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Towards streaming hyperspectral endmember extraction

Dževdet Burazerović, Rob Heylen, Paul ScheundersIBBT-Visionlab, University of Antwerp, Belgium

IGARSS 2011July 24-29, Vancouver, Canada

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Outline

• Prior art and motivation- LMM, N-findr

• The proposed algorithm- Distance-based simplex formulation- Streaming endmember estimation

• Experiments and results• Conclusions

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Linear mixture model

• An observed spectrum x is a (constrained) linear sum of p endmember (EM) spectra ei :

• Then, EMs = vertices of the largest (p-1)-dim. simplex enclosing (most of) the x :

e1

e3

e2

e4

x

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N-findr

• Estimates the largest simplex via repetitive vertex replacements─ “single replacement” (SR) vs. “best replacement” (BR)─ “single iteration” (SI) vs. “full iteration” (FI)

Random initial No replacement Replacement

1 2

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1 2

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Motivation

•Finding the largest simplex is not sufficient/necessary (in real data, un-supervised scenarios)

•Worthwhile to seek efficient implementations

(*) S. Dowler, M. Andrews: “On the convergence of N-findr …”, IEEE GRS Letters, 2011

(*)

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The proposed algorithm

• Extract EMs in 1-pass, streaming (online) fashion1. Reformulate the simplex-vol. measurement to avoid dim. red.2. Grow a suitable initial simplex for a given # of EMs3. Maximize this simplex by subsequent replacements (N-findr)

normally, n > pimageep

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Distance-based simplex formulation

• Via Cayley-Menger determinant, Schur complement

V3 e1 e2

e3e4

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Growing the initial simplex

0

1

V3 h h ~ V4/V3

• Use empirical CDFs to set thresh. for the simplex-vol. increment• E.g., add xk as p-th EM, if FP(Vk/VP-1)≥0.5

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Comparison setup

• Acknowledge the variability of both algorithms─ Streaming: threshold function for growing the initial simplex─ N-findr: random selection of the initial simplex (EMs)

• Compare results (EMs) from multiple runs• Use cluster validation to determine consistent EMs

M – EMs K – runsM x K – data points

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Cluster validation

i = 9(13 spectra)i = 7

Results with N-findr, on Cuprite

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Comparison results

• Ground truth: P EM-cluster centroids from ~40 runs of N-findr• Test data: P EMs from a single streaming pass• Classification: N-Neighbor + visual comparison of the spectra• Accuracy: 13/18 (72.2%) on Cuprite, 4/7 (71.4%) on M.F.

Cuprite, P=18 (350 x 350 x 188)

Moffet Field, P=7 (335 x 370 x 56)

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Conclusions

• The use of dist.-based simplex formulation enables a new paradigm of EM-extraction:─ A streaming (online) implementation based on N-findr─ Avoiding the need to pre-load the entire image into memory

• Tested on diverse data, finds most of the EMs that are found by repetition of the reference methods (N-findr)

• Possible extension to other strategies for streaming-based simplex estimation and measurement