Endmember Extraction from Highly Mixed Data Using MVC-NMF

1. Endmember Extraction from Highly Mixed Data Using MVC-NMF Lidan Miao AICIP Group Meeting Apr. 6, 2006

2. Outline • Motivation • Existing algorithms • Proposed MVC-NMF algorithm • Experimental results • Conclusion and future work

3. 30 m Motivation • In real world applications, mixed signals widely exist • Typical example: remote sensing imagery • Mixed pixel decomposition • Extract source material (endmember) and estimate area proportion • Most algorithms assume the presence of pure pixels, i.e., pixels covering only one type of material • Highly mixed data • All the pixels are mixtures • Low spatial resolution data: MODIS with 500m sampling rate • Specific applications: mineral exploration

4. Mixing Model • Linear mixture model • The measured spectrum is a linear combination of endmember spectra weighted by their area proportions • Two physical constraints: non-negative and sum-to-one • It is a convex combination

5. Existing Algorithms (1) • Convex hull geometry • Based on the convex combination model, each observation is within a simplex whose vertices are endmembers • Without pure pixel assumption • Find a simplex containing the data with minimum volume • Computational prohibitive • With pure pixel assumption • Find extreme pixel from the scene • Sensitive to noise and outliers • Does not consider the approximation error

6. Existing Algorithms (2) • Non-negative matrix factorization (NMF) • Given a non-negative matrix Y, find two matrices such that • Optimization problem • Geometrically, the target is also to find a simplex containing the data but without any constraint on the simplex • Non-unique solution • Need more constraints to confine solution

7. Proposed Idea • Integrate the good aspects of • Convex hull geometry: define criterion for best simplex • NMF: provide goodness-of-fit measure ||X-AS|| • Method used • Incorporate the minimum volume constraint into NMF • Expected advantages • Utilize fast convergence of NMF and eliminate pure pixel assumption • Resistant to noise

8. MVC-NMF Formulation • Problem formulation • First term is the approximation error • Second term is the volume constraint • Internal and external force interpretation • First term serves as external force which force the simplex to expand to enclose all data points • Second term is internal force which makes the simplex as compact as possible

9. Volume determination • Given k affinely independent points in Rk-1, the volume determined by them is • If the k points in Rn, n>k-1, need to transform them to Rk-1 first as the determinant is not defined for non-square matrix. Three points in 2D Three points in 3D

10. Objective function • For c endmembers, the volume is • U consists of c-1 principal components of X using PCA • mu is data mean • Objective function • Regularization factor

11. MVC-NMF learning (1) • Alternating non-negative least squares • Alternatively fix one matrix and improve the other one • Transform original problem to two sub-problems • Projected gradient learning • Follow standard gradient learning, but when the new estimate does not satisfy the constraints, a projective function is used to project the point back to feasible set. • Applied each sub-problem

12. MVC-NMF learning (2) • Gradient calculation

13. MVC-NMF learning (3)

14. Synthetic images • Mixture of four endmembers • Size: 64-by-64, 224 bands • Maximum abundance: 80% • Zero mean Gaussian noise Endmembers Abundances

15. Algorithms Compared • VCA • Convex geometry-based, assume the presence of pure pixels • Only detect endmembers, the abundance is calculated using FCLS, which is a constrained least squares method • PGNMF • Aims at speeding up the convergence of standard NMF algorithm • SCNMF • Incorporate smoothness constraint to standard NMF • Constraint is formulated as J(A) = ||A||2

16. Experimental results (1) Extracted Endmembers using different methods

17. Experimental results (2) Scatterplots using different methods

18. Experimental results (3) Simplex volume and approximation error

19. Conclusion and Future Work • Summary • The introduced volume constraint results in accurate estimates • The algorithm is resistant to noise and outliers • MVC-NMF is an appealing method for mixed pixel decomposition • Future work • Analyze algorithm limitation • Speed up the convergence

20. Thank you!