Variational Methods in Materials Science and Image Processing · Variational Methods in Materials...
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ImagingQuantum Dots
Variational Methods in Materials Science andImage Processing
Irene Fonseca
Department of Mathematical SciencesCenter for Nonlinear Analysis
Carnegie Mellon UniversitySupported by the National Science Foundation (NSF)
Irene Fonseca Variational Methods in Materials Science and Image Processing
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ImagingQuantum Dots
Challenges: Minimize energies involving . . .
bulk and interfacial energies
vector valued fields
higher order derivatives
discontinuities of underlying fields
Irene Fonseca Variational Methods in Materials Science and Image Processing
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ImagingQuantum Dots
Challenges: Minimize energies involving . . .
bulk and interfacial energies
vector valued fields
higher order derivatives
discontinuities of underlying fields
Irene Fonseca Variational Methods in Materials Science and Image Processing
![Page 4: Variational Methods in Materials Science and Image Processing · Variational Methods in Materials Science and Image Processing Irene Fonseca Department of Mathematical Sciences Center](https://reader036.fdocuments.us/reader036/viewer/2022062602/5edc1424ad6a402d66669818/html5/thumbnails/4.jpg)
ImagingQuantum Dots
Challenges: Minimize energies involving . . .
bulk and interfacial energies
vector valued fields
higher order derivatives
discontinuities of underlying fields
Irene Fonseca Variational Methods in Materials Science and Image Processing
![Page 5: Variational Methods in Materials Science and Image Processing · Variational Methods in Materials Science and Image Processing Irene Fonseca Department of Mathematical Sciences Center](https://reader036.fdocuments.us/reader036/viewer/2022062602/5edc1424ad6a402d66669818/html5/thumbnails/5.jpg)
ImagingQuantum Dots
Challenges: Minimize energies involving . . .
bulk and interfacial energies
vector valued fields
higher order derivatives
discontinuities of underlying fields
Irene Fonseca Variational Methods in Materials Science and Image Processing
![Page 6: Variational Methods in Materials Science and Image Processing · Variational Methods in Materials Science and Image Processing Irene Fonseca Department of Mathematical Sciences Center](https://reader036.fdocuments.us/reader036/viewer/2022062602/5edc1424ad6a402d66669818/html5/thumbnails/6.jpg)
ImagingQuantum Dots
Challenges: Minimize energies involving . . .
bulk and interfacial energies
vector valued fields
higher order derivatives
discontinuities of underlying fields
Irene Fonseca Variational Methods in Materials Science and Image Processing
![Page 7: Variational Methods in Materials Science and Image Processing · Variational Methods in Materials Science and Image Processing Irene Fonseca Department of Mathematical Sciences Center](https://reader036.fdocuments.us/reader036/viewer/2022062602/5edc1424ad6a402d66669818/html5/thumbnails/7.jpg)
ImagingQuantum Dots
Why Do We Care?
Imaging
Quantum Dots
Foams
Micromagnetic Materials
Thin Structures
etc.
Irene Fonseca Variational Methods in Materials Science and Image Processing
![Page 8: Variational Methods in Materials Science and Image Processing · Variational Methods in Materials Science and Image Processing Irene Fonseca Department of Mathematical Sciences Center](https://reader036.fdocuments.us/reader036/viewer/2022062602/5edc1424ad6a402d66669818/html5/thumbnails/8.jpg)
ImagingQuantum Dots
Why Do We Care?
Imaging
Quantum Dots
Foams
Micromagnetic Materials
Thin Structures
etc.
Irene Fonseca Variational Methods in Materials Science and Image Processing
![Page 9: Variational Methods in Materials Science and Image Processing · Variational Methods in Materials Science and Image Processing Irene Fonseca Department of Mathematical Sciences Center](https://reader036.fdocuments.us/reader036/viewer/2022062602/5edc1424ad6a402d66669818/html5/thumbnails/9.jpg)
ImagingQuantum Dots
Why Do We Care?
Imaging
Quantum Dots
Foams
Micromagnetic Materials
Thin Structures
etc.
Irene Fonseca Variational Methods in Materials Science and Image Processing
![Page 10: Variational Methods in Materials Science and Image Processing · Variational Methods in Materials Science and Image Processing Irene Fonseca Department of Mathematical Sciences Center](https://reader036.fdocuments.us/reader036/viewer/2022062602/5edc1424ad6a402d66669818/html5/thumbnails/10.jpg)
ImagingQuantum Dots
Why Do We Care?
Imaging
Quantum Dots
Foams
Micromagnetic Materials
Thin Structures
etc.
Irene Fonseca Variational Methods in Materials Science and Image Processing
![Page 11: Variational Methods in Materials Science and Image Processing · Variational Methods in Materials Science and Image Processing Irene Fonseca Department of Mathematical Sciences Center](https://reader036.fdocuments.us/reader036/viewer/2022062602/5edc1424ad6a402d66669818/html5/thumbnails/11.jpg)
ImagingQuantum Dots
Why Do We Care?
Imaging
Quantum Dots
Foams
Micromagnetic Materials
Thin Structures
etc.
Irene Fonseca Variational Methods in Materials Science and Image Processing
![Page 12: Variational Methods in Materials Science and Image Processing · Variational Methods in Materials Science and Image Processing Irene Fonseca Department of Mathematical Sciences Center](https://reader036.fdocuments.us/reader036/viewer/2022062602/5edc1424ad6a402d66669818/html5/thumbnails/12.jpg)
ImagingQuantum Dots
Why Do We Care?
Imaging
Quantum Dots
Foams
Micromagnetic Materials
Thin Structures
etc.
Irene Fonseca Variational Methods in Materials Science and Image Processing
![Page 13: Variational Methods in Materials Science and Image Processing · Variational Methods in Materials Science and Image Processing Irene Fonseca Department of Mathematical Sciences Center](https://reader036.fdocuments.us/reader036/viewer/2022062602/5edc1424ad6a402d66669818/html5/thumbnails/13.jpg)
ImagingQuantum Dots
Here . . .
Imaging
Quantum Dots
Foams
Micromagnetic Materials
Thin Structures
etc.
Irene Fonseca Variational Methods in Materials Science and Image Processing
![Page 14: Variational Methods in Materials Science and Image Processing · Variational Methods in Materials Science and Image Processing Irene Fonseca Department of Mathematical Sciences Center](https://reader036.fdocuments.us/reader036/viewer/2022062602/5edc1424ad6a402d66669818/html5/thumbnails/14.jpg)
ImagingQuantum Dots
Here . . .
Imaging
Quantum Dots
Foams
Micromagnetic Materials
Thin Structures
etc.
Irene Fonseca Variational Methods in Materials Science and Image Processing
![Page 15: Variational Methods in Materials Science and Image Processing · Variational Methods in Materials Science and Image Processing Irene Fonseca Department of Mathematical Sciences Center](https://reader036.fdocuments.us/reader036/viewer/2022062602/5edc1424ad6a402d66669818/html5/thumbnails/15.jpg)
ImagingQuantum Dots
Outline
• black and white – the Mumford-Shah model;
• Rudin-Osher-Fatemi(ROF) model: staircasing;
• second-order models;
• denoising;
• colors – the RGB model;
• reconstructible images – uniformly sparse region.
Irene Fonseca Variational Methods in Materials Science and Image Processing
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ImagingQuantum Dots
“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 LNbΩ + [u]⊗ ν HN−1bS(u) + C (u)De Giorgi, Ambrosio, Bertozzi, Carriero, Chambolle, Chan, Esedoglu, Leaci, P. L.
Lions, Luminita, Y. Meyer, Morel, Osher, et. al.
Irene Fonseca Variational Methods in Materials Science and Image Processing
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ImagingQuantum Dots
“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 LNbΩ + [u]⊗ ν HN−1bS(u) + C (u)De Giorgi, Ambrosio, Bertozzi, Carriero, Chambolle, Chan, Esedoglu, Leaci, P. L.
Lions, Luminita, Y. Meyer, Morel, Osher, et. al.
Irene Fonseca Variational Methods in Materials Science and Image Processing
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ImagingQuantum Dots
“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 LNbΩ + [u]⊗ ν HN−1bS(u) + C (u)De Giorgi, Ambrosio, Bertozzi, Carriero, Chambolle, Chan, Esedoglu, Leaci, P. L.
Lions, Luminita, Y. Meyer, Morel, Osher, et. al.
Irene Fonseca Variational Methods in Materials Science and Image Processing
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ImagingQuantum Dots
The Rudin-Osher-Fatemi Model
ROFλ,f (u) := |u′|(]a, b[) + λ
∫ b
a(u − f )2 dx u ∈ BV (]a, b[)
Lemma [Exact minimizers for ROFλ,f ].
f : [a, b]→ [0, 1] nondecreasing,f+(a) = 0 and f−(b) = 1,The unique minimizer of ROFλ,f is
u(x) :=
c1 if a ≤ x ≤ f −1(c1) ,
f (x) if f −1(c1) < x ≤ f −1(c2) ,
c2 if f −1(c2) < x ≤ b
f −1(c) := infx ∈ [a, b] : f (x) ≥ c, 0 < c1 < c2 < 1 s.t.
2λ∫ f −1(c1)a (c1 − f (x)) dx = 1, 2λ
∫ bf −1(c2)(f (x)− c2) dx = 1.
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ImagingQuantum Dots
The Rudin-Osher-Fatemi Model
ROFλ,f (u) := |u′|(]a, b[) + λ
∫ b
a(u − f )2 dx u ∈ BV (]a, b[)
Lemma [Exact minimizers for ROFλ,f ].
f : [a, b]→ [0, 1] nondecreasing,f+(a) = 0 and f−(b) = 1,The unique minimizer of ROFλ,f is
u(x) :=
c1 if a ≤ x ≤ f −1(c1) ,
f (x) if f −1(c1) < x ≤ f −1(c2) ,
c2 if f −1(c2) < x ≤ b
f −1(c) := infx ∈ [a, b] : f (x) ≥ c, 0 < c1 < c2 < 1 s.t.
2λ∫ f −1(c1)a (c1 − f (x)) dx = 1, 2λ
∫ bf −1(c2)(f (x)− c2) dx = 1.
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ImagingQuantum Dots
The Rudin-Osher-Fatemi Model: staircasingT. Chan, A. Marquina and P. Mulet, SIAM J. Sci. Comput. 22 (2000), 503–516
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ImagingQuantum Dots
The Rudin-Osher-Fatemi Model: staircasing
Staircasing: “ramps” (i.e. affine regions) in the original image yieldstaircase-like structures in the reconstructed image.Original edges are preserved BUT artificial/spurious ones are created. . . “staircasing effect”
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ImagingQuantum Dots
The Rudin-Osher-Fatemi Model: staircasing. An example.
Other examples of staircasing also by Caselles, Chambolle and Novaga
f (x) := x , x ∈ [0, 1] . . . original 1D imageadd “noise”
hn (x) :=i
n− x if
i − 1
n≤ x <
i
n, i = 1, . . . , n
resulting degraded 1D image
fn (x) :=i
nif
i − 1
n≤ x <
i
n, i = 1, . . . , n
Rmk: even though hn → 0 uniformly, the reconstructed image un
preserves the staircase structure of fn.
Theorem.
λ > 4, un . . . unique minimizer of ROFλ,fn in BV (]0, 1[). For nsufficiently large there exist 0 < an < bn < 1,an → 1√
λ, bn → 1− 1√
λ,
un = fn on [an, bn] , un is constant on [0, an) and (bn, 1].
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ImagingQuantum Dots
Second Order Models: The Blake-Zisserman Model
Leaci and Tomarelli, et.al.
E (u) =
∫Ω
W (∇u,∇2u) dx + |u − f |2dx +
∫S(∇u)
γ(ν)dHN−1
Also, Geman and Reynolds, Chambolle and Lions, Blomgren, Chan and Mulet, Kinderman, Osher and Jones, etc.
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ImagingQuantum Dots
Chan et.al. Model
With G. Dal Maso, G. Leoni, M. Morini
Fp(u) =
∫Ω
(|∇u|+ |u − f |2
)dx +
∫Ωψ(|∇u|)|∇2u|p dx
p ≥ 1, ψ ∼ 0 at ∞
∫ ∞∞
(ψ(t))1/p dt < +∞, inft∈K
ψ(t) > 0
for every compact K ⊂ R
All 1D!
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ImagingQuantum Dots
Chan et.al. Model
With G. Dal Maso, G. Leoni, M. Morini
Fp(u) =
∫Ω
(|∇u|+ |u − f |2
)dx +
∫Ωψ(|∇u|)|∇2u|p dx
p ≥ 1, ψ ∼ 0 at ∞
∫ ∞∞
(ψ(t))1/p dt < +∞, inft∈K
ψ(t) > 0
for every compact K ⊂ R
All 1D!
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ImagingQuantum Dots
Chan et.al. Model
With G. Dal Maso, G. Leoni, M. Morini
Fp(u) =
∫Ω
(|∇u|+ |u − f |2
)dx +
∫Ωψ(|∇u|)|∇2u|p dx
p ≥ 1, ψ ∼ 0 at ∞
∫ ∞∞
(ψ(t))1/p dt < +∞, inft∈K
ψ(t) > 0
for every compact K ⊂ R
All 1D!
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ImagingQuantum Dots
p ∈ [1,+∞)
Fp(u) :=
∫ b
a|u′| dx +
∫ b
aψ(|u′|)|u′′|p dx
E.g.
ψ(t) :=1
(1 + t2)12
(3p−1)
the functional becomes
∫ b
a|u′| dx +
∫Graph u
|k |p dH1
k . . . curvature of the graph of uin many computer vision and graphics applications, such as cornerpreserving geometry, denoising and segmentation with depth
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ImagingQuantum Dots
p ∈ [1,+∞)
Fp(u) :=
∫ b
a|u′| dx +
∫ b
aψ(|u′|)|u′′|p dx
E.g.
ψ(t) :=1
(1 + t2)12
(3p−1)
the functional becomes
∫ b
a|u′| dx +
∫Graph u
|k |p dH1
k . . . curvature of the graph of uin many computer vision and graphics applications, such as cornerpreserving geometry, denoising and segmentation with depth
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ImagingQuantum Dots
a few results. . .
framework: minimization problem is well posed;
compactness;
integral representation of the relaxed functional:
Fp (u) := inf
lim infk→+∞
Fp (uk) : uk → u in L1(]a, b[)
higher order regularization eliminates staircasing effectfk := f + hk , f smooth, hk
∗ 0
Is uk smooth for k >> 1 ?Yes: ||uk − u||W 1,p → 0 if p = 1, ||uk − u||C1 → 0 if p > 1
Note: piecewise constant functions are approximable by sequenceswith bounded energy only for p = 1!
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ImagingQuantum Dots
a few results. . .
framework: minimization problem is well posed;
compactness;
integral representation of the relaxed functional:
Fp (u) := inf
lim infk→+∞
Fp (uk) : uk → u in L1(]a, b[)
higher order regularization eliminates staircasing effectfk := f + hk , f smooth, hk
∗ 0
Is uk smooth for k >> 1 ?Yes: ||uk − u||W 1,p → 0 if p = 1, ||uk − u||C1 → 0 if p > 1
Note: piecewise constant functions are approximable by sequenceswith bounded energy only for p = 1!
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ImagingQuantum Dots
a few results. . .
framework: minimization problem is well posed;
compactness;
integral representation of the relaxed functional:
Fp (u) := inf
lim infk→+∞
Fp (uk) : uk → u in L1(]a, b[)
higher order regularization eliminates staircasing effectfk := f + hk , f smooth, hk
∗ 0
Is uk smooth for k >> 1 ?Yes: ||uk − u||W 1,p → 0 if p = 1, ||uk − u||C1 → 0 if p > 1
Note: piecewise constant functions are approximable by sequenceswith bounded energy only for p = 1!
Irene Fonseca Variational Methods in Materials Science and Image Processing
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ImagingQuantum Dots
a few results. . .
framework: minimization problem is well posed;
compactness;
integral representation of the relaxed functional:
Fp (u) := inf
lim infk→+∞
Fp (uk) : uk → u in L1(]a, b[)
higher order regularization eliminates staircasing effectfk := f + hk , f smooth, hk
∗ 0
Is uk smooth for k >> 1 ?Yes: ||uk − u||W 1,p → 0 if p = 1, ||uk − u||C1 → 0 if p > 1
Note: piecewise constant functions are approximable by sequenceswith bounded energy only for p = 1!
Irene Fonseca Variational Methods in Materials Science and Image Processing
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ImagingQuantum Dots
a few results. . .
framework: minimization problem is well posed;
compactness;
integral representation of the relaxed functional:
Fp (u) := inf
lim infk→+∞
Fp (uk) : uk → u in L1(]a, b[)
higher order regularization eliminates staircasing effectfk := f + hk , f smooth, hk
∗ 0
Is uk smooth for k >> 1 ?Yes: ||uk − u||W 1,p → 0 if p = 1, ||uk − u||C1 → 0 if p > 1
Note: piecewise constant functions are approximable by sequenceswith bounded energy only for p = 1!
Irene Fonseca Variational Methods in Materials Science and Image Processing
![Page 35: Variational Methods in Materials Science and Image Processing · Variational Methods in Materials Science and Image Processing Irene Fonseca Department of Mathematical Sciences Center](https://reader036.fdocuments.us/reader036/viewer/2022062602/5edc1424ad6a402d66669818/html5/thumbnails/35.jpg)
ImagingQuantum Dots
Denoising
With R. Choksi and B. Zwicknagl
Given: Measured signal, disturbed by noise
f = f0 + n, n − noise
Want: Reconstruction of clean f0
Tool: Regularized approximation
Minimize J(u) := ||u||kH + λ||u − f ||mW , ; k ,m ∈ N
Questions: “Good” choice of• fidelity measure || · ||W• regularization measure || · ||H• tunning parameter λ
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ImagingQuantum Dots
Denoising
With R. Choksi and B. Zwicknagl
Given: Measured signal, disturbed by noise
f = f0 + n, n − noise
Want: Reconstruction of clean f0
Tool: Regularized approximation
Minimize J(u) := ||u||kH + λ||u − f ||mW , ; k ,m ∈ N
Questions: “Good” choice of• fidelity measure || · ||W• regularization measure || · ||H• tunning parameter λ
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ImagingQuantum Dots
Properties of a “Good” Model
J(u) := ||u||kH + λ||u − f ||mW• consistency: “simple” clean signals f should be recovered exactly
J(f ) ≤ J(u) for all u
• for a sequence of noise hn 0, minimizers of the disturbedfunctionals
Jn(u) := ||u||kH + λ||u − f−hn||mW k,m ∈ N
should converge to minimizers of J
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ImagingQuantum Dots
Properties of a “Good” Model
J(u) := ||u||kH + λ||u − f ||mW• consistency: “simple” clean signals f should be recovered exactly
J(f ) ≤ J(u) for all u
• for a sequence of noise hn 0, minimizers of the disturbedfunctionals
Jn(u) := ||u||kH + λ||u − f−hn||mW k,m ∈ N
should converge to minimizers of J
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ImagingQuantum Dots
Exact Reconstruction - Consistency
Question: For which f can we reconstruct f exactly?
For all u 6= f
J(f ) ≤ J(u)⇔ ||f ||kH ≤ ||u||kH + λ||u − f ||mW
Hence exact reconstruction if and only if
λ ≥ supu 6=f
||f ||kH − ||u||nHλ||u − f ||mW
So . . . when is
supu 6=f
||f ||kH − ||u||kHλ||u − f ||mW
< +∞?
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ImagingQuantum Dots
Bad News if the Fidelity Term Occurs With Power m > 1!
If m > 1, ||f ||kH 6= 0 then
supu 6=f
||f ||kH − ||u||kHλ||u − f ||mW
= +∞
Choose uε := (1− ε)f . Then
supu 6=f
||f ||kH − ||u||kHλ||u − f ||mW
≥ sup0<ε<1
(1− (1− ε)k)||f ||kHεm||f ||mW
= sup0<ε<1
||f ||kH||f ||mW
k∑j=1
(−1)j+1
(k
j
)e j−m =∞
Classical ROF: J(u) = |u|BV + λ||u − f ||2L2(Ω)
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ImagingQuantum Dots
Bad News if the Fidelity Term Occurs With Power m > 1!
If m > 1, ||f ||kH 6= 0 then
supu 6=f
||f ||kH − ||u||kHλ||u − f ||mW
= +∞
Choose uε := (1− ε)f . Then
supu 6=f
||f ||kH − ||u||kHλ||u − f ||mW
≥ sup0<ε<1
(1− (1− ε)k)||f ||kHεm||f ||mW
= sup0<ε<1
||f ||kH||f ||mW
k∑j=1
(−1)j+1
(k
j
)e j−m =∞
Classical ROF: J(u) = |u|BV + λ||u − f ||2L2(Ω)
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ImagingQuantum Dots
Weakly Vanishing Noise
Assume hn 0 weakly in W.Disturbed functionals
Jn(u) := ||u||kH + λ||u − f−hn||mW
Question: What happens in the limit?
• convergence of minimizers to minimizers?• convergence of the energies?
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ImagingQuantum Dots
Γ-convergence
Assume that• H is compactly embedded in W• Brezis-Lieb Type Condition: For all f ∈ W
||f ||kW = limn→∞
(||f−hn||mW − ||hn||mW)
Recall:Jn(u) := ||u||kH + λ||u − f−hn||mW
Theorem.
Jn Γ-converge to
J(u) := ||u||kH + λ||u − f ||mW + λ limn→∞
||hn||mW
with respect to the weak-* topology in H.
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ImagingQuantum Dots
Γ-convergence
Assume that• H is compactly embedded in W• Brezis-Lieb Type Condition: For all f ∈ W
||f ||kW = limn→∞
(||f−hn||mW − ||hn||mW)
Recall:Jn(u) := ||u||kH + λ||u − f−hn||mW
Theorem.
Jn Γ-converge to
J(u) := ||u||kH + λ||u − f ||mW + λ limn→∞
||hn||mW
with respect to the weak-* topology in H.
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ImagingQuantum Dots
Examples: The Brezis-Lieb Condition Holds
• W is a Hilbert space, m = 2
if hn 0 in W then
||f−hn||2W−||hn||2W = ||f ||2W+||hn||2W−2(f , hn)W−||hn||2W → ||f ||2W
E.g., hn 0 in L2(Ω)
Jn(u) := ||u||W 1,2(Ω) + λ||u − f−hn||2L2(Ω)
Then Jn Γ-converge to
J(u) := ||u||W 1,2(Ω) + λ||u − f ||2L2(Ω) + λ limn→∞
||hn||2L2(Ω)
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ImagingQuantum Dots
Examples: The Brezis-Lieb Condition Holds
• W is a Hilbert space, m = 2
if hn 0 in W then
||f−hn||2W−||hn||2W = ||f ||2W+||hn||2W−2(f , hn)W−||hn||2W → ||f ||2W
E.g., hn 0 in L2(Ω)
Jn(u) := ||u||W 1,2(Ω) + λ||u − f−hn||2L2(Ω)
Then Jn Γ-converge to
J(u) := ||u||W 1,2(Ω) + λ||u − f ||2L2(Ω) + λ limn→∞
||hn||2L2(Ω)
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ImagingQuantum Dots
Concentrations: The Brezis-Lieb Condition Holds
• Can handle concentrations
Let hn 0 in Lp(Ω) and pointwise a.e. to 0
Brezis-Lieb Lemma
0 < p <∞, un → u a.e., supn ||un||Lp <∞Then
limn
(||un||pLp(Ω) − ||un − u||pLp(Ω)
)= ||u||pLp(Ω)
E.g.
hn(x) :=
n − n2x 0 ≤ x ≤ 1/n0 1/n < x ≤ 1
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ImagingQuantum Dots
Vector-Valued: Inpainting/Recolorization
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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ImagingQuantum Dots
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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ImagingQuantum Dots
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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ImagingQuantum Dots
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 Science and Image Processing
![Page 52: Variational Methods in Materials Science and Image Processing · Variational Methods in Materials Science and Image Processing Irene Fonseca Department of Mathematical Sciences Center](https://reader036.fdocuments.us/reader036/viewer/2022062602/5edc1424ad6a402d66669818/html5/thumbnails/52.jpg)
ImagingQuantum Dots
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 Science and Image Processing
![Page 53: Variational Methods in Materials Science and Image Processing · Variational Methods in Materials Science and Image Processing Irene Fonseca Department of Mathematical Sciences Center](https://reader036.fdocuments.us/reader036/viewer/2022062602/5edc1424ad6a402d66669818/html5/thumbnails/53.jpg)
ImagingQuantum Dots
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 Science and Image Processing
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ImagingQuantum Dots
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 Science and Image Processing
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ImagingQuantum Dots
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 Science and Image Processing
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ImagingQuantum Dots
inpainting: RGB model
D ⊂ R ⊂ R2 . . . inpainting regionRGBobserved (u0, v0)u0 . . . correct information on R \ Dv0 . . . distorted information . . . only gray level is known on D;v0 = Lu0
L : R3 → R . . . e.g. L(u) := 13 (r + g + b) or L(ξ) := ξ · e for some
e ∈ 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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ImagingQuantum Dots
inpainting: RGB model
D ⊂ R ⊂ R2 . . . inpainting regionRGBobserved (u0, v0)u0 . . . correct information on R \ Dv0 . . . distorted information . . . only gray level is known on D;v0 = Lu0
L : R3 → R . . . e.g. L(u) := 13 (r + g + b) or L(ξ) := ξ · e for some
e ∈ 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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ImagingQuantum Dots
inpainting: RGB model
D ⊂ R ⊂ R2 . . . inpainting regionRGBobserved (u0, v0)u0 . . . correct information on R \ Dv0 . . . distorted information . . . only gray level is known on D;v0 = Lu0
L : R3 → R . . . e.g. L(u) := 13 (r + g + b) or L(ξ) := ξ · e for some
e ∈ 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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ImagingQuantum Dots
inpainting: RGB model
D ⊂ R ⊂ R2 . . . inpainting regionRGBobserved (u0, v0)u0 . . . correct information on R \ Dv0 . . . distorted information . . . only gray level is known on D;v0 = Lu0
L : R3 → R . . . e.g. L(u) := 13 (r + g + b) or L(ξ) := ξ · e for some
e ∈ 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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ImagingQuantum Dots
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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ImagingQuantum Dots
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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ImagingQuantum Dots
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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ImagingQuantum Dots
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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ImagingQuantum Dots
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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ImagingQuantum Dots
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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ImagingQuantum Dots
a couple of questions. . .
“optimal design” : what is the “best” D? How much color dowe need to provide? And where?
are we creating spurious edges?
For a “cartoon” u in SBV , i.e.
Du = ∇uL2bR + (u+ − u−)⊗ νH1bS(u)
its edges are in . . . spt Dsu = S(u)
sptDsui ⊂ sptDs(L(u0))?
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ImagingQuantum Dots
a couple of questions. . .
“optimal design” : what is the “best” D? How much color dowe need to provide? And where?
are we creating spurious edges?
For a “cartoon” u in SBV , i.e.
Du = ∇uL2bR + (u+ − u−)⊗ νH1bS(u)
its edges are in . . . spt Dsu = S(u)
sptDsui ⊂ sptDs(L(u0))?
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ImagingQuantum Dots
a couple of questions. . .
“optimal design” : what is the “best” D? How much color dowe need to provide? And where?
are we creating spurious edges?
For a “cartoon” u in SBV , i.e.
Du = ∇uL2bR + (u+ − u−)⊗ νH1bS(u)
its edges are in . . . spt Dsu = S(u)
sptDsui ⊂ sptDs(L(u0))?
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ImagingQuantum Dots
Two reconstructions by Fornasier-March
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ImagingQuantum Dots
Two reconstructions by Fornasier-March
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ImagingQuantum Dots
Two reconstructions by Fornasier-March
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ImagingQuantum Dots
Two reconstructions by Fornasier-March
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ImagingQuantum Dots
Our analysis
How faithful is the reconstruction in the infinite fidelity limit ?
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ImagingQuantum Dots
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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ImagingQuantum Dots
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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ImagingQuantum Dots
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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ImagingQuantum Dots
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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ImagingQuantum Dots
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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ImagingQuantum Dots
admissible images
Find conditions on the damaged region D which render u0
reconstructible
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ImagingQuantum Dots
admissible images
Find conditions on the damaged region D which render u0
reconstructible
Mathematical simplification: Restrict the analysis to piecewiseconstant images u0
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ImagingQuantum Dots
admissible images
Find conditions on the damaged region D which make u0
reconstructible
Mathematical simplification: Restrict the analysis to piecewiseconstant images u0
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ImagingQuantum Dots
admissible images
Find conditions on the damaged region D which make u0
reconstructible
Mathematical simplification: Restrict the analysis to piecewiseconstant images u0
R = Γ ∪N⋃
k=1
Ωk , u0 =N∑
k=1
ξk1Ωk,
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ImagingQuantum Dots
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 Science and Image Processing
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ImagingQuantum Dots
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 Science and Image Processing
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ImagingQuantum Dots
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 Science and Image Processing
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ImagingQuantum Dots
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 Science and Image Processing
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ImagingQuantum Dots
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 Science and Image Processing
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ImagingQuantum Dots
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 u0
share 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 accuracy
Minimal requirement: must be reconstructible over S = Γ(δ) forsome δ > 0, where
Γ(δ) := x ∈ R : dist(x , Γ) < δ
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ImagingQuantum Dots
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 u0
share 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 accuracy
Minimal requirement: must be reconstructible over S = Γ(δ) forsome δ > 0, where
Γ(δ) := x ∈ R : dist(x , Γ) < δ
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ImagingQuantum Dots
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 u0
share 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 accuracy
Minimal requirement: must be reconstructible over S = Γ(δ) forsome δ > 0, where
Γ(δ) := x ∈ R : dist(x , Γ) < δ
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ImagingQuantum Dots
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 u0
share 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 accuracy
Minimal requirement: must be reconstructible over S = Γ(δ) forsome δ > 0, where
Γ(δ) := x ∈ R : dist(x , Γ) < δ
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ImagingQuantum Dots
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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ImagingQuantum Dots
A simple counterexample when ‖zk‖∞ < 1 is not satisfied
Original image u0:
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ImagingQuantum Dots
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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ImagingQuantum Dots
Adjoint colors have the same gray levels: may createspurious edges
Original image u0:
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ImagingQuantum Dots
Adjoint colors have the same gray levels: may createspurious edges
A simple analytical counterexample
Original image u0:
Resulting image u:
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ImagingQuantum Dots
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 Science and Image Processing
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ImagingQuantum Dots
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 thatdiv M = 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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ImagingQuantum Dots
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 thatdiv M = 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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ImagingQuantum Dots
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 thatdiv M = 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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ImagingQuantum Dots
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 thatdiv M = 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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ImagingQuantum Dots
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 thatdiv M = 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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ImagingQuantum Dots
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 thatdiv M = 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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ImagingQuantum Dots
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 thatdiv M = 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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ImagingQuantum Dots
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 thatdiv M = 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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ImagingQuantum Dots
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 replacedby the calibration M
Hence, the conditions on M can be considered as a weak formulationof the Euler-Lagrange equations of F .
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ImagingQuantum Dots
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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ImagingQuantum Dots
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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ImagingQuantum Dots
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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ImagingQuantum Dots
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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ImagingQuantum Dots
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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ImagingQuantum Dots
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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ImagingQuantum Dots
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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ImagingQuantum Dots
ε-uniformly distributed undamaged regions
Figure: An ε-uniformly distributedundamaged region.
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ImagingQuantum Dots
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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ImagingQuantum Dots
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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ImagingQuantum Dots
A natural assumption
u0 is stably reconstructible over Γ(δ) for some δ > 0.
⇒
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ImagingQuantum Dots
A natural assumption
u0 is stably reconstructible over Γ(δ) for some δ > 0.
⇒
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ImagingQuantum Dots
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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ImagingQuantum Dots
Admissible ε-uniformly distributed undamaged regions
Figure: Denote by Dε the damagedregion
Figure: The original u0.
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ImagingQuantum Dots
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.
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ImagingQuantum Dots
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.
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ImagingQuantum Dots
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.
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ImagingQuantum Dots
uniformly sparse region: scaling ε2 from below for ω(ε) issharp
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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ImagingQuantum Dots
Outline
• wetting and zero contact angle;
• surface diffusion in epitaxially strained solids;
• shapes of islands;
• steps and terraces in epitaxially strained islands.
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ImagingQuantum Dots
The Context
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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ImagingQuantum Dots
The Context
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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ImagingQuantum Dots
The Context
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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ImagingQuantum Dots
The Context
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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ImagingQuantum Dots
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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ImagingQuantum Dots
quantum dots: the profile . . .
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ImagingQuantum Dots
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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ImagingQuantum Dots
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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ImagingQuantum Dots
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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ImagingQuantum Dots
wetting layer and zero contact angle, islands
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ImagingQuantum Dots
Sharp Interface Model
Brian Spencer, Bonnetier and Chambolle, Chambolle and Larsen; Caflish, W. E, Otto, Voorhees, et. al.
Ωh := x = (x , y) : a < x < b, y < h (x)
h : [a, b]→ [0,∞) ... graph of h is the profile of the film
y = 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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ImagingQuantum Dots
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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ImagingQuantum Dots
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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ImagingQuantum Dots
hard to implement . . .
Sharp interface model is difficult to be implemented numerically.Instead: boundary-layer model; discontinuous transition isregularized over a thin transition region of width δ (“smearingparameter”).
Eδ (y) :=1
2e0
(1 + f
(y
δ
))i⊗ i, y ∈ R,
ϕδ (y) := γsub + (γfilm − γsub) f(y
δ
), y ≥ 0,
f (0) = 0, limy→−∞
f (y) = −1, limy→∞
f (y) = 1.
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ImagingQuantum Dots
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 ≥ γsub
γfilm < γsub
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ImagingQuantum Dots
wetting, etc.
asymptotics as δ → 0+
γfilm < γsub
relaxed 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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ImagingQuantum Dots
wetting, etc.
asymptotics as δ → 0+
γfilm < γsub
relaxed 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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ImagingQuantum Dots
wetting, etc.
asymptotics as δ → 0+
γfilm < γsub
relaxed 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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ImagingQuantum Dots
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 vertical cutswhich 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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ImagingQuantum Dots
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 vertical cutswhich 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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ImagingQuantum Dots
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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ImagingQuantum Dots
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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ImagingQuantum Dots
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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ImagingQuantum Dots
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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ImagingQuantum Dots
next . . .
3D case!
surface diffusion in epitaxially strained solids (2D)
shapes of islands
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ImagingQuantum Dots
surface diffusion in epitaxially strained solids
With N. Fusco, G. Leoni, M. Morini
Einstein-Nernst volume preserving evolution law:
V = C ∆Γµ
V . . . normal velocity of evolving interface∆Γ . . . tangential Laplacianµ . . . chemical potential, first variation of the free-energy functional∫
Ωh
W (E(u)) dx +
∫Γh
ϕ(θ)dH1
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ImagingQuantum Dots
ill-posed . . . so add a perturbation
Get (with C = 1)
V = ((ϕθθ + ϕ)k + W (E(u)))σσ
k . . . curvature of Γh
(·)σ . . . tangential derivativeu(·, t) . . . elastic equilibrium in Ωh(·,t) under periodic b. c.
V =
((ϕθθ + ϕ)k + W (E(u))−ε
(kσσ +
1
2k3
))σσ
H−1- gradient flow for (Cahn and Taylor)∫Ωh
W (E(u)) dx +
∫Γh
(ϕ(θ) +
ε
2k2)
dH1
De Giorgi’s minimizing movements: short time existence,uniqueness, regularity
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ImagingQuantum Dots
ill-posed . . . so add a perturbation
Get (with C = 1)
V = ((ϕθθ + ϕ)k + W (E(u)))σσ
k . . . curvature of Γh
(·)σ . . . tangential derivativeu(·, t) . . . elastic equilibrium in Ωh(·,t) under periodic b. c.
V =
((ϕθθ + ϕ)k + W (E(u))−ε
(kσσ +
1
2k3
))σσ
H−1- gradient flow for (Cahn and Taylor)∫Ωh
W (E(u)) dx +
∫Γh
(ϕ(θ) +
ε
2k2)
dH1
De Giorgi’s minimizing movements: short time existence,uniqueness, regularity
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ImagingQuantum Dots
shapes of islands
With A. Pratelli and B. Zwicknagl
We proved that the shape of the island evolves with the size:
small islands always have the half-pyramid shape, and as thevolume increases the island evolves through a sequence of shapesthat include more facets with increasing steepness – half pyramid,pyramid, half dome, dome, half barn, barn
This validates what was experimentally and numerically obtainedin the physics and materials science literature
More in progress! . . .
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ImagingQuantum Dots
shapes of islands
With A. Pratelli and B. Zwicknagl
We proved that the shape of the island evolves with the size:
small islands always have the half-pyramid shape, and as thevolume increases the island evolves through a sequence of shapesthat include more facets with increasing steepness – half pyramid,pyramid, half dome, dome, half barn, barn
This validates what was experimentally and numerically obtainedin the physics and materials science literature
More in progress! . . .
Irene Fonseca Variational Methods in Materials Science and Image Processing
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ImagingQuantum Dots
shapes of islands
With A. Pratelli and B. Zwicknagl
We proved that the shape of the island evolves with the size:
small islands always have the half-pyramid shape, and as thevolume increases the island evolves through a sequence of shapesthat include more facets with increasing steepness – half pyramid,pyramid, half dome, dome, half barn, barn
This validates what was experimentally and numerically obtainedin the physics and materials science literature
More in progress! . . .
Irene Fonseca Variational Methods in Materials Science and Image Processing