PIRE: Science at the Triple Point Between Mathematics ... · detergents, oil recovery, snowboard...
Transcript of PIRE: Science at the Triple Point Between Mathematics ... · detergents, oil recovery, snowboard...
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
FoamsQuantum Dots
Back to Imaging
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
Irene Fonseca Variational Methods in Materials and Image Processing
ImagingMicromagnetics
FoamsQuantum Dots
Back to Imaging
Challenges: Minimize energies involving . . .
bulk and interfacial energies
vector valued fields
higher order derivatives
discontinuities of underlying fields
Irene Fonseca Variational Methods in Materials and Image Processing
ImagingMicromagnetics
FoamsQuantum Dots
Back to Imaging
Challenges: Minimize energies involving . . .
bulk and interfacial energies
vector valued fields
higher order derivatives
discontinuities of underlying fields
Irene Fonseca Variational Methods in Materials and Image Processing
ImagingMicromagnetics
FoamsQuantum Dots
Back to Imaging
Challenges: Minimize energies involving . . .
bulk and interfacial energies
vector valued fields
higher order derivatives
discontinuities of underlying fields
Irene Fonseca Variational Methods in Materials and Image Processing
ImagingMicromagnetics
FoamsQuantum Dots
Back to Imaging
Challenges: Minimize energies involving . . .
bulk and interfacial energies
vector valued fields
higher order derivatives
discontinuities of underlying fields
Irene Fonseca Variational Methods in Materials and Image Processing
ImagingMicromagnetics
FoamsQuantum Dots
Back to Imaging
Why Do We Care?
Imaging
Micromagnetic Materials
Foams
Quantum Dots
Thin Structures
etc.
Irene Fonseca Variational Methods in Materials and Image Processing
ImagingMicromagnetics
FoamsQuantum Dots
Back to Imaging
Why Do We Care?
Imaging
Micromagnetic Materials
Foams
Quantum Dots
Thin Structures
etc.
Irene Fonseca Variational Methods in Materials and Image Processing
ImagingMicromagnetics
FoamsQuantum Dots
Back to Imaging
Why Do We Care?
Imaging
Micromagnetic Materials
Foams
Quantum Dots
Thin Structures
etc.
Irene Fonseca Variational Methods in Materials and Image Processing
ImagingMicromagnetics
FoamsQuantum Dots
Back to Imaging
Why Do We Care?
Imaging
Micromagnetic Materials
Foams
Quantum Dots
Thin Structures
etc.
Irene Fonseca Variational Methods in Materials and Image Processing
ImagingMicromagnetics
FoamsQuantum Dots
Back to Imaging
Why Do We Care?
Imaging
Micromagnetic Materials
Foams
Quantum Dots
Thin Structures
etc.
Irene Fonseca Variational Methods in Materials and Image Processing
ImagingMicromagnetics
FoamsQuantum Dots
Back to Imaging
Why Do We Care?
Imaging
Micromagnetic Materials
Foams
Quantum Dots
Thin Structures
etc.
Irene Fonseca Variational Methods in Materials and Image Processing
ImagingMicromagnetics
FoamsQuantum Dots
Back to Imaging
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)
Irene Fonseca Variational Methods in Materials and Image Processing
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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.]
Irene Fonseca Variational Methods in Materials and Image Processing
ImagingMicromagnetics
FoamsQuantum Dots
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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.]
Irene Fonseca Variational Methods in Materials and Image Processing
ImagingMicromagnetics
FoamsQuantum Dots
Back to Imaging
“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.]
Irene Fonseca Variational Methods in Materials and Image Processing
ImagingMicromagnetics
FoamsQuantum Dots
Back to Imaging
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.]
Irene Fonseca Variational Methods in Materials and Image Processing
ImagingMicromagnetics
FoamsQuantum Dots
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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.]
Irene Fonseca Variational Methods in Materials and Image Processing
ImagingMicromagnetics
FoamsQuantum Dots
Back to Imaging
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.]
Irene Fonseca Variational Methods in Materials and Image Processing
ImagingMicromagnetics
FoamsQuantum Dots
Back to Imaging
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.]
Irene Fonseca Variational Methods in Materials and Image Processing
ImagingMicromagnetics
FoamsQuantum Dots
Back to Imaging
The ContextThe Phase Field ModelSome Conclusions
surfactants and foams: a singular perturbation problem
Irene Fonseca Variational Methods in Materials and Image Processing
ImagingMicromagnetics
FoamsQuantum Dots
Back to Imaging
The ContextThe Phase Field ModelSome Conclusions
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
Irene Fonseca Variational Methods in Materials and Image Processing
ImagingMicromagnetics
FoamsQuantum Dots
Back to Imaging
The ContextThe Phase Field ModelSome Conclusions
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
Irene Fonseca Variational Methods in Materials and Image Processing
ImagingMicromagnetics
FoamsQuantum Dots
Back to Imaging
The ContextThe Phase Field ModelSome Conclusions
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
Irene Fonseca Variational Methods in Materials and Image Processing
ImagingMicromagnetics
FoamsQuantum Dots
Back to Imaging
The ContextThe Phase Field ModelSome Conclusions
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
Irene Fonseca Variational Methods in Materials and Image Processing
ImagingMicromagnetics
FoamsQuantum Dots
Back to Imaging
The ContextThe Phase Field ModelSome Conclusions
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)
Irene Fonseca Variational Methods in Materials and Image Processing
ImagingMicromagnetics
FoamsQuantum Dots
Back to Imaging
The ContextThe Phase Field ModelSome Conclusions
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)
Irene Fonseca Variational Methods in Materials and Image Processing
ImagingMicromagnetics
FoamsQuantum Dots
Back to Imaging
The ContextThe Phase Field ModelSome Conclusions
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)
Irene Fonseca Variational Methods in Materials and Image Processing
ImagingMicromagnetics
FoamsQuantum Dots
Back to Imaging
The ContextThe Phase Field ModelSome Conclusions
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)
Irene Fonseca Variational Methods in Materials and Image Processing
ImagingMicromagnetics
FoamsQuantum Dots
Back to Imaging
The ContextThe Phase Field ModelSome Conclusions
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)
Irene Fonseca Variational Methods in Materials and Image Processing
ImagingMicromagnetics
FoamsQuantum Dots
Back to Imaging
The ContextThe Phase Field ModelSome Conclusions
Plateau borders/struts; nodes/joints
Irene Fonseca Variational Methods in Materials and Image Processing
ImagingMicromagnetics
FoamsQuantum Dots
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The ContextThe Phase Field ModelSome Conclusions
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
Irene Fonseca Variational Methods in Materials and Image Processing
ImagingMicromagnetics
FoamsQuantum Dots
Back to Imaging
The ContextThe Phase Field ModelSome Conclusions
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
Irene Fonseca Variational Methods in Materials and Image Processing
ImagingMicromagnetics
FoamsQuantum Dots
Back to Imaging
The ContextThe Phase Field ModelSome Conclusions
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
Irene Fonseca Variational Methods in Materials and Image Processing
ImagingMicromagnetics
FoamsQuantum Dots
Back to Imaging
The ContextThe Phase Field ModelSome Conclusions
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
Irene Fonseca Variational Methods in Materials and Image Processing
ImagingMicromagnetics
FoamsQuantum Dots
Back to Imaging
The ContextThe Phase Field ModelSome Conclusions
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
Irene Fonseca Variational Methods in Materials and Image Processing
ImagingMicromagnetics
FoamsQuantum Dots
Back to Imaging
The ContextThe Phase Field ModelSome Conclusions
micelle
Irene Fonseca Variational Methods in Materials and Image Processing
ImagingMicromagnetics
FoamsQuantum Dots
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The ContextThe Phase Field ModelSome Conclusions
monolayers of surfactant
Irene Fonseca Variational Methods in Materials and Image Processing
ImagingMicromagnetics
FoamsQuantum Dots
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The ContextThe Phase Field ModelSome Conclusions
surfactants effects
determine macroscopic properties
segregate to interfaces
facilitate formation of interfaces (wetness)
Irene Fonseca Variational Methods in Materials and Image Processing
ImagingMicromagnetics
FoamsQuantum Dots
Back to Imaging
The ContextThe Phase Field ModelSome Conclusions
surfactants effects
determine macroscopic properties
segregate to interfaces
facilitate formation of interfaces (wetness)
Irene Fonseca Variational Methods in Materials and Image Processing
ImagingMicromagnetics
FoamsQuantum Dots
Back to Imaging
The ContextThe Phase Field ModelSome Conclusions
surfactants effects
determine macroscopic properties
segregate to interfaces
facilitate formation of interfaces (wetness)
Irene Fonseca Variational Methods in Materials and Image Processing
ImagingMicromagnetics
FoamsQuantum Dots
Back to Imaging
The ContextThe Phase Field ModelSome Conclusions
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
Irene Fonseca Variational Methods in Materials and Image Processing
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The ContextThe Phase Field ModelSome Conclusions
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
Irene Fonseca Variational Methods in Materials and Image Processing
ImagingMicromagnetics
FoamsQuantum Dots
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The ContextThe Phase Field ModelSome Conclusions
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
Irene Fonseca Variational Methods in Materials and Image Processing
ImagingMicromagnetics
FoamsQuantum Dots
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The ContextThe Phase Field ModelSome Conclusions
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
Irene Fonseca Variational Methods in Materials and Image Processing
ImagingMicromagnetics
FoamsQuantum Dots
Back to Imaging
The ContextThe Phase Field ModelSome Conclusions
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
Irene Fonseca Variational Methods in Materials and Image Processing
ImagingMicromagnetics
FoamsQuantum Dots
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The ContextThe Phase Field ModelSome Conclusions
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
Irene Fonseca Variational Methods in Materials and Image Processing
ImagingMicromagnetics
FoamsQuantum Dots
Back to Imaging
The ContextThe Phase Field ModelSome Conclusions
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
Irene Fonseca Variational Methods in Materials and Image Processing
ImagingMicromagnetics
FoamsQuantum Dots
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The ContextThe Phase Field ModelSome Conclusions
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
Irene Fonseca Variational Methods in Materials and Image Processing
ImagingMicromagnetics
FoamsQuantum Dots
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The ContextThe Phase Field ModelSome Conclusions
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
Irene Fonseca Variational Methods in Materials and Image Processing
ImagingMicromagnetics
FoamsQuantum Dots
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The ContextThe Phase Field ModelSome Conclusions
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
Irene Fonseca Variational Methods in Materials and Image Processing
ImagingMicromagnetics
FoamsQuantum Dots
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The ContextThe Phase Field ModelSome Conclusions
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
Irene Fonseca Variational Methods in Materials and Image Processing
ImagingMicromagnetics
FoamsQuantum Dots
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The ContextThe Phase Field ModelSome Conclusions
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
Irene Fonseca Variational Methods in Materials and Image Processing
ImagingMicromagnetics
FoamsQuantum Dots
Back to Imaging
The ContextThe Phase Field ModelSome Conclusions
Theorem, still
The energy density φ
Irene Fonseca Variational Methods in Materials and Image Processing
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The ContextThe Phase Field ModelSome Conclusions
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
Irene Fonseca Variational Methods in Materials and Image Processing
ImagingMicromagnetics
FoamsQuantum Dots
Back to Imaging
The ContextThe Phase Field ModelSome Conclusions
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
Irene Fonseca Variational Methods in Materials and Image Processing
ImagingMicromagnetics
FoamsQuantum Dots
Back to Imaging
The ContextThe Phase Field ModelSome Conclusions
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
Irene Fonseca Variational Methods in Materials and Image Processing
ImagingMicromagnetics
FoamsQuantum Dots
Back to Imaging
The ContextThe Phase Field ModelSome Conclusions
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
Irene Fonseca Variational Methods in Materials and Image Processing
ImagingMicromagnetics
FoamsQuantum Dots
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The ContextThe Phase Field ModelSome Conclusions
the optimal profile
The solution u to the optimal profile problem and the corresponding ρ
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The ContextThe Phase Field ModelSome Conclusions
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.
Irene Fonseca Variational Methods in Materials and Image Processing
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The ContextThe Phase Field ModelSome Conclusions
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 . . .
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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The ContextThe Phase Field ModelSome Conclusions
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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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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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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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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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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The ContextSharp Interface ModelSome Conclusions
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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The ContextSharp Interface ModelSome Conclusions
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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The ContextSharp Interface ModelSome Conclusions
wetting layer and zero contact angle
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The ContextSharp Interface ModelSome Conclusions
wetting layer and zero contact angle, islands
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The ContextSharp Interface ModelSome Conclusions
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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The ContextSharp Interface ModelSome Conclusions
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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The ContextSharp Interface ModelSome Conclusions
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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The ContextSharp Interface ModelSome Conclusions
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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The ContextSharp Interface ModelSome Conclusions
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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The ContextSharp Interface ModelSome Conclusions
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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The ContextSharp Interface ModelSome Conclusions
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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The ContextSharp Interface ModelSome Conclusions
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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The ContextSharp Interface ModelSome Conclusions
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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Vector-Valued: Inpainting/Recolorization
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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Vector-Valued: Inpainting/Recolorization
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.
Irene Fonseca Variational Methods in Materials and Image Processing
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Vector-Valued: Inpainting/Recolorization
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.
Irene Fonseca Variational Methods in Materials and Image Processing
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Vector-Valued: Inpainting/Recolorization
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.
Irene Fonseca Variational Methods in Materials and Image Processing
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Vector-Valued: Inpainting/Recolorization
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.
Irene Fonseca Variational Methods in Materials and Image Processing
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Vector-Valued: Inpainting/Recolorization
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.
Irene Fonseca Variational Methods in Materials and Image Processing
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FoamsQuantum Dots
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Vector-Valued: Inpainting/Recolorization
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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Vector-Valued: Inpainting/Recolorization
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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Vector-Valued: Inpainting/Recolorization
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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Vector-Valued: Inpainting/Recolorization
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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Vector-Valued: Inpainting/Recolorization
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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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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Vector-Valued: Inpainting/Recolorization
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.
Irene Fonseca Variational Methods in Materials and Image Processing
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Vector-Valued: Inpainting/Recolorization
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.
Irene Fonseca Variational Methods in Materials and Image Processing
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Vector-Valued: Inpainting/Recolorization
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.
Irene Fonseca Variational Methods in Materials and Image Processing
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Vector-Valued: Inpainting/Recolorization
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.
Irene Fonseca Variational Methods in Materials and Image Processing
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Vector-Valued: Inpainting/Recolorization
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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Vector-Valued: Inpainting/Recolorization
Two reconstructions by Fornasier-March
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Vector-Valued: Inpainting/Recolorization
Two reconstructions by Fornasier-March
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Vector-Valued: Inpainting/Recolorization
Two reconstructions by Fornasier-March
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Vector-Valued: Inpainting/Recolorization
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
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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
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
Mathematical simplification: Restrict the analysis to piecewiseconstant images u0
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)
subject to u = u0 in R \ D and u · e = u0 · e in D.
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
Recall that u0 =∑N
k=1 ξk1Ωkis reconstructible over D if it is
the unique minimizer to
minu ∈ BV (R;R3)
|Du|(R) (P)
subject to u = u0 in R \ D and u · e = u0 · e in D.
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
Recall that u0 =∑N
k=1 ξk1Ωkis reconstructible over D if it is
the unique minimizer to
minu ∈ BV (R;R3)
|Du|(R) (P)
subject to u = u0 in R \ D and u · e = u0 · e in D.
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
Recall that u0 =∑N
k=1 ξk1Ωkis reconstructible over D if it is
the unique minimizer to
minu ∈ BV (R;R3)
|Du|(R) (P)
subject to u = u0 in R \ D and u · e = u0 · e in D.
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
Recall that u0 =∑N
k=1 ξk1Ωkis reconstructible over D if it is
the unique minimizer to
minu ∈ BV (R;R3)
|Du|(R) (P)
subject to u = u0 in R \ D and u · e = u0 · e in D.
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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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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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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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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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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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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A simple counterexample when ‖zk‖∞ < 1 is not satisfied
A simple counterexample when ‖zk‖∞ < 1 is not satisfied
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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Adjoint colors have the same gray levels: may createspurious edges
A simple analytical counterexample
Original image u0:
Resulting image u:
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Adjoint colors have the same gray levels: may createspurious edges
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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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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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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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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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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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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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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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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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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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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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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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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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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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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ε-uniformly distributed undamaged regions
Figure: An ε-uniformly distributedundamaged region.
Figure: The damaged regioncontains a δ-neighborhoodΓ(δ) of Γ.
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ε-uniformly distributed undamaged regions
Figure: An ε-uniformly distributedundamaged region.
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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Admissible ε-uniformly distributed undamaged regions
Figure: Denote by Dε the damagedregion
Figure: The original u0.
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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Admissible ε-uniformly distributed undamaged regions
Figure: Denote by Dε the damagedregion
Figure: The original u0.
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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Admissible ε-uniformly distributed undamaged regions
Figure: Denote by Dε the damagedregion
Figure: The original u0.
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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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).
Irene Fonseca Variational Methods in Materials and Image Processing