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# 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

1. 1. Simplex Volume AnalysisBased On TriangularFactorization: A framework forhyperspectral Unmixing Wei Xia, Bin Wang, Liming Zhang, and Qiyong Lu Dept. of Electronic Engineering Fudan University, China
2. 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. 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. 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
5. 5. Different methods under the LMM 5
6. 6. Simplex Volume Analysis (1/2) e3 e2 e1 e0A Simplex of P-vertices is defined by ⎧ P ⎫ ⎨x = s1e 0 + s1e1 + .... + sP −1e P −1 | si > 0, ⎩ ∑ si = 1⎭ i =1 ⎬ 6
7. 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 simplexVolume 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. 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
9. 9. The Proposed Method base on Triangular Factorization (TF) x A s 9
10. 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,” IEEEGeosci. Remote Sens. Lett., vol. 6, no. 2, pp. 219–223, Apr. 2009. 10
11. 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 bedecomposed by Cholesky Factorization Z = LLT 11
12. 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. 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. 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,” IEEETrans. Geosci. Remote Sens., vol. 44, no. 10, pp. 2804-2819, 2006. 14
15. 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. 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,” IEEEGeosci. Remote Sens. Lett., vol. 6, no. 2, pp. 219–223, Apr. 2009. 16
17. 17. The numbers of flops in various endmembersThe 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. 18. The numbers of flops in various pixels Parameters L P N 100 10 100, 1000, …, 1e+8 14 10The 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. 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. 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. 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. 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. 23. • Similarly，obtain A when S is known X = AS X T = ST A TA = 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. 24. A = λ ( inv ( R S ) QS X1 ) + (1 − λ ) A T T T 24
25. 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. 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,” IEEETrans. 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 theSPIE conference on Imaging Spectrometry V, vol. 3753, pp. 266-275, 1999 26
27. 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. 28. Synthetic Data (1/5)x 1 a 0.9 b c 0.8 d e 0.7|| reflectance 0.6 0.5 0.4A 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. 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 DRTime 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
30. 30. Synthetic Data (3/5)Results of the algorithms with different mixing degrees 30
31. 31. Synthetic Data (4/5)Results of the algorithms with Different noise levels 31
32. 32. Synthetic Data (5/5)The effectiveness of AQTF with different mixing degrees 32
33. 33. Real Data-Cuprite(1/3) Cuprite dataset * .• 224 bands• spectral resolution 10nm• captured by AVIRIS in June 1997 33
34. 34. Estimated abundance maps(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. 34
35. 35. The Spectra obtained by SVATF 1 0.5 1 1 0.5 0 0.5 0.5 0 -0.5 0 0 50 100 150 200 50 100 150 200 50 100 150 200 50 100 150 200 (a) (b) (c) (d) 1 1 1 1 0.5 0.5 0.5 0.5 0 0 0 0 50 100 150 200 50 100 150 200 50 100 150 200 50 100 150 200 (e) (f) (g) (h) 1 1 1 1 0.5 0.5 0.5 0.5 0 0 0 0 50 100 150 200 50 100 150 200 50 100 150 200 50 100 150 200 (i) (j) (k) (l) 0.5 1 0.5 Solid line: Reference, 0 0 Dashed line: Estimated result 50 100 150 200 50 100 150 200 (m) (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 35
36. 36. The comparison of SADIndex 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 36
37. 37. Computing time for the Cuprite dataset NFINDR-FCLS SGA-FCLS VCA-FCLS SVATF-AQTFAlgorithms 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. 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. 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
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