Maher Aldeghlawi and Miguel Velez-Reyes Electrical and Computer Engineering Department University of Texas at El Paso 500 West University Avenue | El Paso, Texas 79968 e-mail:
[email protected] [email protected]
Outline • Background
Motivation Linear Mixing Model (LMM) Geometry of the LMM
Hyperspectral Unmixing
Standard two stage Unmixing Geometric Endmember Extraction
Column Subset Selection Problem (CSSP) CSSP for Endmember Extraction
• Experimental Results • Conclusions
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Motivation • Matrix factorizations (SVD, RRQR) are widely used in data analytics – Dimensionality reduction – Feature extraction
• Long history of computational tools (numerical linear algebra) available • Interest in using them for hyperspectral image analysis • Studying column subset selection algorithms application to – Band subset selection (tall matrices) – Endmember extraction (wide matrices)
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Linear Mixing Model 𝑝𝑝
𝒙𝒙𝑗𝑗 = � 𝑎𝑎𝑗𝑗𝑖𝑖 𝒆𝒆𝑖𝑖 𝑖𝑖=1
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where p is the number of endmembers ei endmember spectral signature aji fractional abundance
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Linear Mixing Model: Geometric Interpretation 𝒙𝒙 = 𝑬𝑬𝑬𝑬
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Hyperspectral Unmixing Mixing
Unmixing SPIE, April 17, 2018
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Standard Two-Stage Approach for Unmixing
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Endmember Extraction based on Geometric Properties • Endmembers are the vertices of the simplex – Examples: PPI (Boardman 1994), SMACC (Gruninger 2004), VCA (Nascimento 2005)
• The volume of the simplex formed by the endmembers is larger than the volume formed from any other combination of pixels – Example: NFINDR (Winter 1999)
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Unfolding the Image Cube Into a Matrix
unfolding
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𝑿𝑿 ∈ 𝑅𝑅 𝑚𝑚×𝑵𝑵
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Column Subset Selection Problem (CSSP) • Let 𝑿𝑿 ∈ 𝑅𝑅 𝑚𝑚×𝑵𝑵 be a matrix (e.g. the unfolded hyperspectral cube). • CSSP can be stated as the problem of finding a permutation matrix 𝑷𝑷 such that 𝑿𝑿𝑷𝑷 = 𝑿𝑿1 𝑿𝑿2 where 𝑿𝑿1 ∈ 𝑅𝑅 𝑚𝑚×𝒑𝒑 is the matrix of selected columns and the permutation matrix 𝑷𝑷 is selected to satisfy some optimality criteria. • CSSP is widely studied in linear algebra and data mining SPIE, April 17, 2018
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Column Subset Selection Problem (CSSP)
• The standard CSSP can be stated as the following approximation problem. min
𝑷𝑷 𝑠𝑠.𝑡𝑡 𝑿𝑿𝑿𝑿= 𝑿𝑿1
𝑿𝑿2
𝑿𝑿 −
2 # 𝑿𝑿1 𝑿𝑿1 𝑿𝑿 𝐹𝐹
=
min
𝑠𝑠.𝑡𝑡 𝑿𝑿𝑿𝑿= 𝑿𝑿1
𝑷𝑷
𝑿𝑿2 , 𝑪𝑪=𝑿𝑿# 1 𝑿𝑿
𝑿𝑿 − 𝑿𝑿1 𝑪𝑪
• The optimal 𝑷𝑷 chooses the columns that best predict the other columns in terms of the residual error. • Selected columns 𝑿𝑿1 are called “most representative” columns • This is a combinatorial optimization problem • Algorithms that choose a “good” column subset are proposed in the linear algebra literature. SPIE, April 17, 2018
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2 𝐹𝐹
D2
X
0 D3
𝒆𝒆 = 𝑿𝑿 − 𝑿𝑿𝟏𝟏 𝑪𝑪 X1C
𝟐𝟐 𝑭𝑭
D1
Range space of X1 11/26
CSS, Why and Where? •
•
Motivation Data interpretation through identifying relevant columns Speed-up by performing computationally expensive operations on a small column subset Some applications in data analysis HSI Band Subset Selection (Velez-Reyes 1998) Neuroimaging data (NNCN) (Strauch 2014) Cardiovascular and respiratory modeling (ELLWEIN 2016) Population genetics summarization (Khan 2015) Electronic circuits testing (Abadir 2013) Recommendation systems (Amatriain 2011) Machine learning and statistics (variable selection)(Altschuler 2016)
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CSS for Endmember Extraction • Let 𝑿𝑿 ∈ 𝑅𝑅 𝑚𝑚×𝑵𝑵 be the unfolded hyperspectral cube • CSS can be used to select a subset of pixels that predict the other pixels with a low residual – 𝑿𝑿1 → Representative Pixels • It can be shown that the optimal low residual solution for the CSSP also solves the problem of finding the maximum volume sub-matrix of a matrix (Çivril 2009)
– Relates to the simplex volume maximization problem (e.g. NFINDR)
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Examples of Algorithms to Solve the CSSP DETERMINISTIC ALGORITMS SVD Singular Value decomposition CSS (G. H. GOLUB and C. REINSCH 1970)
We refer to this algorithm as SVDSS
Rank Revealing QR (RRQR) factorization CSS QR Factorization with pivoting CSS (G. H. GOLUB and C. REINSCH 1970) RRQR CSS in (T. F. Chan 1987)
Greedy Nystrom Approximation CSS (a. K. Farahat et al 2011) RANDOMIZED ALGORITHMS A two-stage algorithm for the CSSP (C. Boutsidis et al 2009) Greedy CSS (A. K. Farahat et al 2013) Random Greedy CSS (A. K. Farahat et al 2014) SPIE, April 17, 2018
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Experiments • Multiple Data Sets
– Simulated (4 endmembers) – Hydice Urban
• Endmember extraction using – – – – –
PPI (Boardman 1994) VCA (Nascimento 2005) NFINDR (Winter 1999) SVDSS (G.H. Golub and C. Reinsch 1970) Implemented in Matlab
• Volume of generated p-simplex is computed for extracted endmembers SPIE, April 17, 2018
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Experiments (cont.) • Volume of a p-simplex in an m-dimensional space (p
