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.

Endmember Extraction from Endmember Extraction from Highly Mixed Data Using MVC-Highly Mixed Data Using MVC-

NMFNMF

Lidan MiaoLidan Miao

AICIP Group MeetingAICIP Group Meeting

Apr. 6, 2006Apr. 6, 2006

Lidan MiaoLidan Miao

AICIP Group MeetingAICIP Group Meeting

Apr. 6, 2006Apr. 6, 2006

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OutlineOutline

• 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

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MotivationMotivation

• In real world applications, mixed signals widely existIn real world applications, mixed signals widely exist– Typical example: remote sensing imageryTypical example: remote sensing imagery

• Mixed pixel decompositionMixed pixel decomposition– Extract source material (endmember) and estimate area Extract source material (endmember) and estimate area

proportionproportion

– Most algorithms assume the presence of pure pixels, i.e., pixels Most algorithms assume the presence of pure pixels, i.e., pixels covering only one type of materialcovering only one type of material

• Highly mixed dataHighly mixed data– All the pixels are mixturesAll the pixels are mixtures

– Low spatial resolution data: MODIS with 500m sampling rateLow spatial resolution data: MODIS with 500m sampling rate

– Specific applications: mineral explorationSpecific applications: mineral exploration

• In real world applications, mixed signals widely existIn real world applications, mixed signals widely exist– Typical example: remote sensing imageryTypical example: remote sensing imagery

• Mixed pixel decompositionMixed pixel decomposition– Extract source material (endmember) and estimate area Extract source material (endmember) and estimate area

proportionproportion

– Most algorithms assume the presence of pure pixels, i.e., pixels Most algorithms assume the presence of pure pixels, i.e., pixels covering only one type of materialcovering only one type of material

• Highly mixed dataHighly mixed data– All the pixels are mixturesAll the pixels are mixtures

– Low spatial resolution data: MODIS with 500m sampling rateLow spatial resolution data: MODIS with 500m sampling rate

– Specific applications: mineral explorationSpecific applications: mineral exploration

30 m

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Mixing ModelMixing Model

• Linear mixture modelLinear mixture model– The measured spectrum is a linear combination of The measured spectrum is a linear combination of

endmember spectra weighted by their area proportionsendmember spectra weighted by their area proportions

– Two physical constraints: non-negative and sum-to-oneTwo physical constraints: non-negative and sum-to-one

– It is a convex combinationIt is a convex combination

• Linear mixture modelLinear mixture model– The measured spectrum is a linear combination of The measured spectrum is a linear combination of

endmember spectra weighted by their area proportionsendmember spectra weighted by their area proportions

– Two physical constraints: non-negative and sum-to-oneTwo physical constraints: non-negative and sum-to-one

– It is a convex combinationIt is a convex combination

, 0, T Tl N l c c N X A S S 1 S 1

1 2 1 1 2

| | | | | | | | | | |

| | | | | | | | | | |N c N

x x x a a s s s

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Existing Algorithms (1)Existing Algorithms (1)

• Convex hull geometryConvex hull geometry– Based on the convex combination model, each observation is Based on the convex combination model, each observation is

within a simplex whose vertices are endmemberswithin a simplex whose vertices are endmembers

– Without pure pixel assumptionWithout pure pixel assumption

» Find a simplex containing the data with minimum volumeFind a simplex containing the data with minimum volume

» Computational prohibitiveComputational prohibitive

– With pure pixel assumptionWith pure pixel assumption

» Find extreme pixel from the sceneFind extreme pixel from the scene

– Sensitive to noise and outliersSensitive to noise and outliers

– Does not consider the approximation errorDoes not consider the approximation error

• Convex hull geometryConvex hull geometry– Based on the convex combination model, each observation is Based on the convex combination model, each observation is

within a simplex whose vertices are endmemberswithin a simplex whose vertices are endmembers

– Without pure pixel assumptionWithout pure pixel assumption

» Find a simplex containing the data with minimum volumeFind a simplex containing the data with minimum volume

» Computational prohibitiveComputational prohibitive

– With pure pixel assumptionWith pure pixel assumption

» Find extreme pixel from the sceneFind extreme pixel from the scene

– Sensitive to noise and outliersSensitive to noise and outliers

– Does not consider the approximation errorDoes not consider the approximation error

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Existing Algorithms (2)Existing Algorithms (2)

• Non-negative matrix factorization (NMF)Non-negative matrix factorization (NMF)– Given a non-negative matrix Y, find two matrices such that Given a non-negative matrix Y, find two matrices such that

– Optimization problemOptimization problem

– Geometrically, the target is also to find a simplex containing Geometrically, the target is also to find a simplex containing the data but without any constraint on the simplexthe data but without any constraint on the simplex

– Non-unique solutionNon-unique solution

– Need more constraints to confine solutionNeed 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 Given a non-negative matrix Y, find two matrices such that

– Optimization problemOptimization problem

– Geometrically, the target is also to find a simplex containing Geometrically, the target is also to find a simplex containing the data but without any constraint on the simplexthe data but without any constraint on the simplex

– Non-unique solutionNon-unique solution

– Need more constraints to confine solutionNeed more constraints to confine solution

, 0, 0, ,n m n r r m r m n Y W H W H

1 1( )( ) for any positive matrix ,D D WD D H WH

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Proposed IdeaProposed Idea

• Integrate the good aspects of Integrate the good aspects of – Convex hull geometry: define criterion for best simplexConvex hull geometry: define criterion for best simplex

– NMF: provide goodness-of-fit measure ||X-AS||NMF: provide goodness-of-fit measure ||X-AS||

• Method usedMethod used– Incorporate the minimum volume constraint into NMFIncorporate the minimum volume constraint into NMF

• Expected advantagesExpected advantages– Utilize fast convergence of NMF and eliminate pure pixel Utilize fast convergence of NMF and eliminate pure pixel

assumptionassumption

– Resistant to noiseResistant to noise

• Integrate the good aspects of Integrate the good aspects of – Convex hull geometry: define criterion for best simplexConvex hull geometry: define criterion for best simplex

– NMF: provide goodness-of-fit measure ||X-AS||NMF: provide goodness-of-fit measure ||X-AS||

• Method usedMethod used– Incorporate the minimum volume constraint into NMFIncorporate the minimum volume constraint into NMF

• Expected advantagesExpected advantages– Utilize fast convergence of NMF and eliminate pure pixel Utilize fast convergence of NMF and eliminate pure pixel

assumptionassumption

– Resistant to noiseResistant to noise

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MVC-NMF FormulationMVC-NMF Formulation

• Problem formulationProblem formulation– First term is the approximation errorFirst term is the approximation error

– Second term is the volume constraintSecond 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 First term serves as external force which force the simplex to

expand to enclose all data pointsexpand to enclose all data points

– Second term is internal force which makes the simplex as Second term is internal force which makes the simplex as compact as possiblecompact as possible

• Problem formulationProblem formulation– First term is the approximation errorFirst term is the approximation error

– Second term is the volume constraintSecond 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 First term serves as external force which force the simplex to

expand to enclose all data pointsexpand to enclose all data points

– Second term is internal force which makes the simplex as Second term is internal force which makes the simplex as compact as possiblecompact as possible

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Volume determinationVolume determination

• Given k affinely independent points in RGiven k affinely independent points in Rk-1k-1, the , the volume determined by them is volume determined by them is

• If the k points in RIf the k points in Rnn, n>k-1, need to transform , n>k-1, need to transform them to Rthem to Rk-1k-1 first as the determinant is not first as the determinant is not defined for non-square matrix.defined for non-square matrix.

• Given k affinely independent points in RGiven k affinely independent points in Rk-1k-1, the , the volume determined by them is volume determined by them is

• If the k points in RIf the k points in Rnn, n>k-1, need to transform , n>k-1, need to transform them to Rthem to Rk-1k-1 first as the determinant is not first as the determinant is not defined for non-square matrix.defined for non-square matrix.

Three points in 2D Three points in 3D

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Objective functionObjective function

• For c endmembers, the volume isFor c endmembers, the volume is

– U consists of c-1 principal components of X using PCAU consists of c-1 principal components of X using PCA

– mu is data mean mu is data mean

• Objective functionObjective function– Regularization factor Regularization factor

• For c endmembers, the volume isFor c endmembers, the volume is

– U consists of c-1 principal components of X using PCAU consists of c-1 principal components of X using PCA

– mu is data mean mu is data mean

• Objective functionObjective function– Regularization factor Regularization factor ( 1)!c

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MVC-NMF learning (1)MVC-NMF learning (1)

• Alternating non-negative least squaresAlternating non-negative least squares– Alternatively fix one matrix and improve the other oneAlternatively fix one matrix and improve the other one

– Transform original problem to two sub-problemsTransform original problem to two sub-problems

• Projected gradient learningProjected gradient learning– Follow standard gradient learning, but when the new estimate Follow standard gradient learning, but when the new estimate

does not satisfy the constraints, a projective function is used does not satisfy the constraints, a projective function is used to project the point back to feasible set. to project the point back to feasible set.

– Applied each sub-problemApplied each sub-problem

• Alternating non-negative least squaresAlternating non-negative least squares– Alternatively fix one matrix and improve the other oneAlternatively fix one matrix and improve the other one

– Transform original problem to two sub-problemsTransform original problem to two sub-problems

• Projected gradient learningProjected gradient learning– Follow standard gradient learning, but when the new estimate Follow standard gradient learning, but when the new estimate

does not satisfy the constraints, a projective function is used does not satisfy the constraints, a projective function is used to project the point back to feasible set. to project the point back to feasible set.

– Applied each sub-problemApplied each sub-problem

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MVC-NMF learning (2)MVC-NMF learning (2)

• Gradient calculationGradient calculation• Gradient calculationGradient calculation

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MVC-NMF learning (3)MVC-NMF learning (3)

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Synthetic imagesSynthetic images

EndmembersAbundances

• 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

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Algorithms ComparedAlgorithms Compared

• VCAVCA– Convex geometry-based, assume the presence of pure pixelsConvex geometry-based, assume the presence of pure pixels

– Only detect endmembers, the abundance is calculated using Only detect endmembers, the abundance is calculated using FCLS, which is a constrained least squares methodFCLS, which is a constrained least squares method

• PGNMFPGNMF– Aims at speeding up the convergence of standard NMF Aims at speeding up the convergence of standard NMF

algorithmalgorithm

• SCNMFSCNMF– Incorporate smoothness constraint to standard NMFIncorporate smoothness constraint to standard NMF

– Constraint is formulated as J(A) = ||A||Constraint is formulated as J(A) = ||A||22

• VCAVCA– Convex geometry-based, assume the presence of pure pixelsConvex geometry-based, assume the presence of pure pixels

– Only detect endmembers, the abundance is calculated using Only detect endmembers, the abundance is calculated using FCLS, which is a constrained least squares methodFCLS, which is a constrained least squares method

• PGNMFPGNMF– Aims at speeding up the convergence of standard NMF Aims at speeding up the convergence of standard NMF

algorithmalgorithm

• SCNMFSCNMF– Incorporate smoothness constraint to standard NMFIncorporate smoothness constraint to standard NMF

– Constraint is formulated as J(A) = ||A||Constraint is formulated as J(A) = ||A||22

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Experimental results (1)Experimental results (1)

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Extracted Endmembers using different methods

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Experimental results (2)Experimental results (2)

Scatterplots using different methods

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Experimental results (3)Experimental results (3)

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Conclusion and Future WorkConclusion and Future Work

• SummarySummary– The introduced volume constraint results in accurate The introduced volume constraint results in accurate

estimatesestimates

– The algorithm is resistant to noise and outliersThe algorithm is resistant to noise and outliers

– MVC-NMF is an appealing method for mixed pixel MVC-NMF is an appealing method for mixed pixel decompositiondecomposition

• Future work Future work – Analyze algorithm limitationAnalyze algorithm limitation

– Speed up the convergenceSpeed up the convergence

• SummarySummary– The introduced volume constraint results in accurate The introduced volume constraint results in accurate

estimatesestimates

– The algorithm is resistant to noise and outliersThe algorithm is resistant to noise and outliers

– MVC-NMF is an appealing method for mixed pixel MVC-NMF is an appealing method for mixed pixel decompositiondecomposition

• Future work Future work – Analyze algorithm limitationAnalyze algorithm limitation

– Speed up the convergenceSpeed up the convergence

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