SIMPLEX VOLUME ANALYSIS BASED ON TRIANGULAR FACTORIZATION: A FRAMEWORK FOR HYPERSPECTRAL UNMIXING
1. Simplex Volume Analysis
Based On Triangular
Factorization: A framework for
hyperspectral Unmixing
Wei Xia, Bin Wang, Liming Zhang, and Qiyong Lu
Dept. of Electronic Engineering
Fudan University, China
2. Contents
1. Introduction
2. The Proposed Method
2.1 Endmember extraction
2.2 Abundance Estimation
3. Evaluation with Experiments
3.1 Synthetic data
3.2 Real hyperspectral data
4. Conclusion
2
3. Contents
1. Introduction
2. The Proposed Method
2.1 Endmember extraction
2.2 Abundance Estimation
3. Evaluation with Experiments
3.1 Synthetic data
3.2 Real hyperspectral data
4. Conclusion
3
4. Linear Mixture Model (LMM)
Abundance
fractions
x ∈ R L×1 ,
The
observation
of a pixel x =As +e A ∈ R L× P ,
s ∈ R P× N
endmember
spectra
si ≥ 0, (i = 1, 2,..., P ).
P
∑s
i =1
i =1
4
6. Simplex Volume Analysis (1/2)
e3
e2
e1 e0
A Simplex of P-vertices is defined by
⎧ P
⎫
⎨x = s1e 0 + s1e1 + .... + sP −1e P −1 | si > 0,
⎩
∑ si = 1⎭
i =1
⎬
6
7. Simplex Volume Analysis (2/2)
Related work *
• The observation pixels forms a simplex whose vertices
correspond to the endmembers
• Find the vertices by searching for the pixels which can form the
largest volume of the simplex
Volume formula
1 ⎛ ⎡1T ⎤ ⎞
V= det ⎜ ⎢ ⎥ ⎟
⎜ E ⎟
E = [e 0 , e1 ,..., e P −1 ]
( P − 1)! ⎝ ⎣ ⎦⎠
Disadvantages
• large computing cost caused by calculating volume, hard to be used
for Real-time application
• Require dimensionality reduction (DR), loss of possible information
* M. E. Winter, “N-findr: an algorithm for fast autonomous spectral endmember determination in hyperspectral data,” in: Proc. of the
SPIE conference on Imaging Spectrometry V, vol. 3753, pp. 266–275, 1999.
* C.-I Chang, C-C Wu, W. Liu, and Y-C Ouyang, “A new growing method for simplex-based endmember extraction algorithm,” IEEE
Trans. Geosci. Remote Sens., vol. 44, no. 10, pp. 2804-2819, 2006.
* M. Zortea and A. Plaza, “A quantitative and comparative analysis of different implementations of N-FINDR: A fast endmember
extraction algorithm,” IEEE Geosci. Remote Sens. Lett., vol. 6, no. 4, pp. 787–791,Oct. 2009. 7
8. Contents
1. Introduction
2. The Proposed Method
2.1 Endmember extraction
2.2 Abundance Estimation
3. Evaluation with Experiments
3.1 Synthetic data
3.2 Real hyperspectral data
4. Conclusion
8
10. Proposed Endmember Extraction Framework
SVATF (Simplex Volume Analysis based on Triangular Factorization)
Z = AT A A
Z A
A = [e0 , e1 , e 2 ,..., e P−1 ] A = [e1 − e0 , e 2 − e0 ,..., e P−1 − e0 ]
* X. Tao, B. Wang, and L. Zhang, “Orthogonal Bases Approach for Decomposition of Mixed Pixels for Hyperspectral Imagery,” IEEE
Geosci. Remote Sens. Lett., vol. 6, no. 2, pp. 219–223, Apr. 2009. 10
11. Develop by Cholesky Factorization(1/5)
•Simplex Volume
1
1
V= det( Z ) 2
( P − 1)!
⎡ || α1 ||2 α1 ⋅ α2 ... α1 ⋅ αP−1 ⎤
⎢ ⎥
α2 ⋅ α1 || α2 ||2 ... α2 ⋅ αP−1 ⎥
Z = ATA = ⎢ , (where αi = ei − e0 )
⎢ ⎥
⎢ ⎥
⎣αP−1 ⋅ α1 αP−1 ⋅ α2 ... || αP−1 ||2 ⎦
• Z is a positive definite symmetric matrix, which can be
decomposed by Cholesky Factorization
Z = LLT
11
12. Develop by Cholesky Factorization(2/5)
•Update the Simplex Volume
1
1
V= det( Z ) 2
( P − 1)!
1
det ( LLT )
1 1
= 2
= l11 ⋅ l22 ⋅ ⋅ l( P −1)( P −1)
( P − 1)! ( P − 1)!
•Calculating the simplex volume
Perform the Cholesky factorization
•Maximize the volume V
maximizing diagonal element li , i , (i = 1, 2, ..., P − 1)
12
13. How does SVATF run ?
1/2
⎛ i −1
⎞
li , i = ⎜ zi , i − ∑ li2, k ⎟ ,
⎝ k =1 ⎠
Z = LLT ⎛ j −1
⎞
li , j = ⎜ zi , j − ∑ li , k l j , k ⎟ / l j , j , for i > j.
⎝ k =1 ⎠
• Find the endmember, i.e., search for the pixel which can maximize li , i ,
•1 search for e1 l1, 1 = z1, 1 …… i=1
•2 search for e2 l2, 2 = z2, 2 − l2, 1
2 …… i=2
•3 search for e3 l3, 3 = z3, 3 − l3, 1 − l3, 2
2 2
…… i=3
Easy to realize:
Calculate Cholesky Factorization for N times
to find all the endmembers (N is the number of pixels). 13
14. The benefit of using Cholesky Factorization
• Simplify the searching process
The number of calculated The matrix in calculated
Algorithms
Determinants* Determinant*
N-FINDR
NP P × P size matrix
(after DR)
SGA
Nn ( n starting from 2 to P ) n × n size matrix
(after DR)
SVATF
(With/Without N (P-1) × (P-1) size matrix
DR)
• SVATF calculate the determinants on a smaller matrix using fewer
number
SVATF can perform faster with/without DR
* C.-I Chang, C-C Wu, W. Liu, and Y-C Ouyang, “A new growing method for simplex-based endmember extraction algorithm,” IEEE
Trans. Geosci. Remote Sens., vol. 44, no. 10, pp. 2804-2819, 2006.
14
15. Develop by Cholesky Factorization(5/5)
Given the observation matrix X = [x1 , x 2 ,..., x N ]∈ R L× N, P:endmember number
e 0 = arg max(|| x n ||), ( n = 1, 2,..., N )
xn
e1 = arg max(|| x n ||), (x n = x n − e 0 )
xn
γ n = xn ,
1
id (1) = arg max(γ n )
1
n
t −1
η = ( x n ⋅ α t − ∑ η nkη id ( t ) ) / γ id ( t ) , γ n+1 =
t
n
k
k =1
t t
(γ ) − (η )
t 2
n
t 2
n , αt = et − e0
e t +1 = arg max(γ n+1 ), id (t + 1) = arg max(γ nt +1 )
t
xn n
A = [ e 0 , e 1 , ..., e P − 1 ]
15
16. Computational complexity among N-FINDR, VCA, SGA, OBA, and SVATF
Algorithms Numbers of flops
N-FINDR Pη +1 N
P
SGA (∑ kη ) N
k =2
VCA 2P 2 N …… after dimensionality reduction
2PLN ……without dimensionality reduction
N (3P 2 − 4 P + 1) + P − 1 …… after dimensionality reduction
OBA*
N ( P − 1 + 3PL − 2 L) + L ……without dimensionality reduction
0.5 N (3P 2 − P − 4) …… after dimensionality reduction
SVATF 0.5 N ( P + 2 PL + P − 4)
2
…… without dimensionality reduction
* X. Tao, B. Wang, and L. Zhang, “Orthogonal Bases Approach for Decomposition of Mixed Pixels for Hyperspectral Imagery,” IEEE
Geosci. Remote Sens. Lett., vol. 6, no. 2, pp. 219–223, Apr. 2009.
16
17. The numbers of flops in various endmembers
The flops of dimensionality reduction: > 2NL2
Parameters
L P N
100 3, 4,…, 50 1000
9
10
The number of floating operations
8
10
7
10
VCA
6 SGA
10
NFINDR
OBA
5 SVATF
10
0 10 20 30 40 50
The number of endmembers
17
18. The numbers of flops in various pixels
Parameters
L P N
100 10 100, 1000, …, 1e+8
14
10
The number of floating operations
12
10
10
10
8
10 VCA
SGA
6 NFINDR
10 OBA
SVATF
4
10
2 3 4 5 6 7 8
10 10 10 10 10 10 10
The number of pixels
18
19. The numbers of flops in various bands
Parameters
L P N
100, 200, …, 800 10 1000
10
10
VCA
The number of floating operations
SGA
9
10 NFINDR
OBA
SVATF
8
10
7
10
6
10
100 200 300 400 500 600 700 800
The number of bands
19
20. Contents
1. Introduction
2. The Proposed Method
2.1 Endmember extraction
2.2 Abundance Estimation
3. Evaluation with Experiments
3.1 Synthetic data
3.2 Real hyperspectral data
4. Conclusion
20
21. Abundance Quantification based on TF
•Known the endmembers, the abundances can be given as
• X=AS S = inv( A)X
•Transform into x = QRs Q Tx = Rs
⎡ x1 ⎤ ⎡ r11 r12 r1P ⎤ ⎡ s1 ⎤
⎢x ⎥ ⎢ r22 r2 P ⎥ ⎢ s2 ⎥
QT ⎢ 2 ⎥ = ⎢ ⎥⎢ ⎥
⎢ ⎥ ⎢ ⎥⎢ ⎥
⎢ ⎥ ⎢ ⎥⎢ ⎥
⎣ xL ⎦ ⎣ rPP ⎦ ⎣ sP ⎦
• Estimate the abundance s i , ( i = 1 , 2 , . . . , P )
by solving linear simultaneous equation
21
22. • Resulting formula
⎧ bP
⎪ sP = r
⎪ pp
⎪ b ⎡b1 ⎤ ⎡ x1 ⎤
⎪ sP −1 = P −1 ⎢b ⎥ ⎢x ⎥
⎪ rP −1, P −1 ⎢ 2 ⎥ = QT ⎢ 2 ⎥
⎨ , where
⎪ ⎢ ⎥ ⎢ ⎥
⎪ ⎢ ⎥ ⎢ ⎥
⎣ bP ⎦ ⎣ xL ⎦
⎪ b1 − ∑ i = 2 r1i si
P
⎪ s1 =
⎪
⎩ r11
22
23. • Similarly，obtain A when S is known
X = AS X T = ST A T
A = QR S T =Q SR S
Q Tx = Rs Q S X T = R SA T
T
S = inv(R )Q X A = ( inv(RS ) Q X )
T T
T T
S
23
24. A = λ ( inv ( R S ) QS X1 ) + (1 − λ ) A
T T T
24
25. Contents
1. Introduction
2. The Proposed Method
2.1 Endmember extraction
2.2 Abundance Estimation
3. Evaluation with Experiments
3.1 Synthetic data
3.2 Real hyperspectral data
4. Conclusion
25
26. Experiments
Algorithms
• N-FINDR (M. E. Winter 1999) ***
• SGA (Chang, Wu, Liu, & Ouyang, 2006) **
• VCA (Nascimento & Bioucas-Dias, 2005) *
* J. Nascimento and J. Bioucas-Dias, “Vertex Component Analysis: A Fast Algorithm to Unmix Hyperspectral Data,” IEEE Trans.
Geosci. Remote Sens., vol. 43, no. 4, pp. 898-910, Apr. 2005.
* * C.-I Chang, C-C Wu, W. Liu, and Y-C Ouyang, “A new growing method for simplex-based endmember extraction algorithm,” IEEE
Trans. Geosci. Remote Sens., vol. 44, no. 10, pp. 2804-2819, 2006.
* * * M. E. Winter, “N-findr: an algorithm for fast autonomous spectral endmember determination in hyperspectral data,” in: Proc. of the
SPIE conference on Imaging Spectrometry V, vol. 3753, pp. 266-275, 1999
26
27. Criteria
aiT ai
ˆ
• SAD SADi = arccos
ai ⋅ aiˆ
ai ∈RL×1 The spectral of the ith endmember
ai ∈RL×P
ˆ The estimated spectral of the ith endmember
N
• RMSE RMSEi = ∑ ( yij − sij ) 2 / N
j =1
sij The abundance of the ith endmember according
to the jth pixel
yij The estimated abundance of the ith endmember
according to the jth pixel
27
28. Synthetic Data (1/5)
x
1
a
0.9 b
c
0.8 d
e
0.7
|| reflectance
0.6
0.5
0.4
A 0.3
0.2
0.1
×
0
0 50 100 150 200
band
Endmember spectra from USGS. a. Andradite_WS488, b. Kaolinite_CM9, c.
Montmorillonite_CM20, d. Desert_Varnish_GDS141, and e. Muscovite_GDS116.
s The abundances fractions are subject to Dirichlet distribution.
28
29. Synthetic Data (2/5)
Results of the algorithms with Different image sizes
CPU memory OS Software
Intel(R) Xeon CPU
48 GBytes 64-bit Window7 Matlab 2010
X5667 3.07GHZ
Parameters
L P N
224 5 100×100, 200×200, …, 600×600
2
10
1
10
0
SVATF without DR
Time
10
Other Methods: use DR
VCA
-1 N-FINDR
10
SGA
SVATF
-2
10
100×100 200×200 300×300 400×400 500×500 600×600
Image Size
29
37. Computing time for the Cuprite dataset
NFINDR-FCLS SGA-FCLS VCA-FCLS SVATF-AQTF
Algorithms
NFINDR FCLS SGA FCLS VCA FCLS SVATF AQTF
Time
28.27780 16.63869 3.13832 16.38314 0.85985 16.97822 0.24133 0.22719
(seconds)
The computer environment
CPU memory OS Software
Intel(R) Xeon CPU
48 GBytes 64-bit Window7 Matlab 2010
X5667 3.07GHZ
37
38. Contents
1. Introduction
2. The Proposed Method
2.1 Endmember extraction
2.2 Abundance Estimation
3. Evaluation with Experiments
3.1 Synthetic data
3.2 Real hyperspectral data
4. Conclusion
38
39. Conclusion
• Proposed a new method based on triangular factorization for
the simplex analysis of hyperspectral unmixing.
• A framework including a group of algorithms.
• Dimensionality reduction (DR) is optional .
• Efficiency and accuracy. Both the theoretical analysis and
experimental results show that the proposed methods can
perform faster than the state-of-the-art methods, with precise
results.
Should be very useful for Real-time application.
• Steady. always outputs the same results in the same
sequence when being applied to a certain dataset.
39
