MINIMUM ENDMEMBER-WISE DISTANCE CONSTRAINED NONNEGATIVE MATRIX FACTORIZATION FOR SPECTRAL MIXTURE ANALYSIS OF HYPERSPECTRAL IMAGES
Minimum endmember-wise Distance Constrained NMF for SMA of Hyperspectral H perspectral Images Shaohui Mei & Mingyi He Center for Earth Observation C t f E th Ob ti School of Electronics and Information Northwestern P l t h i l University N th t Polytechnical U i it
1. IntroductionGenerally, spectral sensors are deployed oneither aircrafts or satellites and images are Tutorial talk, IGARSS’11acquired in low spatial resolution. As a result, Special Issue, TGRS/11pixels in a spectral remote sensing imageoften contain more than one type of groundobjects and are known as mixed pixels ormixtures. The existence of mixed pixels notonly influences the performance of imageclassification and target recognition, but alsois an obstacle to quantitative analysis of q yspectral remote sensing images.
(1) Endmember Extraction/Detection -- Pattern recognition Problem Spectral Library Spectral EE algorithm Geometric algorithms (N-FINDR,VCA,SGA,…) Least unmixing error ( g (IEA, UFCLS, UGDME) ) machine learning approach self-organizing neural network Morphological associative memories Spatial-Spectral EE algorithm Automated Morphological Endmember Extraction (AMEE) Spatial spectral Spatial-spectral EE tool Spatial Purity based EE (SPEE) (IEEE TGRS, Vol.48, No.9, 2010)
(2) Abundance Estimation -- Optimization Problem min M r min M r M r T i Ma i Ma Ma c s.t. aj 1 j 1 aj 0 j 1, 2, , c Nonnegative least square method Gradient Descent Maximum Entropy py augmented Lagrangian approach M lti h Multi-channel Hopfield Neural Network (IEEE GRSL, l H fi ld N lN t k Vol.7, No.3, 2010)……
( )(3) EE + AE simultaneously ( y (Unsupervised) p ) -- Blind Signal Decomposition Problem Independent component analysis – independent set Nonnegative matrix factorization -- constraints …… UFCLS, UFCLS GDME
Problems and Motivation The Linear Mixture Model (LMM) has been widely utilized in SMA due to its effectiveness and simplicity. However, However endmember must be determined previously. previously Recently, unsupervised SMA has been proposed to simultaneously extract endmembers and estimate their corresponding fractional abundance.—UFCLS, GDME However, pure pixels for each endmember are assumed to present in the data and the strong requirement does not hold in many cases.Therefore, a robust unsupervised SMA algorithm must be , p g capable to handle highly mixed hyperspectral data.
Many nonnegative matrix factorization (NMF) based algorithms have been proposed for this highly mixed situation: MiniDisCo-NMF (IEEE TGRS, Vol.48, No.6, 2010): minimum the variance of each endmember spectra -- over-smooth the spectra over smooth MVC-NMF (IEEE TGRS, Vol.45 No.3, 2007): minimum the volume of simplex determined by endmembers, however: endmembers • DR is required • numerical instabilityMinimum endmember-wise distance constrained NMF(MewDC-NMF) endmember wise endmember-wise distance volume of simplex Triangular (2D): Perimeter area Ti l (2D) P i t
2. MewDC-NMF Nonnegative Matrix Factorization (NMF): R MA s.t. M 0, A 0 Linear mixture model: R MAN Abundance Non-negative constraint: A0 Abundance Sum-to-one constraint – pseudo band: R M R , M 11o 11c
However, minimizing the representation error in LMM byNMF is not sufficient for SMA since the unmixing result ofNMF is not unique. Therefore, extra constraint must beimposed on NMF to ensure satisfying unmixing results ofhyperspectral images: 1 f M, A R M A 2 1J1 M 2 J2 A 2 s.t. M 0 A 0 0,It is very difficult to enforce a constraint on A since this maybe problem-dependent. Therefore, in this paper, we enforceextra constraint on M only. y
An endmember-wise distance constraint ( enforcecompactness of endmembers ) is proposed 1 c c J M mi m j mi m j T 2 i1 j1or in matrix form 1 c J M trace M MDi M MDi 2 i1 T in hi h i which Di ei 1T c
Alternating optimization algorithm Ak 1 argmin f Mk , Ak , A0 Mk 1 argmin f Mk , Ak 1 g M0Therefore A k 1 max 0, A k k A f M k , A k , Armijos technique M k 1 max 0, M k k M f M k , A k 1 in which A f M, A MT MA R M f M, A MA R AT M J M M J M M I b Di DT DT Di c i i i 1
3. Experiments Experiments with Synthetic hyperspectral pixels • Spectral liberty: Five spectra of minerals – USGS • LMM • The abundance -- generated randomly based on the Sum to one constraint and the Nonnegative constraint • In order to simulated highly mixed pixels, all the pixels pixels whose abundance is larger than 80% are g regenerated • zero-mean white Gaussian noise is added to simulate possible errors and sensor noises.
SAD for EE RMSE for AEObviously, no matter how noisy the data is, the proposedMewD-NMF algorithm outperforms the other two constrainedNMF algorithms – MVC NMF and MiniDisco NMF MVC-NMF MiniDisco-NMF.
Experiments with AVIRIS dateset • acquired on June 19th, 1997 by AVIRIS • 224 channels covering from 370 nm to 2510 nm with a Ground Instantaneous Field of View of 20 m. • A 200 200 pixels subset of the eastern hydrothermal alteration zone is selected for evaluation. • Of the 224 atmospherically corrected channels, only channels 185 bands are adopted by removing the channels associated with H2O and OH absorption features near 1400 and 1900 nm.
Abundance maps pure white denotes that the percentage of the ground objects in the i t th mixture is 100%, i 100% while pure black denotes 0. 0 Alunite l i Buddingtonite ddi i Visually consistent it t with the results presented by previousChalcedony Kaolinite Montrnorillonite researchers
Endmember spectraFourteen endmembers areextracted (Hysim algorithm). algorithm)The right figure shows theextracted spectra (five highlyrepresentative minerals) and theircorresponding library spectra inUSGS spectral library.Endmembers extracted by theproposed MewDC-NMFalgorithm have similar absorptionand reflection characteristics with d fl ti h t i ti iththeir corresponding ground-truthspectra
4. ConclusionsMewDC-NMF (minimum endmember-wise constrained NMF) algorithm is proposed for unsupervised unmixing of highlymixed hyperspectral data, simultaneously in spectral and spatial The proposed algorithm utilizes cumulative distance betweenendmembers to optimize endmember spectra as compact aspossible, convex function → convergence guaranteed As a result, the non-uniqueness problems in NMF basedunmixing can be alleviated. Experiments on both synthetic and real data havedemonstrated its effectiveness