SIMPLEX VOLUME ANALYSIS BASED ON TRIANGULAR FACTORIZATION: A FRAMEWORK FOR HYPERSPECTRAL UNMIXING

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





3.1 Synthetic 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





3.1 Synthetic 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


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


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


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


ParametersL P N

100, 200, …, 800 10 1000

Contents

20

1. Introduction





3.1 Synthetic 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





3.1 Synthetic 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.


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


30

Results of the algorithms with different mixing degrees


31

Results of the algorithms with Different noise levels


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





3.1 Synthetic 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

SIMPLEX VOLUME ANALYSIS BASED ON TRIANGULAR FACTORIZATION: A FRAMEWORK FOR HYPERSPECTRAL UNMIXING

Documents

Transcript of SIMPLEX VOLUME ANALYSIS BASED ON TRIANGULAR FACTORIZATION: A FRAMEWORK FOR HYPERSPECTRAL UNMIXING