Supervised Planetary Unmixing with Optimal Transport

Supervised Planetary Unmixing withOptimal Transport

August 23, 2016

Sina Nakhostin, Nicolas Courty, Remi Flamary and Thomas Corpetti

Contact: [email protected]

IRISAUniversité de Bretagne-SUD

France

[email protected]

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Problem Definition

Optimal Transport(OT)

Unmixing with OT

Experiments andresults

Dept. IRISAUniversité de Bretagne-SUD

France

Agenda

Problem Definition

Optimal Transport (OT)

Unmixing with OT

Experiments and results

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Unmixing with OT



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Supervised UnmixingIt is about a projection

Given:I A multi/hyper-spectral

dataset.I A dictionary of reference

signatures.Goal:

I Producing a set ofabundance mapsrepresenting distribution ofdifferent materials withinthe scene.

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PredicamentEndmember Variability

I Signature profile of thesame material is usuallycharacterized by morethan one signature due to:

I Sensing device accuracyI Reflectance angleI Shading effectI etc.

I Exploiting Overcomplete Dictionaries is a way to accountfor endmember variability.

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PredicamentEndmember Variability

I Signature profile of thesame material is usuallycharacterized by morethan one signature due to:

I Sensing device accuracyI Reflectance angleI Shading effectI etc.

I Exploiting Overcomplete Dictionaries is a way to accountfor endmember variability.

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PredicamentChoice of Distance

I What is the best distance measure for comparingdictionary atoms ?

Conventional Distance Measures

I Euclidean DistanceI Spectral Angle MapperI Spectral Information Divergence

Proposed Measure

I A distance measure based on Optimal Transport (OT).I Wasserstein Distance (a.k.a. Earth Mover Distance)I Defined between probability distributions.I Can be designed to be mostly sensitive to shifts in

frequency domain.

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Proposed Measure


frequency domain.

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Proposed Measure


frequency domain.

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Why optimal Transport after all?

I To see spectra as probability distributions.I Each spectrum should to be normalized along spectral

values.I Normalization makes the analysis less sensitive to the

total power of spectra in each pixel.I This improves robustness against shadows or other large

radiance changes and thus can prevent degeneratesolutions.

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Contributions

Figure : Courtesy of Cuturi. Transporting 2D probability distributions

In this work we:

I Introduce an original Unmixing Algorithm based onOptimal Transport Theory.

I Use an efficient optimization scheme based on iterativeBregman projections for solving the underlying problem.

I Our formulation allows one to input an eventual prior aboutthe abundances.

I We give preliminary results on the challenging asteroid4-Vesta dataset.

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Problem Definition

7Optimal Transport(OT)

Unmixing with OT



France

What is Optimal Transport?

I Lets µs and µt be two discreteprobability distributions in R+.

I Let a transport plan be anassociation (a coupling) betweeneach bins of µs and µt .

I The Kantorovitch formulation ofOT looks for an optimal couplingbetween the two probabilitydistributions wrt. to a givenmetric (see Figure)

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Discreet Optimal Transport

I Knowing that distributions are available through a finitenumber of bins (i.e. spectral bands) in R+, we can writethem as:

µs =

ns∑i=1

psi δ

sxi

;µt =

nt∑i=1

pti δ

txi

Where δxi is the Dirac at location xi ∈ R+. psi and pt

i areprobability masses associated to the i-th bins.

I The set of probability couplings (joint probabilitydistributions) between µs and µt is defined as:∏

= {γ ∈ (R+)ns×nt |γ1nt = µs; γ1ns = µt}

Where ns and nt are the number of bins in µs and µt .

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Wasserstein Distance

I OT seeks for γ minimizing the quantity:

WC(µs, µt) = minγ∈

∏(µs,µt )

< γ,C >F , (1)

Where < ., . >F is the Frobenius norm and C(d×d) ≥ 0 is thecost matrix (pairwise distance wrt. a given metric).

I Here, WC(µs, µt) is called the Wasserstein distance.

What about Scalability?

The solution of (1) is a linear program with equality constraints.Its resolution can be very time consuming.

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Entropic Regularization

I In order to control the smoothness of the coupling, [Cuturi,2013] proposes an Entropy-based regularization term overγ which reads:WC,ε(µs, µt) = min

γ∈∏

(µs,µt )< γ,C >F − εh(γ)︸︷︷︸

Entropy Regularizer

, (2)

I This allows to draw a parallel between OT and a Bregmanprojection:

γ? = arg minγ∈

∏(µs,µt )

KL(γ, ζ), (3)

Where ζ = exp(−Cε ).

I This version of OT admits a simpler resolution method,based on successive projections over the two marginalconstraints.

We use this closed form projection to solve forUnmixing problem

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Problem Definition


11Unmixing with OT



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Unmixing of the spectrum µ

I Lets assume a linear mixture : µ = Eα.Where E(d×q) is the overcomplete dictionary and α > 0 isa q-vector of abundance values and α>1 = 1.

I We seek for p abundance values for each pixel and(p ≤ q)→ Endmember variability.

I We also assume to have a prior knowledge α0(p×1) overthe abundances.

I The unmixing of µ is then the solution of the followingoptimization:

α = arg minα

WC0,ε0(µ,Eα)︸︷︷︸data fitting

+τ WC1,ε1(α, α0)︸︷︷︸prior

. (4)

Data fitting part searches for the best decomposition fromthe observations. Regularization part enforces thecompliance of the solution with the priors, balanced byparameter τ ∈ R+.

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Problem Definition


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France

Unmixing of the spectrum µ

α = arg minα

WC0,ε0(µ,Eα)︸︷︷︸data fitting

+τ WC1,ε1(α, α0)︸︷︷︸prior

. (5)

I C0(d×d) and C1(q×p) are respectively the cost functionmatrix in the spectral domain and the cost function whichcontains information about the endmember groups.

I The resolution of the optimization is also an algorithmbased on iterative Bregman projections. See details in thepaper.

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Problem Definition


Unmixing with OT

13Experiments andresults


France

4-Vesta dataset

I We do unmixing on a portion of 4-Vesta northernhemispher.

I The VIR image has 383 bands covering the ranges:I 0.55 − 1.05µm with spectral sampling of 1.8nm.I 1.0 − 2.5µm with spectral sampling of 9.8nm.

I We look for three main lithologies : Eucrite, Orthopyroxeneand Olivine.

I A dictionary of 10 atoms formed by the signatures ofdifferent lithologies which found in meteorites was used.

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Cost metric for the captor (C0)

I In order to tailor our cost matrix C0 in alignment to thecharacteristics of the dataset, we build C0(383×383) as thesquare euclidean distance over the spectral values.

I This clearly reflects the characteristic of the spectra andthe level of (dis)similarity among them.

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Cost metric for the materials (C1)

I We manually construct C1(10×3)to reflect the informationregarding the groups ofendmembers belonging to thesame material.

I Two endmembers belonging tothe same material share a verylow cost with the correspondingmaterial in α0, C1(i,j) = 0.

Priors over material groups α

We can also encode our prior knowledge about the dominationof one or another material through the vector α(3×1). In casethere is no such prior knowledge, we can set all the priorsequal value eg here 1/3.

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Comparison with other method

Abundance maps by OT

Abundance maps by constrained LS

I Unmixing based on OT reveals interesting patterns fordistribution of each material.

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Abundance maps with varying priors

I More extensive tests should be conducted, for finding thebest parametrization.

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Conclusion/Perspectives

Conclusion

I An unmixing algorithm based on Optimal Transport.I The metric devoted to distributions is mostly sensitive to

shifts in the frequency domain.I Endmember variability is addressed through the use of

overcomplete dictionary.I Through an iterative Bregman projection a cost function is

to be optimized.

Perspectives

I Introducing new regularization term that will account forsparsity in the groupings.

I Possible candidate could be sparse Group Lasso (or FuseLasso).

Thank you!

For questions please [email protected]

[email protected]

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Supervised Planetary Unmixing with Optimal Transport

Data & Analytics