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1
Towards streaming hyperspectral endmember extraction
Dževdet Burazerović, Rob Heylen, Paul ScheundersIBBT-Visionlab, University of Antwerp, Belgium
IGARSS 2011July 24-29, Vancouver, Canada
2
Outline
• Prior art and motivation
- LMM, N-findr
• The proposed algorithm
- Distance-based simplex formulation- Streaming endmember estimation
• Experiments and results
• Conclusions
3
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
4
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
3
1 2
5
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
(*)
6
The proposed algorithm
• Extract EMs in 1-pass, streaming (online) fashion
1. Reformulate the simplex-vol. measurement to avoid dim. red.
2. Grow a suitable initial simplex for a given # of EMs
3. Maximize this simplex by subsequent replacements (N-findr)
normally, n > pimageep
7
Distance-based simplex formulation
• Via Cayley-Menger determinant, Schur complement
V3 e1
e2
e3
e4
8
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
9
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
10
Cluster validation
i = 9(13 spectra)i = 7
Results with N-findr, on Cuprite
11
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)
12
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