Variational Methods in Materials Science and Image Processing Irene - PowerPoint PPT Presentation

Imaging Quantum Dots Chan et.al. Model With G. Dal Maso, G. Leoni, M. Morini � � ψ ( |∇ u | ) |∇ 2 u | p dx |∇ u | + | u − f | 2 � � F p ( u ) = dx + Ω Ω p ≥ 1, ψ ∼ 0 at ∞ � ∞ ( ψ ( t )) 1 / p dt < + ∞ , t ∈ K ψ ( t ) > 0 inf ∞ for every compact K ⊂ R All 1D! Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots p ∈ [1 , + ∞ ) � b � b ψ ( | u ′ | ) | u ′′ | p dx | u ′ | dx + F p ( u ) := a a E.g. 1 ψ ( t ) := 1 2 (3 p − 1) (1 + t 2 ) the functional becomes � b � | k | p d H 1 | u ′ | dx + a Graph u k . . . curvature of the graph of u in many computer vision and graphics applications, such as corner preserving geometry, denoising and segmentation with depth Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots p ∈ [1 , + ∞ ) � b � b ψ ( | u ′ | ) | u ′′ | p dx | u ′ | dx + F p ( u ) := a a E.g. 1 ψ ( t ) := 1 2 (3 p − 1) (1 + t 2 ) the functional becomes � b � | k | p d H 1 | u ′ | dx + a Graph u k . . . curvature of the graph of u in many computer vision and graphics applications, such as corner preserving geometry, denoising and segmentation with depth Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots a few results. . . framework: minimization problem is well posed; compactness; integral representation of the relaxed functional: � � k → + ∞ F p ( u k ) : u k → u in L 1 (] a , b [) F p ( u ) := inf lim inf higher order regularization eliminates staircasing effect ∗ f k := f + h k , f smooth, h k ⇀ 0 Is u k smooth for k >> 1 ? Yes: || u k − u || W 1 , p → 0 if p = 1, || u k − u || C 1 → 0 if p > 1 Note: piecewise constant functions are approximable by sequences with bounded energy only for p = 1! Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots a few results. . . framework: minimization problem is well posed; compactness; integral representation of the relaxed functional: � � k → + ∞ F p ( u k ) : u k → u in L 1 (] a , b [) F p ( u ) := inf lim inf higher order regularization eliminates staircasing effect ∗ f k := f + h k , f smooth, h k ⇀ 0 Is u k smooth for k >> 1 ? Yes: || u k − u || W 1 , p → 0 if p = 1, || u k − u || C 1 → 0 if p > 1 Note: piecewise constant functions are approximable by sequences with bounded energy only for p = 1! Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots a few results. . . framework: minimization problem is well posed; compactness; integral representation of the relaxed functional: � � k → + ∞ F p ( u k ) : u k → u in L 1 (] a , b [) F p ( u ) := inf lim inf higher order regularization eliminates staircasing effect ∗ f k := f + h k , f smooth, h k ⇀ 0 Is u k smooth for k >> 1 ? Yes: || u k − u || W 1 , p → 0 if p = 1, || u k − u || C 1 → 0 if p > 1 Note: piecewise constant functions are approximable by sequences with bounded energy only for p = 1! Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots a few results. . . framework: minimization problem is well posed; compactness; integral representation of the relaxed functional: � � k → + ∞ F p ( u k ) : u k → u in L 1 (] a , b [) F p ( u ) := inf lim inf higher order regularization eliminates staircasing effect ∗ f k := f + h k , f smooth, h k ⇀ 0 Is u k smooth for k >> 1 ? Yes: || u k − u || W 1 , p → 0 if p = 1, || u k − u || C 1 → 0 if p > 1 Note: piecewise constant functions are approximable by sequences with bounded energy only for p = 1! Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots a few results. . . framework: minimization problem is well posed; compactness; integral representation of the relaxed functional: � � k → + ∞ F p ( u k ) : u k → u in L 1 (] a , b [) F p ( u ) := inf lim inf higher order regularization eliminates staircasing effect ∗ f k := f + h k , f smooth, h k ⇀ 0 Is u k smooth for k >> 1 ? Yes: || u k − u || W 1 , p → 0 if p = 1, || u k − u || C 1 → 0 if p > 1 Note: piecewise constant functions are approximable by sequences with bounded energy only for p = 1! Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots Denoising With R. Choksi and B. Zwicknagl Given: Measured signal, disturbed by noise f = f 0 + n , n − noise Want: Reconstruction of clean f 0 Tool: Regularized approximation J ( u ) := || u || k H + λ || u − f || m W , ; k , m ∈ N Minimize Questions: “Good” choice of • fidelity measure || · || W • regularization measure || · || H • tunning parameter λ Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots Denoising With R. Choksi and B. Zwicknagl Given: Measured signal, disturbed by noise f = f 0 + n , n − noise Want: Reconstruction of clean f 0 Tool: Regularized approximation J ( u ) := || u || k H + λ || u − f || m W , ; k , m ∈ N Minimize Questions: “Good” choice of • fidelity measure || · || W • regularization measure || · || H • tunning parameter λ Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots Properties of a “Good” Model J ( u ) := || u || k H + λ || u − f || m W • consistency: “simple” clean signals f should be recovered exactly J ( f ) ≤ J ( u ) for all u • for a sequence of noise h n ⇀ 0, minimizers of the disturbed functionals J n ( u ) := || u || k H + λ || u − f − h n || m k , m ∈ N W should converge to minimizers of J Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots Properties of a “Good” Model J ( u ) := || u || k H + λ || u − f || m W • consistency: “simple” clean signals f should be recovered exactly J ( f ) ≤ J ( u ) for all u • for a sequence of noise h n ⇀ 0, minimizers of the disturbed functionals J n ( u ) := || u || k H + λ || u − f − h n || m k , m ∈ N W should converge to minimizers of J Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots Exact Reconstruction - Consistency Question: For which f can we reconstruct f exactly? For all u � = f J ( f ) ≤ J ( u ) ⇔ || f || k H ≤ || u || k H + λ || u − f || m W Hence exact reconstruction if and only if || f || k H − || u || n H λ ≥ sup λ || u − f || m u � = f W So . . . when is || f || k H − || u || k H sup < + ∞ ? λ || u − f || m u � = f W Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots Bad News if the Fidelity Term Occurs With Power m > 1! If m > 1, || f || k H � = 0 then || f || k H − || u || k H sup = + ∞ λ || u − f || m u � = f W Choose u ε := (1 − ε ) f . Then || f || k H − || u || k (1 − (1 − ε ) k ) || f || k H H sup ≥ sup λ || u − f || m ε m || f || m 0 <ε< 1 u � = f W W k || f || k � k � e j − m = ∞ H � ( − 1) j +1 = sup || f || m j 0 <ε< 1 W j =1 Classical ROF: J ( u ) = | u | BV + λ || u − f || 2 L 2 (Ω) Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots Bad News if the Fidelity Term Occurs With Power m > 1! If m > 1, || f || k H � = 0 then || f || k H − || u || k H sup = + ∞ λ || u − f || m u � = f W Choose u ε := (1 − ε ) f . Then || f || k H − || u || k (1 − (1 − ε ) k ) || f || k H H sup ≥ sup λ || u − f || m ε m || f || m 0 <ε< 1 u � = f W W k || f || k � k � e j − m = ∞ H � ( − 1) j +1 = sup || f || m j 0 <ε< 1 W j =1 Classical ROF: J ( u ) = | u | BV + λ || u − f || 2 L 2 (Ω) Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots Weakly Vanishing Noise Assume h n ⇀ 0 weakly in W . Disturbed functionals J n ( u ) := || u || k H + λ || u − f − h n || m W Question: What happens in the limit? • convergence of minimizers to minimizers? • convergence of the energies? Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots Γ-convergence Assume that • H is compactly embedded in W • Brezis-Lieb Type Condition : For all f ∈ W || f || k n →∞ ( || f − h n || m W − || h n || m W = lim W ) Recall: J n ( u ) := || u || k H + λ || u − f − h n || m W Theorem. J n Γ-converge to ˜ J ( u ) := || u || k H + λ || u − f || m n →∞ || h n || m W + λ lim W with respect to the weak-* topology in H . Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots Γ-convergence Assume that • H is compactly embedded in W • Brezis-Lieb Type Condition : For all f ∈ W || f || k n →∞ ( || f − h n || m W − || h n || m W = lim W ) Recall: J n ( u ) := || u || k H + λ || u − f − h n || m W Theorem. J n Γ-converge to ˜ J ( u ) := || u || k H + λ || u − f || m n →∞ || h n || m W + λ lim W with respect to the weak-* topology in H . Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots Examples: The Brezis-Lieb Condition Holds • W is a Hilbert space, m = 2 if h n ⇀ 0 in W then || f − h n || 2 W −|| h n || 2 W = || f || 2 W + || h n || 2 W − 2( f , h n ) W −|| h n || 2 W → || f || 2 W E.g., h n ⇀ 0 in L 2 (Ω) J n ( u ) := || u || W 1 , 2 (Ω) + λ || u − f − h n || 2 L 2 (Ω) Then J n Γ-converge to ˜ J ( u ) := || u || W 1 , 2 (Ω) + λ || u − f || 2 n →∞ || h n || 2 L 2 (Ω) + λ lim L 2 (Ω) Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots Examples: The Brezis-Lieb Condition Holds • W is a Hilbert space, m = 2 if h n ⇀ 0 in W then || f − h n || 2 W −|| h n || 2 W = || f || 2 W + || h n || 2 W − 2( f , h n ) W −|| h n || 2 W → || f || 2 W E.g., h n ⇀ 0 in L 2 (Ω) J n ( u ) := || u || W 1 , 2 (Ω) + λ || u − f − h n || 2 L 2 (Ω) Then J n Γ-converge to ˜ J ( u ) := || u || W 1 , 2 (Ω) + λ || u − f || 2 n →∞ || h n || 2 L 2 (Ω) + λ lim L 2 (Ω) Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots Concentrations: The Brezis-Lieb Condition Holds • Can handle concentrations Let h n ⇀ 0 in L p (Ω) and pointwise a.e. to 0 Brezis-Lieb Lemma 0 < p < ∞ , u n → u a.e., sup n || u n || L p < ∞ Then � � || u n || p L p (Ω) − || u n − u || p = || u || p lim L p (Ω) L p (Ω) n E.g. � n − n 2 x 0 ≤ x ≤ 1 / n h n ( x ) := 0 1 / n < x ≤ 1 Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum 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 from a significantly incomplete information Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots inpainting; recovery of damaged frescos Figure: A fresco by Mantegna damaged during Second World War. RGB model: u 0 : R → R 3 color image, u 0 = ( u 1 0 , u 2 0 , u 3 0 ) channels L : R 3 → R L ( y ) = L ( e · y ) projection on gray levels L increasing function, e ∈ S 2 L ( u 0 ) : R → R gray level associated with u 0 . Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots inpainting; recovery of damaged frescos Figure: A fresco by Mantegna damaged during Second World War. RGB model: u 0 : R → R 3 color image, u 0 = ( u 1 0 , u 2 0 , u 3 0 ) channels L : R 3 → R L ( y ) = L ( e · y ) projection on gray levels L increasing function, e ∈ S 2 L ( u 0 ) : R → R gray level associated with u 0 . Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots inpainting; recovery of damaged frescos Figure: A fresco by Mantegna damaged during Second World War. RGB model: u 0 : R → R 3 color image, u 0 = ( u 1 0 , u 2 0 , u 3 0 ) channels L : R 3 → R L ( y ) = L ( e · y ) projection on gray levels L increasing function, e ∈ S 2 L ( u 0 ) : R → R gray level associated with u 0 . Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots inpainting; recovery of damaged frescos Figure: A fresco by Mantegna damaged during Second World War. RGB model: u 0 : R → R 3 color image, u 0 = ( u 1 0 , u 2 0 , u 3 0 ) channels L : R 3 → R L ( y ) = L ( e · y ) projection on gray levels L increasing function, e ∈ S 2 L ( u 0 ) : R → R gray level associated with u 0 . Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots inpainting; recovery of damaged frescos Figure: A fresco by Mantegna damaged during Second World War. RGB model: u 0 : R → R 3 color image, u 0 = ( u 1 0 , u 2 0 , u 3 0 ) channels L : R 3 → R L ( y ) = L ( e · y ) projection on gray levels L increasing function, e ∈ S 2 L ( u 0 ) : R → R gray level associated with u 0 . Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots inpainting; recovery of damaged frescos Figure: A fresco by Mantegna damaged during Second World War. RGB model: u 0 : R → R 3 color image, u 0 = ( u 1 0 , u 2 0 , u 3 0 ) channels L : R 3 → R L ( y ) = L ( e · y ) projection on gray levels L increasing function, e ∈ S 2 L ( u 0 ) : R → R gray level associated with u 0 . Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots inpainting; recovery of damaged frescos Figure: A fresco by Mantegna damaged during Second World War. RGB model: u 0 : R → R 3 color image, u 0 = ( u 1 0 , u 2 0 , u 3 0 ) channels L : R 3 → R L ( y ) = L ( e · y ) projection on gray levels L increasing function, e ∈ S 2 L ( u 0 ) : R → R gray level associated with u 0 . Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots inpainting: RGB model D ⊂ R ⊂ R 2 . . . inpainting region RGB observed ( u 0 , v 0 ) u 0 . . . correct information on R \ D v 0 . . . distorted information . . . only gray level is known on D ; v 0 = L u 0 L : R 3 → R . . . e.g. L ( u ) := 1 3 ( r + g + b ) or L ( ξ ) := ξ · e for some e ∈ S 2 Goal to produce a new color image that extends colors of the fragments to the gray region, constrained to match the known gray level Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots inpainting: RGB model D ⊂ R ⊂ R 2 . . . inpainting region RGB observed ( u 0 , v 0 ) u 0 . . . correct information on R \ D v 0 . . . distorted information . . . only gray level is known on D ; v 0 = L u 0 L : R 3 → R . . . e.g. L ( u ) := 1 3 ( r + g + b ) or L ( ξ ) := ξ · e for some e ∈ S 2 Goal to produce a new color image that extends colors of the fragments to the gray region, constrained to match the known gray level Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots inpainting: RGB model D ⊂ R ⊂ R 2 . . . inpainting region RGB observed ( u 0 , v 0 ) u 0 . . . correct information on R \ D v 0 . . . distorted information . . . only gray level is known on D ; v 0 = L u 0 L : R 3 → R . . . e.g. L ( u ) := 1 3 ( r + g + b ) or L ( ξ ) := ξ · e for some e ∈ S 2 Goal to produce a new color image that extends colors of the fragments to the gray region, constrained to match the known gray level Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots inpainting: RGB model D ⊂ R ⊂ R 2 . . . inpainting region RGB observed ( u 0 , v 0 ) u 0 . . . correct information on R \ D v 0 . . . distorted information . . . only gray level is known on D ; v 0 = L u 0 L : R 3 → R . . . e.g. L ( u ) := 1 3 ( r + g + b ) or L ( ξ ) := ξ · e for some e ∈ S 2 Goal to produce a new color image that extends colors of the fragments to the gray region, constrained to match the known gray level Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots The variational approach by Fornasier-March Problem: Reconstruct u 0 from the knowledge of L ( u 0 ) in the damaged region D and of u 0 on R \ D . Fornasier (2006) proposes to solve: � � |L ( u ) −L ( u 0 ) | 2 dx + λ 2 | u − u 0 | 2 dx u ∈ BV ( R ; R 3 ) | D u | ( R )+ λ 1 min D R \ D λ 1 , λ 2 > 0 are fidelity parameters. Studied by Fornasier-March (2007) Related work by Kang-March (2007), using the Brightness/Chromaticity decomposition model. Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots The variational approach by Fornasier-March Problem: Reconstruct u 0 from the knowledge of L ( u 0 ) in the damaged region D and of u 0 on R \ D . Fornasier (2006) proposes to solve: � � |L ( u ) −L ( u 0 ) | 2 dx + λ 2 | u − u 0 | 2 dx u ∈ BV ( R ; R 3 ) | D u | ( R )+ λ 1 min D R \ D λ 1 , λ 2 > 0 are fidelity parameters. Studied by Fornasier-March (2007) Related work by Kang-March (2007), using the Brightness/Chromaticity decomposition model. Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots The variational approach by Fornasier-March Problem: Reconstruct u 0 from the knowledge of L ( u 0 ) in the damaged region D and of u 0 on R \ D . Fornasier (2006) proposes to solve: � � |L ( u ) −L ( u 0 ) | 2 dx + λ 2 | u − u 0 | 2 dx u ∈ BV ( R ; R 3 ) | D u | ( R )+ λ 1 min D R \ D λ 1 , λ 2 > 0 are fidelity parameters. Studied by Fornasier-March (2007) Related work by Kang-March (2007), using the Brightness/Chromaticity decomposition model. Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots The variational approach by Fornasier-March Problem: Reconstruct u 0 from the knowledge of L ( u 0 ) in the damaged region D and of u 0 on R \ D . Fornasier (2006) proposes to solve: � � |L ( u ) −L ( u 0 ) | 2 dx + λ 2 | u − u 0 | 2 dx u ∈ BV ( R ; R 3 ) | D u | ( R )+ λ 1 min D R \ D λ 1 , λ 2 > 0 are fidelity parameters. Studied by Fornasier-March (2007) Related work by Kang-March (2007), using the Brightness/Chromaticity decomposition model. Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots The variational approach by Fornasier-March Problem: Reconstruct u 0 from the knowledge of L ( u 0 ) in the damaged region D and of u 0 on R \ D . Fornasier (2006) proposes to solve: � � |L ( u ) −L ( u 0 ) | 2 dx + λ 2 | u − u 0 | 2 dx u ∈ BV ( R ; R 3 ) | D u | ( R )+ λ 1 min D R \ D λ 1 , λ 2 > 0 are fidelity parameters. Studied by Fornasier-March (2007) Related work by Kang-March (2007), using the Brightness/Chromaticity decomposition model. Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots The variational approach by Fornasier-March Problem: Reconstruct u 0 from the knowledge of L ( u 0 ) in the damaged region D and of u 0 on R \ D . Fornasier (2006) proposes to solve: � � |L ( u ) −L ( u 0 ) | 2 dx + λ 2 | u − u 0 | 2 dx u ∈ BV ( R ; R 3 ) | D u | ( R )+ λ 1 min D R \ D λ 1 , λ 2 > 0 are fidelity parameters. Studied by Fornasier-March (2007) Related work by Kang-March (2007), using the Brightness/Chromaticity decomposition model. Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots a couple of questions. . . “optimal design” : what is the “best” D ? How much color do we need to provide? And where? are we creating spurious edges? For a “cartoon” u in SBV , i.e. Du = ∇ u L 2 ⌊ R + ( u + − u − ) ⊗ ν H 1 ⌊ S ( u ) its edges are in . . . spt D s u = S ( u ) spt D s u i ⊂ spt D s ( L ( u 0 ))? Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots a couple of questions. . . “optimal design” : what is the “best” D ? How much color do we need to provide? And where? are we creating spurious edges? For a “cartoon” u in SBV , i.e. Du = ∇ u L 2 ⌊ R + ( u + − u − ) ⊗ ν H 1 ⌊ S ( u ) its edges are in . . . spt D s u = S ( u ) spt D s u i ⊂ spt D s ( L ( u 0 ))? Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots a couple of questions. . . “optimal design” : what is the “best” D ? How much color do we need to provide? And where? are we creating spurious edges? For a “cartoon” u in SBV , i.e. Du = ∇ u L 2 ⌊ R + ( u + − u − ) ⊗ ν H 1 ⌊ S ( u ) its edges are in . . . spt D s u = S ( u ) spt D s u i ⊂ spt D s ( L ( u 0 ))? Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots Two reconstructions by Fornasier-March Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots Two reconstructions by Fornasier-March Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots Two reconstructions by Fornasier-March Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots Two reconstructions by Fornasier-March Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots Our analysis How faithful is the reconstruction in the infinite fidelity limit ? Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots Our analysis How faithful is the reconstruction in the infinite fidelity limit ? Sending λ 1 and λ 2 → ∞ in � � |L ( u ) −L ( u 0 ) | 2 dx + λ 2 | u − u 0 | 2 dx u ∈ BV ( R ; R 3 ) | D u | ( R )+ λ 1 min D R \ D Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots Our analysis How faithful is the reconstruction in the infinite fidelity limit ? the problem becomes min | D u | ( R ) (P) u ∈ BV ( R ; R 3 ) subject to u = u 0 on R \ D and L ( u · e ) = L ( u 0 · e ) in D . Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots Our analysis How faithful is the reconstruction in the infinite fidelity limit ? the problem becomes min | D u | ( R ) (P) u ∈ BV ( R ; R 3 ) subject to u = u 0 on R \ D and u · e = u 0 · e in D . Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots Our analysis How faithful is the reconstruction in the infinite fidelity limit ? the problem becomes: min | D u | ( R ) (P) u ∈ BV ( R ; R 3 ) subject to u = u 0 in R \ D and u · e = u 0 · e in D . Definition u 0 is reconstructible over D if it is the unique minimizer of (P). Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots λ 1 = λ 2 = ∞ | Du | ( R ) : u ∈ BV ( R ; R 3 ) , Lu = Lu 0 � � ( P ) inf in D , u = u 0 on R \ D Theorem u 0 ∈ BV ( R ; R 3 ) and D open Lipschitz domain. Then (P) has a minimizer. isoperimetric inequality → boundedness in BV Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots admissible images Find conditions on the damaged region D which render u 0 reconstructible Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots admissible images Find conditions on the damaged region D which render u 0 reconstructible Mathematical simplification: Restrict the analysis to piecewise constant images u 0 Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots admissible images Find conditions on the damaged region D which make u 0 reconstructible Mathematical simplification: Restrict the analysis to piecewise constant images u 0 Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots admissible images Find conditions on the damaged region D which make u 0 reconstructible Mathematical simplification: Restrict the analysis to piecewise constant images u 0 N N � � R = Γ ∪ Ω k , u 0 = ξ k 1 Ω k , k =1 k =1 Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots Our analysis Recall that u 0 = � N k =1 ξ k 1 Ω k is reconstructible over D if it is the unique minimizer to min | D u | ( R ) (P) u ∈ BV ( R ; R 3 ) subject to u = u 0 in R \ D and u · e = u 0 · e in D . Strengthened notion of reconstructibility: Definition u 0 is stably reconstructible over D if there exists ε > 0 such that all u of the form N � ξ ′ 1 ≤ k ≤ N | ξ ′ u = k 1 Ω k , with max k − ξ k | < ε , k =1 are reconstructible over D . Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots Our analysis Recall that u 0 = � N k =1 ξ k 1 Ω k is reconstructible over D if it is the unique minimizer to min | D u | ( R ) (P) u ∈ BV ( R ; R 3 ) subject to u = u 0 in R \ D and u · e = u 0 · e in D . Strengthened notion of reconstructibility: Definition u 0 is stably reconstructible over D if there exists ε > 0 such that all u of the form N � ξ ′ 1 ≤ k ≤ N | ξ ′ u = k 1 Ω k , with max k − ξ k | < ε , k =1 are reconstructible over D . Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots Our analysis Recall that u 0 = � N k =1 ξ k 1 Ω k is reconstructible over D if it is the unique minimizer to min | D u | ( R ) (P) u ∈ BV ( R ; R 3 ) subject to u = u 0 in R \ D and u · e = u 0 · e in D . Strengthened notion of reconstructibility: Definition u 0 is stably reconstructible over D if there exists ε > 0 such that all u of the form N � ξ ′ 1 ≤ k ≤ N | ξ ′ u = k 1 Ω k , with max k − ξ k | < ε , k =1 are reconstructible over D . Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots Our analysis Recall that u 0 = � N k =1 ξ k 1 Ω k is reconstructible over D if it is the unique minimizer to min | D u | ( R ) (P) u ∈ BV ( R ; R 3 ) subject to u = u 0 in R \ D and u · e = u 0 · e in D . Strengthened notion of reconstructibility: Definition u 0 is stably reconstructible over D if there exists ε > 0 such that all u of the form N � ξ ′ 1 ≤ k ≤ N | ξ ′ u = k 1 Ω k , with max k − ξ k | < ε , k =1 are reconstructible over D . Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots Our analysis Recall that u 0 = � N k =1 ξ k 1 Ω k is reconstructible over D if it is the unique minimizer to min | D u | ( R ) (P) u ∈ BV ( R ; R 3 ) subject to u = u 0 in R \ D and u · e = u 0 · e in D . Strengthened notion of reconstructibility: Definition u 0 is stably reconstructible over D if there exists ε > 0 such that all u of the form N � ξ ′ 1 ≤ k ≤ N | ξ ′ u = k 1 Ω k , with max k − ξ k | < ε , k =1 are reconstructible over D . Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots reconstructible images when is an admissible image u 0 reconstructible over a damaged region S ? Answer: NO when a pair of neighboring colors ξ h and ξ k in u 0 share the same gray level, i.e., if H 1 ( ∂ Ω k ∩ ∂ Ω h ) > 0 and L ξ h = L ξ k Answer: YES if an algebraic condition involving the values of the colors and the angles of the corners possibly present in Γ is satisfied . . . quantitative validation of the model’s accuracy Minimal requirement: must be reconstructible over S = Γ( δ ) for some δ > 0, where Γ( δ ) := { x ∈ R : dist ( x , Γ) < δ } Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots reconstructible images when is an admissible image u 0 reconstructible over a damaged region S ? Answer: NO when a pair of neighboring colors ξ h and ξ k in u 0 share the same gray level, i.e., if H 1 ( ∂ Ω k ∩ ∂ Ω h ) > 0 and L ξ h = L ξ k Answer: YES if an algebraic condition involving the values of the colors and the angles of the corners possibly present in Γ is satisfied . . . quantitative validation of the model’s accuracy Minimal requirement: must be reconstructible over S = Γ( δ ) for some δ > 0, where Γ( δ ) := { x ∈ R : dist ( x , Γ) < δ } Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots reconstructible images when is an admissible image u 0 reconstructible over a damaged region S ? Answer: NO when a pair of neighboring colors ξ h and ξ k in u 0 share the same gray level, i.e., if H 1 ( ∂ Ω k ∩ ∂ Ω h ) > 0 and L ξ h = L ξ k Answer: YES if an algebraic condition involving the values of the colors and the angles of the corners possibly present in Γ is satisfied . . . quantitative validation of the model’s accuracy Minimal requirement: must be reconstructible over S = Γ( δ ) for some δ > 0, where Γ( δ ) := { x ∈ R : dist ( x , Γ) < δ } Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots reconstructible images when is an admissible image u 0 reconstructible over a damaged region S ? Answer: NO when a pair of neighboring colors ξ h and ξ k in u 0 share the same gray level, i.e., if H 1 ( ∂ Ω k ∩ ∂ Ω h ) > 0 and L ξ h = L ξ k Answer: YES if an algebraic condition involving the values of the colors and the angles of the corners possibly present in Γ is satisfied . . . quantitative validation of the model’s accuracy Minimal requirement: must be reconstructible over S = Γ( δ ) for some δ > 0, where Γ( δ ) := { x ∈ R : dist ( x , Γ) < δ } Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots u 0 does not have neighboring colors with the same gray level � ξ k − ξ h � z k ( x ) := P if x ∈ ∂ Ω k ∩ ∂ Ω h ∩ R , h � = k , | ξ k − ξ h | where P is the orthogonal projection on � e � ⊥ P ( ξ ) := ξ − ( ξ · e ) e u 0 does not have neighboring colors with the same gray level IFF || z k || L ∞ < 1 sup 1 ≤ K ≤ N Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots A simple counterexample when � z k � ∞ < 1 is not satisfied Original image u 0 : Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots A simple counterexample when � z k � ∞ < 1 is not satisfied A simple counterexample when � z k � ∞ < 1 is not satisfied Original image u 0 : Resulting image u : Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots Adjoint colors have the same gray levels: may create spurious edges Original image u 0 : Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots Adjoint colors have the same gray levels: may create spurious edges A simple analytical counterexample Original image u 0 : Resulting image u : Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots Adjoint colors have the same gray levels: may create spurious edges A simple analytical counterexample Original image u 0 : Resulting image u : A spurious contour appears! Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots Minimality conditions Theorem (Necessary and sufficient minimality conditions) D ⊂ R Lipschitz, H 1 ( ∂ D ∩ Γ) = 0 . Then the following two conditions are equivalent: (i) u 0 is stably reconstructible over D; (ii) there exists a tensor field M : D → � e � ⊥ ⊗ R 2 such that div M = 0 in D � M � ∞ < 1 M [ ν Ω k ] = − z k on D ∩ ∂ Ω k . and The tensor field M is called a calibration for u 0 in D . Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots Minimality conditions Theorem (Necessary and sufficient minimality conditions) D ⊂ R Lipschitz, H 1 ( ∂ D ∩ Γ) = 0 . Then the following two conditions are equivalent: (i) u 0 is stably reconstructible over D; (ii) there exists a tensor field M : D → � e � ⊥ ⊗ R 2 such that div M = 0 in D � M � ∞ < 1 M [ ν Ω k ] = − z k on D ∩ ∂ Ω k . and The tensor field M is called a calibration for u 0 in D . Irene Fonseca Variational Methods in Materials Science and Image Processing

Imaging Quantum Dots Minimality conditions Theorem (Necessary and sufficient minimality conditions) D ⊂ R Lipschitz, H 1 ( ∂ D ∩ Γ) = 0 . Then the following two conditions are equivalent: (i) u 0 is stably reconstructible over D; (ii) there exists a tensor field M : D → � e � ⊥ ⊗ R 2 such that div M = 0 in D � M � ∞ < 1 M [ ν Ω k ] = − z k on D ∩ ∂ Ω k . and The tensor field M is called a calibration for u 0 in D . Irene Fonseca Variational Methods in Materials Science and Image Processing