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Total (gradient) variation regularization: exact support recovery and grid-free numerical methods Romain Petit, joint work with Yohann De Castro and Vincent Duval Paris-Saclay Signal Seminar, 16 November 2023

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Inverse problems in imaging Unknown image u0 : R2 → R 1

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Inverse problems in imaging Unknown image u0 : R2 → R Obs. y0 = Φu0 ∈ H 1

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Inverse problems in imaging Unknown image u0 : R2 → R Obs. y0 = Φu0 ∈ H Noisy obs. y = y0 + w 1

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Inverse problems in imaging Unknown image u0 : R2 → R Inverse problem Recover u0 from y Obs. y0 = Φu0 ∈ H Noisy obs. y = y0 + w 1

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Variational approach [Rudin et al., 1992, Chambolle and Lions, 1997] The total (gradient) variation TV(u) def. = sup − R2 u divφ φ ∈ C∞ c (R2, R2), φ ∞ ≤ 1 “ = ” R2 |∇u| 2

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Variational approach [Rudin et al., 1992, Chambolle and Lions, 1997] The total (gradient) variation TV(u) def. = sup − R2 u divφ φ ∈ C∞ c (R2, R2), φ ∞ ≤ 1 “ = ” R2 |∇u| Solve min u∈L2(R2) 1 2 Φu − y 2 + λ TV(u) (Pλ(y)) 2

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Variational approach [Rudin et al., 1992, Chambolle and Lions, 1997] The total (gradient) variation TV(u) def. = sup − R2 u divφ φ ∈ C∞ c (R2, R2), φ ∞ ≤ 1 “ = ” R2 |∇u| noisy obs. y Solve min u∈L2(R2) 1 2 Φu − y 2 + λ TV(u) (Pλ(y)) 2

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Variational approach [Rudin et al., 1992, Chambolle and Lions, 1997] The total (gradient) variation TV(u) def. = sup − R2 u divφ φ ∈ C∞ c (R2, R2), φ ∞ ≤ 1 “ = ” R2 |∇u| noisy obs. y solution (small λ) Solve min u∈L2(R2) 1 2 Φu − y 2 + λ TV(u) (Pλ(y)) 2

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Variational approach [Rudin et al., 1992, Chambolle and Lions, 1997] The total (gradient) variation TV(u) def. = sup − R2 u divφ φ ∈ C∞ c (R2, R2), φ ∞ ≤ 1 “ = ” R2 |∇u| noisy obs. y solution (small λ) solution (large λ) Solve min u∈L2(R2) 1 2 Φu − y 2 + λ TV(u) (Pλ(y)) 2

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Variational approach [Rudin et al., 1992, Chambolle and Lions, 1997] Representer th. [Boyer et al., 2019, Bredies and Carioni, 2019] + [Fleming, 1957] Some sol. of (Pλ(y)) are linear combinations of 1E with E simple Solve min u∈L2(R2) 1 2 Φu − y 2 + λ TV(u) (Pλ(y)) 2

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Variational approach [Rudin et al., 1992, Chambolle and Lions, 1997] Representer th. [Boyer et al., 2019, Bredies and Carioni, 2019] + [Fleming, 1957] Some sol. of (Pλ(y)) are linear combinations of 1E with E simple Solve min u∈L2(R2) 1 2 Φu − y 2 + λ TV(u) (Pλ(y)) 2

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Variational approach [Rudin et al., 1992, Chambolle and Lions, 1997] Representer th. [Boyer et al., 2019, Bredies and Carioni, 2019] + [Fleming, 1957] Some sol. of (Pλ(y)) are linear combinations of 1E with E simple The sparse objects associated to TV are the piecewise constant functions Solve min u∈L2(R2) 1 2 Φu − y 2 + λ TV(u) (Pλ(y)) 2

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Variational approach [Rudin et al., 1992, Chambolle and Lions, 1997] Representer th. [Boyer et al., 2019, Bredies and Carioni, 2019] + [Fleming, 1957] Some sol. of (Pλ(y)) are linear combinations of 1E with E simple The sparse objects associated to TV are the piecewise constant functions Solve min u∈L2(R2) 1 2 Φu − y 2 + λ TV(u) (Pλ(y)) 2

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Considered problems im. inconnue u0 obs. y0 = Φu0 obs. bruit´ ees y = y0 + w 3

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Considered problems im. inconnue u0 obs. y0 = Φu0 obs. bruit´ ees y = y0 + w Noise robustness Conv. of sol. to (Pλ(y0 +w)) when λ, w → 0 3

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Considered problems im. inconnue u0 obs. y0 = Φu0 obs. bruit´ ees y = y0 + w Noise robustness Conv. of sol. to (Pλ(y0 +w)) when λ, w → 0 Reconstruction algorithm Numerical resolution of (Pλ(y)) 3

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Considered problems im. inconnue u0 obs. y0 = Φu0 obs. bruit´ ees y = y0 + w Noise robustness Conv. of sol. to (Pλ(y0 +w)) when λ, w → 0 Reconstruction algorithm Numerical resolution of (Pλ(y)) Assumptions

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Considered problems im. inconnue u0 obs. y0 = Φu0 obs. bruit´ ees y = y0 + w Noise robustness Conv. of sol. to (Pλ(y0 +w)) when λ, w → 0 Reconstruction algorithm Numerical resolution of (Pλ(y)) Assumptions • u0 piecewise constant

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Considered problems unknown im. u0 obs. y0 = Φu0 noisy obs. y = y0 + w Noise robustness Conv. of sol. to (Pλ(y0 +w)) when λ, w → 0 Reconstruction algorithm Numerical resolution of (Pλ(y)) Assumptions • u0 piecewise constant

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Considered problems unknown im. u0 obs. y0 = Φu0 noisy obs. y = y0 + w Noise robustness Conv. of sol. to (Pλ(y0 +w)) when λ, w → 0 Reconstruction algorithm Numerical resolution of (Pλ(y)) Assumptions • u0 piecewise constant • Im(Φ∗) ⊂ C1(R2) 3

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Noise robustness: exact support recovery

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What kind of convergence? 4

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First convergence result (Pλ (y0 + w)) min u∈L2(R2) TV(u) + 1 2λ Φu −(y0 +w) 2 (P0 (y0 )) min u∈L2(R2) TV(u) s.t. Φu = y0 5

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First convergence result (Pλ (y0 + w)) min u∈L2(R2) TV(u) + 1 2λ Φu −(y0 +w) 2 (P0 (y0 )) min u∈L2(R2) TV(u) s.t. Φu = y0 Proposition [Chambolle et al., 2016, Iglesias et al., 2018] If λn → 0 and wn = O(λn) (+ source cond.) then un → u0 (strictly in BV(R2)) and U(t) n ∆U(t) 0 → 0 and ∂U(t) n Hausdorff − − − − − − − A ∂U(t) 0 with U(t) = {u ≥ t} if t ≥ 0 {u ≤ t} otherwise. 5

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First convergence result (Pλ (y0 + w)) min u∈L2(R2) TV(u) + 1 2λ Φu −(y0 +w) 2 (P0 (y0 )) min u∈L2(R2) TV(u) s.t. Φu = y0 u0 un U(t) n , t ∈ R 5

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The prescribed curvature problem Optimality condition (regularized pb.) If u solves (Pλ (y0 + w)) then • ∀t > 0, {u ≥ t} solves (Q(+ηλ,w )) • ∀t < 0, {u ≤ t} solves (Q(−ηλ,w )) for some ηλ,w u 6

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The prescribed curvature problem Optimality condition (regularized pb.) If u solves (Pλ (y0 + w)) then • ∀t > 0, {u ≥ t} solves (Q(+ηλ,w )) • ∀t < 0, {u ≤ t} solves (Q(−ηλ,w )) for some ηλ,w u 6

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The prescribed curvature problem Prescribed curvature problem min E⊂R2, |E|<+∞ Per(E) − E η (Q(η)) Optimality condition (regularized pb.) If u solves (Pλ (y0 + w)) then • ∀t > 0, {u ≥ t} solves (Q(+ηλ,w )) • ∀t < 0, {u ≤ t} solves (Q(−ηλ,w )) for some ηλ,w u 6

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The prescribed curvature problem Prescribed curvature problem min E⊂R2, |E|<+∞ Per(E) − E η (Q(η)) Optimality condition (regularized pb.) If u solves (Pλ (y0 + w)) then • ∀t > 0, {u ≥ t} solves (Q(+ηλ,w )) • ∀t < 0, {u ≤ t} solves (Q(−ηλ,w )) for some ηλ,w u 6

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The prescribed curvature problem Prescribed curvature problem min E⊂R2, |E|<+∞ Per(E) − E η (Q(η)) Optimality condition (regularized pb.) If u solves (Pλ (y0 + w)) then • ∀t > 0, {u ≥ t} solves (Q(+ηλ,w )) • ∀t < 0, {u ≤ t} solves (Q(−ηλ,w )) for some ηλ,w u Convergence of curvature functionals If λn → 0 and wn λn → 0 (+ source cond.) then ηλn,wn → η0 6

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The prescribed curvature problem Prescribed curvature problem min E⊂R2, |E|<+∞ Per(E) − E η (Q(η)) Optimality condition (constrained pb.) We have that • ∀t > 0, {u0 ≥ t} solves (Q(+η0)) • ∀t < 0, {u0 ≤ t} solves (Q(−η0)) for some η0 u Convergence of curvature functionals If λn → 0 and wn λn → 0 (+ source cond.) then ηλn,wn → η0 6

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Stability of the prescribed curvature problem Assumptions • η close to η0 in L2(R2) and C1(R2) • (Q(η0)) has finitely many minimizers, all strictly stable 7

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Stability of the prescribed curvature problem Assumptions • η close to η0 in L2(R2) and C1(R2) • (Q(η0)) has finitely many minimizers, all strictly stable Proposition (informal) Around each sol. E of (Q(η0)) there is ex- actly one sol. F of (Q(η)) and ∂F = (Id + ϕνE )(∂E) with ϕ ∈ C2(∂E) J0 J [ ] [ ] 7

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Exact support recovery Assumptions • u0 = N i=1 ai 1Ei with Ei simple and ∂Ei ∩ ∂Ej = ∅ for every i = j • non-degenerate source cond. + injectivity cond. 8

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Exact support recovery Assumptions • u0 = N i=1 ai 1Ei with Ei simple and ∂Ei ∩ ∂Ej = ∅ for every i = j • non-degenerate source cond. + injectivity cond. u0 uλ,w 8

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Exact support recovery Assumptions • u0 = N i=1 ai 1Ei with Ei simple and ∂Ei ∩ ∂Ej = ∅ for every i = j • non-degenerate source cond. + injectivity cond. u0 uλ,w Theorem There exists α, λ0 ∈ R∗ + s.t., if λ ≤ λ0 and w /λ ≤ α, then • uλ,w = N i=1 bi 1Fi with ∂Fi = (Id + ϕi νE )(∂E) • bi → ai and ϕi C2(∂Ei ) → 0 when λ, w → 0 8

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Numerical verif. of the non-degenerate source cond. Φu = h u with h(x) = exp(− x 2/(2σ2)) github.com/rpetit/2023-support-recovery-tv

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Numerical verif. of the non-degenerate source cond. Φu = h u with h(x) = exp(− x 2/(2σ2)) Deconvolution of a disk: u0 = a 1B(0,R) Condition satisfied if σ ≤ σ0(R) github.com/rpetit/2023-support-recovery-tv

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Numerical verif. of the non-degenerate source cond. Φu = h u with h(x) = exp(− x 2/(2σ2)) Deconvolution of a disk: u0 = a 1B(0,R) Condition satisfied if σ ≤ σ0(R) Deconvolution of radial images: u0 = a1 1B(0,R1) + a2 1B(0,R2) github.com/rpetit/2023-support-recovery-tv

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Numerical verif. of the non-degenerate source cond. Φu = h u with h(x) = exp(− x 2/(2σ2)) Deconvolution of a disk: u0 = a 1B(0,R) Condition satisfied if σ ≤ σ0(R) Deconvolution of radial images: u0 = a1 1B(0,R1) + a2 1B(0,R2) • If signe(a1) = −signe(a2): need |R1 − R2| > ∆ ∆ github.com/rpetit/2023-support-recovery-tv

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Numerical verif. of the non-degenerate source cond. Φu = h u with h(x) = exp(− x 2/(2σ2)) Deconvolution of a disk: u0 = a 1B(0,R) Condition satisfied if σ ≤ σ0(R) Deconvolution of radial images: u0 = a1 1B(0,R1) + a2 1B(0,R2) • If signe(a1) = −signe(a2): need |R1 − R2| > ∆ • If signe(a1) = signe(a2): super-resolution ∆ github.com/rpetit/2023-support-recovery-tv 9

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Numerical resolution: a grid-free approach

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Numerical resolution of (Pλ (y)) Solve min u∈L2(R2) 1 2 Φu − y 2 + λ TV(u) (Pλ(y)) 10

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Numerical resolution of (Pλ (y)) Solve min u∈L2(R2) 1 2 Φu − y 2 + λ TV(u) (Pλ(y)) Fixed grid approximation u = i j uij 1Cij 10

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Discretizations of the total variation (images: [Tabti et al., 2018]) Anisotropic • ij |(Dx u)ij | + |(Dy u)ij | • Sharp edges, grid bias Isotropic • ij (Dx u)2 ij + (Dy u)2 ij • Blur State of the art review: [Chambolle and Pock, 2021] 11

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Numerical representation of simple images Fixed grid • O 1/h2 pixels • O (1/h) “relevant” pixels • u → TV(u) convex 12

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Numerical representation of simple images Fixed grid • O 1/h2 pixels • O (1/h) “relevant” pixels • u → TV(u) convex Boundary discretization • More efficient for simple img. • Numerically more involved • E → TV(1E ) “non convex” 12

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Frank-Wolfe based algorithm Algorithm [Bredies and Pikkarainen, 2013] [Boyd et al., 2017, Denoyelle et al., 2019] 13

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Frank-Wolfe based algorithm Algorithm • ηk = − 1 λ Φ∗(Φuk − y) [Bredies and Pikkarainen, 2013] [Boyd et al., 2017, Denoyelle et al., 2019] 13

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Frank-Wolfe based algorithm Algorithm • ηk = − 1 λ Φ∗(Φuk − y) • Ek+1 ∈ Argmax E simple 1 P(E) E ηk [Bredies and Pikkarainen, 2013] [Boyd et al., 2017, Denoyelle et al., 2019] 13

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Frank-Wolfe based algorithm Algorithm • ηk = − 1 λ Φ∗(Φuk − y) • Ek+1 ∈ Argmax E simple 1 P(E) E ηk [Bredies and Pikkarainen, 2013] [Boyd et al., 2017, Denoyelle et al., 2019] Generalized Cheeger pb. Max. E⊂R2 1 P(E) E η s.t. |E| < +∞, 0 < P(E) < +∞ 13

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Frank-Wolfe based algorithm Algorithm • ηk = − 1 λ Φ∗(Φuk − y) • Ek+1 ∈ Argmax E simple 1 P(E) E ηk [Bredies and Pikkarainen, 2013] [Boyd et al., 2017, Denoyelle et al., 2019] Generalized Cheeger pb. Max. E⊂R2 1 P(E) E η s.t. |E| < +∞, 0 < P(E) < +∞ 13

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Frank-Wolfe based algorithm Algorithm • ηk = − 1 λ Φ∗(Φuk − y) • Ek+1 ∈ Argmax E simple 1 P(E) E ηk [Bredies and Pikkarainen, 2013] [Boyd et al., 2017, Denoyelle et al., 2019] Generalized Cheeger pb. Max. E⊂R2 1 P(E) E η s.t. |E| < +∞, 0 < P(E) < +∞ 13

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Frank-Wolfe based algorithm Algorithm • ηk = − 1 λ Φ∗(Φuk − y) • Ek+1 ∈ Argmax E simple 1 P(E) E ηk • uk+1 = αk uk + βk 1Ek+1 [Bredies and Pikkarainen, 2013] [Boyd et al., 2017, Denoyelle et al., 2019] Generalized Cheeger pb. Max. E⊂R2 1 P(E) E η s.t. |E| < +∞, 0 < P(E) < +∞ 13

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Frank-Wolfe based algorithm Algorithm • ηk = − 1 λ Φ∗(Φuk − y) • Ek+1 ∈ Argmax E simple 1 P(E) E ηk • uk+1 = αk uk + βk 1Ek+1 [Bredies and Pikkarainen, 2013] [Boyd et al., 2017, Denoyelle et al., 2019] Generalized Cheeger pb. Max. E⊂R2 1 P(E) E η s.t. |E| < +∞, 0 < P(E) < +∞ Iterates are linear combinations of indicator functions of simple sets 13

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Frank-Wolfe based algorithm Algorithm • ηk = − 1 λ Φ∗(Φuk − y) • Ek+1 ∈ Argmax E simple 1 P(E) E ηk • uk+1 = αk uk + βk 1Ek+1 • Loc. opt. (a, E) → F i ai 1Ei [Bredies and Pikkarainen, 2013] [Boyd et al., 2017, Denoyelle et al., 2019] Generalized Cheeger pb. Max. E⊂R2 1 P(E) E η s.t. |E| < +∞, 0 < P(E) < +∞ Iterates are linear combinations of indicator functions of simple sets 13

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Numerical results github.com/rpetit/tvsfw Unknown image u0 Observations y = Φu0 + w 14

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Numerical results github.com/rpetit/tvsfw u[1] (before sliding) −2 0 2 u[1] u0 − u[1] Supp(Du[1]), Supp(Du0 ) u[2] (before sliding) −2 0 2 u[2] u0 − u[2] Supp(Du[2]), Supp(Du0 ) u[3] (before sliding) −2 0 2 u[3] u0 − u[3] Supp(Du[3]), Supp(Du0 ) 15

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Numerical results github.com/rpetit/tvsfw Left to right: observations, signal, ours, isotropic TV, “Condat’s” TV 16

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Numerical results github.com/rpetit/tvsfw Left to right: observations, signal, ours, isotropic TV, “Condat’s” TV Signal u0 Isotropic TV Ours ˆ uλ,w Observations “Condat’s” TV Supp(Du0), Supp(Dˆ uλ,w ) 16

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Perspectives (i) u0 Φu0 + w ˆ uλ,w ˆ uλ,w − u0 Du0, Dˆ uλ,w 17

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Perspectives (i) u0 Φu0 + w ˆ uλ,w ˆ uλ,w − u0 Du0, Dˆ uλ,w Recovery guarantees • Non-degenerate source cond. • Implicit bias of TV reg. 17

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Perspectives (i) u0 Φu0 + w ˆ uλ,w ˆ uλ,w − u0 Du0, Dˆ uλ,w Recovery guarantees • Non-degenerate source cond. • Implicit bias of TV reg. Numerical resolution • Robust sliding step (topology changes) 17

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Perspectives (i) u0 Φu0 + w ˆ uλ,w ˆ uλ,w − u0 Du0, Dˆ uλ,w Recovery guarantees • Non-degenerate source cond. • Implicit bias of TV reg. Numerical resolution • Robust sliding step (topology changes) • “Continuous” TV reg. benchmark 17

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Perspectives (i) u0 Φu0 + w ˆ uλ,w ˆ uλ,w − u0 Du0, Dˆ uλ,w Recovery guarantees • Non-degenerate source cond. • Implicit bias of TV reg. Numerical resolution • Robust sliding step (topology changes) • “Continuous” TV reg. benchmark • Applications (cell im., piecewise homog. textures) 17

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Perspectives (ii) OT reg. for dynamic inverse problems [Bredies et al., 2022] • Recover t → i ai (t)δxi (t) • Support stability, implicit bias, fast reconstruction? 0.0 0.5 1.0 x1 0.0 0.5 1.0 x2 w0 t at time t = 0.0 0.0 0.5 1.0 x1 0.0 0.5 1.0 x2 w0 t at time t = 0.5 0.0 0.5 1.0 x1 0.0 0.5 1.0 x2 w0 t at time t = 1.0 0.0 0.5 1.0 x1 0.0 0.5 1.0 x2 w0 t at time t = 0.0 0.0 0.5 1.0 x1 0.0 0.5 1.0 x2 w0 t at time t = 0.5 0.0 0.5 1.0 x1 0.0 0.5 1.0 x2 w0 t at time t = 1.0 18

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References i Boyd, N., Schiebinger, G., and Recht, B. (2017). The Alternating Descent Conditional Gradient Method for Sparse Inverse Problems. SIAM Journal on Optimization, 27(2):616–639. Boyer, C., Chambolle, A., Castro, Y. D., Duval, V., de Gournay, F., and Weiss, P. (2019). On Representer Theorems and Convex Regularization. SIAM Journal on Optimization, 29(2):1260–1281. Bredies, K. and Carioni, M. (2019). Sparsity of solutions for variational inverse problems with finite-dimensional data. Calculus of Variations and Partial Differential Equations, 59(1):14. 19

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References ii Bredies, K., Carioni, M., Fanzon, S., and Romero, F. (2022). A Generalized Conditional Gradient Method for Dynamic Inverse Problems with Optimal Transport Regularization. Foundations of Computational Mathematics. Bredies, K. and Pikkarainen, H. K. (2013). Inverse problems in spaces of measures. ESAIM: Control, Optimisation and Calculus of Variations, 19(1):190–218. Carlier, G., Comte, M., and Peyr´ e, G. (2009). Approximation of maximal Cheeger sets by projection. ESAIM: Mathematical Modelling and Numerical Analysis, 43(1):139–150. 20

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References iii Chambolle, A., Duval, V., Peyr´ e, G., and Poon, C. (2016). Geometric properties of solutions to the total variation denoising problem. Inverse Problems, 33(1):015002. Chambolle, A. and Lions, P.-L. (1997). Image recovery via total variation minimization and related problems. Numerische Mathematik, 76(2):167–188. Chambolle, A. and Pock, T. (2021). Approximating the total variation with finite differences or finite elements. In Bonito, A. and Nochetto, R. H., editors, Handbook of Numerical Analysis, volume 22 of Geometric Partial Differential Equations - Part II, pages 383–417. Elsevier. 21

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References iv Denoyelle, Q., Duval, V., Peyre, G., and Soubies, E. (2019). The Sliding Frank-Wolfe Algorithm and its Application to Super-Resolution Microscopy. Inverse Problems. Fleming, W. H. (1957). Functions with generalized gradient and generalized surfaces. Annali di Matematica Pura ed Applicata, 44(1):93–103. Iglesias, J. A., Mercier, G., and Scherzer, O. (2018). A note on convergence of solutions of total variation regularized linear inverse problems. Inverse Problems, 34(5):055011. Rudin, L. I., Osher, S., and Fatemi, E. (1992). Nonlinear total variation based noise removal algorithms. Physica D: Nonlinear Phenomena, 60(1):259–268. 22

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References v Tabti, S., Rabin, J., and Elmoata, A. (2018). Symmetric Upwind Scheme for Discrete Weighted Total Variation. In 2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pages 1827–1831. 23

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Two-step approximation of generalized Cheeger sets Fixed grid initialization [Carlier et al., 2009] Solve min u∈Eh ηh, u s.t. TVh(u) ≤ 1 Shape gradient algorithm • θn ∈ Argmax θ∈Θ lim →0+ 1 [J ((Id + θ)(En)) − J (En)] • En+1 = (Id + n θn)(En) Implementation: github.com/rpetit/PyCheeger 24

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Topology changes during the local descent 25