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

2. 2/20 Outline MotivationMotivation Existing algorithmsExisting algorithms Proposed MVC-NMF algorithmProposed MVC-NMF algorithm Experimental resultsExperimental results Conclusion and future workConclusion and future work MotivationMotivation Existing algorithmsExisting algorithms Proposed MVC-NMF algorithmProposed MVC-NMF algorithm Experimental resultsExperimental results Conclusion and future workConclusion and future work

3. 3/20 Motivation In real world applications, mixed signals widely existIn real world applications, mixed signals widely exist –Typical example: remote sensing imagery Mixed pixel decompositionMixed 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 dataHighly mixed data –All the pixels are mixtures –Low spatial resolution data: MODIS with 500m sampling rate –Specific applications: mineral exploration In real world applications, mixed signals widely existIn real world applications, mixed signals widely exist –Typical example: remote sensing imagery Mixed pixel decompositionMixed 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 dataHighly mixed data –All the pixels are mixtures –Low spatial resolution data: MODIS with 500m sampling rate –Specific applications: mineral exploration 30 m

4. 4/20 Mixing Model Linear mixture modelLinear 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 Linear mixture modelLinear 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. 5/20 Existing Algorithms (1) Convex hull geometryConvex 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 Convex hull geometryConvex 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. 6/20 Existing Algorithms (2) Non-negative matrix factorization (NMF)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 Non-negative matrix factorization (NMF)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. 7/20 Proposed Idea Integrate the good aspects ofIntegrate the good aspects of –Convex hull geometry: define criterion for best simplex –NMF: provide goodness-of-fit measure ||X-AS|| Method usedMethod used –Incorporate the minimum volume constraint into NMF Expected advantagesExpected advantages –Utilize fast convergence of NMF and eliminate pure pixel assumption –Resistant to noise Integrate the good aspects ofIntegrate the good aspects of –Convex hull geometry: define criterion for best simplex –NMF: provide goodness-of-fit measure ||X-AS|| Method usedMethod used –Incorporate the minimum volume constraint into NMF Expected advantagesExpected advantages –Utilize fast convergence of NMF and eliminate pure pixel assumption –Resistant to noise

8. 8/20 MVC-NMF Formulation Problem formulationProblem formulation –First term is the approximation error –Second term is the volume constraint Internal and external force interpretationInternal 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 Problem formulationProblem formulation –First term is the approximation error –Second term is the volume constraint Internal and external force interpretationInternal 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. 9/20 Volume determination Given k affinely independent points in R k-1, the volume determined by them isGiven k affinely independent points in R k-1, the volume determined by them is If the k points in R n, n>k-1, need to transform them to R k-1 first as the determinant is not defined for non-square matrix.If the k points in R n, n>k-1, need to transform them to R k-1 first as the determinant is not defined for non-square matrix. Given k affinely independent points in R k-1, the volume determined by them isGiven k affinely independent points in R k-1, the volume determined by them is If the k points in R n, n>k-1, need to transform them to R k-1 first as the determinant is not defined for non-square matrix.If the k points in R n, n>k-1, need to transform them to R k-1 first as the determinant is not defined for non-square matrix. Three points in 2D Three points in 3D

10. 10/20 Objective function For c endmembers, the volume isFor c endmembers, the volume is –U consists of c-1 principal components of X using PCA –mu is data mean Objective functionObjective function –Regularization factor For c endmembers, the volume isFor c endmembers, the volume is –U consists of c-1 principal components of X using PCA –mu is data mean Objective functionObjective function –Regularization factor

11. 11/20 MVC-NMF learning (1) Alternating non-negative least squaresAlternating non-negative least squares –Alternatively fix one matrix and improve the other one –Transform original problem to two sub-problems Projected gradient learningProjected 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 Alternating non-negative least squaresAlternating non-negative least squares –Alternatively fix one matrix and improve the other one –Transform original problem to two sub-problems Projected gradient learningProjected 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. 12/20 MVC-NMF learning (2) Gradient calculationGradient calculation

13. 13/20 MVC-NMF learning (3)

14. 14/20 Synthetic images Endmembers Abundances Mixture of four endmembersMixture of four endmembers Size: 64-by-64, 224 bandsSize: 64-by-64, 224 bands Maximum abundance: 80%Maximum abundance: 80% Zero mean Gaussian noiseZero mean Gaussian noise Mixture of four endmembersMixture of four endmembers Size: 64-by-64, 224 bandsSize: 64-by-64, 224 bands Maximum abundance: 80%Maximum abundance: 80% Zero mean Gaussian noiseZero mean Gaussian noise

15. 15/20 Algorithms Compared VCAVCA –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 PGNMFPGNMF –Aims at speeding up the convergence of standard NMF algorithm SCNMFSCNMF –Incorporate smoothness constraint to standard NMF –Constraint is formulated as J(A) = ||A|| 2 VCAVCA –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 PGNMFPGNMF –Aims at speeding up the convergence of standard NMF algorithm SCNMFSCNMF –Incorporate smoothness constraint to standard NMF –Constraint is formulated as J(A) = ||A|| 2

16. 16/20 Experimental results (1) Extracted Endmembers using different methods

17. 17/20 Experimental results (2) Scatterplots using different methods

18. 18/20 Experimental results (3) Simplex volume and approximation error

19. 19/20 Conclusion and Future Work SummarySummary –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 workFuture work –Analyze algorithm limitation –Speed up the convergence SummarySummary –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 workFuture work –Analyze algorithm limitation –Speed up the convergence

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