week 7
Ron Resmini v 703-735-3899 ronald.g.resmini_at_boein
g.com Office hours by appointment
Put EOS740 in the subject line of e-mails to
me...Thanks!
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Outline

• How are your projects going?
• Review of ELM
• Other techniques
• Principal Components Analysis (PCA)
• Some more ENVI Functionality
• Spectral Mixture Analysis (SMA)
• The mixed pixel
• Matched Filters Introduction

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Principal Components Analysis (PCA)

• Reading from two weeks ago...
• What is PCA?
• Why apply it?
• Applying PCA in ENVI

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The Mixed Pixel Spectral Mixture Analysis
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• The mixed pixel
• a basis concept a very important
  conceptdescription of a mixed pixel
• determining quantity of material
• building a mixed pixel - spectral math
• building a mixed pixel - MS Excel
• mixing trends in hyperspace

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• Linear spectral mixture analysis
• applications
• scene characterization
• material mapping
• anomaly detection
• other...

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Linear Spectral Unmixing
The reflectance of an image pixel is a linear
combination of reflectances from (typically)
several pure substances (or endmembers)
contained within the ground-spot sampled by
the remote sensing system
where Ri is the reflectance of a pixel in band
i, fj is the fractional abundance of endmember j
in the pixel, Mj,i is the reflectance of
endmember substance j in band i, ri is the
unmodeled reflectance for the pixel in band i,
and n is the number of endmembers.
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Linear Spectral Unmixing Theory
Spectral unmixing theory states that the
reflectance of an image pixel is a linear
combination of reflectances from the (typically)
several pure substances (or endmembers)
contained within the ground-spot sampled by the
remote sensing system. This is indicated below
where Ri is the reflectance of a pixel in band
i, fj is the fractional abundance of substance
(or endmember) j in the pixel, and Mj,i is the
reflectance of endmember substance j in band i.
ri is the band-residual or unmodeled reflectance
for the pixel in band i, and n is the number of
endmembers. A spectral unmixing analysis results
in n fraction-plane images showing the
quantitative areal distribution of each of the
endmember substances and one root mean squared
(RMS) image showing an overall or global goodness
of fit of the suite of endmembers for each pixel.
The RMS image is formed, on a pixel-by-pixel
basis, by
Objects may also be detected as anomalies in the
RMS image.
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Spectral Mixture Analysis (SMA)

• An area of ground of, say 1.5 m by 1.5 m may
  contain 3 materials A, B. and C.
• An HSI sensor with a GSD of 1.5 m would measure
  the Mixture spectrum
• SMA is an inversion technique to determine the
  quantities of A, B, and Cin the Mixture
  spectrum
• SMA is physically-based on the spectral
  interaction of photons of light and matter
• SMA is in widespread use today in all sectors
  utilizing spectral remote sensing
• Variations include different constraints on the
  inversion linear SMA nonlinear SMA

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• Endmember selection
• manual
• convex hull
• Pixel-Purity Index (PPI)
• adaptive/updating/pruning...
• other (e.g., N-FINDR, ORASIS,URSSA, etc...)
• wild outliers

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• Unmixing inversion
• interpretation of results
• RMS, RSS, algebraic and geometricinterpretations
• are RMS, RSS, etc. the best measures?
• residual correlation analysis (RCA)
• band residual cube
• can we build a band residual cubewith ENVI?

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• iterative process
• the inversion in Matlab a closer lookat the
  math
• fraction-plane color-composites
• change detection with fraction planes
• constraints on inversion
• is it a linear mixture?

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• application strategies (i.e., in-scenespectra/lib
  rary spectra)
• directed search? anomaly detection?
• Shade/shadow
• shade endmember
• shade removal
• Other...
• objective endmember determinationTompkins et al.

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• Non-linear spectral mixture analysis
• checker-board mixtures
• intimate mixtures
• building a non-linear mixture
• spectral transformations (e.g., SSA)and use of
  ENVI

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A linear equation...
x
A
b
5 endmembers in a 7-band spectral data set
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MS Excel and Matlab Commands
s1, s2, and s3 are column vectors Ms1, s2,
s3 x0.25 0.35 0.40T b0.25s10.35s20.40s3
Copy/paste s1, s2, s3, b into Matlab As1 s2
s3 xinv(AA)Ab
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Prelude to Other Algorithms Statistical Spectral
Matched Filters OSP OBS

• Orthogonal Subspace Projection (OSP)
• Derivation in detail (next several slides...)
• Application of the filter
• Endmembers
• Statistics
• Interpretation of results
• OSP w/endmembers unconstrained SMA
• Different ways to apply the filter/application
  strategies(i.e., in-scene spectra/library
  spectra)

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OSP/LPD/DSR Scene-Derived Endmembers
(Harsanyi et al., 1994 see also ch. 3 of Chang,
2003)
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The value of xT which maximizes l is given by xT
dT
This is equivalent to Unconstrained SMA
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MS Excel and Matlab Commands
s1, s2, and d are column vectors Ms1, s2,
d x0.25 0.35 0.40T r0.25s10.35s20.40d Co
py/paste s1, s2, d, r into Matlab Us1
s2 Ieye(348) P(I-Upinv(U)) qdP this
is actually qT qr qd (qr)/qd)
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Statistical Characterization of the
Background (LPD/DSR)
(Harsanyi et al., 1994)
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• The statistical spectral matched filter (SSMF)
• Derivation in detail
• Application of the filter
• Statistics
• Endmembers (FBA/MCEM)
• Interpretation of results
• Many algorithms are actually the basic SSMF
• Different ways to apply the filter/application
  strategies(i.e., in-scene spectra/library
  spectra)
• Matched filter in ENVI

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Constrained Energy Minimization (CEM)

• The description of CEM is similar to that of
  OSP/DSR (previous slides)
• Like OSP and DSR, CEM is an Orthogonal Subspace
  Projection (OSP)family algorithm
• CEM differs from OSP/DSR in the following,
  important ways
• CEM does not simply project away the first n
  eigenvectors
• The CEM operator is built using a weighted
  combination of theeigenvectors (all or a subset)
• Though an OSP algorithm, the structure of CEM is
  equally readily observed bya formal derivation
  using a Lagrange multiplier

• CEM is a commonly used statistical spectral
  matched filter
• CEM for spectral remote sensing has been
  published on for over 10 years
• CEM has a much longer history in the
  multi-dimensional/array signalprocessing
  literature
• Just about all HSI tools today contain CEM or a
  variant of CEM
• If an algorithm is using M-1d as the heart of its
  filter kernel (where M is thedata covariance
  matrix and d is the spectrum of the target of
  interest), thenthat algorithm is simply a CEM
  variant

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Derivation taken from
Stocker, A.D., Reed, I.S., and Yu, X., (1990).
Multi-dimensional signal processing for
electro-Optical target detection. In Signal
and Data Processing of Small Targets 1990,
Proceedingsof the SPIE, v. 1305, pp. 218-231.
J of Bands
Form the log-likelihood ratio test of Hº and H1
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Some algebra...
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A trick...recast as a univariable problem
After lots of simple algebra applied to the r.h.s
Now, go back to matrix-vector notation
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Take the natural log
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The vector QTx is a projection of the original
spectraldata onto the eigenvectors of the
covariance matrix, M, which corresponds to the
principal axes of clutterdistribution. Stocker
et al., 1990.
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Further SCR gain is obtained by forming the
optimumweighted combination of principal
components usingthe weight vector
From Stocker et al., 1990.
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Constrained Energy Minimization (CEM)
(Harsanyi et al., 1994)
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An Endnote...

• Previous techniques exploit shape and albedo
• this can cause problems...
• Sub-classes of algorithms developed to mitigate
  this
• shape, only, operators
• MED, RSD of ASIT, Inc.
• MTMF of ENVI (ITT/RSI)

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Another Endnote...

• Performance prediction/scoring/NP-Theory, etc...
• Hybrid techniques
• still some cream to be skimmed...
• Caveat emptor...
• lots of reproduction of work already accomplished
• who invented what? when?
• waste of resources
• please do your homework!read the lit.!

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Backup Slides
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Lagrange Multiplier Derivation of CEM Filter
Minimizing E is equivalent to minimizing each yi2
(for k 1, 2, 3, ... )
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In Matrix Notation
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