simplex volume analysis based on triangular factorization: a framework for hyperspectral unmixing
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Simplex Volume Analysis Based On Triangular Factorization: A framework for hyperspectral Unmixing
Wei Xia, Bin Wang, Liming Zhang, and Qiyong LuDept. of Electronic EngineeringFudan University, China
Contents
2
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
Contents
3
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
Linear Mixture Model (LMM)
4
= +x A s e
0, ( 1, 2,..., ).is i P≥ =
1
1P
ii
s=
=∑
1R ,L×∈x
R ,L P×∈A
RP N×∈s
The observation
of a pixel
Abundance fractions
endmember spectra
Different methods under the LMM
5
Simplex Volume Analysis (1/2)
6
1 0 1 1 1 11
.... | 0, 1P
P P i ii
s s s s s− −=
⎧ ⎫= + + + > =⎨ ⎬
⎩ ⎭∑x e e e
A Simplex of P-vertices is defined by
3e
0e1e
2e
• The observation pixels forms a simplex whose vertices correspond to the endmembers
• Find the vertices by searching for the pixels which can form thelargest volume of the simplex
7
Simplex Volume Analysis (2/2)
* 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.
Related work *
• large computing cost caused by calculating volume, hard to be used for Real-time application
• Require dimensionality reduction (DR), loss of possible information
T1 det( 1)!
VP
⎛ ⎞⎡ ⎤= ⎜ ⎟⎢ ⎥⎜ ⎟− ⎣ ⎦⎝ ⎠
1E
Disadvantages
0 1 1[ , ,..., ]P−=E e e eVolume formula
Contents
8
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
The Proposed Method base on Triangular Factorization (TF)
9
x
A
s
Proposed Endmember Extraction Framework
Z A
0 1 2 1[ , , ,..., ]P−=A e e e e
10
A
* 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.
T=Z A A
SVATF (Simplex Volume Analysis based on Triangular Factorization)
1 0 2 0 1 0[ , ,..., ]P−= − − −A e e e e e e
Develop by Cholesky Factorization(1/5)
11
21 1 2 1 1
2T 2 1 2 2 1
0
21 1 1 2 1
|| || ...|| || ...
, ( )
... || ||
P
Pi i
P P P
where
−
−
− − −
⎡ ⎤⋅ ⋅⎢ ⎥⋅ ⋅⎢ ⎥= = = −⎢ ⎥⎢ ⎥
⋅ ⋅⎣ ⎦
α α α α αα α α α α
Z A A α e e
α α α α α
•Simplex Volume
• Z is a positive definite symmetric matrix, which can be decomposed by Cholesky Factorization
TZ = LL
12
1 det( )( 1)!
VP
=−
Z
Develop by Cholesky Factorization(2/5)
12
•Update the Simplex Volume
•Calculating the simplex volume
( )
12
1T 2
11 22 ( 1)( 1)
1 det( )( 1)!
1 1det( 1)! ( 1)! P P
VP
l l lP P − −
=−
= = ⋅ ⋅ ⋅− −
Z
LL
, ,i il ( 1,2, ..., 1)i P= −maximizing diagonal element
•Maximize the volume
Perform the Cholesky factorization
V
How does SVATF run ?
13
2 23, 3 3, 3 3,1 3, 2l z l l= − −
22, 2 2, 2 2,1l z l= −
1,1 1,1 l z=
• Find the endmember, i.e., search for the pixel which can maximize
…… i=1
…… i=2
…… i=3
1/212
, , ,1
,i
i i i i i kk
l z l−
=
⎛ ⎞= −⎜ ⎟⎝ ⎠
∑1
, , , , ,1
/ , .j
i j i j i k j k j jk
l z l l l for i j−
=
⎛ ⎞= − >⎜ ⎟⎝ ⎠
∑
, ,i il
•1 search for 1e
•2 search for
•3 search for
2e
3e
Easy to realize:Calculate Cholesky Factorization for N timesto find all the endmembers (N is the number of pixels).
TZ = LL
The benefit of using Cholesky Factorization
14
Algorithms The number of calculated Determinants*
The matrix in calculated Determinant*
N-FINDR (after DR) NP P × P size matrix
SGA (after DR) Nn ( n starting from 2 to P ) n × n size matrix
SVATF(With/Without
DR)N (P-1) × (P-1) size matrix
* 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.
• Simplify the searching process
• SVATF calculate the determinants on a smaller matrix using fewer number
SVATF can perform faster with/without DR
Develop by Cholesky Factorization(5/5)
15
1 2[ , ,..., ]N=X x x x L NR ×∈Given the observation matrix
0 1 1[ , , ..., ]P −=A e e e
1 0arg max(|| ||), ( )n
n n n= = −x
e x x x e
1(1) arg max( )nn
id γ=
( ) ( )2 21 ,t t tn n nγ γ η+ = − 0t t= −α e e
11 arg max( ),
n
tt nγ
++ =
xe 1( 1) arg max( )t
nn
id t γ ++ =
1
( ) ( )1
( ) / ,t
t k k tn n t n id t id t
k
η η η γ−
=
= ⋅ −∑x α
0 arg max(|| ||), ( 1,2,..., )n
n n N= =x
e x
1 , n nγ = x
, P:endmember number
Computational complexity among N-FINDR, VCA, SGA, OBA, and SVATF
16
Algorithms Numbers of flops
N-FINDR
SGA
VCA …… after dimensionality reduction……without dimensionality reduction
OBA* …… after dimensionality reduction……without dimensionality reduction
SVATF …… after dimensionality reduction…… without dimensionality reduction
1NPη+
2
( )P
k
k Nη
=∑
22P N
20.5 (3 4)N P P− −
* 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.
20.5 ( 2 4)N P PL P+ + −
( 1 3 2 )N P PL L L− + − +
2(3 4 1) 1N P P P− + + −
2PLN
The numbers of flops in various endmembers
17
The flops of dimensionality reduction: 22NL>
0 10 20 30 40 5010
5
106
107
108
109
The number of endmembers
The
num
ber
of f
loat
ing
oper
atio
ns
VCASGANFINDROBASVATF
ParametersL P N
100 3, 4,…, 50 1000
The numbers of flops in various pixels
18
102
103
104
105
106
107
108
104
106
108
1010
1012
1014
The number of pixels
The
num
ber
of f
loat
ing
oper
atio
ns
VCASGANFINDROBASVATF
ParametersL P N
100 10 100, 1000, …, 1e+8
The numbers of flops in various bands
19
100 200 300 400 500 600 700 80010
6
107
108
109
1010
The number of bands
The
num
ber
of f
loat
ing
oper
atio
ns
VCASGANFINDROBASVATF
ParametersL P N
100, 200, …, 800 10 1000
Contents
20
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
Abundance Quantification based on TF
•Known the endmembers, the abundances can be given as • X=AS
•Transform into
• Estimate the abundance by solving linear simultaneous equation
21
T =Q x R s
1 111 12 1
2 22 2 2T
P
P
PPL P
x sr r rx r r s
rx s
⎡ ⎤ ⎡ ⎤⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥=⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥
⎣ ⎦⎣ ⎦ ⎣ ⎦
Q
, ( 1 , 2 , . . . , )is i P=
=x QRs
( )inv=S A X
• Resulting formula
22
1 11
1 2 2T1, 1
1 121
11
,
PP
pp
PP
P P
P P Li ii
bsr
b xbs b xr where
b xb r s
sr
−−
− −
=
⎧ =⎪⎪ ⎡ ⎤ ⎡ ⎤⎪ ⎢ ⎥ ⎢ ⎥=⎪⎪ ⎢ ⎥ ⎢ ⎥=⎨ ⎢ ⎥ ⎢ ⎥⎪ ⎢ ⎥ ⎢ ⎥⎪ ⎣ ⎦ ⎣ ⎦⎪ −
=⎪⎪⎩
∑
Q
• Similarly,obtain when is known
23
=X AS T T T=X S A
T =Q x R s T T TS S =Q X R A
T( )inv=S R Q X ( )TT TS S( ) inv=A R Q X
=A Q R TS S=S Q R
SA
24
( ) ( )TT TS S 1( ) 1invλ λ= + −A R Q X A
Contents
25
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
Experiments
• 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
Algorithms
Criteria
27
• SAD
• RMSE
T ˆarccos
ˆi i
ii i
SAD =⋅
a aa a
2
1( ) /
N
i ij ijj
RMSE y s N=
= −∑
1RLi
×∈a
ˆ RL Pi
×∈a
The spectral of the ith endmember
The estimated spectral of the ith endmember
ijs The abundance of the ith endmember according to the jth pixel
ijy The estimated abundance of the ith endmember according to the jth pixel
Synthetic Data (1/5)
28
||
×
x
A
s
0 50 100 150 2000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
band
refle
ctan
ce
abcde
Endmember spectra from USGS. a. Andradite_WS488, b. Kaolinite_CM9, c. Montmorillonite_CM20, d. Desert_Varnish_GDS141, and e. Muscovite_GDS116.
The abundances fractions are subject to Dirichlet distribution.
Synthetic Data (2/5)
29
Results of the algorithms with Different image sizes
100×100 200×200 300×300 400×400 500×500 600×60010
-2
10-1
100
101
102
Image Size
Tim
e
VCAN-FINDRSGASVATF
CPU memory OS SoftwareIntel(R) Xeon CPU X5667 3.07GHZ 48 GBytes 64-bit Window7 Matlab 2010
ParametersL P N
224 5 100×100, 200×200, …, 600×600
SVATF without DR Other Methods: use DR
Synthetic Data (3/5)
30
Results of the algorithms with different mixing degrees
Synthetic Data (4/5)
31
Results of the algorithms with Different noise levels
Synthetic Data (5/5)
32
The effectiveness of AQTF with different mixing degrees
Real Data-Cuprite(1/3)
33
Cuprite dataset * .• 224 bands • spectral resolution 10nm• captured by AVIRIS in June 1997
Estimated abundance maps
34
(a) Muscovite #1, (b) Desert_Varnish, (c) Alunite, (d) Kaolinite #1, (e) Montmorillonite #1, (f) Kaolinite #2, (g) Buddingtonite, (h) Jarosite, (i) Nontronite, (j) Chalcadony, (k) Kaolinite #3, (l) Muscovite #2, (m) Sphene, (n) Montmorillonite #2.
The Spectra obtained by SVATF
35
Solid line: Reference,
Dashed line: Estimated result
50 100 150 2000
0.5
1
(a)50 100 150 200
-0.5
0
0.5
(b)50 100 150 200
0
0.5
1
(c)50 100 150 200
0
0.5
1
(d)
50 100 150 2000
0.5
1
(e)50 100 150 200
0
0.5
1
(f)50 100 150 200
0
0.5
1
(g)50 100 150 200
0
0.5
1
(h)
50 100 150 2000
0.5
1
(i)50 100 150 200
0
0.5
1
(j)50 100 150 200
0
0.5
1
(k)50 100 150 200
0
0.5
1
(l)
50 100 150 2000
0.5
(m)50 100 150 200
0
0.5
1
(n)
(a)Muscovite #1, (b)Desert_Varnish, (c)Alunite, (d)Kaolinite #1, (e)Montmorillonite, (f)Jarosite, (g)Buddingtonite, (h)Kaolinite #2, (i)Nontronite, (j)Chalcadony, (k)Kaolinite #3, (l) Muscovite#2, (M) Sphene, (n)Montmorillonite #2
The comparison of SAD
36
Index Reference Spectra N-FINDR SGA VCA SVATF
1 Muscovite GDS108 0.0900 0.0724 0.1631 0.0801
2 Desert_Varnish GDS141 0.2252 0.1599 0.2454 0.1595
3 Alunite GDS82 Na82 0.0690 0.0690 0.2172 0.0714
4 Kaolinite KGa-2 0.2574 0.2577 0.2201 0.2586
5 Montmorillonite+Illi CM37 0.1519 0.1259 0.0544 0.0501
6 Kaolinite CM7 0.2530 0.2550 0.1769 0.0814
7 Buddingtonite GDS85 D-206 0.0761 0.1598 0.1053 0.0674
8 Jarosite GDS98 0.2812 0.2113 0.2368 0.2163
9 Nontronite NG-1.a 0.0717 0.1374 0.0741 0.0682
10 Chalcedony CU91 0.1241 0.1666 0.1317 0.0727
11 Kaolinite GDS11 <63um 0.1870 0.1896 0.2376 0.1752
12 Muscovite IL107 0.1019 0.0995 0.0888 0.0801
13 Sphene HS189.3B 0.2128 0.1502 0.0970 0.0677
14 Montmorillonite Sca2b 0.1298 0.1206 0.1103 0.1674
sum SAD values 2.2311 2.1749 2.1587 1.6161
37
Algorithms NFINDR-FCLS SGA-FCLS VCA-FCLS SVATF-AQTF
NFINDR FCLS SGA FCLS VCA FCLS SVATF AQTF
Time(seconds) 28.27780 16.63869 3.13832 16.38314 0.85985 16.97822 0.24133 0.22719
Computing time for the Cuprite dataset
The computer environment
CPU memory OS SoftwareIntel(R) Xeon CPU X5667 3.07GHZ 48 GBytes 64-bit Window7 Matlab 2010
Contents
38
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
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
40
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