Review of Spectral Unmixing for Hyperspectral Imagery Lidan Miao Sept. 29, 2005

Motivation The wide existence of mixed signals –In hyperspectral image, the measurement of a single pixel is usually a contribution from several materials (endmembers). What is spectral unmixing? –A process by which mixed pixel spectra are decomposed into endmember signatures and their fractional abundances. Applications –Subpixel detection. –Classification –Material quantification

Data Mixing Model Linear mixing –Mixing scale is macroscopic and there is negligible interaction among distinct endmembers. Nonlinear mixing –Mixing scale is microscopic. The incident radiations scattered through multiple bounces involves several endmembers. Linear model Nonlinear model

Linear Mixing Model Measurement model –Observation vector –Material signature matrix –Abundance fractions Nonnegative and sum-to-one constraints

Unmixing Algorithms Supervised –Obtain M from laboratory data or training samples. –With known M, unmixing is a least square problem. –Not reliable due to spatial and temporal variation in illumination and atmospheric conditions. Unsupervised –Linear spectral mixture analysis (LSMA) Two steps: estimate M from given data + inversing. –Independent component analysis (ICA) How to interpret the system model? –Lagrange constrained neural network (LCNN) Pixel-based algorithm.

LSMA Assumptions –There exist at least one pure pixel for each class. –The material signature matrix is the same for all image pixels in the scene. –The number of endmembers can be determined. Two-step process –Endmember detection Convex geometry-based approach (MVT, PPI, N-FINDR, VCA) LSE-based approach (UFCLS, USCLS, UNCLS) –Abundance estimation Least square method (LSE, SCLS, NCLS, FCLS) Orthogonal subspace projection (OPS) Quadratic programming (QP) Target constrained method (CEM)

Convex Geometry Convex hull –The set of all convex combinations of point in C. Simplex –Convex hull of k+1 affinely independent points. Strong parallelism between LSMA and CG. –Endmember detection is equivalent to identifying vertices. Convex hull in R 2 Simplex in R 2

LSE-based Approach Minimize the LSE between the linear mixture model and estimated measurements. Select the brightest pixel as the first endmember,after each iteration, select the most distinct pixel as new endmember. Use currently selected endmembers to unmix and calculate LSE.

ICA System model Application –Blind source separation (BSS) and deconvolution. Assumption –Mutually independent sources. Permutation and scaling problem: –This compromises its application in spectral unmixing

ICA in Spectral Unmixing Two interpretations of system model –Vector x i is the stack notation of image of band i. –Column of M is endmember signature. –Vector s i is the abundance of endmember i at all pixel positions. –Sources are not mutually independent. –Vector x i is the spectrum of pixel i. –Vector m i is endmember signature. –Row vector s i is the abundance of pixel i. –Sources are mutually independent.

LCNN System model –Pixel-by-pixel processing Two principles –Maximum entropy Given incomplete information, maximum entropy is the least bias estimate Closed system, no energy exchange –Minimum Helmholtz free energy The minimum free energy is achieved at thermal equilibrium state. Open system, nonzero energy exchange

Minimum Energy-based LCNN Objective function Nonlinear programming formulation The problem in objective function cannot be solved by learning algorithm.

Evaluation System Selection of spectral signatures Generation of simulated scene Unmixing algorithm Ground truth Abundance map Endmember signatures Estimated abundance Extracted endmembers 1. Evaluation of abundance fraction –Root mean square error (RMSE) –Fractional abundance angle distance (FAAD) 2. Evaluation of endmember signature –Spectral angle distance (SAD) –Spectral information divergence (SID) 1 2

Simulated Data

Unmixing Results (1)

Unmixing Result (2) VCAUSCLS There exist pure pixels in the scene (SNR=30dB)

Unmixing Result (3) VCAUSCLS There are no pure pixels in the scene (SNR=30dB)

Unmixing Result (4)

Experiments on Real Data (1) N-FINDR VCA

Experiments on Real Data (2) FastICA USCLS

Conclusion Convex geometry-based methods can successfully extract endmembers. ICA is not a robust algorithm for spectral unmixing. More works on LCNN. –Spatial information? –Redefine the objective function?