A posteriori error control for the binary Mumford Shah model

Transcription

1 A posteriori error control for the binary Mumford Shah model Benjamin Berkels 1, Alexander Effland 2, Martin Rumpf 2 1 AICES Graduate School, RWTH Aachen University, Germany 2 Institute for Numerical Simulation, University of Bonn Minisymposium on Recent Advances in Convex Relaxations SIAM Conference on IMAGING SCIENCE May 23rd 26th, Hotel Albuquerque at Old Town

2 1 The binary Mumford Shah model u 0 O Binary Mumford Shah model θ 1, θ 2 L 1 () E[O] = θ 1 dx + θ 2 dx + Per[O] O \O

3 1 The binary Mumford Shah model u 0 χ BV (, {0, 1}) Binary Mumford Shah model θ 1, θ 2 L 1 () E[χ] = θ 1 χ + θ 2 (1 χ) dx + Dχ ()

4 1 The binary Mumford Shah model u 0 χ BV (, {0, 1}) Binary Mumford Shah model θ 1, θ 2 L 1 () E[χ, c 1, c 2 ] = θ 1 χ + θ 2 (1 χ) dx + Dχ () θ i = 1 ν (c i u 0 ) 2 (i = 1, 2) for weight parameter ν > 0

5 1 The binary Mumford Shah model u 0 χ BV (, {0, 1}) Binary Mumford Shah model θ 1, θ 2 L 1 () E[χ, c 1, c 2 ] = θ 1 χ + θ 2 (1 χ) dx + Dχ () θ i = 1 ν (c i u 0 ) 2 (i = 1, 2) for weight parameter ν > 0 c 1 = ( χ dx) 1 χu 0 dx, c 2 = ( 1 χ dx) 1 (1 χ)u 0 dx

6 2 The binary Mumford Shah model (cont.) E[χ, c 1, c 2 ] = θ 1 χ + θ 2 (1 χ) dx + Dχ () minimize E over the set χ BV (, {0, 1}) (c 1, c 2 fixed)

7 2 The binary Mumford Shah model (cont.) E[χ, c 1, c 2 ] = θ 1 χ + θ 2 (1 χ) dx + Dχ () minimize E over the set χ BV (, {0, 1}) (c 1, c 2 fixed) Goal: derive functional a posteriori error estimates for χ χ h L 1 ()

8 2 The binary Mumford Shah model (cont.) E[χ, c 1, c 2 ] = θ 1 χ + θ 2 (1 χ) dx + Dχ () minimize E over the set χ BV (, {0, 1}) (c 1, c 2 fixed) Goal: derive functional a posteriori error estimates for Major problems: χ χ h L 1 () optimization over a non convex set

9 2 The binary Mumford Shah model (cont.) E[χ, c 1, c 2 ] = θ 1 χ + θ 2 (1 χ) dx + Dχ () minimize E over the set χ BV (, {0, 1}) (c 1, c 2 fixed) Goal: derive functional a posteriori error estimates for χ χ h L 1 () Major problems: optimization over a non convex set no uniform convexity of E

10 2 The binary Mumford Shah model (cont.) E[χ, c 1, c 2 ] = θ 1 χ + θ 2 (1 χ) dx + Dχ () minimize E over the set χ BV (, {0, 1}) (c 1, c 2 fixed) Goal: derive functional a posteriori error estimates for χ χ h L 1 () Major problems: optimization over a non convex set no uniform convexity of E Strategy: use a suitable uniformly convex relaxation

11 2 The binary Mumford Shah model (cont.) E[χ, c 1, c 2 ] = θ 1 χ + θ 2 (1 χ) dx + Dχ () minimize E over the set χ BV (, {0, 1}) (c 1, c 2 fixed) Goal: derive functional a posteriori error estimates for χ χ h L 1 () Major problems: optimization over a non convex set no uniform convexity of E Strategy: use a suitable uniformly convex relaxation error estimate for the relaxation based on the duality gap

12 2 The binary Mumford Shah model (cont.) E[χ, c 1, c 2 ] = θ 1 χ + θ 2 (1 χ) dx + Dχ () minimize E over the set χ BV (, {0, 1}) (c 1, c 2 fixed) Goal: derive functional a posteriori error estimates for χ χ h L 1 () Major problems: optimization over a non convex set no uniform convexity of E Strategy: use a suitable uniformly convex relaxation error estimate for the relaxation based on the duality gap thresholding of the relaxed solution and cut out argument

13 3 Convex relaxation and thresholding - CEN Idea Relax range of χ to [0, 1] [Nikolova, Esedoḡlu, Chan 06] E CEN [u] = uθ 1 dx + (1 u) θ 2 dx + Du ().

14 3 Convex relaxation and thresholding - CEN Idea Relax range of χ to [0, 1] [Nikolova, Esedoḡlu, Chan 06] E CEN [u] = uθ 1 dx + (1 u) θ 2 dx + Du (). Theorem u argmin E CEN [u] [u > s] argmin E[χ] u BV (,[0,1]) χ BV (,{0,1}) for all a.e. s [0, 1], where [u > s] is the s-superlevel set of u, i.e. {x : u(x) > s}.

15 3 Convex relaxation and thresholding - CEN Idea Relax range of χ to [0, 1] [Nikolova, Esedoḡlu, Chan 06] E CEN [u] = uθ 1 dx + (1 u) θ 2 dx + Du (). Theorem u argmin E CEN [u] [u > s] argmin E[χ] u BV (,[0,1]) χ BV (,{0,1}) for all a.e. s [0, 1], where [u > s] is the s-superlevel set of u, i.e. {x : u(x) > s}. BV (, [0, 1]) convex E CEN convex, but not uniformly convex in u

16 4 Convex relaxation and thresholding - UC Uniformly convex approach E rel [u] = u 2 θ 1 dx + (1 u) 2 θ 2 dx + Du ()

17 4 Convex relaxation and thresholding - UC Uniformly convex approach E rel [u] = u 2 θ 1 dx + (1 u) 2 θ 2 dx + Du () Theorem [Chambolle/Darbon 08][B. SSVM 09] Let θ 1, θ 2 L 1 (), θ 1, θ 2 0 a. e.. Then, E rel has a unique minimizer on BV (). u = argmin E rel [v] χ [u>0.5] argmin E[χ]. v BV () χ BV (,{0,1}) Moreover, u(x) [0, 1] for a.e. x.

18 4 Convex relaxation and thresholding - UC Uniformly convex approach E rel [u] = u 2 θ 1 dx + (1 u) 2 θ 2 dx + Du () Theorem [Chambolle/Darbon 08][B. SSVM 09] Let θ 1, θ 2 L 1 (), θ 1, θ 2 0 a. e.. Then, E rel has a unique minimizer on BV (). u = argmin E rel [v] χ [u>0.5] argmin E[χ]. v BV () χ BV (,{0,1}) Moreover, u(x) [0, 1] for a.e. x. BV () convex (even unconstrained) E UC uniformly convex in u

19 5 Convex relaxation and thresholding - UC (cont.) Proof sketch: [Chambolle/Darbon 09] E rel [min{max{0, v}, 1}] E rel [v] for any v BV ().

20 5 Convex relaxation and thresholding - UC (cont.) Proof sketch: [Chambolle/Darbon 09] E rel [min{max{0, v}, 1}] E rel [v] for any v BV (). Ψ(x, t) := t 2 θ 1 (x) + (1 t) 2 θ 2 (x), C Ψ := Ψ(x, 0) dx. 1 Ψ(x, u) dx = Ψ(x, 0) dx + t Ψ(x, s)χ [u>s] dx ds 0

21 5 Convex relaxation and thresholding - UC (cont.) Proof sketch: [Chambolle/Darbon 09] E rel [min{max{0, v}, 1}] E rel [v] for any v BV (). Ψ(x, t) := t 2 θ 1 (x) + (1 t) 2 θ 2 (x), C Ψ := Ψ(x, 0) dx. 1 Ψ(x, u) dx = Ψ(x, 0) dx + t Ψ(x, s)χ [u>s] dx ds h 1 h 2 a. e., χ i argmin χ BV (,{0,1}) 0 h iχ dx + Dχ ()

22 5 Convex relaxation and thresholding - UC (cont.) Proof sketch: [Chambolle/Darbon 09] E rel [min{max{0, v}, 1}] E rel [v] for any v BV (). Ψ(x, t) := t 2 θ 1 (x) + (1 t) 2 θ 2 (x), C Ψ := Ψ(x, 0) dx. 1 Ψ(x, u) dx = Ψ(x, 0) dx + t Ψ(x, s)χ [u>s] dx ds h 1 h 2 a. e., χ i argmin χ BV (,{0,1}) 0 h iχ dx + Dχ () χ 1 χ 2

23 5 Convex relaxation and thresholding - UC (cont.) Proof sketch: [Chambolle/Darbon 09] E rel [min{max{0, v}, 1}] E rel [v] for any v BV (). Ψ(x, t) := t 2 θ 1 (x) + (1 t) 2 θ 2 (x), C Ψ := Ψ(x, 0) dx. 1 Ψ(x, u) dx = Ψ(x, 0) dx + t Ψ(x, s)χ [u>s] dx ds h 1 h 2 a. e., χ i χ s argmin χ BV (,{0,1}) 0 argmin h iχ dx + Dχ () χ 1 χ 2 χ BV (,{0,1}) { E rel s [χ] := sψ(x, s)χ dx + Dχ () }

24 5 Convex relaxation and thresholding - UC (cont.) Proof sketch: [Chambolle/Darbon 09] E rel [min{max{0, v}, 1}] E rel [v] for any v BV (). Ψ(x, t) := t 2 θ 1 (x) + (1 t) 2 θ 2 (x), C Ψ := Ψ(x, 0) dx. 1 Ψ(x, u) dx = Ψ(x, 0) dx + t Ψ(x, s)χ [u>s] dx ds h 1 h 2 a. e., χ i χ s argmin χ BV (,{0,1}) 0 argmin h iχ dx + Dχ () χ 1 χ 2 χ BV (,{0,1}) { E rel s [χ] := sψ(x, s)χ dx + Dχ () } Ψ(x, ) strictly convex t Ψ(x, ) is strictly increasing χ s 1 χ s 2 a. e. for s 1 s 2.

25 5 Convex relaxation and thresholding - UC (cont.) Proof sketch: [Chambolle/Darbon 09] E rel [min{max{0, v}, 1}] E rel [v] for any v BV (). Ψ(x, t) := t 2 θ 1 (x) + (1 t) 2 θ 2 (x), C Ψ := Ψ(x, 0) dx. 1 Ψ(x, u) dx = Ψ(x, 0) dx + t Ψ(x, s)χ [u>s] dx ds h 1 h 2 a. e., χ i χ s argmin χ BV (,{0,1}) 0 argmin h iχ dx + Dχ () χ 1 χ 2 χ BV (,{0,1}) { E rel s [χ] := sψ(x, s)χ dx + Dχ () } Ψ(x, ) strictly convex t Ψ(x, ) is strictly increasing χ s 1 χ s 2 a. e. for s 1 s 2. graph reconstruction u (x) = sup{s χ s (x) = 1} is well-defined

26 5 Convex relaxation and thresholding - UC (cont.) Proof sketch: [Chambolle/Darbon 09] E rel [min{max{0, v}, 1}] E rel [v] for any v BV (). Ψ(x, t) := t 2 θ 1 (x) + (1 t) 2 θ 2 (x), C Ψ := Ψ(x, 0) dx. 1 Ψ(x, u) dx = Ψ(x, 0) dx + t Ψ(x, s)χ [u>s] dx ds h 1 h 2 a. e., χ i χ s argmin χ BV (,{0,1}) 0 argmin h iχ dx + Dχ () χ 1 χ 2 χ BV (,{0,1}) { E rel s [χ] := sψ(x, s)χ dx + Dχ () } Ψ(x, ) strictly convex t Ψ(x, ) is strictly increasing χ s 1 χ s 2 a. e. for s 1 s 2. graph reconstruction u (x) = sup{s χ s (x) = 1} is well-defined E rel [u] = C Ψ E rel s = E rel [u ] E rel [u] [ χ[u>s] ] ds CΨ Es rel [χ s ] ds

27 6 Predual functional The dual of BV () is difficult to handle.

28 6 Predual functional The dual of BV () is difficult to handle. Use predual of E rel.

29 6 Predual functional The dual of BV () is difficult to handle. Use predual of E rel. Starting point: D rel [q] = F [q] + G[Λq] [Hintermüller/Kunisch 04] { 0 if q 1 a.e. F [q] = I B1 [q] = + else 1 4 G[v] = v2 + vθ 2 θ 1 θ 2 dx θ 1 + θ 2

30 6 Predual functional The dual of BV () is difficult to handle. Use predual of E rel. Starting point: D rel [q] = F [q] + G[Λq] [Hintermüller/Kunisch 04] { 0 if q 1 a.e. F [q] = I B1 [q] = + else 1 4 G[v] = v2 + vθ 2 θ 1 θ 2 dx θ 1 + θ 2 D rel is the predual functional of E rel, i. e. (D rel ) = E rel.

31 6 Predual functional The dual of BV () is difficult to handle. Use predual of E rel. Starting point: D rel [q] = F [q] + G[Λq] [Hintermüller/Kunisch 04] { 0 if q 1 a.e. F [q] = I B1 [q] = + else 1 4 G[v] = v2 + vθ 2 θ 1 θ 2 dx θ 1 + θ 2 D rel is the predual functional of E rel, i. e. (D rel ) = E rel. Λ L(Q, V), Q = H N (div, ) and V = L 2 ()

32 6 Predual functional The dual of BV () is difficult to handle. Use predual of E rel. Starting point: D rel [q] = F [q] + G[Λq] [Hintermüller/Kunisch 04] { 0 if q 1 a.e. F [q] = I B1 [q] = + else 1 4 G[v] = v2 + vθ 2 θ 1 θ 2 dx θ 1 + θ 2 D rel is the predual functional of E rel, i. e. (D rel ) = E rel. Λ L(Q, V), Q = H N (div, ) and V = L 2 () H(div, ) = {q L 2 (, R n ) : div q L 2 ()} and H N (div, ) = H(div, ) {q ν = 0 on }

33 6 Predual functional The dual of BV () is difficult to handle. Use predual of E rel. Starting point: D rel [q] = F [q] + G[Λq] [Hintermüller/Kunisch 04] { 0 if q 1 a.e. F [q] = I B1 [q] = + else 1 4 G[v] = v2 + vθ 2 θ 1 θ 2 dx θ 1 + θ 2 D rel is the predual functional of E rel, i. e. (D rel ) = E rel. Λ L(Q, V), Q = H N (div, ) and V = L 2 () H(div, ) = {q L 2 (, R n ) : div q L 2 ()} and H N (div, ) = H(div, ) {q ν = 0 on } Λ = div, Λ = holds in the sense Λ v, q = (v, div q) L 2 () v V, q Q

34 7 Predual functional (cont.) From general theory one knows (D rel ) [v] = F [ Λ v] + G [v]. F [q] = I B1 [q], G[v] = 1 4 v2 + vθ 2 θ 1 θ 2 dx θ 1 + θ 2

35 7 Predual functional (cont.) From general theory one knows (D rel ) [v] = F [ Λ v] + G [v]. The Fenchel conjugates can be computed as follows: F [ Λ v] = sup( Λ v, q F [q]) q Q F [q] = I B1 [q], G[v] = 1 4 v2 + vθ 2 θ 1 θ 2 dx θ 1 + θ 2

36 7 Predual functional (cont.) From general theory one knows (D rel ) [v] = F [ Λ v] + G [v]. The Fenchel conjugates can be computed as follows: ( ) F [ Λ v] = sup( Λ v, q F [q]) = sup v div q dx I B1 [q] q Q q Q = sup v div q dx = Dv () q Q, q 1 F [q] = I B1 [q], G[v] = 1 4 v2 + vθ 2 θ 1 θ 2 dx θ 1 + θ 2

37 7 Predual functional (cont.) From general theory one knows (D rel ) [v] = F [ Λ v] + G [v]. The Fenchel conjugates can be computed as follows: ( ) F [ Λ v] = sup( Λ v, q F [q]) = sup v div q dx I B1 [q] q Q q Q = sup v div q dx = Dv () q Q, q 1 G ( [v] = (v, w)l 2 () G[w] ) sup w L 2 () F [q] = I B1 [q], G[v] = 1 4 v2 + vθ 2 θ 1 θ 2 dx θ 1 + θ 2

38 7 Predual functional (cont.) From general theory one knows (D rel ) [v] = F [ Λ v] + G [v]. The Fenchel conjugates can be computed as follows: ( ) F [ Λ v] = sup( Λ v, q F [q]) = sup v div q dx I B1 [q] q Q q Q = sup v div q dx = Dv () q Q, q 1 G ( [v] = (v, w)l 2 () G[w] ) = v 2 θ 1 + (1 v) 2 θ 2 dx sup w L 2 () where last supremum is attained for w = 2v(θ 1 + θ 2 ) 2θ 2. F [q] = I B1 [q], G[v] = 1 4 v2 + vθ 2 θ 1 θ 2 dx θ 1 + θ 2

39 7 Predual functional (cont.) From general theory one knows (D rel ) [v] = F [ Λ v] + G [v]. The Fenchel conjugates can be computed as follows: ( ) F [ Λ v] = sup( Λ v, q F [q]) = sup v div q dx I B1 [q] q Q q Q = sup v div q dx = Dv () q Q, q 1 G ( [v] = (v, w)l 2 () G[w] ) = v 2 θ 1 + (1 v) 2 θ 2 dx sup w L 2 () where last supremum is attained for w = 2v(θ 1 + θ 2 ) 2θ 2. Thus, (D rel ) [v] = F [ Λ v] + G [v] = E rel [v]. F [q] = I B1 [q], G[v] = 1 4 v2 + vθ 2 θ 1 θ 2 dx θ 1 + θ 2

40 8 Predual functional (cont.) Central insight: Minimizers p of D rel and u of (D rel ) fulfill D rel [p] = (D rel ) [u]

41 8 Predual functional (cont.) Central insight: Minimizers p of D rel and u of (D rel ) fulfill D rel [p] = (D rel ) [u] Can be seen by formally exchanging inf and sup: D rel [p] = inf q H N (div,) (F [q] + G[Λq])

42 8 Predual functional (cont.) Central insight: Minimizers p of D rel and u of (D rel ) fulfill D rel [p] = (D rel ) [u] Can be seen by formally exchanging inf and sup: D rel [p] = inf q H N (div,) (F [q] + G[Λq]) = inf (F [q] + q H N (div,) G [Λq])

43 8 Predual functional (cont.) Central insight: Minimizers p of D rel and u of (D rel ) fulfill D rel [p] = (D rel ) [u] Can be seen by formally exchanging inf and sup: D rel [p] = inf q H N (div,) = inf q H N (div,) v L 2 () (F [q] + G[Λq]) = inf sup (F [q] + q H N (div,) G [Λq]) (F [q] + v, Λq G [v])

44 8 Predual functional (cont.) Central insight: Minimizers p of D rel and u of (D rel ) fulfill D rel [p] = (D rel ) [u] Can be seen by formally exchanging inf and sup: D rel [p] = inf q H N (div,) = inf q H N (div,) v L 2 () = sup v L 2 () ( (F [q] + G[Λq]) = inf sup sup q H N (div,) (F [q] + q H N (div,) G [Λq]) (F [q] + v, Λq G [v]) ( Λ v, q F [q]) G [v] )

45 8 Predual functional (cont.) Central insight: Minimizers p of D rel and u of (D rel ) fulfill D rel [p] = (D rel ) [u] Can be seen by formally exchanging inf and sup: D rel [p] = inf q H N (div,) = inf q H N (div,) v L 2 () = sup v L 2 () ( (F [q] + G[Λq]) = inf sup sup q H N (div,) = sup ( F [ Λ v] G [v]) v L 2 () (F [q] + q H N (div,) G [Λq]) (F [q] + v, Λq G [v]) ( Λ v, q F [q]) G [v] )

46 8 Predual functional (cont.) Central insight: Minimizers p of D rel and u of (D rel ) fulfill D rel [p] = (D rel ) [u] Can be seen by formally exchanging inf and sup: D rel [p] = inf q H N (div,) = inf q H N (div,) v L 2 () = sup v L 2 () ( (F [q] + G[Λq]) = inf sup sup q H N (div,) = sup ( F [ Λ v] G [v]) v L 2 () (F [q] + q H N (div,) G [Λq]) (F [q] + v, Λq G [v]) ( Λ v, q F [q]) G [v] = sup ( (D rel ) [v]) = inf v L 2 () v L 2 () (Drel ) [v] = (D rel ) [u]. )

47 9 Functional a posteriori error estimates for E rel Two measures of uniform convexity for J : X R

48 9 Functional a posteriori error estimates for E rel Two measures of uniform convexity for J : X R J [ x 1 +x 2 ] 2 + ΦJ (x 2 x 1 ) 1 2 (J[x 1] + J[x 2 ]) for all x 1, x 2 X

49 9 Functional a posteriori error estimates for E rel Two measures of uniform convexity for J : X R J [ x 1 +x 2 ] 2 + ΦJ (x 2 x 1 ) 1 2 (J[x 1] + J[x 2 ]) for all x 1, x 2 X x, x 2 x 1 + Ψ J (x 2 x 1 ) J[x 2 ] J[x 1 ] for all x J[x 1 ]

50 9 Functional a posteriori error estimates for E rel Two measures of uniform convexity for J : X R J [ x 1 +x 2 ] 2 + ΦJ (x 2 x 1 ) 1 2 (J[x 1] + J[x 2 ]) for all x 1, x 2 X x, x 2 x 1 + Ψ J (x 2 x 1 ) J[x 2 ] J[x 1 ] for all x J[x 1 ] Theorem [Repin 00]: Let u argminṽ V E rel [ṽ] and q Q, v V = V = L 2 (). Then, Φ G (v u) + Φ F ( Λ (v u)) + Ψ E rel( v u 2 ) 1 2 (Erel [v] + D rel [q]).

51 10 Repin theorem - Proof sketch The convexity property of Φ G and Φ F gives: Φ G (v u) + Φ F ( Λ (v u)) 1 2 (F [ Λ v] + G [v] + F [ Λ u] + G [u]) ( F [ Λ u+v ] 2 + G [ ]) u+v 2

52 10 Repin theorem - Proof sketch The convexity property of Φ G and Φ F gives: Φ G (v u) + Φ F ( Λ (v u)) 1 2 (F [ Λ v] + G [v] + F [ Λ u] + G [u]) ( F [ Λ u+v ] 2 + G [ ]) u+v 2 = 1 2 (Erel [v] + E rel [u]) E rel [ u+v 2 ]

53 10 Repin theorem - Proof sketch The convexity property of Φ G and Φ F gives: Φ G (v u) + Φ F ( Λ (v u)) 1 2 (F [ Λ v] + G [v] + F [ Λ u] + G [u]) ( F [ Λ u+v ] 2 + G [ ]) u+v 2 = 1 2 (Erel [v] + E rel [u]) E rel [ u+v 2 ] Since u is a minimizer of E rel and thus 0 E rel [u], we get

54 10 Repin theorem - Proof sketch The convexity property of Φ G and Φ F gives: Φ G (v u) + Φ F ( Λ (v u)) 1 2 (F [ Λ v] + G [v] + F [ Λ u] + G [u]) ( F [ Λ u+v ] 2 + G [ ]) u+v 2 = 1 2 (Erel [v] + E rel [u]) E rel [ u+v 2 ] Since u is a minimizer of E rel and thus 0 E rel [u], we get Ψ E rel( v u ) ( 2 = u+v ΨE rel 2 u ) E rel [ u+v 2 ] Erel [u].

55 10 Repin theorem - Proof sketch The convexity property of Φ G and Φ F gives: Φ G (v u) + Φ F ( Λ (v u)) 1 2 (F [ Λ v] + G [v] + F [ Λ u] + G [u]) ( F [ Λ u+v ] 2 + G [ ]) u+v 2 = 1 2 (Erel [v] + E rel [u]) E rel [ u+v 2 ] Since u is a minimizer of E rel and thus 0 E rel [u], we get Ψ E rel( v u ) ( 2 = u+v ΨE rel 2 u ) E rel [ u+v 2 ] Erel [u]. Adding both inequalities gives Φ G (v u) + Φ F ( Λ (v u)) + Ψ E rel( v u 2 ) 1 2 (Erel [v] E rel [u]).

56 10 Repin theorem - Proof sketch The convexity property of Φ G and Φ F gives: Φ G (v u) + Φ F ( Λ (v u)) 1 2 (F [ Λ v] + G [v] + F [ Λ u] + G [u]) ( F [ Λ u+v ] 2 + G [ ]) u+v 2 = 1 2 (Erel [v] + E rel [u]) E rel [ u+v 2 ] Since u is a minimizer of E rel and thus 0 E rel [u], we get Ψ E rel( v u ) ( 2 = u+v ΨE rel 2 u ) E rel [ u+v 2 ] Erel [u]. Adding both inequalities gives Φ G (v u) + Φ F ( Λ (v u)) + Ψ E rel( v u 2 ) 1 2 (Erel [v] E rel [u]). The claim follows using the weak complementarity principle E rel [u] D rel [q].

57 11 Functional a posteriori error estimates for E rel Theorem [Repin 00]: Let u argminṽ V E rel [ṽ] and q Q, v V = V = L 2 (). Then, Φ G (v u) + Φ F ( Λ (v u)) + Ψ E rel( v u 2 ) 1 2 (Erel [v] + D rel [q]).

58 11 Functional a posteriori error estimates for E rel Theorem [Repin 00]: Let u argminṽ V E rel [ṽ] and q Q, v V = V = L 2 (). Then, Φ G (v u) + Φ F ( Λ (v u)) + Ψ E rel( v u 2 In our case, one gets Φ F 0, Φ G (v) = 1 4 v 2 (θ 1 + θ 2 ) dx, Ψ E rel(v) = ) 1 2 (Erel [v] + D rel [q]). v 2 (θ 1 + θ 2 ) dx.

59 11 Functional a posteriori error estimates for E rel Theorem [Repin 00]: Let u argminṽ V E rel [ṽ] and q Q, v V = V = L 2 (). Then, Φ G (v u) + Φ F ( Λ (v u)) + Ψ E rel( v u 2 In our case, one gets Φ F 0, Φ G (v) = 1 4 v 2 (θ 1 + θ 2 ) dx, Ψ E rel(v) = ) 1 2 (Erel [v] + D rel [q]). v 2 (θ 1 + θ 2 ) dx. Using 1 2ν (c 1 c 2 ) 2 θ 1 + θ 2 this leads to: Theorem [A posteriori error estimate for E rel ]: Let u V be the minimizer of E rel. Then, for any v V and q Q it holds that u v 2 L 2 () 2ν ( ) err2 u[v, q] := (c 1 c 2 ) 2 E rel [v] + D rel [q]

60 11 Functional a posteriori error estimates for E rel Theorem [Repin 00]: Let u argminṽ V E rel [ṽ] and q Q, v V = V = L 2 (). Then, Φ G (v u) + Φ F ( Λ (v u)) + Ψ E rel( v u 2 In our case, one gets Φ F 0, Φ G (v) = 1 4 v 2 (θ 1 + θ 2 ) dx, Ψ E rel(v) = ) 1 2 (Erel [v] + D rel [q]). v 2 (θ 1 + θ 2 ) dx. Using 1 2ν (c 1 c 2 ) 2 θ 1 + θ 2 this leads to: Theorem [A posteriori error estimate for E rel ]: Let u V be the minimizer of E rel. Then, for any v V and q Q it holds that u v 2 L 2 () 2ν ( ) err2 u[v, q] := (c 1 c 2 ) 2 E rel [v] + D rel [q] = 2ν (c 1 c 2 ) 2 v 2 θ 1 + (1 v) 2 θ 2 + v (div q)2 +div qθ 2 θ 1 θ 2 θ 1 +θ 2 dx.

61 12 A posteriori error estimates for the binary model Key observation: solutions of E rel are characterized by steep profiles

62 12 A posteriori error estimates for the binary model Key observation: solutions of E rel are characterized by steep profiles S η = [ 1 2 η v η] (η ( 0, 1 2) ) set of non properly identified phase regions and a[v, η] = S η

63 12 A posteriori error estimates for the binary model Key observation: solutions of E rel are characterized by steep profiles S η = [ 1 2 η v η] (η ( 0, 1 2) ) set of non properly identified phase regions and a[v, η] = S η η u 0 u χ [u>0.5] 0 1 a[v, η] and err χ [v, q, η]

64 12 A posteriori error estimates for the binary model Key observation: solutions of E rel are characterized by steep profiles S η = [ 1 2 η v η] (η ( 0, 1 2) ) set of non properly identified phase regions and a[v, η] = S η u 0 u χ [u>0.5] 0 1 a[v, η] and err χ [v, q, η] Theorem [A posteriori error estimate for E]: Let χ BV (, {0, 1}) be the minimizer of the binary Mumford Shah functional obtained from E rel. Then for all v V = L 2 () and q Q = H N (div, ) we have { χ χ [v> 1 L 2 ] 1 () inf err χ [v, q, η] = η (0, 1 2 ) ( a[v, η] + 1 η 2 err 2 u[v, q] η )}.

65 12 A posteriori error estimates for the binary model Theorem [A posteriori error estimate for E]: Let χ BV (, {0, 1}) be the minimizer of the binary Mumford Shah functional obtained from E rel. Then for all v V = L 2 () and q Q = H N (div, ) we have { χ χ [v> 1 L 2 ] 1 () inf err χ [v, q, η] = η (0, 1 2 ) ( a[v, η] + 1 η 2 err 2 u[v, q] Proof: [ u > 1 2] [ v > 1 2] [ 1 2 η v η] [ u v > η] η η )}. u minimizer of E rel v V

66 12 A posteriori error estimates for the binary model Theorem [A posteriori error estimate for E]: Let χ BV (, {0, 1}) be the minimizer of the binary Mumford Shah functional obtained from E rel. Then for all v V = L 2 () and q Q = H N (div, ) we have { χ χ [v> 1 L 2 ] 1 () inf err χ [v, q, η] = η (0, 1 2 ) ( a[v, η] + 1 η 2 err 2 u[v, q] Proof: [ u > 1 [ [ 2] v > 1 2] 1 2 η v η] [ u v > η] Using the estimate for u v 2 L 2 (), we get L n 1 ([ u v > η]) u v 2 dx 1 err 2 η 2 η u[v, q]. 2 { u v >η} The above holds for any η (0, 1 2 ), so also for inf. η (0, 1 2 ) )}.

67 13 Two discretizations and primal-dual algorithm T adaptive simplicial mesh Vh 0 space of piecewise constant finite element functions on T space of continuous and affine finite element functions on T V 1 h

68 13 Two discretizations and primal-dual algorithm T adaptive simplicial mesh Vh 0 space of piecewise constant finite element functions on T Vh 1 space of continuous and affine finite element functions on T 1 4 G h [v h ] = v2 h + v h θ 1,h θ 2,h dx, F h [q h ] = I B1 [q h ], θ 1,h + θ 2,h G h [v h] = vh 2 θ 1,h + (1 v h ) 2 θ 2,h dx, Fh [q h] = I h ( q h ) dx, for θ i,h = I h ( 1 ν (c i u 0 ) 2 ) for i = 1, 2, I h Lagrange interpolation

69 13 Two discretizations and primal-dual algorithm T adaptive simplicial mesh Vh 0 space of piecewise constant finite element functions on T Vh 1 space of continuous and affine finite element functions on T 1 4 G h [v h ] = v2 h + v h θ 1,h θ 2,h dx, F h [q h ] = I B1 [q h ], θ 1,h + θ 2,h G h [v h] = vh 2 θ 1,h + (1 v h ) 2 θ 2,h dx, Fh [q h] = I h ( q h ) dx, for θ i,h = I h ( 1 ν (c i u 0 ) 2 ) for i = 1, 2, I h Lagrange interpolation (FE) discretization V h = Vh 1 with L2 -scalar product and Q h = (Vh 1)2 with lumped mass scalar product, Λ h : V h Q h implicitly defined via (cf. [Bartels 14]) I h ( Λ h v h q h ) dx = v h P h div q h dx q h Q h, v h V h P h : L 2 () V h denotes L 2 -projection

70 13 Two discretizations and primal-dual algorithm T adaptive simplicial mesh Vh 0 space of piecewise constant finite element functions on T Vh 1 space of continuous and affine finite element functions on T 1 4 G h [v h ] = v2 h + v h θ 1,h θ 2,h dx, F h [q h ] = I B1 [q h ], θ 1,h + θ 2,h G h [v h] = vh 2 θ 1,h + (1 v h ) 2 θ 2,h dx, Fh [q h] = q h dx, for θ i,h = I h ( 1 ν (c i u 0 ) 2 ) for i = 1, 2, I h Lagrange interpolation (FE ) discretization V h = Vh 1 and Q h = (Vh 0)2 with L 2 -scalar product Λ h : Q h V h given by I h (Λ h q h v h ) dx = q h v h dx q h Q h, v h V h Λ h v h = v h To evaluate err 2 u an L 2 -projection of the dual solution is performed

71 14 The primal-dual algorithm Primal-dual algorithm [Chambolle, Pock 11] while u k+1 h u k h >THRESHOLD do p k+1 h = (Id + σ F h ) 1 (p k h σλ hūk h ); u k+1 h = (Id + τ G h ) 1 (u k h + τλ hp k+1 h ); ū k+1 h = 2u k+1 h u k h ; end

72 14 The primal-dual algorithm Primal-dual algorithm [Chambolle, Pock 11] while u k+1 h u k h >THRESHOLD do p k+1 h = (Id + σ F h ) 1 (p k h σλ hūk h ); u k+1 h = (Id + τ G h ) 1 (u k h + τλ hp k+1 h ); ū k+1 h = 2u k+1 h u k h ; end The resolvent (Id + σ F h ) 1 for F h is well known.

73 14 The primal-dual algorithm Primal-dual algorithm [Chambolle, Pock 11] while u k+1 h u k h >THRESHOLD do p k+1 h = (Id + σ F h ) 1 (p k h σλ hūk h ); u k+1 h = (Id + τ G h ) 1 (u k h + τλ hp k+1 h ); ū k+1 h = 2u k+1 h u k h ; end The resolvent (Id + σ F h ) 1 for F h is well known. The resolvent (Id + τ G h ) 1 for G h [v h] = vh 2 θ 1,h + (1 v h ) 2 θ 2,h dx can be computed in a straightforwardly, but involves mass matrices.

74 The primal-dual algorithm Primal-dual algorithm [Chambolle, Pock 11] while u k+1 h u k h >THRESHOLD do p k+1 h = (Id + σ F h ) 1 (p k h σλ hūk h ); u k+1 h = (Id + τ G h ) 1 (u k h + τλ hp k+1 h ); ū k+1 h = 2u k+1 h u k h ; end The resolvent (Id + σ F h ) 1 for F h is well known. The resolvent (Id + τ G h ) 1 for G h [v h] = vh 2 θ 1,h + (1 v h ) 2 θ 2,h dx can be computed in a straightforwardly, but involves mass matrices. The operator norm of Λ can be estimated as follows Λ h 2 48( )h 2 min for (FE ) and n = 2 Λ h 2 96( )h 2 min for (FE) 14

75 15 Numerical results - Flower u 0 u h χ [uh >0.5]

76 15 Numerical results - Flower u 0 u h χ [uh >0.5] mesh after 2 nd, 5 th, 10 th iteration

77 16 Numerical results - Flower (cont.) degrees of freedom err 2 u estimator for u u h 2 L 2 () ((FE) (black), (FE ) (red)) err χ estimator for χ χ [uh >0.5] L 1 () ((FE) (orange), (FE ) (blue))

78 17 Numerical results - Cameraman u 0 u h χ [uh >0.5] Image source: MATLAB

79 17 Numerical results - Cameraman u 0 u h χ [uh >0.5] mesh after 5 th, 10 th iteration Image source: MATLAB

80 18 Numerical results - Cameraman (cont.) degrees of freedom err 2 u estimator for u u h 2 L 2 () ((FE) (black), (FE ) (red)) err χ estimator for χ χ [uh >0.5] L 1 () ((FE) (orange), (FE ) (blue))

81 19 Numerical results - Gaussians using (FE ) u 0 u h χ [uh >0.5]

82 19 Numerical results - Gaussians using (FE ) u 0 u h χ [uh >0.5] (p h ) 1 (p h ) 2 0 1

83 20 Numerical results - Gaussians using (FE ) degrees of freedom err 2 u estimator for u u h 2 L 2 () err χ estimator for χ χ [uh >0.5] L 1 ()

84 21 Conclusion and outlook Summary A posteriori error estimates for the binary Mumford Shah model

85 21 Conclusion and outlook Summary A posteriori error estimates for the binary Mumford Shah model Natural error quantity: area of the non-properly segmented region

86 21 Conclusion and outlook Summary A posteriori error estimates for the binary Mumford Shah model Natural error quantity: area of the non-properly segmented region Key ingredients:

87 21 Conclusion and outlook Summary A posteriori error estimates for the binary Mumford Shah model Natural error quantity: area of the non-properly segmented region Key ingredients: Uniformly convex, unonstrained relaxation of the original problem

88 21 Conclusion and outlook Summary A posteriori error estimates for the binary Mumford Shah model Natural error quantity: area of the non-properly segmented region Key ingredients: Uniformly convex, unonstrained relaxation of the original problem Repin s theorem for a posteriori error estimation of the relaxation

89 21 Conclusion and outlook Summary A posteriori error estimates for the binary Mumford Shah model Natural error quantity: area of the non-properly segmented region Key ingredients: Uniformly convex, unonstrained relaxation of the original problem Repin s theorem for a posteriori error estimation of the relaxation Cut out argument to estimate the area mismatch

90 21 Conclusion and outlook Summary A posteriori error estimates for the binary Mumford Shah model Natural error quantity: area of the non-properly segmented region Key ingredients: Uniformly convex, unonstrained relaxation of the original problem Repin s theorem for a posteriori error estimation of the relaxation Cut out argument to estimate the area mismatch Estimate can be used for adaptive mesh refinement

91 21 Conclusion and outlook Summary A posteriori error estimates for the binary Mumford Shah model Natural error quantity: area of the non-properly segmented region Key ingredients: Uniformly convex, unonstrained relaxation of the original problem Repin s theorem for a posteriori error estimation of the relaxation Cut out argument to estimate the area mismatch Estimate can be used for adaptive mesh refinement Two different adaptive primal-dual finite element schemes

92 21 Conclusion and outlook Summary A posteriori error estimates for the binary Mumford Shah model Natural error quantity: area of the non-properly segmented region Key ingredients: Uniformly convex, unonstrained relaxation of the original problem Repin s theorem for a posteriori error estimation of the relaxation Cut out argument to estimate the area mismatch Estimate can be used for adaptive mesh refinement Two different adaptive primal-dual finite element schemes Numerical experiments show qualitative and quantitative properties

93 21 Conclusion and outlook Summary A posteriori error estimates for the binary Mumford Shah model Natural error quantity: area of the non-properly segmented region Key ingredients: Uniformly convex, unonstrained relaxation of the original problem Repin s theorem for a posteriori error estimation of the relaxation Cut out argument to estimate the area mismatch Estimate can be used for adaptive mesh refinement Two different adaptive primal-dual finite element schemes Numerical experiments show qualitative and quantitative properties Outlook Transfer the approach to more general computer vision problems

94 21 Conclusion and outlook Summary A posteriori error estimates for the binary Mumford Shah model Natural error quantity: area of the non-properly segmented region Key ingredients: Uniformly convex, unonstrained relaxation of the original problem Repin s theorem for a posteriori error estimation of the relaxation Cut out argument to estimate the area mismatch Estimate can be used for adaptive mesh refinement Two different adaptive primal-dual finite element schemes Numerical experiments show qualitative and quantitative properties Outlook Transfer the approach to more general computer vision problems Develop/use minimization algorithms that are tailored to the adaptive structure

95 22 References S. Bartels, Error control and adaptivity for a variational model problem defined on functions of bounded variation, Math. Comp. (online first), (2014). B. Berkels, A. Effland and M. Rumpf, A Posteriori Error Control for the Binary Mumford Shah Model, Math. Comp. (accepted) B. Berkels, An unconstrained multiphase thresholding approach for image segmentation. In Proceedings of the Second SSVM, A. Chambolle and J. Darbon, On total variation minimization and surface evolution using parametric maximum flows, International Journal of Computer Vision, 84 (2009), pp A. Chambolle and T. Pock, A first-order primal-dual algorithm for convex problems with applications to imaging, Journal of Mathematical Imaging and Vision, 40 (2011), pp M. Hintermüller and K. Kunisch, Total bounded variation regularization as a bilaterally constrained optimization problem, SIAM J. Appl. Math., 64 (2004), pp M. Nikolova, S. Esedoḡlu and T. F. Chan, Algorithms for finding global minimizers of image segmentation and denoising models, SIAM Journal on Applied Mathematics, 66 (2006), pp S. Repin, A posteriori error estimation for variational problems with uniformly convex functionals, Mathematics of Computation, 69 (2000), pp

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