Optimality Bounds for a Variational Relaxation of the Image Partitioning Problem

Journal man uscript No. (will be inserted b y the editor) Optimali t y Bounds for a V ariatio nal Relaxati on of the Image P artitioning Problem Jan Lellmann · F rank Lenzen · Chris toph Sc hn¨ orr Receiv ed: da te / Accepted: date Abstract W e consider a v ariationa l co n vex rela xation of a class of optimal par titioning and m ulticlass labe l- ing problems, which has recently proven quite success- ful and can be seen as a contin uous ana logue of Lin- ear P r ogramming (LP) rela xation metho ds for finite- dimensional problems. While for the la tter case several optimality bo unds are known, to our knowledge no s uch bo unds exist in the con tinu ous setting. W e pro vide s uc h a b ound by analyzing a pr obabilistic r ounding metho d, showing that it is p ossible to obtain an integral so lution of the origina l partitioning problem from a solution o f the relaxed problem with an a priori upper b ound on the ob jective, ensuring the qua lity of the result from the viewp oint of optimization. The appro ach has a natural int erpreta tion as an approximate, multiclass v a riant of the celebrated coare a formula. Keyw ords Conv ex Relaxa tio n · Multiclas s Lab eling · Approximation Bound · Com binatorial Optimization · T otal V a riation · Linea r P rogr a mming Relaxation J. Lellmann Image and P attern Analy sis Group & HCI Dept. of M athematics and Computer Science, Univ ersity of Heidelb erg Curr ent A d dr ess: Dept. of Applied Mathem atics and Theo- retical Physics, Uni ver sity of Cam b ridge, United Kingdom E-mail: j.l ellmann@dam tp. cam.ac .uk F. Lenzen · C. Schn¨ orr Image and P attern Analy sis Group & HCI Dept. of Mathematics and Computer Science Universit y of Heidelb erg E-mail: lenzen@iwr.uni-heidelb erg.de, sc h noerr@math.uni- heidelb erg.de 1 Introduction and Bac kground 1.1 Co nvex Relaxations of Partitioning Problems In this work, we will b e concerne d with a class of vari- ational problems used in image pro ces s ing and a nalysis for fo rmulating multiclass image par titio ning pro blems, which a re of the for m inf u ∈C E f ( u ) := Z Ω h u ( x ) , s ( x ) i dx + Z Ω dΨ ( Du ) , (1) C E := BV( Ω , E ) (2) = { u ∈ BV ( Ω ) l | u ( x ) ∈ E for a.e. x ∈ Ω } , (3) E := { e 1 , . . . , e l } . (4) The la b eling function u : Ω → R l assigns to each po in t in the image do main Ω := (0 , 1) d a label i ∈ I := { 1 , . . . , l } , which is represented by one of the l - dimensional unit vectors e 1 , . . . , e l . Since it is piecewise constant and therefore canno t be assumed to b e dif- ferentiable, the problem is formulated as a fr e e disc on- tinuity pr oblem in the space BV( Ω , E ) of functions o f bo unded v ariatio n; w e refer to [2 ] for a gener al ov er view. The ob jectiv e function f consists of a da ta term and a regula r izer. The data term is given in terms of the L 1 function s ( x ) = ( s 1 ( x ) , . . . , s l ( x )) ∈ R l , and assigns to the choice u ( x ) = e i the “p enalty” s i ( x ), in the sense that Z Ω h u ( x ) , s ( x ) i dx = l X i =1 Z Ω i s i ( x ) dx, (5) where Ω i := u − 1 ( { e i } ) = { x ∈ Ω | u ( x ) = e i } is the class r e gion for lab el i , i.e., the set of p oints that are assigned the i - th lab el. The data term g e ne r ally dep ends o n the input data – such as color v a lue s of a re c o rded imag e, depth measur ement s, or other features – and promo tes a g o o d fit of the minimizer to the input data . Wh ile it 2 Jan Lellmann et al. is purely lo cal, there a re no further restrictions such as contin uity , conv exity etc., therefor e it covers many in- teresting applica tio ns such as seg men tation, mult i-view reconstructio n, stitching, and inpainting [24]. 1.2 Conv ex Regularize rs The r e gularizer is defined b y the p os itiv ely homoge- neous, c ont inuous and conv ex function Ψ : R d × l → R > 0 acting on the distributional deriv ative D u of u , a nd in- corp ora tes additional prior kno wle dg e ab out the “typ- ical” app earance of the desir ed output. F or piece wise constant u , it ca n b e seen that th e definition in (1) amounts to a weigh ted penaliz a tion of the disc ontinu- ities of u : Z Ω dΨ ( Du ) = (6) Z J u Ψ ( ν u ( x )( u + ( x ) − u − ( x )) ⊤ ) d H d − 1 ( x ) , where J u is the jump set of u , i.e., the set of p oints where u ha s well-defined right-hand and left-hand lim- its u + and u − and (in an infinitesimal sens e) jumps betw een the v a lues u + ( x ) , u − ( x ) ∈ R l across a hyper- plane with normal ν u ( x ) ∈ R d , k ν u ( x ) k 2 = 1 (see [2] for the precise definitions). A pa rticular case is to s et Ψ = (1 / √ 2) k · k 2 , i.e ., the scaled F rob enius norm. In this case J ( u ) is just the (scaled) total v ar ia tion of u , and, since u + ( x ) a nd u − ( x ) assume v a lue s in E and cannot be equal on the jump set J u , it holds that J ( u ) = 1 √ 2 Z J u k u + ( x ) − u − ( x ) k 2 d H d − 1 ( x ) , (7) = H d − 1 ( J u ) . (8) Therefore, for Ψ = (1 / √ 2) k · k 2 the regulariz e r just amounts to pe nalizing the total length of the inter- faces b et ween cla ss regions as measur e d by the ( d − 1 )- dimensional Hausdorff measure H d − 1 , which is known as uniform met ric or Pott s regularizer . A g eneral regular izer was prop osed in [18], based on [5]: Given a metric ( distanc e ) d : { 1 , . . . , l } 2 → R > 0 (not to b e confused with the a m bient spa ce dimension), define Ψ d ( z = ( z 1 , . . . , z l )) := sup v ∈ D d loc h z , v i , (9) D d lo c := {  v 1 , . . . , v l  ∈ R d × l | . . . (10) k v i − v j k 2 6 d ( i, j ) ∀ i, j ∈ { 1 , . . . , l } , . . . l X k =1 v k = 0 } . It was then sho wn that Ψ d ( ν ( e j − e i ) ⊤ ) = d ( i, j ) , (11) therefore in view of (7) the corres po nding regular iz er is non-uniform : the boundary b etw een the class regions Ω i and Ω j is p enalize d by its length, multiplied by the weigh t d ( i, j ) dep ending on t he lab els of b oth r e gions . How ever, even for the compar ably simple regula rizer (7), the mo del (1 ) is a (spatially contin uo us) c ombinato- rial pro ble m due to the integral nature of the constra int set C E , there fo re optimization is nontrivial. In the con- text of m ulticlass imag e partitio ning , a fir s t a pproach can be fo und in [2 0], where the problem w as p osed in a level set-for mu lation in terms o f a labeling function φ : Ω → { 1 , . . . , l } , which is subseq uen tly rela xed to R . Then φ is replaced by p olynomials in φ , whic h coin- cide with the indica to r functions u i for the ca se where φ a ssumes integral v alues . Howev er , the n umer ical a p- proach inv olves several nonlinearities and r equires to solve a se quence o f no n trivial subproblems. In contrast, r epresentation (1) directly suggests a more straight forward relaxation to a conv ex pro blem: replace E by its conv ex hull, which is just the unit sim- plex in l dimensions, ∆ l := conv { e 1 , . . . , e l } (12) = { a ∈ R l | a > 0 , l X i =1 a i = 1 } , and s olve the r elaxe d problem inf u ∈C f ( u ) , (13) C := BV( Ω , ∆ l ) (14) = { u ∈ B V( Ω ) l | u ( x ) ∈ ∆ l for a.e. x ∈ Ω } . (15) Sparked by a series of paper s [29, 5, 17], recently ther e has b een m uch interest in problems of t his form, since they – a lthough generally nonsmo o th – are conv ex and therefore can b e so lved to global optimality , e .g ., using primal-dual techniques. The approa c h has prov en useful for a wide range of a pplications [14, 11 , 10, 28]. 1.3 Finite-Dimensiona l vs. Contin uous Approaches Many of these applications hav e been tackled b efore in a finite-dimensio nal setting, where they can b e form u- lated as combinatorial problems on a g rid graph, and solved using com binatorial optimization meth o ds such as α - e xpansion and related integer linear pr ogramming (ILP) metho ds [4, 15]. Thes e metho ds hav e b e e n shown to yield a n integral labeling u ′ ∈ C E with the a priori bo und f ( u ′ ) 6 2 max i 6 = j d ( i, j ) min i 6 = j d ( i, j ) f ( u ∗ E ) , (16) where u ∗ E is the (unknown) solution o f the in tegra l prob- lem (1). They therefore permit to compute a subopti- mal so lutio n to the – or iginally NP-ha rd [4] – co m bina- torial problem with an upp er b ound on the o b jective. Optimalit y Bounds for a V ariational Relaxation of the Image Partitioning Problem 3 Figure 1 Segmentat ion of an image into 12 classes using a com b inatorial metho d . Left: Input image, Right: Result ob- tained by solving a c ombinato rial d iscretized problem with 4-neigh b orhoo d. The b ottom row shows detailed views of the mark ed parts of the image. The minimizer of the com bin ato- rial problem exhib its bl o cky artifacts caused by th e cho ice of discretization. Figure 2 Segmen tation obtained by solvin g a fin ite- differences discretiza tion of the r elax e d spatially con tinuous problem. L eft: Non-integr al s olution obtained as a minimizer of the discretized relaxed p roblem. Right: Integ ral labeling obtained by round ing the fractional lab els in th e s olution of the relaxed problem to the nearest in tegral lab el. The rounded result con tains almost no s tructural artifacts. No such bo und is y et a v ailable for methods base d o n the spatially contin uo us problem (13). Despite these strong theoretical and practical re- sults a v ailable for the finite-dimensional com bina torial energies, the function-base d, spatially contin uo us for- m ulation (1) has several unique adv a nt ages: – The energy (1) is truly isotropic, in the sense that for a pro per choice of Ψ it is inv ariant under r otation of the co ordinate system. Pursuing finite-dimensional “discretize-fir st” a pproaches generally introduces ar- tifacts due to the inheren t anisotropy , whic h can only b e av o ided b y increas ing the neighbor ho o d size, thereby reducing spa rsity and severely slowing down the graph cut-based metho ds. In con trast, prop erly discretizing the r elaxe d prob- lem (13) and solv ing it a s a c onvex problem with subsequent thresholding yields m uch better r e s ults without compromising sparsity (Fig. 1 and 2 , [13]) . This can b e attributed to the fact that so lving the discretized problem as a c ombinatorial pro blem in effect discar ds m uch of the information ab out the problem structure that is contained in the nonlin- ear terms of the discretized ob jectiv e. – Present combinatorial optimization methods [4, 15] are inherently sequential and difficult to par allelize. On the other hand, par allelizing primal-dual meth- o ds for solving the relaxed problem (13) is straig ht - forward, and GPU implementations hav e been shown to outper form state-of-the-ar t gr aph cut metho ds [29 ]. – Analyzing the pro blem in a fully functiona l- analytic setting g ives v alua ble insig ht int o the pr oblem struc- ture, and is of theoretical interest in itself. 1.4 Optimality Bounds How ever, one p ossible drawback of the spatially co n- tin uous approa ch is that the solution of the relaxed problem (13) ma y as sume fr actional v alues, i.e., v al- ues in ∆ l \ E . Ther efore, in a pplications that req uire a true par tition of Ω , some rounding pro ces s is needed in order to generate an integral lab eling ¯ u ∗ . This may in- crease the ob jectiv e , and lead to a subo ptimal so lution of the o riginal pr oblem (1). The regular izer Ψ d as defined in (9) enjoys the prop- erty that it ma jorizes all other r egularizers that ca n b e written in in tegral form and satisfy (11). Therefore it is in a sense “optimal”, since it in tro duce s as few fr a c- tional solutions as p ossible. In pr actice, this forces solu- tions of the relaxed problem to assume in tegra l v alues in most p oints, and rounding is in practice only re q uired in sma ll regions. How ever, the rounding step may still increase the ob jectiv e and genera te sub optimal in tegral so lutions. Therefore the question a rises whether this approach al- lows to re c ov er “g o o d” integral solutions of the o riginal problem (1). In the following, w e are c oncerned with the ques- tion whether it is p ossible to obtain, using the conv ex relaxatio n (13), inte gr al solutions with an upper b ound on the ob jectiv e. Sp ecifically , w e focus on ineq ua lities of the for m f ( ¯ u ∗ ) 6 (1 + ε ) f ( u ∗ E ) (17) for some constant ε > 0, which provide an upp er b ound on the obje ctive of the r oun de d integral s o lution ¯ u ∗ with resp ect to the ob j ective of the (unkno wn) optimal in- tegral solution u ∗ E of (1). Note that gener ally it is not po ssible to show that (1 7) holds for any ε > 0 . The reverse ine q uality f ( u ∗ E ) 6 f ( ¯ u ∗ ) (18) alwa y s holds since ¯ u ∗ ∈ C E and u ∗ E is an optimal int egra l solution. The sp ecific form (1 7 ) can be attributed to the 4 Jan Lellmann et al. alternative int erpreta tion f ( ¯ u ∗ ) − f ( u ∗ E ) f ( u ∗ E ) 6 ε, (19) which provides a b ound for the r elative gap to the opti- mal ob jective of the combinatorial pro blem. Such ε can be obtained a p osteriori by actually computing (or ap- proximating) ¯ u ∗ and a dual feasible p oint: Ass ume tha t a feas ible primal- dual pair ( u, v ) ∈ C × D is kno wn, where u a pproximates u ∗ , and a ssume that some in- tegral feasible ¯ u ∈ C E has b een obtained from u b y a rounding pro c e ss. Then the pa ir ( ¯ u, v ) is feasible as well since C E ⊆ C , and we obtain an a p osteriori optimal- it y bo und of the form (19) with r esp ect to the o ptimal inte gr al s olution u ∗ E : f ( ¯ u ) − f D ( u ∗ E ) f D ( u ∗ E ) 6 f ( ¯ u ) − f D ( u ∗ E ) f D ( v ) 6 f ( ¯ u ) − f D ( v ) f D ( v ) =: ε ′ . (20) How ever, this require s that the the primal and dual ob jectiv es f and f D can b e acc urately ev aluated, and requires to co mpute a minimizer of the pro blem for the sp ecific input d ata, which is generally difficult, espe- cially in the spatially contin uo us formulation. In contrast, true a priori bounds do not require knowledge o f a solution and apply unifor mly to all prob- lems of a cla ss, irresp ective of the particular input. When consider ing rounding metho ds, o ne generally ha s to disc riminate b etw een – deterministic vs. pr ob abilistic methods, a nd – sp atial ly discr ete ( fi nite-dimensional) vs. s p atial ly c ontinuous metho ds. Most known a priori approximation results only hold in the finite-dimensional s e tting, and a re usua lly pr oven using gra ph-based pair wise for m ulations. In contrast, we as s ume a n “optimize first” persp ective due to the reasons outlined in the introduction. Unfortunately , the pro ofs for the finite-dimensio na l results often r ely o n po in twise arguments that ca nnot directly b e tra nsferred to the contin uous setting. Deriving similar results for contin uous pro blems therefore requir es consider able a d- ditional w ork. 1.5 Contribution and Main Results In this work we prove that using the reg ularizer (9), the a priori bo und (16) can be car ried over to the spa- tially co n tinuous setting. Preliminary versions o f these results with excerpts o f the pro ofs have been announced as conference pro ceeding s [18]. W e extend these results to pr ovide the exa ct b ound (16), and supply the full pro ofs. As the main r esult, w e show that it is pos sible to construct a rounding metho d para metrized b y a pa- rameter γ ∈ Γ , where Γ is an appro priate para meter space: R : C × Γ → C E , (21) u ∈ C 7→ ¯ u γ := R γ ( u ) ∈ C E , (22) such that for a suitable pr obability distr ibution on Γ , the following theorem holds for the exp ectation E f ( ¯ u ) := E γ f ( ¯ u γ ): Theorem 1 L et u ∈ C , s ∈ L 1 ( Ω ) l , s > 0 , and let Ψ : R d × l → R > 0 b e p ositively h omo gene ous, c onvex and c ontinuous. Ass u me ther e exists a lower b ound λ l > 0 such that, for z = ( z 1 , . . . , z l ) , Ψ ( z ) > λ l 1 2 l X i =1 k z i k 2 ∀ z ∈ R d × l , l X i =1 z i = 0 . (23) Mor e over, assume ther e exists an upp er b ound λ u < ∞ such that, for every ν ∈ R d satisf y i ng k ν k 2 = 1 , Ψ ( ν ( e i − e j ) ⊤ ) 6 λ u ∀ i, j ∈ { 1 , . . . , l } . (24) Then Al g. 1 (se e b elow) gener ates an inte gr al lab eling ¯ u ∈ C E almost su r ely, a nd E f ( ¯ u ) 6 2 λ u λ l f ( u ) . (25) Note that λ u > λ l alwa y s holds if b oth are defined, since (2 3) implies, fo r ν with k ν k 2 = 1, λ u > Ψ ( ν ( e i − e j ) ⊤ ) > λ l 2 ( k ν k 2 + k ν k 2 ) = λ l . (26) The pro o f of Thm. 1 (Sect. 4) is based on the work of Kleinberg and T ardos [12], which is set in an LP r e lax- ation fr amework. Howev er their r esults a re restr ic ted in that they a ssume a gr aph-based repr esentation and ex- tensively rely o n the finite dimens io nality . In con trast, our results hold in the con tin uous setting without as- suming a pa r ticular pro blem discretization. Theorem 1 gua r antees that – in a proba bilistic sens e – the r ounding pr o c ess may only increa se the energ y in a controlled wa y , with an upp er b ound de p ending o n Ψ . An immediate consequence is Corollary 1 U nder the c onditions of Thm. 1 , if u ∗ minimizes f over C , u ∗ E minimizes f over C E , and ¯ u ∗ denotes t he output of Alg. 1 a pplie d to u ∗ , t hen E f ( ¯ u ∗ ) 6 2 λ u λ l f ( u ∗ E ) . (27) Therefore the prop ose d approach allows to recover, from the solution u ∗ of the conv ex r elaxe d pro blem (13), an approximate inte gra l solution ¯ u ∗ of the noncon vex Optimalit y Bounds for a V ariational Relaxation of the Image Partitioning Problem 5 original pro blem (1) with an upp er b ound on the ob- jective. In particula r, for the tig h t re la xation of the regular- izer as in (9), we obtain (cf. P rop. 10) E f ( ¯ u ∗ ) 6 2 λ u λ l = 2 max i 6 = j d ( i, j ) min i 6 = j d ( i, j ) , (28) which is exactly the sa me b ound a s has b een pr ov en fo r the combinatorial α -expansio n metho d (16). T o o ur knowledge, this is the first b ound av a ilable for the fully spatially conv ex r elaxed pr oblem (13). Re- lated is the work of Olsson et al. [22, 23], where the au- thors consider a contin uous analog ue to the α -ex pa nsion metho d k nown a s contin uous binary fusion [2 7], a nd claim that a b ound similar to (16 ) holds for the corre - sp onding fixed points when using the separable regu- larizer Ψ A ( z ) := l X j =1 k Az j k 2 , z ∈ R d × l , (29) for s o me A ∈ R d × d , whic h implemen ts an a nis otropic v ariant of the unifor m metric. Howev er , a rigor ous pro of in the BV framework w as not given. In [3], the autho rs prop ose to solve the problem (1 ) by consider ing the dual pro ble m to (13) consisting of l coupled max im um-flow problems, whic h ar e so lved us- ing a log -sum-exp smo othing technique and gra dien t descent. In case the dual solution a llows to unam big u- ously rec over an (integral) primal s olution, the latter is necess arily the unique minimizer o f f , a nd therefore a global inte gr al minimizer of the combinatorial prob- lem (1). This provides an a p osteriori bo und, which ap- plies if a dual solution can be computed. While useful in pr actice a s a certificate for global optimality , in the spatially contin uous setting it requir es explicit k nowl- edge of a dual solution, whic h is rarely av a ilable since it dep ends on the r e gularizer Ψ as well a s the input data s . In contrast, the a priori b ound (27) holds uniformly ov er all problem instances, do es not req uire knowledge of any prima l or dual solutions and cov ers also non- uniform r egularizer s. 2 A Probabil istic View of the Coarea F orm ula 2.1 The Two-Class Case As a motiv ation for the following sections , we fir st pro- vide a pro babilistic interpretation of a to ol often used in geometric mea sure theory , the c o ar e a formula (cf. [2]). Assuming u ′ ∈ BV( Ω ) and u ′ ( x ) ∈ [0 , 1] for a.e. x ∈ Ω , the coarea form ula states that the total v a r iation o f u can be represented by summing the b oundary lengths of its s uper -levelsets: TV( u ′ ) = Z 1 0 TV(1 { u ′ >α } ) dα . (30) Here 1 A denotes the characteristic function of a set A , i.e., 1 A ( x ) = 1 iff x ∈ A and 1 A ( x ) = 0 o therwise. The coarea formula provides a connection b etw een prob- lem (1) a nd the relaxation (13) in the t wo-class case, where E = { e 1 , e 2 } , u ∈ C E and u 1 = 1 − u 2 : A s noted in [16 ], TV( u ) = k e 1 − e 2 k 2 TV( u 1 ) = √ 2 TV( u 1 ) , (31) therefore the co area formula (3 0 ) can be r ewritten a s TV( u ) = √ 2 Z 1 0 TV(1 { u 1 >α } ) dα (32) = Z 1 0 TV( e 1 1 { u 1 >α } + e 2 1 { u 1 6 α } ) dα (33) = Z 1 0 TV( ¯ u α ) dα, (34) ¯ u α := e 1 1 { u 1 >α } + e 2 1 { u 1 6 α } . (35) Consequently , the tota l v ariatio n of u ca n b e ex pressed as the me an o ver the total v ar iations of a s e t of int e gr al lab elings { ¯ u α ∈ C E | α ∈ [0 , 1] } , obtained b y r ounding u at differ ent thr esholds α . W e no w a do pt a pr ob abilistic view o f (35): W e regar d the mapping R : ( u, α ) ∈ C × [0 , 1] 7→ ¯ u α ∈ C E (a.e. α ∈ [0 , 1]) (36) as a p ar ametrize d, deterministic rounding alg orithm that depe nds on u and on a n additio na l par ameter α . F ro m this we obtain a pr ob abilistic (randomized) rounding al- gorithm b y assuming α to b e a unif ormly distributed random v a r iable. Under these assumptions the co area formula (3 5) can be written as TV( u ) = E α TV( ¯ u α ) . (37) This has the pr o babilistic interpretation that apply ing the proba bilistic rounding to (arbitrar y , but fixed) u do es – in a pr obabilistic sense, i.e., in the mean – not change the ob jectiv e. It can b e shown that this pr o pe r t y extends to the full functional f in (13): In the tw o-c la ss case, the “ c oarea- like” pr op erty f ( u ) = E α f ( ¯ u α ) (38) holds. F unctions with prop erty (38) are also kno wn a s levelable functions [8, 9] or discr ete total variatio ns [6] and have b een studied in [26]. A well-known implication is that if u = u ∗ , i.e., u minimizes the rela xed pr oblem (13), then in the tw o-clas s case almost ev ery ¯ u ∗ = ¯ u ∗ α is an integral minimizer o f the o riginal problem (1), i.e., the o ptimalit y b ound (17) holds with ε = 0 [7]. 6 Jan Lellmann et al. 2.2 The Multi-Cla ss C a se a nd Generalized Co area F ormulas Generalizing these o bs erv ations to mo r e than tw o la- bels hinges on a prop er ty similar to (38) tha t holds for ve ctor-value d u . In a gener al s e tting, the question is whether ther e exist – a pr obability s pace ( Γ , µ ), and – a p ar ametrize d r ounding metho d , i.e., for µ -almost every γ ∈ Γ : R : C × Γ → C E , (39) u ∈ C 7→ ¯ u γ := R γ ( u ) ∈ C E (40) satisfying R γ ( u ′ ) = u ′ for all u ′ ∈ C E , such that a “m ulticlass coarea-like property” (or gen- er alize d c o ar e a f ormula ) f ( u ) = Z Γ f ( ¯ u γ ) dµ ( γ ) (41) holds. In a pr obabilistic sense this corres po nds to f ( u ) = Z Γ f ( ¯ u γ ) dµ ( γ ) = E γ f ( ¯ u γ ) . (42) F or l = 2 and Ψ ( x ) = k · k 2 , (37) shows that (42) holds with γ = α , Γ = [0 , 1], µ = L 1 , and R : C × Γ → C E as defined in (36). Unfortunately , prop erty (37) is int rinsically restricted to the tw o-class ca s e with TV regular iz er. In the m ulticlass case, the difficulty lies in provid- ing a suitable co m bination of a pr obability s pace ( Γ , µ ) and a parametrize d rounding step ( u, γ ) 7→ ¯ u γ . U nfor- tunately , obtaining a relation such as (37) for the full functional (1) is unlikely , a s it would mean that so- lutions to the (after discre tiza tion) NP-hard problem (1) could b e obtained by solv ing the co n vex relax ation (13) a nd subsequent rounding, which can be ac hie ved in po lynomial time. Therefore we restrict ourse lves to an appr oximate v ariant of the genera liz ed coare a formula: (1 + ε ) f ( u ) > Z Γ f ( ¯ u γ ) dµ ( γ ) = E γ f ( ¯ u γ ) . (43) While (43) is not sufficient to provide a b ound on f ( ¯ u γ ) for p articular γ , it p er mits a pr ob abilistic bo und: for any minimizer u ∗ of the r elaxed problem (13), e q . (43 ) implies E γ f ( ¯ u ∗ γ ) 6 (1 + ε ) f ( u ∗ ) 6 (1 + ε ) f ( u ∗ E ) , (44) i.e., the ratio betw een the o b jectiv e of the r ounde d r e- laxe d solut ion and the optimal inte gr al s olution is b ounded – in a probabilistic sense – by (1 + ε ). In the following sec tions we construct a suitable parametrized rounding metho d a nd pro ba bilit y space in order to obtain an approximate g eneralized coarea formula of the form (4 3). Algorithm 1 Contin uous Probabilis tic Rounding 1: u 0 ← u , U 0 ← Ω , c 0 ← (1 , . . . , 1) ∈ R l . 2: for k = 1 , 2 , . . . do 3: Randomly choose γ k = ( i k , α k ) ∈ I × [0 , 1] uniformly . 4: M k ← U k − 1 ∩ { x ∈ Ω | u k − 1 i k ( x ) > α k } . 5: u k ← e i k 1 M k + u k − 1 1 Ω \ M k . 6: U k ← U k − 1 \ M k . 7: c k j ← ( min { c k − 1 j , α k } , j = i k , c k − 1 j , otherwise . 8: end for 3 Probabil istic Rounding for Multiclass Image P artitio ns 3.1 Appro a ch W e consider the pr obabilistic r ounding approach based on [1 2] a s defined in Alg. 1. The algorithm proce eds in a num b er of phases. A t each iter ation, a label and a threshold γ k := ( i k , α k ) ∈ Γ ′ := I × [0 , 1] are r a ndomly chosen (step 3), a nd lab el i k is a ssigned to all y et unassigned points x where u k − 1 i k ( x ) > α k holds (step 5). In contrast to the tw o- class case consider ed ab ov e, the randomness is provided by a se qu enc e ( γ k ) of uniformly distributed random v ar iables, i.e., Γ = ( Γ ′ ) N . After iteration k , all p oints in the set U k ⊆ Ω are still un assigne d , while all p oints in Ω \ U k hav e b een as- signed a n (integral) lab el in itera tion k or in a previous iteration. Iteration k + 1 p otentially mo difies p oints only in the s et U k . The v ariable c k j stores the low es t thresh- old α c hosen for la bel j up to and including iteration k , a nd is o nly re quired fo r the pr o ofs. While the algorithm is defined using point wise op- erations, it is well-defined in the sense that for fixe d γ , the sequence ( u k ), viewed as elements in L 1 , do es not dep end on the sp ecific r epresentativ e of u in its equiv alence cla ss in L 1 . The seq uenc e s ( M k ) and ( U k ) depe nd on the repre s en tative, but are unique up to L d - negligible sets. In an actual implementation, the algor ithm c o uld b e terminated as so on as all p o int s in Ω have b een a ssigned a labe l, i.e., U k = ∅ . Ho wev er , in our framework used for analysis the alg orithm never terminates ex plicitly . Instead, for fix ed input u we rega r d the a lgorithm a s a mapping b etw een se qu en c es of parameters (or instances of random v ariables ) γ = ( γ k ) ∈ Γ and s e quenc es o f states ( u k γ ), ( U k γ ) and ( c k γ ). W e drop the subscr ipt γ if it do es not create ambiguities. The elements of the se- quence ( γ ( k ) ) are indep enden tly unifor mly distributed, and b y the Kolmogor ov extension theo r em [21, Thm. 2.1.5] ther e exists a proba bilit y s pace and a sto chas- Optimalit y Bounds for a V ariational Relaxation of the Image Partitioning Problem 7 tic pr o cess on the s e t of sequences γ with compatible marginal distributions. In order to define the parametrized rounding step ( u, γ ) 7→ ¯ u γ , we observe that once U k ′ γ = ∅ o ccurs for some k ′ ∈ N , the sequence ( u k γ ) becomes stationary at u k ′ γ . In this case the algorithm may b e terminated, with output ¯ u γ := u k ′ γ : Definition 1 Le t u ∈ BV ( Ω ) l and f : BV( Ω ) l → R . F or some γ ∈ Γ , if U k ′ γ = ∅ in Alg. 1 for some k ′ ∈ N , we denote ¯ u γ := u k ′ γ . W e define f ( ¯ u ( · ) ) : Γ → R ∪ { + ∞} , γ ∈ Γ 7→ f ( ¯ u γ ) , (45) f ( ¯ u γ ) :=  f ( u k ′ γ ) , ∃ k ′ ∈ N : U k ′ γ = ∅ ∧ u k ′ γ ∈ B V( Ω ) l , + ∞ , otherwise . W e denote by f ( ¯ u ) the corr esp onding ra ndom v a riable induced by assuming γ to b e uniformly distr ibuted on Γ . As indicated ab ove, f ( ¯ u γ ) is well-defined: if U k ′ γ = ∅ for some ( γ , k ′ ) then u k ′ γ = u k ′′ γ for all k ′′ > k ′ . Instead of fo cusing on lo ca l properties of the random sequence ( u k γ ) as in the pr o ofs for the finite-dimensiona l ca se, we derive our res ults directly for the sequence ( f ( u k γ )). In particular, we show that the expec tation of f ( ¯ u ) ov er all s equences γ can b e b ounded accor ding to E f ( ¯ u ) = E γ f ( ¯ u γ ) 6 (1 + ε ) f ( ¯ u ) (46) for some ε > 0, cf. (43). Conseque ntly , the r o unding pro cess may only increase the av e rage ob jectiv e in a controlled wa y . 3.2 T ermina tio n Pr op erties Theoretically , the algor ithm may pro duce a se q uence ( u k γ ) that do e s not become stationary , or b ecomes sta- tionary with a solution that is not an element of BV( Ω ) l . In Thm. 2 b elow we show that this happ ens only with zero probability , i.e., a lmost surely Alg. 1 generates (in a finite num b er of iterations) an inte gr al lab eling function ¯ u γ ∈ C E . The fo llowing tw o prop ositions ar e required for the pro of. Prop ositio n 1 F or the se quenc e ( c k ) gener ate d by Al- gorithm 1, P ( e ⊤ c k < 1) > (47) X p ∈{ 0 , 1 } l ( − 1) e ⊤ p   l X j =1 1 l  1 − 1 l  p j    k holds. In p articular, P ( e ⊤ c k < 1) k →∞ → 1 . (48) Pr o of Denote by n k j ∈ N 0 the num b er of k ′ ∈ { 1 , . . . , k } such that i k ′ = j , i.e., the num b er o f times lab el j w a s selected up to and inc luding the k -th step. Then ( n k 1 , . . . , n k l ) ∼ Multinomial  k ; 1 l , . . . , 1 l  , (49) i.e., the pr obability of a sp ecific insta nce is P (( n k 1 , . . . , n k l )) = ( k ! n k 1 ! · ... · n k l !  1 l  k , P j n k j = k , 0 , otherwise . (50) Therefore, P ( e ⊤ c k < 1) = X n k 1 ,...,n k l P ( e ⊤ c k < 1 | ( n k 1 , . . . , n k l )) · . . . P (( n k 1 , . . . , n k l )) (51) = X n k 1 + ... + n k l = k k ! n k 1 ! · . . . · n k l !  1 l  k · . . . P ( e ⊤ c k < 1 | ( n k 1 , . . . , n k l )) . (52) Since c k 1 , . . . , c k l < 1 l is a sufficient condition for e ⊤ c < 1, we may bo und the probability according to P ( e ⊤ c < 1) > X n k 1 + ... + n k l = k k ! n k 1 ! · . . . · n k l !  1 l  k · . . . P  c k j < 1 l ∀ j ∈ I | ( n k 1 , . . . , n k l )  . (5 3) W e now consider the distributions of the comp onents c k j of c k conditioned on the v ector ( n k 1 , . . . , n k l ). Given n k j , the pro ba bilit y of { c k j > t } is the probabilit y that in eac h of the n k j steps where lab el j was selected the threshold α was randomly chosen to b e at le ast as lar ge as t . F or 0 < t < 1, w e conclude P ( c k j < t | ( n k 1 , . . . , n k l )) = P ( c k j < t | n k j ) (54) = 1 − P ( c k j > t | n k j ) (55) 0 X n k 1 + ... + n k l = k k ! n k 1 ! · . . . · n k l !  1 l  k · l Y j =1 P  c k j < 1 l | ( n k 1 , . . . , n k l )  (57) ( 56 ) = X n k 1 + ... + n k l = k k ! n k 1 ! · . . . · n k l !  1 l  k + l Y j =1 1 −  1 − 1 l  n k j ! . (58) 8 Jan Lellmann et al. Expanding the product and swapping the summation order, w e derive P ( e ⊤ c k < 1) (59) > X n k 1 + ... + n k l = k k ! n k 1 ! · . . . · n k l !  1 l  k · X p ∈{ 0 , 1 } l l Y j =1 −  1 − 1 l  n k j ! p j (60) = X p ∈{ 0 , 1 } l ( − 1) e ⊤ p X n k 1 + ... + n k l = k k ! n k 1 ! · . . . · n k l ! · l Y j =1  1 l  1 − 1 l  p j  n k j . (61) Using the m ultinomial summatio n formula, w e conclude P ( e ⊤ c k < 1) > X p ∈{ 0 , 1 } l ( − 1) e ⊤ p        l X j =1 1 l  1 − 1 l  p j | {z } =: q p        k , ( 62) which pro ves (47). At ( ∗ ) the m ultinomial s ummation formula w as inv oked. N ote that in (62) the n k j do not o ccur ex plic itly a nymore. T o show the second assertion (48), we use the fact that, for any p 6 = (0 , . . . , 0), q p can be bounded by 0 < q p < 1. Therefore P ( e ⊤ c k < 1) > q 0 + X p ∈{ 0 , 1 } l ,p 6 =0 ( − 1) e ⊤ p ( q p ) k (63) = 1 + X p ∈{ 0 , 1 } l ,p 6 =0 ( − 1) e ⊤ p ( q p ) k | {z } k →∞ → 0 (64) k →∞ → 1 , (65) which pr ov es (48). W e now s how that Alg. 1 gener ates a seq ue nc e in BV( Ω ) l almost s urely . The p erimeter of a set A is de- fined as the total v aria tio n of its characteristic function Per( A ) := TV (1 A ). Prop ositio n 2 F or the se qu enc es ( u k ) , ( U k ) gener ate d by Alg. 1, define A := ∞ \ k =1 { γ ∈ Γ | Per ( U k γ ) < ∞} . (66) Then P ( A ) = 1 . (67) If Per( U k γ ) < ∞ for al l k , then u k γ ∈ BV( Ω ) l for al l k as wel l. Mor e over, P ( u k ∈ BV( Ω ) l ∧ Per ( U k ) < ∞∀ k ∈ N ) = 1 , (68) i.e., the algorithm almost sur ely gener ates a se quenc e of BV functions ( u k ) and a se quen c e of sets of fin ite p erimeter ( U k ) . Pr o of W e fir st show that if Per( U k ′ ) < ∞ for all k ′ 6 k , then u k ∈ B V( Ω ) l for all k ′ 6 k as well. F or k = 0, the assertion holds, since u 0 = u ∈ B V ( Ω ) l by assumption. F or k > 1, u k = e i k 1 M k + u k − 1 1 Ω \ M k . (69) Since M k = U k − 1 ∩ ( Ω \ U k ), and U k , U k − 1 are a s sumed to ha ve finite p erimeter, M k also has finit e pe rimeter. Applying [2, T hm. 3 .84] together with the b oundedness of u k − 1 and u k − 1 ∈ BV( Ω ) l by induction then provides u k ∈ B V( Ω ) l . W e now denote I k := { γ ∈ Γ | Per( U k γ ) = ∞} , (70) and the even t that the first s et with non-finite p e rimeter is enco un tered at step k ∈ N 0 by B k := I k ∩  Γ \ I k − 1  ∩ . . . ∩  Γ \ I 0  . (71) Then P ( A ) = 1 − P ∞ [ k =0 B k ! . (72) As the sets B k are pairwise disjoint, a nd due to the countable additivity o f the probability measur e , we have P ( A ) = 1 − ∞ X k =0 P ( B k ) . (73) Now U 0 = Ω , therefore Per( U 0 ) = TV (1 U 0 ) = 0 < ∞ and P ( B 0 ) = 0. F o r k > 1 , we hav e P ( B k ) 6 P  Per( U k ) = ∞ ∧ Per( U k ′ ) < ∞ ∀ k ′ < k  6 P  Per( U k ) = ∞| Per( U k ′ ) < ∞ ∀ k ′ < k  = P  Per( U k − 1 ∩ { x ∈ Ω | u k − 1 i k ( x ) 6 α k } ) = ∞| Per( U k ′ ) < ∞ ∀ k ′ < k  . (74) By the a r gument from the b eginning of the pro of, we know that u k − 1 ∈ BV( Ω ) l under the condition on the per imeter Per( U k ′ ), therefore fr om [2, Thm. 3.4 0] we conclude that Per( { x ∈ Ω | u k − 1 i k ( x ) 6 α k } ) is finite for L 1 -a.e. α k and all i k . As the sets of finite p erimeter a re closed under finite intersection, and since the α k are drawn fr om an unif orm distribution, this implies that P (Per ( U k ) < ∞| Per( U k − 1 ) < ∞ ) = 1 . (75) T oge ther with (74) we ar r ive at P ( B k ) = 0 . (76) Substituting this res ult into (73) leads to the ass e rtion, P ( A ) = 1 . (77) Equation (6 8) follows immediately . Optimalit y Bounds for a V ariational Relaxation of the Image Partitioning Problem 9 Using these prop ositions , we now formulate the main result of this section: Alg. 1 almost sur e ly g enerates an int egra l lab eling that is of b ounded v ar ia tion. Theorem 2 L et u ∈ BV ( Ω ) l and f ( ¯ u ) as in Def. 1. Then P ( f ( ¯ u ) < ∞ ) = 1 . (78) Pr o of The first pa rt is to show that ( u k ) b ecomes sta- tionary a lmost s urely , i.e., P ( ∃ k ∈ N : U k = ∅ ) = 1 . (79) Assume ther e exists k s uch that e ⊤ c k < 1 , and a ssume further tha t U k 6 = ∅ , i.e., ther e exis ts so me x ∈ U k . Then u j ( x ) 6 c k j for all labels j . But then e ⊤ u ( x ) 6 e ⊤ c k < 1, which is a contradiction to u ( x ) ∈ ∆ l . There- fore U k m ust b e empty . F rom this observ ation and P rop. 1 we conclude, for all k ′ ∈ N , 1 > P ( ∃ k ∈ N : U k = ∅ ) > P ( e ⊤ c k ′ < 1 ) k ′ →∞ → 1 , (80) which pr ov es (79). In or der to show that f ( ¯ u γ ) < ∞ with probability 1, it remains to show that the result is almost surely in BV( Ω ) l . A sufficient c o ndition is that almos t surely al l iterates u k are elements of B V ( Ω ) l , i.e., P  u k ∈ B V( Ω ) l ∀ k ∈ N  = 1 . (81) This is shown by Prop. 2. Then P ( f ( ¯ u ) < ∞ ) (8 2 ) > P ( {∃ k ∈ N : U k = ∅} ∧ { u k ∈ BV( Ω ) l ∀ k ∈ N } ) = P ( { u k ∈ B V( Ω ) l ∀ k ∈ N } ) (83 ) − P ( {∀ k ∈ N : U k 6 = ∅} ∧ { u k ∈ B V( Ω ) l ∀ k ∈ N } ) ( 81 ) = P ( { u k ∈ B V( Ω ) l ∀ k ∈ N } ) − 0 (84) = 1 . (85) Thu s P ( f ( ¯ u ) < ∞ ) = 1, which pr ov es the a ssertion. ⊓ ⊔ 4 Pro o f of the Main Theorem In order to show the b ound (46) and Thm. 1, we first need sev e r al technical prop ositions regarding the com- po sition of t wo BV functions along a set of finite pe rime- ter. W e denote by ( E ) 1 and ( E ) 0 the measur e - theoretic int erior and exterio r of a set E , see [2], ( E ) t := { x ∈ Ω | lim ρ ց 0 |B ρ ( x ) ∩ E | |B ρ ( x ) | = t } , t ∈ [0 , 1 ] . (8 6) Here B ρ ( x ) denotes the ball with radius ρ centered in x , and | A | := L d ( A ) the Le b esg ue conten t of a set A ⊆ R d . Prop ositio n 3 L et Ψ b e p ositively homo gene ous and c onvex, and satisfy the upp er-b ounde dness c ondition (24). Then Ψ ( ν ( z 1 − z 2 ) ⊤ ) 6 λ u ∀ z 1 , z 2 ∈ ∆ l . (87) Mor e over, ther e ex ists a c onstant C < ∞ such t hat Ψ ( w ) 6 C k w k 2 ∀ w ∈ W , (88) W := { w = ( w 1 | . . . | w l ) ∈ R d × l | l X i =1 w i = 0 } . (89 ) Pr o of See app endix. Prop ositio n 4 L et E , F ⊆ Ω d b e L d -me asur able sets. Then ( E ∩ F ) 1 = ( E ) 1 ∩ ( F ) 1 . (90) Pr o of See app endix. Prop ositio n 5 L et u, v ∈ B V ( Ω , ∆ l ) and E ⊆ Ω s uch that Per( E ) < ∞ . Define w := u 1 E + v 1 Ω \ E . (91) Then w ∈ BV ( Ω , ∆ l ) l , and D w = D u x ( E ) 1 + D v x ( E ) 0 + ν E  u + F E − v − F E  ⊤ H d − 1 x ( F E ∩ Ω ) , (92) wher e u + F E and v − F E denote the one-side d appr oximate limits of u and v on the r e duc e d b oundary F E , and ν E is the gener alize d inner normal of E [2]. Mor e over, for c ontinuous, c onvex and p ositively homo gene ous Ψ sat- isfying the u pp er-b ounde dness c ondition (24) and some Bor el set A ⊆ Ω , Z A dΨ ( Dw ) 6 Z A ∩ ( E ) 1 dΨ ( Du ) + Z A ∩ ( E ) 0 dΨ ( Dv ) + λ u Per( E ) . (93) Pr o of See app endix. Prop ositio n 6 L et u, v ∈ BV( Ω , ∆ l ) , E ⊆ Ω such that Per( E ) < ∞ , and u | ( E ) 1 = v | ( E ) 1 L d -a.e. (94) Then ( D u ) x ( E ) 1 = ( D v ) x ( E ) 1 , and Ψ ( D u ) x ( E ) 1 = Ψ ( Dv ) x ( E ) 1 . In p articular, Z ( E ) 1 dΨ ( Du ) = Z ( E ) 1 dΨ ( Dv ) . (95) The r esult also holds when ( E ) 1 is re plac e d by ( E ) 0 . Mor e over, the c ondition (94) is e quivalent to u | E = v | E L d - a.e. (96) Pr o of See app endix. R emark 1 No te that tak ing the measure-theor etic inte- rior ( E ) 1 is of cen tral imp ortance. The coro llary do es not hold when r eplacing the int egra l ov er ( E ) 1 with the int egra l ov er E , as can be seen from the exa mple of the closed unit ball, i.e., E = B 1 (0), u = 1 E and v ≡ 1. 10 Jan Lellmann et al. 4.1 Pro of o f Theorem 1 In Sect. 3.2 w e have s hown that the rounding pro cess induced by Alg. 1 is well-defined in the se nse that it re - turns an in tegral solution ¯ u γ ∈ BV ( Ω ) l almost surely . W e no w return to proving an upper bound for th e ex- pec tation of f ( ¯ u ) as in the approximate coar ea for - m ula (43). W e firs t s how that the exp e ctation o f the lin- e ar p art (data term) of f is in v aria nt under the round- ing pr o cess. Prop ositio n 7 The se quenc e ( u k ) gener ate d by A lg. 1 satisfies E ( h u k , s i ) = h u, s i ∀ k ∈ N . (9 7) Pr o of In Alg. 1, instead of step 5 we co nsider the sim- pler update u k ← e i k 1 { u k − 1 i k >α k } + u k − 1 1 { u k − 1 i k 6 α k } . (98) This yields exactly the same sequence ( u k ), since if u k − 1 i k ( x ) > α k for a n y α k > 0, then either x ∈ U k − 1 , or u k − 1 i k ( x ) = 1. In b oth alg orithms, po in ts tha t a re assigned a label e i k at some p oint in the pro cess will never b e ass ig ned a differ ent lab e l a t a later p oint. This is made explicit in Alg. 1 b y keeping track o f the set U k of yet unas signed p oints. In contrast, using the step (98), a p oint ma y formally be assig ne d the same lab el m ultiple times. Denote γ ′ := ( γ 1 , . . . , γ k − 1 ) and u γ ′ := u k − 1 γ . W e apply induction on k : F or k > 1 , E γ h u k γ , s i (99) = E γ ′ 1 l l X i =1 Z 1 0 l X j =1 s j ·  e i 1 { u γ ′ i >α } + u γ ′ 1 { u γ ′ i 6 α }  j dα = E γ ′ 1 l l X i =1 Z 1 0  s i · 1 { u γ ′ i >α } + u γ ′ 1 { u γ ′ i 6 α } h u γ ′ , s i  dα = E γ ′ 1 l l X i =1 Z 1 0  s i · 1 { u γ ′ i >α } +  1 − 1 { u γ ′ i >α }  h u γ ′ , s i  dα . (100) Now w e take into account the pr op erty [2, Prop. 1 .78], which is a direct cons e q uence of F ubini’s t heorem, and also used in the proo f of the thresholding theorem for the t wo-class case [7]: Z 1 0 Z Ω s i ( x ) · 1 { u i >α } ( x ) dxdα (101) = Z Ω s i ( x ) u i ( x ) dx = h u i , s i i . (102) This lea ds to E γ h u k γ , s i = E γ ′ 1 l l X i =1  s i u γ ′ i + h u γ ′ , s i − u γ ′ i h u γ ′ , s i  dα (103) and ther efore, using u γ ′ ( x ) ∈ ∆ l , E γ h u k γ , s i = E γ ′ h u γ ′ , s i = E γ h u k − 1 γ , s i . (104) Since h u 0 , s i = h u, s i , the as sertion follows by induction. ⊓ ⊔ R emark 2 P rop. 7 shows that the data term is – in the mean – not affected by the probabilistic rounding pro cess, i.e., it s atisfies an exact co area-like formula, even in the mu lticlass case. Bounding the r egularizer is more inv olved: F or γ k = ( i k , α k ), define U γ k := { x ∈ Ω | u i k ( x ) 6 α k } , (105) V γ k :=  U γ k  1 , (106) V k := ( U k ) 1 . (107) As the measur e-theoretic interior is inv a riant under L d - negligible mo difications, given some fixed s equence γ the sequence ( V k ) is inv a riant under L d -negligible mo d- ifications of u = u 0 , i.e., it is uniquely defined when viewing u a s an elemen t of L 1 ( Ω ) l . Some ca lculations yield U k = U γ 1 ∩ . . . ∩ U γ k , k > 1 , (108) U k − 1 \ U k = U γ 1 ∩   U γ 2 ∩ . . . ∩ U γ k − 1  \  U γ 2 ∩ . . . ∩ U γ k   , k > 2 . (109) F rom these observ a tions and P r op. 4, V k = V γ 1 ∩ . . . ∩ V γ k , k > 1 , (110) V k − 1 \ V k = V γ 1 ∩   V γ 2 ∩ . . . ∩ V γ k − 1  \  V γ 2 ∩ . . . ∩ V γ k   , k > 2 , (11 1) Ω \ V k = k [ k ′ =1  V k ′ − 1 \ V k ′  , k > 1 . (112) Moreov er, since V k is the measure- theoretic interior of U k , b oth sets are equal up to an L d -negligible s e t (cf. (174)). W e now pr e pare for an induction argument on the exp ectation of the reg ularizing term when re stricted to the sets V k − 1 \ V k . The following pr op osition provides the initial step ( k = 1). Prop ositio n 8 Ass ume that Ψ satisfies the lower- and upp er-b oun de dness c onditio ns (23) and (24). Then E Z V 0 \ V 1 dΨ ( D ¯ u ) 6 2 l λ u λ l Z Ω dΨ ( Du ) . (113) Optimalit y Bounds for a V ariational Relaxation of the Image Partitioning Problem 11 Pr o of Denote ( i , α ) = γ 1 . Since 1 U ( i,α ) = 1 V ( i,α ) L d -a.e., we hav e ¯ u γ = 1 V ( i,α ) e i + 1 Ω \ V ( i,α ) ¯ u γ L d - a .e. (114) Therefore, since V 0 = ( U 0 ) 1 = ( Ω ) 1 = Ω , Z V 0 \ V 1 dΨ ( D ¯ u γ ) = Z Ω \ V ( i,α ) dΨ ( D ¯ u γ ) = Z Ω \ V ( i,α ) dΨ  D  1 V ( i,α ) e i + 1 Ω \ V ( i,α ) ¯ u γ  . (115) Since u ∈ BV ( Ω ) l , we know that Per ( V ( i,α ) ) < ∞ holds for L 1 -a.e. α and any i [2, Thm. 3 .40]. There fore we conclude from Pro p. 5 that for L 1 -a.e. α , Z Ω \ V ( i,α ) dΨ ( D ¯ u γ ) 6 λ u Per  V ( i,α )  + Z  Ω \ V ( i,α )  ∩  Ω \ V ( i,α )  1 dΨ  D e i  + Z  Ω \ V ( i,α )  ∩  Ω \ V ( i,α )  0 dΨ ( D ¯ u γ ) . (116) Both o f the in tegra ls are zero , since D e i = 0 and ( Ω \ V ( i,α ) ) 0 = ( V ( i,α ) ) 1 = V ( i,α ) , (117) therefore Z Ω \ V ( i,α ) dΨ ( D ¯ u γ ) 6 λ u Per ( V ( i,α ) ) . (118) Carrying the b ound ov e r to the exp ectation yields E γ Z Ω \ V ( i,α ) dΨ ( D ¯ u γ ) 6 1 l l X i =1 Z 1 0 λ u Per( V ( i,α ) ) dα . Also, Per( V ( i,α ) ) = Per( U ( i,α ) ) since the p erimeter is inv ar iant under L d -negligible mo difica tions. The asser- tion then follows using V 0 = Ω , V 1 = V ( i,α ) and the coarea for m ula: E γ Z V 0 \ V 1 dΨ ( D ¯ u γ ) (119) 6 1 l l X i =1 Z 1 0 λ u Per ( U ( i,α ) ) dα (120) coarea = λ u l l X i =1 TV( u i ) = λ u l Z Ω l X i =1 d k D u i k 2 (121) (23) 6 2 l λ u λ l Z Ω dΨ ( Du ) . (122) W e now take care of the induction step fo r the reg- ularizer bo und. Prop ositio n 9 L et Ψ satisfy t he upp er-b ounde dness c on- dition (24). Then, for any k > 2 , F := E Z V k − 1 \ V k dΨ ( D ¯ u ) (123) 6 ( l − 1) l E Z V k − 2 \ V k − 1 dΨ ( D ¯ u ) . (124) Pr o of Define the shifted sequence γ ′ = ( γ ′ k ) ∞ k =1 by γ ′ k := γ k +1 , and let W γ ′ := V k − 2 γ ′ \ V k − 1 γ ′ (125) =  V γ 2 ∩ . . . ∩ V γ k − 1  \  V γ 2 ∩ . . . ∩ V γ k  . (126) By Pro p. 2 we may assume that, under the exp ectation, ¯ u γ exists and is an element o f BV( Ω ) l . W e denote γ 1 = ( i, α ), then V k − 1 \ V k = V ( i,α ) ∩ W γ ′ due to (1 11). F o r each pa ir ( i, α ) we denote by (( i, α ) , γ ′ ) the seq ue nce obtained by prep ending ( i, α ) to the seq uence γ ′ . T hen F = 1 l l X i =1 Z 1 0 E γ ′ Z V ( i,α ) ∩ W γ ′ dΨ ( D ¯ u (( i,α ) ,γ ′ ) ) dα. (1 27) Since in the fir st iteration of the alg orithm no points in U ( i,α ) are as s igned a lab el, ¯ u (( i,α ) ,γ ′ ) = ¯ u γ ′ holds on U ( i,α ) , and therefore L d -a.e. on V ( i,α ) . Therefore we may apply P rop. 6 a nd substitute D ¯ u (( i,α ) ,γ ′ ) by D ¯ u γ ′ in (12 7): F = 1 l l X i =1 Z 1 0 E γ ′ Z V ( i,α ) ∩ W γ ′ dΨ ( D ¯ u γ ′ ) ! dα (128) = 1 l l X i =1 Z 1 0 E γ ′ Z W γ ′ 1 V ( i,α ) dΨ ( D ¯ u γ ′ ) ! dα. (1 29) By definition of the measure-theo retic interior (86), the indicator function 1 V ( i,α ) is bo unded from ab ov e by the density function Θ U ( i,α ) of U ( i,α ) , 1 V ( i,α ) ( x ) 6 Θ ( i,α ) ( x ) := lim δ ց 0 |B δ ( x ) ∩ U ( i,α ) | |B δ ( x ) | , (130) which exists H d − 1 -a.e. on Ω by [2, Prop. 3.61]. There- fore, denoting b y B δ ( · ) the mapping x ∈ Ω 7→ B δ ( x ), F 6 1 l l X i =1 Z 1 0 E γ ′ Z W γ ′ lim δ ց 0 |B δ ( · ) ∩ U ( i,α ) | |B δ ( · ) | dΨ ( D ¯ u γ ′ ) dα. Rearra ng ing the integrals and the limit, whic h can b e justified by TV( ¯ u γ ′ ) < ∞ almo st s urely a nd dominated conv er gence using (24), we get F 6 1 l E γ ′ lim δ ց 0 Z W γ ′ l X i =1 Z 1 0 |B δ ( · ) ∩ U ( i,α ) | |B δ ( · ) | dα dΨ ( D ¯ u γ ′ ) = 1 l E γ ′ lim δ ց 0 Z W γ ′ 1 |B δ ( · ) | · (131) l X i =1 Z 1 0 Z B δ ( · ) 1 { u i ( y ) 6 α } dy dα ! dΨ ( D ¯ u γ ′ ) . 12 Jan Lellmann et al. W e ag a in apply [2, Prop. 1 .78] to the tw o innermo st int egra ls (alternatively , us e F ubini’s theorem), which leads to F 6 1 l E γ ′ lim δ ց 0 Z W γ ′ 1 |B δ ( · ) | · ( 132 ) l X i =1 Z B δ ( · ) (1 − u i ( y )) dy ! dΨ ( D ¯ u γ ′ ) . (133) Using the fact that u ( y ) ∈ ∆ l , this colla pses accor ding to F 6 1 l E γ ′ lim δ ց 0 Z W γ ′ 1 |B δ ( · ) | Z B δ ( · ) ( l − 1) dy ! dΨ ( D ¯ u γ ′ ) = 1 l E γ ′ lim δ ց 0 Z W γ ′ ( l − 1) dΨ ( D ¯ u γ ′ ) (134) = l − 1 l E γ ′ Z W γ ′ dΨ ( D ¯ u γ ′ ) ( 135 ) = l − 1 l E γ ′ Z V k − 2 γ ′ \ V k − 1 γ ′ dΨ ( D ¯ u γ ′ ) . (136) Reverting the index shift and using ¯ u γ ′ = ¯ u γ concludes the pro of: F 6 l − 1 l E γ Z V k − 1 γ \ V k γ dΨ ( D ¯ u γ ) . (137) W e a re now ready to prove the ma in r esult, Thm. 1, as stated in the intro duction. Pr o of (Theorem 1) The fa ct that the algorithm pr o - vides ¯ u ∈ C E almost surely follows from Thm. 2. There- fore there almost s urely ex ists k ′ := k ′ ( γ ) > 1 s uch that U k ′ = ∅ and ¯ u γ = u k ′ γ . On one ha nd, this implies h ¯ u γ , s i = h u k ′ γ , s i = lim k →∞ h u k γ , s i (138) almost surely . On the other hand, V k ′ = ( U k ′ ) 1 = ∅ and ther efore k ′ [ k =1 V k − 1 \ V k ( ∗ ) = Ω \ V k ′ = Ω (139) almost s urely . The equality ( ∗ ) can b e shown by induc- tion: F or the bas e case k ′ = 1 , w e hav e V 0 = ( U 0 ) 1 = ( Ω ) 1 = Ω , since Ω is the op en unit b ox. F or k ′ > 2 , k ′ [ k =1 V k − 1 \ V k (140) =  V k ′ − 1 \ V k ′  ∪ k ′ − 1 [ k =1  V k − 1 \ V k  (141) =  V k ′ − 1 \ V k ′  ∪  Ω \ V k ′ − 1  (142) V k ′ − 1 ⊆ Ω = Ω \ V k ′ − 1 . (143) almost surely (cf. (112)). F r om (138) and (139) we ob- tain E γ f ( ¯ u γ ) = E γ  lim k →∞ h u k γ , s i  + E γ ∞ X k =1 Z V k − 1 \ V k dΨ ( D ¯ u γ ) ! (144) = lim k →∞  E γ h u k γ , s i  + ∞ X k =1 E γ Z V k − 1 \ V k dΨ ( D ¯ u γ ) (14 5) The first term in (145) is equal to h u, s i due to Prop. 7. An induction a r gument using P rop. 8 and P rop. 9 shows that the sec o nd term ca n b e b ounded acco rding to ∞ X k =1 E γ Z V k − 1 \ V k dΨ ( D ¯ u γ ) ( 146 ) 6 ∞ X k =1  l − 1 l  k − 1 2 l λ u λ l Z Ω dΨ ( Du ) (147 ) = 2 λ u λ l Z Ω dΨ ( Du ) , (148) therefore E γ f ( ¯ u γ ) 6 h u, s i + 2 λ u λ l Z Ω dΨ ( Du ) . (149) Since s > 0 and λ u > λ l , and therefor e the linear term is bo unded by h u, s i 6 2 ( λ u /λ l ) h u, s i , this proves the a s- sertion. Swapping the int egra l and limit in (145) can be justified r etrosp ectively by the dominated conv er gence theorem, using 0 6 h u, s i 6 ∞ and R Ω dΨ ( Du ) < ∞ due to the upp er- bo undedness co ndition and P rop. 3. ⊓ ⊔ Corollar y 1 (see introduction) follows immediately using f ( u ∗ ) 6 f ( u ∗ E ), cf. (44). W e ha ve demonstrated that the prop osed approach allows to recover, fro m the solution u ∗ of the convex r elaxe d problem (13), an ap- proximate inte gr al so lution ¯ u ∗ of the no nc o nv ex original problem (1) with an upp er b ound on the ob jectiv e. F or the sp ecific cas e Ψ = Ψ d , we hav e Prop ositio n 10 L et d : I 2 → R > 0 b e a metric and Ψ = Ψ d . Then one may set λ u = max i,j ∈{ 1 ,...,l } d ( i, j ) and λ l = min i 6 = j d ( i, j ) . Pr o of F ro m the remarks in the intro ductio n we obtain (cf. [19]) Ψ d ( ν ( e i − e j )) = d ( i, j ) , which shows the uppe r b o und. F or the lower b ound, s et c := min i 6 = j d ( i, j ), v ′ i := c 2 w i k w i k 2 and v := v ′ ( I − 1 l ee ⊤ ). Optimalit y Bounds for a V ariational Relaxation of the Image Partitioning Problem 13 Then v ∈ D d lo c , since k v i − v j k 2 = k v ′ i − v ′ j k 2 6 c and v e = v ′ ( I − 1 l ee ⊤ ) e = 0. Therefore, for w ∈ R d × l satisfying we = 0, Ψ d ( w ) > h w , v i = h w, v ′ i (150) = l X i =1 h w i , c 2 w i k w i k 2 i = c 2 l X i =1 k w i k 2 , (151) proving the low er b ound. Finally , for Ψ d we obtain the factor 2 λ u λ l = 2 max i,j d ( i, j ) min i 6 = j d ( i, j ) , (152 ) determining the optimality b ound, as cla imed in the int ro duction (28). The b ound in (27) is the s a me as the k nown bounds fo r finite-dimensional metric lab el- ing [12] and α -expansion [4], how ever it extends these results to pr oblems on co n tinuous domains for a broad class of regularizer s. 5 Co nclusion In this work we consider ed a method for r ecov e r ing approximate solutions o f image partitioning pro ble ms from solutions of a conv ex relaxation. W e propo sed a probabilistic r ounding metho d motiv ated by the finite- dimensional f ramework, and show ed that it is p ossible to obtain a priori b ounds on the optimality o f the in- tegral solution obtained by rounding a solution of the conv ex relaxa tion. The obtained bo unds are co mpatible with k nown bo unds for the finite- dimens io nal setting. Ho wever, to our knowledge, this is the first fully conv ex approach that is b oth for m ulated in the spatia lly contin uous set- ting and provides a true a priori b ound. W e show ed that the appro ach can a lso b e interpreted a s an approximate v ariant of the coare a formula. While the res ults apply to a quite general class of regular iz ers, they are formulated for the homog eneous case. Non-ho mogeneous r egularizer s co ns titute an in- teresting direction for future w ork. In particular, suc h regular iz ers naturally o ccur when applying co n vex re- laxation techniques [1, 25] in or der to solve nonco n vex v ariatio na l problems. With the increasing computational p ow er, suc h tech - niques ha ve b ecome quite popular recently . F or prob- lems where the convexit y is confined to the data t erm, they p ermit to find a global minimizer. A prop er ex- tension of the results outlined in this work may provide a w ay to find go o d approximate so lutions of pr oblems where also the r e gularizer is nonconv ex. 6 App endix Pr o of (Pr op. 3) In order to prov e the first ass ertion (87), note that the mapping w 7→ Ψ ( ν w ⊤ ) is conv ex , therefore it must assume its maximum on the p oly to pe ∆ l − ∆ l := { z 1 − z 2 | z 1 , z 2 ∆ l } in a vertex o f the poly- top e.. Since the p olytop e ∆ l − ∆ l is the differenc e of t wo p olytop es, its vertex s et is at most the difference of their vertex sets, V := { e i − e j | i, j ∈ { 1 , . . . , l }} . On this set, the b ound Ψ ( ν w ⊤ ) 6 λ u holds for w ∈ V due to the upper -b oundedness condition (24), which prov es (87). The second equality (89) follows fr om the fact tha t G := { b ik := e k ( e i − e i +1 ) ⊤ | 1 6 k 6 d, 1 6 i 6 l − 1 } is a basis of the linear subspace W , satisfying Ψ ( b ik ) 6 λ u , and Ψ is po sitively homo geneous a nd conv ex, a nd thus subadditive. Sp ecifically , there is a linear tr ansform T : W → R d × ( l − 1) such that w = P i,k b ik α ik for α = T ( w ). Then Ψ ( w ) = Ψ   X i,k b ik α ik   (153) 6 Ψ X ik | α ik | sgn( α ik ) b ik ! (154) 6 X ik | α ik | Ψ  sgn( α ik ) b ik  . (155) Since (24 ) provides Ψ ( ± b ik ) 6 λ u , we obtain Ψ ( w ) 6 λ u X ik | α ik | 6 λ u k T kk w k 2 (156) for some suitable o per ator no rm k · k a nd any w ∈ W . Pr o of (Pr op. 4) W e pr ov e m utual inclusion: ′′ ⊆ ′′ : F rom the definition of the mea s ure-theoretic int erior , x ∈ ( E ∩ F ) 1 ⇒ lim δ ց 0 |B δ ( x ) ∩ E ∩ F | |B δ ( x ) | = 1 . (157) Since |B δ ( x ) | > |B δ ( x ) ∩ E | > |B δ ( x ) ∩ E ∩ F | (and vice versa for |B δ ( x ) ∩ F | ), it follows by the “sand- wich” cr iterion that b oth lim δ ց 0 |B δ ( x ) ∩ E | / |B δ ( x ) | and lim δ ց 0 |B δ ( x ) ∩ F | / |B δ ( x ) | exist and ar e equal to 1, which s hows x ∈ E 1 ∩ F 1 . ′′ ⊇ ′′ : Assume tha t x ∈ E 1 ∩ F 1 . Then 1 > lim δ ց 0 sup |B δ ( x ) ∩ E ∩ F | |B δ ( x ) | (158) > lim δ ց 0 inf |B δ ( x ) ∩ E ∩ F | |B δ ( x ) | (159) = lim δ ց 0 inf |B δ ( x ) ∩ E | + |B δ ∩ F | − |B δ ∩ ( E ∪ F ) | |B δ ( x ) | . 14 Jan Lellmann et al. W e obtain equality , 1 > lim δ ց 0 inf |B δ ( x ) ∩ E ∩ F | |B δ ( x ) | (160) > lim δ ց 0 inf |B δ ( x ) ∩ E | |B δ ( x ) | + lim δ ց 0 inf |B δ ( x ) ∩ F | |B δ ( x ) | + lim δ ց 0 inf  − |B δ ∩ ( E ∪ F ) | |B δ ( x ) |  (161) = 2 − lim δ ց 0 sup |B δ ∩ ( E ∪ F ) | |B δ ( x ) | | {z } 6 1 > 1 , (162) from whic h we conclude that lim δ ց 0 sup |B δ ( x ) ∩ E ∩ F | |B δ ( x ) | = lim δ ց 0 inf |B δ ( x ) ∩ E ∩ F | |B δ ( x ) | = 1 , i.e., x ∈ ( E ∩ F ) 1 . Pr o of (Pr op. 5) First note that Z F E ∩ Ω k w + F E − w − F E k 2 d H d − 1 (163) 6 sup {k w + F E ( x ) − w − F E ( x ) k 2 | x ∈ F E ∩ Ω } · H d − 1 ( F E ∩ Ω ) (164) ( ∗ ) 6 sup {k w ( x ) − w ( y ) k 2 | x, y ∈ Ω } · TV (1 E ) (165) ( ∗∗ ) 6 √ 2 TV(1 E ) (166) = √ 2 Per( E ) < ∞ . (167 ) The inequality ( ∗ ) is a consequence of the definition of w ± F E and [2, Thm. 3.5 9], and ( ∗∗ ) follows directly from w ( x ) , w ( y ) ∈ ∆ l . The upp er bound (167) p ermits ap- plying [2, Thm. 3.84] o n w , which provides w ∈ BV ( Ω ) l and (92). Due to [2, Pr op. 3.61], the sets ( E ) 0 , ( E ) 1 and F E form a (pair wise disjoint) partition o f Ω , up to a n H d − 1 -zero s et. Mo reov er, since Ψ ( D u ) ≪ | D u | ≪ H d − 1 by construction, we hav e, for some Bor e l set A , Z A Ψ ( Dw ) (168) = Z A ∩ ( E ) 1 dΨ ( Dw ) + Z A ∩ ( E ) 0 dΨ ( Dw ) + (169) Z A ∩F E ∩ Ω Ψ  ν E  w + F E ( x ) − w − F E ( x )  ⊤  d H d − 1 ( ∗∗ ) 6 Z A ∩ ( E ) 1 dΨ ( Dw ) + Z A ∩ ( E ) 0 dΨ ( Dw ) + Z A ∩F E ∩ Ω λ u d H d − 1 (170) ( 167 ) 6 Z A ∩ ( E ) 1 dΨ ( Dw ) + Z A ∩ ( E ) 0 dΨ ( Dw ) + λ u Per ( E ) . (171) The inequa lit y ( ∗∗ ) holds due to the upp er b oundedness and Prop. 3. F rom [2, Prop. 2.37] w e obtain that Ψ is additive on mutually singular Radon measur es µ, ν , i.e., if | µ |⊥| ν | , then Z B dΨ ( µ + ν ) = Z B dΨ ( µ ) + Z B dΨ ( ν ) (172) for any Bo rel set B ⊆ Ω . Substituting D w in (17 1) ac- cording to (92) and using the fact that the three mea- sures that for m D w in (92) are m utually singular, the additivity prop erty (172) lea ds to th e rema ining as ser- tion, Z A dΨ ( Dw ) 6 (173) Z A ∩ ( E ) 1 dΨ ( Du ) + Z A ∩ ( E ) 0 dΨ ( Dv ) + λ u Per( E ) . Pr o of (Pr op. 6) W e first show (96). It suffices to sho w that  x ∈ ( E ) 1 ⇔ x ∈ E } for L d -a.e. x ∈ Ω . (174 ) This can be seen by cons idering the precise repres e n ta- tive f 1 E of 1 E [2, Def. 3.6 3]: Starting with the definition, x ∈ ( E ) 1 ⇔ lim δ ց 0 | E ∩ B δ ( x ) | |B δ ( x ) | = 1 , (175) the fact that lim δ ց 0 | Ω ∩B δ ( x ) | |B δ ( x ) | = 1 implies x ∈ ( E ) 1 ⇔ lim δ ց 0 | ( Ω \ E ) ∩ B δ ( x ) | |B δ ( x ) | = 0 (1 76) ⇔ lim δ ց 0 1 |B δ ( x ) | Z B δ ( x ) | 1 E − 1 | dy = 0 (177) ⇔ f 1 E ( x ) = 1 . (178) Substituting E by Ω \ E , the same equiv alence shows that x ∈ ( E ) 0 ⇔ ] 1 Ω \ E ( x ) = 1 ⇔ f 1 E ( x ) = 0. As L d ( Ω \ (( E ) 0 ∪ ( E ) 1 )) = 0, this shows that 1 E 1 = f 1 E L d -a.e. Using the fact that f 1 E = 1 E [2, P rop. 3 .6 4], we conclude that 1 ( E ) 1 = 1 E L d -a.e., whic h pro ves (174) and ther efore the assertion (96). Since the measure-theo retic int erior ( E ) 1 is defined ov er L d -integrals, it is inv ariant under L d -negligible mo d- ifications of E . T og ether with (17 4) this implies (( E ) 1 ) 1 = ( E ) 1 , F ( E ) 1 = F E , (( E ) 1 ) 0 = ( E ) 0 . (179) T o s how the r elation ( D u ) x ( E ) 1 = ( D v ) x ( E ) 1 , cons ide r D u x ( E ) 1 = D  1 Ω \ ( E ) 1 u + 1 ( E ) 1 u  x ( E ) 1 (180) ( ∗ ) = D  1 Ω \ ( E ) 1 u + 1 ( E ) 1 v  x ( E ) 1 . (181) The equality ( ∗ ) holds due to the ass umption (94), and due to the fact that Df = D g if f = g L d -a.e. (see, Optimalit y Bounds for a V ariational Relaxation of the Image Partitioning Problem 15 e.g., [2, Prop. 3.2]). W e contin ue fro m (18 1) via D u x ( E ) 1 (182) Prop . 5 = { D u x (( E ) 1 ) 0 + D v x (( E ) 1 ) 1 + (183 ) ν ( E ) 1  u + F E 1 − v − F E 1  ⊤ H d − 1 x ( F ( E ) 1 ∩ Ω ) } x ( E ) 1 ( 179 ) =  D u x ( E ) 0 + D v x ( E ) 1  x ( E ) 1 + (184)  ν ( E ) 1  u + F E 1 − v − F E 1  ⊤ H d − 1 x ( F E ∩ Ω )  x ( E ) 1 = D u x  ( E ) 0 ∩ ( E ) 1  + D v x  ( E ) 1 ∩ ( E ) 1  + ( 185 ) ν ( E ) 1  u + F E 1 − v − F E 1  ⊤ H d − 1 x ( F E ∩ Ω ∩ ( E ) 1 ) = D v x ( E ) 1 . (186 ) Therefore D u x ( E ) 1 = D v x ( E ) 1 . Then, Ψ ( Du ) x ( E ) 1 = Ψ ( Du x ( E ) 1 + D u x ( Ω \ ( E ) 1 )) x ( E ) 1 (187) ( ∗ ) = Ψ  D u x ( E ) 1  x ( E ) 1 + Ψ  D u x ( Ω \ ( E ) 1 )  x ( E ) 1 . (1 88) In the equality ( ∗ ) we used the a dditivity of Ψ on mutu- ally singular Radon measure s [2, Prop. 2 .37]. By defini- tion o f the total v ariation, | µ x A | = | µ | x A holds for any measure µ , therefor e | D u x ( Ω \ ( E ) 1 ) | = | D u | x ( Ω \ ( E ) 1 ) and | D u x ( Ω \ ( E ) 1 ) | (( E ) 1 ) = 0 , which together with (again by definition) Ψ ( µ ) ≪ | µ | implies that the second term in (188) v a nishes. Since all obser v ations equally hold for v instead of u , we conclude Ψ ( Du ) x ( E ) 1 = Ψ ( D u x ( E ) 1 ) x ( E ) 1 (189) ( 186 ) = Ψ ( D v x ( E ) 1 ) x ( E ) 1 (190) = Ψ ( D v ) x ( E ) 1 . (19 1 ) Equation (95) follows immediately . References 1. Alb erti, G., Bouchitt ´ e, G ., Dal Maso, G.: The calibra- tion method for the Mumford-Sh ah functional and free- discontin u ity problems. Calc. V ar. Pa rt. Diff. Eq. 16 (3), 299–3 33 (2003) 2. Ambrosio, L., F usco, N., P allara, D.: F un ctions of Bounded V ariation and F ree Discontin uity Problems. Clarendon Press (2000) 3. 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