PIRE: Science at the Triple Point Between Mathematics ... · detergents, oil recovery, snowboard wax, hair conditioner scaﬀolds for bone tissue engineering, biotechnology lightweight

PIRE: Science at the Triple Point Between Mathematics,Mechanics and Materials Sciencehttp://www.math.cmu.edu/PIRE/index.html

Research and Training

Pattern formation from energy minimization

Challenges in atomistic to continuum modeling and computing

Prediction of hysteresis

Pattern dynamics and evolution of material microstructure

Micostructure of thermoelectric PbTe − Sb2Te3 composites

Scientific Activities

PhD and Postdoctoral Training

seminars

graduate courses

workshops

summer schools

working groups

outreach

MPI ISA UoA

UoB UoO RBG

ImagingMicromagnetics

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Variational Methods in Materials and ImageProcessing

Irene Fonseca

Department of Mathematical SciencesCenter for Nonlinear AnalysisCarnegie Mellon University

Supported by the National Science Foundation (NSF)

Irene Fonseca Variational Methods in Materials and Image Processing


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Challenges: Minimize energies involving . . .

bulk and interfacial energies

vector valued fields

higher order derivatives

discontinuities of underlying fields



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Why Do We Care?

Imaging

Micromagnetic Materials

Foams

Quantum Dots

Thin Structures

etc.



FoamsQuantum Dots

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Why Do We Care?

Imaging


Foams

Quantum Dots

Thin Structures

etc.



FoamsQuantum Dots

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Why Do We Care?

Imaging


Foams

Quantum Dots

Thin Structures

etc.



FoamsQuantum Dots

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Why Do We Care?

Imaging


Foams

Quantum Dots

Thin Structures

etc.



FoamsQuantum Dots

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Why Do We Care?

Imaging


Foams

Quantum Dots

Thin Structures

etc.



FoamsQuantum Dots

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Why Do We Care?

Imaging


Foams

Quantum Dots

Thin Structures

etc.



FoamsQuantum Dots

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Outline

• sharp interface models; diffuse interface models:Imaging, Micromagnetics

• diffuse interface model: Foams• sharp interface model: Quantum Dots• back to Imaging: Starcasing (black and white)

Recolorization (colors)



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“sharp interface” model

Mumford-Shah model

E (u) =

∫

Ω

(

|∇u|p + |u − f |2)

dx +

∫

S(u)γ(ν)dHN−1

|u − f |2 . . . fidelity termp ≥ 1, p = 1 . . .TV model

u ∈ BV (bounded variation)Du = ∇u LN⌊Ω+ [u]⊗ ν HN−1⌊S(u) + C (u)[De Giorgi, Ambrosio, Carriero, Leaci, Chan, Osher, et. al.]



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Mumford-Shah model

E (u) =

∫

Ω

(

|∇u|p + |u − f |2)

dx +

∫

S(u)γ(ν)dHN−1





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Mumford-Shah model

E (u) =

∫

Ω

(

|∇u|p + |u − f |2)

dx +

∫

S(u)γ(ν)dHN−1





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diffuse interface modelMaxwell’s equations

Maxwell’s equations; Cahn-Hilliard type singularperturbations/phase field models

With G. Bouchitte, G. Leoni and V. Millot

div (h + χΩm) = 0 in D′(R3)curl h = 0 in D′(R3)|m| = 1 a.e. inΩ

∫

Ωεα|∇mε|

2 +1

εβϕ(mε) +

∫

R3

1

εγ|hε|

2

. . .∇mε ∼ ∇2 . . .α = β = γ = 1[DeSimone, Kohn, Muller, Otto, Riviere, Serfaty, etc.]



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∫

Ωεα|∇mε|

2 +1

εβϕ(mε) +

∫

R3

1

εγ|hε|

2




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∫

Ωεα|∇mε|

2 +1

εβϕ(mε) +

∫

R3

1

εγ|hε|

2




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∫

Ωεα|∇mε|

2 +1

εβϕ(mε) +

∫

R3

1

εγ|hε|

2




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The ContextThe Phase Field ModelSome Conclusions

surfactants and foams: a singular perturbation problem



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foams

With M. Morini and V. Slastikov

aequous foams, polymeric foams, metallic foams (AL2O3)

metal foams: cellular structure of solid material (e.g.Aluminium) w/ large (≥ 80 % volume fraction of gas-filledpores); size of pores: 1mm–8mm

open-cell foams (interconnected network): lightweight optics,advanced aerospace technology

closed-cell foams (sealed pores): high impact loads



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foams








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foams








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foams








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what for

detergents, oil recovery, snowboard wax, hair conditioner

scaffolds for bone tissue engineering, biotechnology

lightweight structural materials

. . .

little liquid in thin film that separates adjacent bubbles; most liquidin Plateau Borders/Struts (region between 3 touching bubbles)



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what for




. . .




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what for




. . .




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what for




. . .




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what for




. . .




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Plateau borders/struts; nodes/joints



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surfactants

Surf(ace) Act(ive) A(ge)nt

organic compounds - wetting agentsAmphiphiles: hydrophobic tail and hydrophilic head decreasesurface tension; allow easier spreading; lower interfacial tensionbetween two liquids

stabilizing effect of multiple interfaces configurations –Gibbs-Marangoni effect : mass transfer due to gradients in surfacetension

Typical mixtures:– water-oil micelles of surfactants– water-air monolayers of surfactants at the interface



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surfactants







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surfactants







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surfactants







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surfactants







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micelle



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monolayers of surfactant



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surfactants effects

determine macroscopic properties

segregate to interfaces

facilitate formation of interfaces (wetness)



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surfactants effects






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surfactants effects






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diffuse interfacial energy model

[R. Perkins, R. Sekerka, J. Warren and S. Langer]

Modification of the Cahn-Hilliard model for fluid-fluid phasetransitions (recall micromagnetics!)

FCHε (u) :=

∫

Ω

(

1

εf (u) + ε|∇u|2

)

dx

f . . . double well potential . . . f = 0 = 0, 1

u : Ω → [0, 1] order parameter,∫

Ω u = a



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the phase field model

To this we add

F surfα(ε)(u, ρs) := α(ε)

∫

Ω(ρs − |∇u|)2 + β(ε)

∫

Ω|∇ρs |

2

Fε(u, ρ) :=

∫

Ω

(

1

εf (u) + ε|∇u|2 + ε (ρ− |∇u|)2

)

dx

ρs ≥ 0 . . . surfactant density

∫

Ω ρs = b



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To this we add


∫

Ω(ρs − |∇u|)2 + β(ε)

∫

Ω|∇ρs |

2

Fε(u, ρ) :=

∫

Ω

(

1

εf (u) + ε|∇u|2 + ε (ρ− |∇u|)2

)

dx


∫

Ω ρs = b



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To this we add


∫

Ω(ρs − |∇u|)2 + β(ε)

∫

Ω|∇ρs |

2

Fε(u, ρ) :=

∫

Ω

(

1

εf (u) + ε|∇u|2 + ε (ρ− |∇u|)2

)

dx


∫

Ω ρs = b



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asymptotic behavior

Goal: Study the asymptotics as ε → 0+ (β(ε) = 0)

Fε(u, ρs) := FCHε (u) + F surf

α(ε)(u, ρs)

minima of Fε converge to minima of F . . .

Theorem

Γ-limit Fε(u, µ) = F (u, µ) in BV (Ω; 0, 1)×M(Ω)

Different regimes:

α(ε) ∼ ε

ε << α(ε) << 1 . . . surfactant plays no role

α(ε) << ε . . . creation of interfaces needs ∞ energy



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asymptotic behavior



α(ε)(u, ρs)


Theorem


Different regimes:

α(ε) ∼ ε





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asymptotic behavior



α(ε)(u, ρs)


Theorem


Different regimes:

α(ε) ∼ ε





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Theorem

α(ε) = ε

F (u, µ) :=

∫

S(u) Φ(

dµ

dHN−1⌊S(u)(x)

)

dHN−1

if u ∈ BV (Ω; 0, 1)+∞ otherwise

Φ described by optimal profile problem

Φ nonincreasing, convex

Φ is constant on [1,+∞) . . . saturation



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Theorem

α(ε) = ε

F (u, µ) :=

∫

S(u) Φ(

dµ

dHN−1⌊S(u)(x)

)

dHN−1







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Theorem

α(ε) = ε

F (u, µ) :=

∫

S(u) Φ(

dµ

dHN−1⌊S(u)(x)

)

dHN−1







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Theorem

α(ε) = ε

F (u, µ) :=

∫

S(u) Φ(

dµ

dHN−1⌊S(u)(x)

)

dHN−1







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Theorem

α(ε) = ε

F (u, µ) :=

∫

S(u) Φ(

dµ

dHN−1⌊S(u)(x)

)

dHN−1







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Theorem, still

The energy density φ



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surfactant and location of interfaces

surfactant segregates to interface

prescribed distribution of surfactant dictates location ofinterfaces

macroscopic energy F unchanged is density of surfactant onthe interfaces exceeds 1 . . . saturation threshold

stability implies uniform distribution of surfactant. . .Marangoni effect



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the optimal profile

The solution u to the optimal profile problem and the corresponding ρ



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next . . .

1D metastability of multiple interfaces configurations

the impact of surfactants on the dynamics of phase separation

vectorial models . . . more complicated interaction between uand ρs , e.g. [Acerbi and Bouchitte]

Fε(u, ρs) =1

ε

∫

Ωf (x , u, ε∇u, ερs) dx

stability/critical points via Jerrard and StenbergΓ-convergence technique; ongoing work with Jerrard andMorini.



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next . . .




Fε(u, ρs) =1

ε

∫





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next . . .




Fε(u, ρs) =1

ε

∫





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next . . .




Fε(u, ρs) =1

ε

∫





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The ContextSharp Interface ModelSome Conclusions

Quantum Dots: Lattice Misfit

With N. Fusco, G. Leoni, M. Morini

Strained epitaxial films on a relatively thick substrate

plane linear elasticity (In-GaAs/GaAs or SiGe/Si)

free surface of film is flat until reaching a critical thikness

lattice misfits between substrate and film induce strains in thefilm

Complete relaxation to bulk equilibrium ⇒ crystallinestructure would be discontinuous at the interface

Strain ⇒ flat layer of film morphologically unstable ormetastable after a critical value of the thickness is reached(competition between surface and bulk energies)



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islands

To release some of the elastic energy due to the strain: atoms onthe free surface rearrange and morphologies such as formation ofisland (quatum dots) of pyramidal shapes are energetically moreeconomical



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quantum dots: the profile . . .



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some potential applications

optical and optoelectric devices (quantum dot laser), informationstorage, . . .

electronic properties depend on the regularity of the dots, size,spacing, etc.



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some questions

explain how isolated islands are separated by a wetting layer

validate the zero contact angle between wetting layer and theisland



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some questions

explain how isolated islands are separated by a wetting layer

validate the zero contact angle between wetting layer and theisland



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wetting layer and zero contact angle



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wetting layer and zero contact angle, islands



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the model

[Brian Spencer, Bonnetier and Chambolle, Chambolle and Larsen]

Ωh := x = (x , y) : a < x < b, y < h (x)

h : [a, b] → [0,∞) ... graph of h is the profile of the filmy = 0 . . . film/substrate interface

mismatch strain (at which minimum energy is attained)

E0 (y) =

e0i⊗ i if y ≥ 0,0 if y < 0,



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more on the model

e0 > 0i the unit vector along the x direction

elastic energy per unit area: W (E− E0 (y))

W (E) :=1

2E · C [E] , E (u) :=

1

2(∇u + (∇u)T )

C . . . positive definite fourth-order tensor

film and the substrate have similar material properties, share the same

homogeneous elasticity tensor C



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sharp interface model

ϕ0 (y) :=

γfilm if y > 0,γsub if y = 0.

Total energy of the system:

F (u,Ωh) :=

∫

Ωh

W (E (u) (x)− E0 (y)) dx+

∫

Γh

ϕ0 (y) dH1 (x) ,

Γh := ∂Ωh ∩ ((a, b)× R) . . . free surface of the film



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hard to implement . . .

Sharp interface model is difficult to be implemented numerically.Replace it with a boundary-layer model: discontinuous transition isregularized over a thin transition region of width δ (“smearingparameter”).Regularized mismatch strain

Eδ (y) :=1

2e0

(

1 + f(y

δ

))

i⊗ i, y ∈ R,

regularized surface energy density

ϕδ (y) := γsub + (γfilm − γsub) f(y

δ

)

, y ≥ 0,

f . . . smooth increasing function such that

f (0) = 0, limy→−∞

f (y) = −1, limy→∞

f (y) = 1.



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regularized energy

Regularized total energy of the system

Fδ (u,Ωh) :=

∫

Ωh

W (E (u) (x)− Eδ (y)) dx+

∫

Γh

ϕδ (y) dH1 (x)

Two regimes :

γfilm ≥ γsubγfilm < γsub



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wetting, etc.

asymptotics as δ → 0+

γfilm < γsubrelaxed surface energy density is no longer discontinuous: it isconstantly equal to γfilm. . .WETTING!

more favorable to cover the substrate with an infinitesimallayer of film atoms (and pay surface energy with density γfilm)rather than to leave any part of the substrate exposed (andpay surface energy with density γsub)

wetting regime: regularity of local minimizers (u,Ω) of thelimiting functional F∞ under a volume constraint.



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wetting, etc.







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wetting, etc.







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cusps and vertical cuts

The profile h of the film for a locally minimizing configuration isregular except for at most a finite number of cusps and verticalcuts which correspond to vertical cracks in the film.

[Spencer and Meiron]: steady state solutions exhibit cuspsingularities, time-dependent evolution of small disturbances of theflat interface result in the formation of deep grooved cusps (also[Chiu and Gao]); experimental validation of sharp cusplike featuresin SI0.6 Ge0.4

zero contact-angle condition between the wetting layer and islands



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cusps and vertical cuts

The profile h of the film for a locally minimizing configuration isregular except for at most a finite number of cusps and verticalcuts which correspond to vertical cracks in the film.

[Spencer and Meiron]: steady state solutions exhibit cuspsingularities, time-dependent evolution of small disturbances of theflat interface result in the formation of deep grooved cusps (also[Chiu and Gao]); experimental validation of sharp cusplike featuresin SI0.6 Ge0.4

zero contact-angle condition between the wetting layer and islands



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regularization . . .

conclude that the graph of h is a Lipschitz continuous curveaway from a finite number of singular points (cusps, verticalcuts).

. . . and more: Lipschitz continuity of h +blow upargument+classical results on corner domains for solutions ofLame systems of h ⇒ decay estimate for the gradient of thedisplacement u near the boundary ⇒ C 1,α regularity of h and∇u; bootstrap.

this takes us to linearly isotropic materials



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regularization . . .

conclude that the graph of h is a Lipschitz continuous curveaway from a finite number of singular points (cusps, verticalcuts).

. . . and more: Lipschitz continuity of h +blow upargument+classical results on corner domains for solutions ofLame systems of h ⇒ decay estimate for the gradient of thedisplacement u near the boundary ⇒ C 1,α regularity of h and∇u; bootstrap.

this takes us to linearly isotropic materials



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Linearly isotropic elastic materials

W (E) =1

2λ [tr (E)]2 + µ tr

(

E2)

λ and µ are the (constant) Lame moduli

µ > 0 , µ+ λ > 0 .

Euler-Lagrange system of equations associated to W

µ∆u+ (λ+ µ)∇ (div u) = 0 in Ω.



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Regularity of Γ: No corners

Γsing := Γcusps ∪ (x , h(x)) : h(x) < h−(x)

Already know that Γsing is finite.

Theorem (u,Ω) ∈ X . . . δ-local minimizer for the functional F∞.Then Γ \ Γsing is of class C 1,σ for all 0 < σ < 1

2 .

As an immediate corollary, get the zero contact-angle condition

Corollary (u,Ω) ∈ X . . . local minimizer for the functional F∞.If z0 = (x0, 0) ∈ Γ \ Γsing then h′(x0) = 0.



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next . . .

3D case!

(evolution of) material voids under highly anisotropic surfaceenergy: surface motion by surface diffusion via De Giorgi’sminimizing movements; ongoing work with N. Fusco, G.Leoni, V. Millot, and M. Morini.



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Vector-Valued: Inpainting/Recolorization

The Context

With G. Leoni, F. Maggi, M. Morini

Restoration of color images by vector-valued BV functions

Recovery is obtained from few, sparse complete samples and froma significantly incomplete information



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inpainting; recovery of damaged frescos

Figure: A fresco by Mantegna damaged during Second World War.

RGB model: u0 : R → R3 color image, u0 = (u10 , u

20 , u

30) channels

L : R3 → R L(y) = L(e · y) projection on gray levels

L increasing function, e ∈ S2

L(u0) : R → R gray level associated with u0.



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20 , u

30) channels






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20 , u

30) channels






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20 , u

30) channels






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20 , u

30) channels






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20 , u

30) channels






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20 , u

30) channels






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inpainting: RGB model

D ⊂ R ⊂ R2 . . . inpainting region

RGBobserved (u0, v0)u0 . . . correct information on R \ Dv0 . . . distorted information . . . only gray level is known on D;v0 = Lu0L : R3 → R . . . e.g. L(u) := 1

3(r + g + b) or L(ξ) := ξ · e for somee ∈ S2

Goal

to produce a new color image that extends colors of the fragmentsto the gray region, constrained to match the known gray level



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Goal




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Goal




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Goal




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The variational approach by Fornasier-March

Problem: Reconstruct u0 from the knowledge of L(u0) in thedamaged region D and of u0 on R \ D.

Fornasier (2006) proposes to solve:

minu∈BV (R;R3)

|Du|(R)+λ1

∫

D

|L(u)−L(u0)|2 dx+λ2

∫

R\D|u−u0|

2 dx

λ1, λ2 > 0 are fidelity parameters.

Studied by Fornasier-March (2007)

Related work by Kang-March (2007), using theBrightness/Chromaticity decomposition model.



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minu∈BV (R;R3)

|Du|(R)+λ1

∫

D

|L(u)−L(u0)|2 dx+λ2

∫

R\D|u−u0|

2 dx






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minu∈BV (R;R3)

|Du|(R)+λ1

∫

D

|L(u)−L(u0)|2 dx+λ2

∫

R\D|u−u0|

2 dx






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minu∈BV (R;R3)

|Du|(R)+λ1

∫

D

|L(u)−L(u0)|2 dx+λ2

∫

R\D|u−u0|

2 dx






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minu∈BV (R;R3)

|Du|(R)+λ1

∫

D

|L(u)−L(u0)|2 dx+λ2

∫

R\D|u−u0|

2 dx






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minu∈BV (R;R3)

|Du|(R)+λ1

∫

D

|L(u)−L(u0)|2 dx+λ2

∫

R\D|u−u0|

2 dx






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Two reconstructions by Fornasier-March



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Our analysis

How faithful is the reconstruction in the infinite fidelity limit ?



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Our analysis

How faithful is the reconstruction in the infinite fidelity limit ?

Sending λ1 and λ2 → ∞ in

minu∈BV (R;R3)

|Du|(R)+λ1

∫

D

|L(u)−L(u0)|2 dx+λ2

∫

R\D|u−u0|

2 dx



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Our analysis

How faithful is the reconstruction in the infinite fidelity limit ?the problem becomes

minu ∈ BV (R;R3)

|Du|(R) (P)

subject to u = u0 on R \ D and L(u · e) = L(u0 · e) in D.



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Our analysis

How faithful is the reconstruction in the infinite fidelity limit ?the problem becomes

minu ∈ BV (R;R3)

|Du|(R) (P)

subject to u = u0 on R \ D and u · e = u0 · e in D.



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Our analysis

How faithful is the reconstruction in the infinite fidelity limit ?the problem becomes:

minu ∈ BV (R;R3)

|Du|(R) (P)

subject to u = u0 in R \ D and u · e = u0 · e in D.

Definition

u0 is reconstructible over D if it is the unique minimizer of (P).



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λ1 = λ2 = ∞

(P) inf

|Du|(R) : u ∈ BV (R ;R3), Lu = Lu0 in D, u = u0 on R \ D

Theorem

u0 ∈ BV (R ;R3) and D open Lipschitz domain. Then (P) has aminimizer.

isoperimetric inequality → boundedness in BV



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admissible images

Find conditions on the damaged region D which render u0reconstructible



FoamsQuantum Dots

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admissible images

Find conditions on the damaged region D which render u0reconstructible

Mathematical simplification: Restrict the analysis to piecewiseconstant images u0



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admissible images

Find conditions on the damaged region D which make u0reconstructible




FoamsQuantum Dots

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admissible images

Find conditions on the damaged region D which make u0reconstructible


R = Γ ∪

N⋃

k=1

Ωk , u0 =

N∑

k=1

ξk1Ωk,



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Our analysis

Recall that u0 =∑N

k=1 ξk1Ωkis reconstructible over D if it is

the unique minimizer to

minu ∈ BV (R;R3)

|Du|(R) (P)


Strengthened notion of reconstructibility:

Definition

u0 is stably reconstructible over D if there exists ε > 0 such thatall u of the form

u =

N∑

k=1

ξ′k1Ωk, with max

1≤k≤N|ξ′k − ξk | < ε ,

are reconstructible over D.Irene Fonseca Variational Methods in Materials and Image Processing


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Our analysis




minu ∈ BV (R;R3)

|Du|(R) (P)



Definition


u =N∑

k=1


1≤k≤N|ξ′k − ξk | < ε ,



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Our analysis




minu ∈ BV (R;R3)

|Du|(R) (P)



Definition


u =N∑

k=1


1≤k≤N|ξ′k − ξk | < ε ,



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Our analysis




minu ∈ BV (R;R3)

|Du|(R) (P)



Definition


u =

N∑

k=1


1≤k≤N|ξ′k − ξk | < ε ,



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Our analysis




minu ∈ BV (R;R3)

|Du|(R) (P)



Definition


u =

N∑

k=1


1≤k≤N|ξ′k − ξk | < ε ,



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reconstructible images

when is an admissible image u0 reconstructible over a damagedregion S?Answer: NO when a pair of neighboring colors ξh and ξk in u0share the same gray level, i.e., if H1(∂Ωk ∩ ∂Ωh) > 0 andLξh = LξkAnswer: YES if an algebraic condition involving the values of thecolors and the angles of the corners possibly present in Γ issatisfied . . . quantitative validation of the model’s accuracyMinimal requirement: must be reconstructible over S = Γ(δ) forsome δ > 0, where

Γ(δ) := x ∈ R : dist(x , Γ) < δ



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Γ(δ) := x ∈ R : dist(x , Γ) < δ



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Γ(δ) := x ∈ R : dist(x , Γ) < δ



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Γ(δ) := x ∈ R : dist(x , Γ) < δ



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u0 does not have neighboring colors with the same graylevel

zk(x) := P

(

ξk − ξh

|ξk − ξh|

)

if x ∈ ∂Ωk ∩ ∂Ωh ∩ R , h 6= k ,

where P is the orthogonal projection on 〈e〉⊥

P(ξ) := ξ − (ξ · e)e

u0 does not have neighboring colors with the same gray level IFF

sup1≤K≤N

||zk ||L∞ < 1



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A simple counterexample when ‖zk‖∞ < 1 is not satisfied

Original image u0:



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Original image u0:

Resulting image u:



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Adjoint colors have the same gray levels: may createspurious edges

Original image u0:



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A simple analytical counterexample

Original image u0:

Resulting image u:



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A simple analytical counterexample

Original image u0:

Resulting image u:

A spurious contour appears!Irene Fonseca Variational Methods in Materials and Image Processing


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Minimality conditions

Theorem (Necessary and sufficient minimality conditions)

D ⊂ R Lipschitz, H1(∂D ∩ Γ) = 0. Then the following twoconditions are equivalent:

(i) u0 is stably reconstructible over D;

(ii) there exists a tensor field M : D → 〈e〉⊥ ⊗ R2 such that

divM = 0 in D

‖M‖∞ < 1 and M[νΩk] = −zk on D ∩ ∂Ωk .

The tensor field M is called a calibration for u0 in D.



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divM = 0 in D





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divM = 0 in D





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divM = 0 in D





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divM = 0 in D





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divM = 0 in D





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divM = 0 in D





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divM = 0 in D





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1-Laplacian . . .

Reformulate the minimization problem (P) as

inf

F (u,D) : u ∈ BV (D;R3) , u · e = u0 · e L2-a.e. in D

,

where

F (u,D) := |Du|(D) +

N∑

k=1

∫

∂D∩Ωk

|u − ξk | dH1 .

Euler-Lagrange equation: formally given by the 1-LaplacianNeumann problem

div Du|Du| ‖ e in D ,

P(

Du|Du| [νD ]

)

= −z on ∂D , z := P(

u−ξk|u−ξk |

)

Since this equation is in general not well-defined, Du|Du| is replaced

by the calibration M

Hence, the conditions on M can be considered as a weakIrene Fonseca Variational Methods in Materials and Image Processing


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Necessary and sufficient minimality conditions

Writing M = (M(1),M(2)), locally there exists a Lipschitzfunction f = (f (1), f (2)) such that ‖∇f ‖∞ < 1,

[M(i)]⊥ = −∇f i and ∂τΩkf = M[νΩk

] = −zk on D∩∂Ωk .

Hence, the construction of the calibration can be oftenreduced to a Lipschitz extension problem



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] = −zk on D∩∂Ωk .




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] = −zk on D∩∂Ωk .




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] = −zk on D∩∂Ωk .




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] = −zk on D∩∂Ωk .




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When is u0 stably reconstructible over D?

Recall the reconstruction

Question: what happens when the exact information on colorsis known only in a region of possibly small total area butuniformly (randomly) distributed?



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When is u0 stably reconstructible over D?

Recall the reconstruction

Question: what happens when the exact information on colorsis known only in a region of possibly small total area butuniformly (randomly) distributed?



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ε-uniformly distributed undamaged regions

Figure: An ε-uniformly distributedundamaged region.



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Figure: The damaged regioncontains a δ-neighborhoodΓ(δ) of Γ.



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Figure: The damaged regioncontains a δ-neighborhoodΓ(δ) of Γ.

It is natural to assume that u0 is stably reconstructible over Γ(δ)for some δ > 0.Can treat more general non-periodic geometries, e.g. Q(x , ω(ε)) isreplaced by a closed connected set with diameter of order ω(ε)



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A natural assumption

u0 is stably reconstructible over Γ(δ) for some δ > 0.

⇒



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A natural assumption

u0 is stably reconstructible over Γ(δ) for some δ > 0.

⇒



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uniformly sparse region: an asymptotic result

The TV model provides asymptotically exact reconstruction ongeneric color images . . . No info on gray levels!!!

Theorem

u0 ∈ BV (R ;R3) ∩ L∞(R ;R3)

Dε ⊂ R ∩

⋃

x∈εZ2

Q(x , ε) \ Q(x , ω(ε))

,

Let uε be minimizer of

inf |Du|(R) : u = u0 on R \ Dε

Thenuε → u0 in L1



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Admissible ε-uniformly distributed undamaged regions

Figure: Denote by Dε the damagedregion

Figure: The original u0.



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Theorem

Let u0 be stably reconstructible over Γ(δ) for some δ > 0. Assumethat

limε→0+

ω(ε)

ε= 0 , lim

ε→0+

ω(ε)

ε2= ∞ .

Then, there exists ε0 > 0 such that u0 is stably reconstructibleover Dε for all ε ≤ ε0. Irene Fonseca Variational Methods in Materials and Image Processing


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Theorem


limε→0+

ω(ε)

ε= 0 , lim

ε→0+

ω(ε)

ε2= ∞ .



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Theorem


limε→0+

ω(ε)

ε= 0 , lim

ε→0+

ω(ε)

ε2= ∞ .



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uniformly sparse region: scaling ε2 from below for ω(ε) is

sharp

if ω(ε) ≤ cε2 cannot expect exact reconstruction.

Counterexample withω(ε) ≤ cε2

for c small enough

u0 = χΩξ0, R := (0, 3)× (0, 3), Ω := (1, 2)× (1, 2).


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