An Augmented Lagrangian Model for Signal Segmentation

c The Author(s) 2022

An Augmented Lagrangian Model for Signal Segmentation

Salvador Moll and Vicent Pallard´o

Abstract.In this paper, we provide a new insight to the two-phase sig- nal segmentation problem. We propose an augmented Lagrangian vari- ational model based on Chan–Vese’s original one. Using both energy methods and PDE methods, we show, in the one-dimensional case, that the set of minimizers to the proposed functional contains only binary functions and it coincides with the set of minimizers to Chan–Vese’s one.

This fact allows us to obtain two important features of the minimizers as a by-product of our analysis. First of all, for a piecewise constant initial signal, the jump set of any minimizer is a subset of the jump set of the given signal. Second, all of the jump points of the minimizer belong to the same level set of the signal, in a multivalued sense. This last property permits to design a trivial algorithm for computing the minimizers.

Mathematics Subject Classification. 35G60, 35Q68, 35J92, 49J10.

Keywords. Segmentation, signal processing, total variation, minimizers, one dimensional.

1. Introduction

Segmentation is the task of partitioning an object into its constituent parts.

In signal or image processing, it consists in decomposing the domain Ω of a given input: a signal (an interval in which case Ω = [a, b] ⊂ R) or an image (Ω⊂R²) into some regions of interest. In the particular case of two- phase segmentation, the aim is to find an optimal partition into two disjoint subsets, the foreground domain Ω_F and the background domain Ω_B, such that Ω = Ω_F ∪Ω_B.

After the seminal work by Mumford and Shah [17], in which the authors introduced a celebrated variational model for image segmentation, Chan and

Salvador Moll has been partially supported by the Spanish Ministerio de Ciencia e Inno- vaci´on, and FEDER, Project PGC2018-094775-B-I00.

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Vese rewrote it in the two–phase framework [7]. They propose to obtain the optimal partition by minimizing the following energy functional:

E,cmin1,c2Per(E; Ω) +λ1

E(c1−f)²dx+λ2

Ω\E(c2−f)²dx (1.1) among all sets of finite perimeter E ⊂ Ω and all constants c1, c2 ∈ [0,1], for some given parametersλ1, λ2 ≥ 0. From now on, we do not distinguish between the weights of the foreground and background, and thus, we take λ1 = λ2 = λ. A minimizer E ⊂ Ω can be considered as the foreground domain and Ω\E as the background domain. In this case, the constants c1

andc2turn out to be the average off inE and the average in Ω\E, respec- tively. The authors proposed in [7] an iterative two-step algorithm for finding the minimizers of the energy based on the level-set formulation developed by Osher and Sethian [18]. Basically, after initialization of the constants, the first step consists in finding the corresponding minimizing set with fixed con- stants as a steady-state solution of the correspondentL²-gradient flow of the functional associated with the level-set formulation. Then, one recomputes the constants and comes back to the first step until convergence has been reached.

The main problem with this algorithm is that the energy functional is not convex. Therefore, the gradient descent scheme is prone to get stuck at critical points other than global minima. This issue was fixed by Chan–

Esedoglou and Nikolova, who proved that minimizers to Chan–Vese’s level-set functional with fixed constants (i.e., solutions to Step 2) are solutions to the following constraint convex energy minimization problem (and vice versa):

{0≤u≤1}min |Du|(Ω) +λ

Ω

u(c1−f)²+ (1−u)(c2−f)²

dx. (1.2) Observe that, if the solutionuis the characteristic function of a set with finite perimeter;u=χ_E, then, the energy in (1.2) coincides with that in (1.1).

There are still two main problems: the main one remains at the noncon- vex nature of the original energy functional, i.e., convergence of the algorithm to a global minima is not ensured, and it heavily depends on the initialization.

Moreover, it is not known if Chan–Vese’s algorithm (with Chan–Esedoglu–

Nikolova modification) could lead to non-binary solutions (see [9]).

The main objective of this work is to give another approach to original Chan–Vese’s minimizers in the easiest possible case, the one-dimensional case, which corresponds to signal segmentation. In the context of signal segmenta- tion, Chan–Vese’s algorithm was already proposed in [8] and has been used in some works (see [15] or [16]). Our starting point is the functional appearing in Problem (1.2). We aim at minimizing, simultaneously the functionuand the constantsc₁, c₂. To do that, we introduce an augmented Lagrangian version of the functional, coupled with the constraint 0≤u≤1. In this new func- tional, we replace the constants by BV functions while highly penalizing their variation. For anyε >0, we define the functionalF_ε: (L²(0,1))³→[0,+∞[

by letting

F_ε(u, v1, v2) = |Du|(Ω) +1

ε(|Dv1|(Ω) +|Dv2|(Ω)) +λ

Ω

u(v1−f)² dx +λ

Ω

(1−u) (v2−f)²dx+

Ω

I_[0,1](u) dx, (1.3) whereI[0,1](·) denotes the indicator function of the interval [0,1], that is

I_[0,1](x) =

0 if x∈[0,1]

+∞ otherwise.

The second term is implemented for penalizing the variation of the pair of functions (v1, v2). Observe that, letting ε → 0, we are forcing v1, v2 to become constants. With this addition, the functionalF_ε fails to be convex.

However, we can use standard PDE methods to obtain some features of the set of minimizers via its correspondent system of Euler–Lagrange equations.

In particular, we prove the following results in the case that Ω is an interval ofR:

Theorem 1.1. Letε < ₄¹_λ. Then, if(u, v1, v2)is a minimizer ofF_ε, thenv1, v2

are constants.

To characterize the first component of the minimizer, we need to assume further that the datum is not too oscillatory in the following sense:

(H)f ∈BV(0,1) satisfies that for everyc∈(0,1) L¹(∂{x∈(0,1)\J_f :f(x) =c}) = 0.

Observe that with this assumption, we exclude some pathologies on the data such as having a fat Cantor set as a level set.

Theorem 1.2. Given ε < _4λ¹ , forf ∈BV(Ω) satisfying(H), any minimizer (u, v₁, v₂) of F_ε is independent of ε and it satisfies that either uis constant oru(Ω)⊂ {0,1}, i.e., uis a binary function inBV(Ω).

With this last result, we can show that the set of minimizers of Chan–

Vese’s problem (1.1) coincides with the set of minimizers ofF_ε (independent ofεunder the size condition above expressed). As a by-product of our anal- ysis, we obtain two important properties of solutions to (1.1)

(a) The jump set of any solution is concentrated in the topological boundary of a sole level set (in a multivalued sense, see (4.2) for the proper statement).

(b) Iff is piecewise constant, then the jump set of any solution is contained in the jump set off.

These two properties, though quite intuitive in the one-dimensional setting, were not known in the literature, to the best of our knowledge. Moreover, property (a) allows to build a trivial algorithm to find the minimizers of Chan–Vese’s problem in the one-dimensional case.

The plan of the paper is the following one: In Sect. 2, we obtain the system of PDE’s that minimizers toF_εsatisfy, i.e., the corresponding Euler–

Lagrange equations in this non-smooth case. In Sect. 3, we prove Theo- rems 1.1 and 1.2. Section 4 is devoted to the proof of properties (a) and

(b). In Sect. 4.2, we explain the trivial algorithm to compute the minimiz- ers. We finish the paper with some conclusions and with an Appendix, in which we collect the existence of minimizers toF_ε as well as the proof of an auxiliary result we need.

Notations. Throughout the paper, Ω denotes an open bounded set in R^N with Lipschitz boundary andL^N denotes the Lebesgue measure in R^N. We denote by L^p(Ω), 1 ≤ p ≤ ∞ the Lebesgue space of functions which are integrable with power p with respect to L^N. We use the notation ·,· to denote the scalar product between two L² functions. We denote by H¹(Ω) the Hilbert spaceW^1,2(Ω) and byH₀¹(Ω) the completion inH¹(Ω) of smooth functions with compact support in Ω. We use standard notation for functions of bounded variation (BV functions) as in [2]. In particular, givenu∈BV(Ω), we writeu_x,D^cuandD^jufor the absolutely continuous part of the measure Duwith respect toL^N, for the Cantor part ofDu and for the jump part of Du, respectively. We use the notationu^±(x) for the left and right approximate limits ofuatx∈Ω (we use only this convention for the case of Ω = (a, b)⊂R, a ≤ b ∈ R), J_u for its jump set and ν^u will denote the Radon–Nikodym derivative of Du with respect to |Du|. Given a set E ⊆ Ω, we say that it is a set of finite perimeter in Ω if χ_E ∈ BV(Ω), where χ_E denotes the characteristic function of the setE. In this case, its perimeter is defined as P er(E; Ω) :=|Dχ_E|. Finally, unless otherwise specified, we always identify a function (inH₀¹(0,1) or inBV(Ω)) by its precise representative.

2. System of Euler–Lagrange Equations

In this section, we derive the system of equations that minimizers ofF_εmust satisfy. Although F_ε is not a convex functional in (L²(Ω))³, it is a convex functional in each of their coordinates when one fixes the other two ones.

Therefore, by standard results in convex analysis, we obtain that the Euler–

Lagrange system of PDE’s is the following one:

⎧⎪

⎨

⎪⎩ λ

(v1−f)²−(v2−f)²

+∂(Φ + Ψ)(u)0 2ελu(v1−f) +∂Φ(v1)0 2ελ(1−u)(v₂−f) +∂Φ(v₂)0,

(2.1)

where the symbol ∂ denotes the subdifferential (in L²(Ω)) of the following two extended real-valued convex functions:

Φ(g) :=

|Dg|(Ω) if g∈L²(Ω)∩BV(Ω) +∞ if g∈L²(Ω)\BV(Ω) , and

Ψ(g) =

Ω

I[0,1](g) dx.

We easily note that a.e. ∂Ψ(g) = ∂I_[0,1](g), for any g ∈ L²(Ω). Next result is crucial and permits to reformulate∂(Φ + Ψ)

Theorem 2.1. Let g∈L²(Ω). Then,

∂(Φ + Ψ) (g) =∂Φ(g) +∂Ψ(g).

Proof. We point out that standard results to decompose the subdifferential of the sum, such as those in [6], cannot be applied, since the interior of the domains of both functionals is empty. We follow the strategy of [19, Thm.

3.1], consisting in an ad-hoc proof by approximating the subdifferential of the indicator function by its Yoshida’s regularization. First of all, we state the following claim, whose proof we postpone to the Appendix.

Claim. Leth:R→Rbe a Lipschitz non decreasing function. Then, it holds v, h(u)=|Dh(u)|(Ω), ∀v∈∂Φ(u).

Since we know that the inclusion

∂(Φ + Ψ) (u)⊇∂Φ(u) +∂Ψ(u), ∀u∈L²(Ω)

is satisfied, it is sufficient to show the converse inclusion. Letu∈L²(Ω) be such that∂(Φ + Ψ)(u)=∅, and letv∈∂(Φ + Ψ)(u). We defineϕ_u:L²(Ω)→ R∪ {+∞}by

ϕ_u(w) = (Φ + Ψ)(w) +1

2w²2− v+u, w.

We note thatϕ_uis coercive, strictly convex and lower semicontinuous. Then, it is easy to see thatuis the unique minimizer ofϕ_u.

Now, for any 0 < η < 1, let β_η be the Yoshida regularization of ∂I[0,1], that is

β_η(t) = (t−1)+−t₋

η , ∀t∈R;

here, subindexes±represent, respectively, the positive and the negative part of the function. We next considerγ:L²(Ω)→R∪+∞defined by

γ(w) = Φ(w) +1

2w²₂− v+u, w+

Ω

β_η(w) dx ,

whereβ_ηis a primitive ofβ_η. We note thatγis coercive, strictly convex over Dom(Φ + Ψ) and lower semicontinuous. Thus,γhas a unique minimizer u_η which satisfies the corresponding Euler–Lagrange equation

−β_η(u_η)−u_η+ (v+u)∈∂Φ(u_η).

In consequence, we know that the following equation has a unique solution:

v_η+β_η(u_η) +u_η=v+u , wherev_η ∈∂Φ(u_η).

Multiplying byβ_η(u_η) both sides of the previous equation, we have, for any 0< η <1

v_η, β_η(u_η)+β_η(u_η)²2+ u_η, β_η(u_η)= v+u, β_η(u_η).

Here, sincetβ_η(t)≥0 for anyt∈R, we note that u_η, β_η(u_η) ≥0. Moreover, by the previous Claim, we have that

v_η, β_η(u_η) ≥0.

Then, we see that{β_η(u_η)}_η and{v_η}_η are bounded in L²(Ω) and {u_η}_η is bounded inL²(Ω) andBV(Ω). Therefore, there is a sequence{η_n}_n⊂(0,1), such thatη_n→0 and there exists a functionu0∈L²(Ω), such that

u_ηn^n→∞−→ u0 in L¹(Ω).

Moreover, we can assume that there existv0, ξ ∈ L²(Ω), such that u_ηn, v_ηn andβ_ηn(u_ηn) converge weakly inL²(Ω) tou0, v0 andξ, respectively.

In addition, we note that 1

η(u_η−1)₊≤ |β_η(u_η)| and 1

η(u_η)₋ ≤ |β_η(u_η)|, for any 0< η <1, and thus, we have

(u_η−1)₊→0 and (u_η)₋→0 inL²(Ω) asη→0. Hence, we can show thatu0(x)∈[0,1] a.e. in Ω. We conclude that

ξ∈∂I_[0,1](u0) a.e. in Ω,

v₀∈∂Φ(u₀) and v₀+ξ+u₀=v+uinL²(Ω).

This implies that u0 is a minimizer of ϕ_u. Then, u0 = u, and conse- quently,v=ξ+v0 in L²(Ω), which finishes the proof.

Up to this point, we have worked without imposing any restriction in the dimension of the domain. Now, we introduce the characterization of the subdifferential of the total variation inL²(Ω), proposed by Andreu, Ballester, Caselles, and Maz´on in [3] (see also [4]), in the specific case of the domain being an interval in 1D, which we take as (0,1) without loose of generality (see [10] for a proof).

Theorem 2.2. (TV characterization in 1D) Let u be in BV(0,1), such that

∂Φ(u)=∅. Then, v∈∂Φ(u) if and only if there existsz_u ∈H₀¹(0,1), such that a.e.|z_u| ≤1and

v=−(zu)_x, |Du|=zu·Du, where the measurez_u·Du∈ M(0,1) is defined as

(zu·Du)(U) :=

Uzu·u_xdx+

Uzu·ν^ud|D^cu|

+

x∈Ju∩U

(u⁺(x)−u⁻(x))zu(x)·ν^u(x), for any Borel setU ⊂(0,1).

With this characterization in mind, the system of Euler–Lagrange equa- tions can be rephrased in the following way:

Proposition 2.3. Let (u, v1, v2) be a minimizer of F_ε. Then, there exist z_u, z_v1,z_v2∈H₀¹(0,1), corresponding to∂Φ(u),∂Φ(v1)and∂Φ(v2), respectively,

as given by Theorem2.2, andg∈∂I_[0,1](u), such that

⎧⎪

⎨

⎪⎩

(zu)_x=λ

(v₁−f)²−(v₂−f)² +g (zv1)_x= 2ελu(v₁−f) (z_v2)_x= 2ελ(1−u)(v2−f).

(2.2)

3. Proofs of the Main Results

3.1. Proof of Theorem1.1

We need to show first the following auxiliary result.

Lemma 3.1. (Behaviour of z_u) Let u∈BV(0,1) and z_u ∈H₀¹(0,1), corre- sponding to∂Φ(u)as provided by Theorem 2.2. Then

|zu|= 1, |Du| − a.e.

Proof. We decompose both measures|Du|andzu·Duin the following way:

z_u·Du= (z_u·u_x)L¹+z_u·D^ju+z_u·D^cu

|Du|=|u_x|L¹+|D^ju|+|D^cu|.

Since both decompositions are mutually singular, we have

zu =

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩ u_x

|u_x|, |ux|L¹ -a.e u⁺−u⁻

|u⁺−u⁻|, |D^ju|-a.e D^cu

|D^cu|, |D^cu|-a.e,

and thus, we have|zu|= 1, |Du| −a.e.

Proof of Thm.1.1. First, we note that if (u, v1, v2) is a minimizer ofF_ε, then it is easy to see that all variables take values in [0,1] a.e in (0,1) as shown in LemmaA.1in the Appendix. Suppose that there exists a Borel setU ⊂(0,1), such thatDv₁(U)= 0. By Lemma 3.1, we know that there is x₁ ∈U, such that zv1(x₁) ∈ {−1,+1}, where zv1 is given by Proposition 2.3. Then, by (2.2)₂, we have the next inequality

1 = _x1

0

(z_v1)_xdx

= 2ελ _x1

0

u(v1−f) dx

≤2ελ 1

0

|u||v1−f|dx≤4ελ,

and in consequence,ε≥1/(4λ) and, thus, a contradiction by hypothesis.

To get thatv₂ is also constant, we apply the same argument to (2.2)₃.

Under the size condition ε < ₄¹_λ, which we assume from now on, we integrate the second and third Eq. (2.2) in (0,1) and we obtain

v1= 1

0

ufdx 1

0

udx

, v2= 1

0

(1−u)fdx 1

0

(1−u) dx

, (3.1)

in the case thatuis not constant with values 0 or 1. Ifu≡0 (resp.u≡1), thenv₁ (resp.v₂) can be any constant value in (0,1). We finally note that, beingv₁, v₂constants, the energy functional does not depend onε. From now on, we will assume the size condition ε < ₄¹_λ. We rename the constants as c₁, c₂ for consistency and remove theεdependence by letting

F(u, c₁, c₂) =|Du|((0,1)) + ₁

0

I[0,1](u) dx +λ

₁

0

(u(c₁−f)²+ (1−u)(c₂−f)²) dx . 3.2. Proof of Theorem1.2

This section is devoted to prove that the first coordinate of the minimizer is necessarily a binary BV function. Hereinafter, we assume that the datum f satisfies (H). The proof will be done in two different steps. First of all, we show that if (u, c1, c2) is a minimizer, then there is a “quasi–piecewise constant”competitoruwith lower energy than (or equal to) the corresponding tou. Then, we prove that the competitor cannot be a minimizer in case it is not binary using the PDE system (2.2).

We start by defining our concept of quasi–piecewise constant function.

Definition 3.2. We say thatu∈BV(0,1) is quasi–piecewise constant if there exist a piecewise constant functionu_s and an a.e binary function u_b which fulfill that they are not non-simultaneously and

u(x) =u_s(x) +u_b(x), a.e. x∈(0,1).

Theorem 3.3. Let c₁, c₂ ∈ [0,1]. Given u a minimizer of F(·, c1, c₂), there exists a quasi–piecewise constantu∈BV(0,1) satisfying

F(u, c₁, c₂)≤F(u, c₁, c₂). (3.2) Proof. Let x0 be in (0,1), such that u(x0) ∈ {0,/ 1} and x0 ∈/ J_u, and let z_u ∈H₀¹(0,1) be the vector field associated with∂Φ(u) as given by Theorem 2.2. We distinguish two cases:

(i) The casex0∈(|z_u|)⁻¹({1}): We assume without loss of generality that z_u(x0) = 1 (forz_u(x0) =−1, the argument is analogous). Under this assumption, we know that u is continuous at x0 (remember that we always identify a BV function with its precise representative) and thus, u /∈ {0,1}in a neighbourhood ofx0denoted byEx0. Hence, using (2.2)1, we know that

(z_u)_x=λ

(c1−f)²−(c2−f)²

inE_x0. (3.3)

Since f ∈ BV(0,1), we have by the above expression that (z_u)_x ∈ BV(E_x0), and thus, its lateral traces (z_u)⁺_x(x0) and (z_u)⁻_x(x0) are well defined. In addition, sincezu(x0) = 1 andzu_L^∞(Ω)≤1, this implies that

(z_u)⁺_x(x₀)≤0≤(z_u)⁻_x(x₀), and we have by (3.3) that

f⁻(x₀)≤c₁+c₂

2 ≤f⁺(x₀). Then, we note that the set

{x∈(0,1)\(J_u∪J_f) :|z_u(x)|= 1∧u(x)∈(0,1)} is a subset of the following set:

A:=

x∈(0,1)\J_f :u(x)∈(0,1) ∧ f(x) =c1+c2

2

.

We note thatA=A^o∪(∂A∩A) whereA^oand∂Aare the (topological) interior and boundary ofA, respectively. Particularly, we note that

A^o=

k=1

I_k,

where {I_k = (a_k, b_k)}_k is a disjoint collection of open intervals; and L¹(∂A∩A) = 0 by assumption on f and the fact that ∂A∩A ⊂ J_f∪∂{f =^c1+c₂ ²}.

Next, we will modifyuin each interval I_k to decrease the energy.

Before doing it, we point out that, for c₁, c₂ fixed, minimizing F is equivalent to minimize

G(w) :=|Dw|((0,1)) + ₁

0

I[0,1](w) dx +λ

₁

0

w(c₁−c₂)(c₁+c₂−2f) dx . SinceI_k ⊂A, we observe that

G(uχ_Ik) =|Du|((a_k, b_k)).

Therefore, if we take u_k :=u(a_k)⁺χ_Ik+uχ_(0,1)\Ik, it is easy to show that

F(u_k, c₁, c₂)≤F(u, c₁, c₂).

(ii) Ifx0∈/(|z_u|)⁻¹({1}): Sincez_u ∈H₀¹(0,1)⊂C(0,1), we take the largest interval I containing x0, such that |z_u|(I) ⊆ [0,1). By Lemma 3.1,

|Du|= 0 in this interval, and thus,uis constant inI.

Consequently, we note that (0,1)\(∂A∩A) can be decomposed as (0,1)∩(u)⁻¹({0,1})

B

∪

(0,1)∩(u)⁻¹((0,1))

C

,

such thatCis a subset of

k=1

I_k

∪

C∩(|zu|)⁻¹([0,1)) .

Defining

u(x) :=

u_k(x) ifx∈I_k for anyk u(x) ifx /∈I_k for allk , we obtain that the inequality (3.2) is clearly satisfied.

Note thatuis a piecewise constant function inCbecause of the reason- ing in (i) and (ii), and thus, we can take

u_s:=uXC.

Finally, asBandCare disjoint, we obtain thatuis a quasi-piecewise constant

function.

We introduce now two useful remarks:

Remark 3.4. Let (u, c^u₁, c^u₂) be a minimizer ofF, such that c^u₁ ≤c^u₂. Defin- ing w := 1 −u, it is easy to show that c^w₁ = c^u₂, c^w₂ = c^u₁ and that F(w, c^w₁, c^w₂) = F(u, c^u₁, c^u₂). On account of it, we assume hereinafter that c2≤c1. Furthermore, ifc^u₁ =c^u₂, we note that

F(u, c^u₁, c^u₂) =|Du|((0,1)) +λ ₁

0

(c^u₂−f)²dx, and thus, to be a minimizer,uis a necessarily a constant function.

Remark 3.5. Let (u, c₁, c₂) be a minimizer ofF and suppose that there exist a < b ∈ J_u, such that u⁺(a), u⁻(b), u(x) ∈ {0,/ 1}, for any x ∈ (a, b). By integrating (2.2)1 on (a, b), we have that

λ _b

a

(c1−f)²−(c2−f)²

dx=z_u(b)−z_u(a),

wherezu corresponds to∂Φ(u). Note that the right-hand side term is equal to 2, −2 or 0 because of the fact that |zu| = 1 in the jump set J_u as a consequence of Lemma3.1; aszu ∈H₀¹(0,1), ifx₀∈J_u, then

• z_u(x0) =−1 when ujumps toward a lower step inx0.

• zu(x0) = 1 whenujumps toward an upper step inx0. After these remarks, we prove the following statement:

Proposition 3.6. Let (u, c1, c2)be a minimizer of F, such that uis a quasi–

piecewise constant function. Then, eitheruis constant oruis an a.e. binary function.

Proof. This statement is proved by contradiction. We suppose thatuis not an a.e binary function and not constant, i.e., u_s has some non-binary step (u_sbeing the piecewise constant part). Leta, b∈J_ube such thatu_s((a, b)) = {β}∈ {0,/ 1}. In addition, letα:=u⁻(a) andγ:=u⁺(b).

According to the relationα=β =γ, there are three different cases to study:

(i) α < β < γ (orγ < β < α, resp.): We define v_τ(x) :=

u(x) ifx /∈(a, b) τ ifx∈(a, b),

where τ ∈ (α, β) (or τ ∈ (γ, β), resp.). In any case, according to Remarks3.4and3.5, we can suppose thatc1> c2and

λ _b

a

(c1−f)²−(c2−f)²

dx=λ(c1−c2) _b

a (c1+c2−2f) dx= 0. Sinceλ(c1−c2)>0, we obtain

_b

a

fdx=

c₁+c₂ 2

(b−a).

It is clear thatF(u, c1, c2) =F(v_τ, c1, c2). Then, (v_τ, c1, c2) is a mini- mizer too. Consequently, by (3.1), we have

c1= ₁

0

ufdx ₁

0

udx

= ₁

0

v_τfdx ₁

0

v_τdx

i.e.

Iufdx +βM L

I

udx +βL

=

Iufdx +τ M L

I

udx +τ L ,

where we denoted byI := [0, a)∪(b,1], M := ^c1+c₂ ² and L := b−a.

Then βM L

Iudx+τ L

Iufdx =τ M L

Iudx+βL

Iufdx Sinceβ=τ andL= 0, we have

M

Iudx =

Iufdx . This yields

M 1

0

u −βL

= 1

0

uf −βM L i.e M = ₁

0

ufdx 1

0

udx

=c1,

thus leading toc1=c2, i.e, to a contradiction by Remark3.4.

(ii) β < α≤γ (orβ < γ≤α, resp.): Similarly as before, we consider v_τ(x) :=

u(x) ifx /∈(a, b) τ ifx∈(a, b),

whereτ =α(or τ =γ, resp.). In any case, according to Remarks 3.4 and3.5, we can suppose thatc1> c2 and

λ _b

a

(c1−f)²−(c2−f)²

dx= 2.

We observe that

_b

a fdx=M L−A,

where M and L are the constants defined in the previous case and A:= _λ(c¹

1−c2). Then, it is easy to check thatF(u, c1, c2) =F(v_τ, c1, c2).

Again, (v_τ, c₁, c₂) is a minimizer, and thus, it satisfies (3.1). Repeating the reasoning in the previous case (replacingM L−Ainstead ofM L), we obtain

M−A L =c₁,

and if we redo the same computations forc2, we obtain the same equa- tion, thus leading toc1=c2, i.e., to a contradiction as before.

(iii) α≤γ < β (orγ≤α < β, resp.): In this case, we define v_τ(x) :=

u(x) ifx /∈(a, b) τ ifx∈(a, b),

whereτ=γ(or τ=α, resp.). Once again, repeating the computations in the previous case, we end up withc1=c2, thus finishing the proof.

We finish this section by pointing out that Theorem 3.3and Proposi- tion3.6already prove Theorem 1.2.

4. Properties of Minimizers

4.1. Proof of Properties(a)and(b)

In this section, we show that the set of minimizers ofF coincides with that of minimizers to Chan–Vese and we prove Properties (a) and (b) of the min- imizers.

Remark 4.1. It is obvious that givenE⊂Ω of finite perimeter F(χ_E, c1, c2) = Per(E; (0,1))+λ

E(c1−f)²dx+λ

(0,1)\E(c2−f)²dx.

Then, we note that

u∈L²min(0,1),c1,c2

F(u, c₁, c₂)≤ min

E,c1,c2

Per(E; Ω) +λ

E(c₁−f)²dx +λ

Ω\E(c2−f)²dx. (4.1) On the other hand, by Theorem 1.2, we now that the minimum of F is achieved in a constant or binary BV function u, for the first coordinate.

Then, u = χ_E (or u = k ∈ [0,1])for a set E ⊆ (0,1) of finite perimeter, which proves the reverse inequality in (4.1). This shows that minimizers of F coincide with Chan–Vese’s minimizers.

We now prove Properties (a) and (b):

Suppose thatx0 ∈J_u and thatu⁻(x0) = 1,u⁺(x0) = 0. Then, as explained in Remark 3.5, z_u(x0) = −1, with z_u corresponding to ∂Φ(u) as given by Theorem2.2. Moreover, since∂I_[0,1](x) =]− ∞,0]δ0+ [0,+∞[δ1, by (2.2)1, as in the Proof of Theorem3.3that

f⁺(x0)≤c1+c2

2 ≤f⁻(x0).

If insteadu⁻(x0) = 0,u⁺(x0) = 1, one gets f⁺(x0)≥c1+c2

2 ≥f⁻(x0).

Therefore

J_u⊆C_1,2:={x∈(0,1) :f(x) c₁+c₂

2 }, (4.2)

where the image of a jump point is understood in a multivalued sense (i.e., f(x) = [min{f⁻(x), f⁺(x)},max{f⁻(x), f⁺(x)}]). Note that this fact almost proves Properties (a) and (b). The only remaining thing to prove is that a jump point cannot exist in the interior ofC1,2. Suppose, by contradiction that x0∈J_u∩C₁^o_,2 and leta < x0 be such thatJ_u∩[a, x0[=∅. Without loosing generality, we can suppose thatu= 1 in [a, x0[ and that u⁺(x0) = 0. Then, considering v := uχ_(0,1)\[a,x0[, we easily get that F(v, c1, c2) = F(u, c1, c2) and, therefore, (v, c1, c2) is a minimizer too. Then, repeating the reasoning in Proposition3.6, in case (ii), we arrive at a contradiction, which finishes the proof of Properties (a) and (b).

Remark 4.2. We note that Properties (a) and (b) of the minimizers are only expected to hold in the one-dimensional case. In fact, an easy counterexample in two dimensions is provided byf =χ_EwithE= [−¹₂,¹₂]²and Ω = [−1,1]². In this case,J_f is precisely the boundary of the square [−¹₂,¹₂]² while it can be proved that, for any λ > ¹⁶₃ Chan–Vese’s minimizer cannot be either u=χΩ oru=χ_E , thus showing that (a) and (b) do not hold (Fig.1).

In fact, Chan–Vese’s energy for these two candidates is exactly ³₄λ(for u=χΩ, c1=¹₄) and 4 (foru=χ_E, c1= 1,c2= 0). Therefore, for λ > ¹⁶₃, u=χΩis not a minimizer. On the other hand, it is easy to modify the corners of the square by reducing the perimeter and not changing to much the fidelity

Figure 1. Left:∂E and∂Ω represented by solid and dashed lines, respectively. Right: Chan–Vese segmentation, where black and white colors represent 1 and 0 numerical values, respectively

Figure 2. ∂E_δ, i.e.,J_vδ

term in the energy. We modify E, calling the new set E_δ, by removing the corners through the use of δ-radius circumferential arches tangent to every two contiguous sides of∂E (see Fig.2).

Then, one can check that v_δ := χ_Eδ, c1 = 1, c2 = ₃₊₍₄^(4−π)δ_−π)²_δ2 for δ satisfying

3δ

3 + (4−π)δ² < 2 λ,

has strictly less energy. This fact is related to the non-calibrability of the set Ewith respect to the isotropic norm in the total variation (see [1]).

Therefore, if one wants to obtain similar results to Properties (a) and (b) in higher dimensions, the total variation term needs to be changed by an anisotropic version of it as in the case of the anisotropic Rudin–Osher–Fatemi

functional, for which stability of piecewise constant functions on rectangles has been recently shown in [14] and [13]. We will investigate this issue further in a subsequent paper.

4.2. Application of the Properties

In this section, we propose a trivial way to approach the solution of the 1D Chan–Vese problem using properties (a) and (b) of the minimizer. We will also comment on the pros of this trivial algorithm in front of those based on a Gradient Descent (GD) scheme. We remark that those algorithms are applied in an 1D version of alternating scheme proposed by Chan and Vese in [7]. Hereinafter, we will assume that the boundary of each level of the datum f set has a finite number of points.

We start with the general case. In this case, the idea is:

(1) Take a discretization of the range, thus defining the working level sets.

(2) In each level set, compute the binary candidate with the least energy with jumps in the boundary of the level set.

(3) Compare between the solutions and choose one with the smallest energy.

Besides, in the case off being a step function, we can further simplify the previous idea thanks to the implementation of the inclusionJ_u⊆J_f. This reduction is based on trying all the possible combinations of characteristic solutions whose jump set is a subset ofJ_f.

In Fig.3, we sketch how to perform Step 2 in the general algorithm for a fixed level set. First, we obtain all possible jump points for the possible min- imizeru, (a_i, i∈ {1, ..., m}). Then, we know that the candidate to minimizer takes the form

u=χ_{∪j∈I[aj,aj+1]} where I ⊆ {0, ..., m−1}.

Computing all the possibilities, we keep the candidate with the least energy among them. We compare its energy with that of the candidate obtained from a previous bigger level set and the procedure is repeated until we reach the lowest level set in the discretization.

To show the suitability of this trivial approach in some situations, we present the following example:

Example. Suppose we segment by the 1D Chan–Vese model a signal whose shape is similar to the one of the Weierstrass function. This kind of signals has the property of exhibiting abrupt variations of its slope on the whole domain, which causes problems for GD-based algorithms. This intuitive idea can be seen in Fig. 4, where we compute an approximation to the minimizer using two different approaches: in one of them, we use the alternating Chan–Vese scheme with a GD-based method (ADAGRAD algorithm, see [11]); while on the other one, we use the trivial scheme explained above.

We note how the GD-based approach suffers from the variations of the signal, thus affecting to the performance of the alternating Chan–Vese scheme. In contrast, the trivial approach, based on properties (a) and (b), provides an adequate minimizer approximation. Moreover, we note that even

Figure 3. Given the level set (illustrated by the dashed blue line), the candidates take a constant (0 or 1) value between each couple of vertical lines

Figure 4. Comparison of different approaches in the segmen- tation of a Weierstrass type function by 1D Chan–Vese (orig- inal signal in black; in colors the result of the segmentation).

Top: GD-based approach. Bottom: Trivial approach. Left:

Result in [0,1].Right: Result zoomed in [0.1,0.23]

if we chose a particular case of an algorithm based on GD, this type of behaviour is a trend in any of these algorithms. Therefore, the use of the

scheme presented in this paper is beneficial in situations where the applica- tion of GD (or variants) gives a wrong segmentation of the signal, as shown in the figure above.

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Appendix A

In this Appendix, we show that the functional F_ε always has a minimizer regardless of the dimension of the domain. Note that, for ε ≤ ₄¹_λ, in the 1–dimensional case, according to Remark 3, existence of minimizers follows directly from existence of minimizers to Chan–Vese’s functional. Here, we give a direct proof for anyε >0, which, in turn, provides an alternative proof of existence of solutions to (1.1). Existence of minimizers will be shown through the study of the following auxiliary energy:G_ε: (L²(0,1))³→[0,∞],

G_ε(u, v₁, v₂,) :=F_ε(u, v₁, v₂) +

Ω

I[0,1](v₁) +I[0,1](v₂) dx .

Existence of minimizers to functionalG_ε is guaranteed by the Direct Method in the Calculus of Variations, since the functional is easily seen to be lower semicontinuous inL²(Ω) and coercive. Now, we relate both functionals F_ε andG_ε in the following result:

Lemma A.1. Let (u, v1, v2)∈

L²(Ω)3

. Then

F_ε(u, T(v1), T(v2))≤F_ε(u, v1, v2), (A.1) whereT is the truncation function in[0,1]; i.e., T(t) := (t−1)+−t₋. Proof. First, note that we can suppose thatu(Ω)∈[0,1] a.e. (otherwise, both terms in the inequality are equal to +∞). Now, we observe that the following inequality is easily seen to be true (sincef(Ω)∈[0,1]):

(T(v_i))−f)²≤(v_i−f)², ∀i∈ {1,2}.

In consequence

Ω

u(T(v1)−f)²+ (1−u) (T(v2)−f)²

dx

≤

Ω

u(v1−f)²+ (1−u) (v2−f)²

dx . (A.2)

Moreover, applying the Chain Rule in the composition of aBV function and a Lipschitz function ([2, Theorem 101]), we have

|D(T(v_i))|(Ω)≤ |Dvi|(Ω), ∀i∈ {1,2}. (A.3) Therefore, by (A.2) and (A.3), we conclude (A.1).

Proposition A.2. For any ε > 0, the functional F_ε has a minimizer (u, v1, v2)∈BV(Ω)³.

Proof. The proof follows directly from LemmaA.1, since we obtain that (u, v₁, v₂) is a minimizer ofF_ε←→ it is a minimizer ofG_ε. Therefore, by the existence of minimizers toG_ε, we can conclude that F_ε admits (at least) one minimizer [which moreover belongs to (BV(Ω))³].

We finish this Appendix with the proof of the

Claim. Let h : R → R be a Lipschitz non decreasing function. Then, the following equality holds:

v, h(u)=|Dh(u)|(Ω), ∀v∈∂Φ(u).

Proof. Since this Claim is stated in the multidimensional case, we need to recall (see [3]) thatv∈∂Φ(u) if, and only if, there exists a vector fieldzwith

||z||_L^∞(Ω)≤1, divz∈L²(Ω), such thatv=−divz, [z, ν^Ω] = 0 and v, u=|Du|(Ω) =

Ω

d(z, Du) =

Ω

θ(z, Du)d|Du|,

with (z, Du) being the Radon measure defined by (see [5] for precise defini- tions and results here stated)

(z, Du)(ϕ) =−

Ω

uϕdivzdx−

Ω

uz· ∇ϕdx , forϕ∈ D(Ω), θ(z, Du) being the Radon– Nikodym derivative of (z, Du) over |Du| and [z, ν^Ω] being the weak normal trace of z on the boundary. Note that this implies, in particular, that

θ(z, Du) = 1, |Du| −a.e. (A.4) Since h is Lipschitz, by Chain’s rule, we have that h(u) ∈ BV(Ω).

Suppose now that h ∈ C¹(R) and h is increasing. By [5, Proposition 2.8], we have that θ(z, Dh(u)) = θ(z, Du), |Du|-a.e. Observe that in this case, the measure |Dh(u)| is absolutely continuous with respect to|Du|and vice

versa. Then, a property holds|Du|-a.e. iff it holds|Dh(u)|a.e. Therefore, by integration by parts (see [5]), we obtain

v, h(u) = −

Ω

h(u)divzdx=

Ω

d(z, Dh(u))

=

Ω

θ(z, Dh(u))d|Dh(u)|=

Ω

θ(z, Du)d|Dh(u)|

(A.4)

= |Dh(u)|(Ω). (A.5)

In the general case, we approximatehby a sequence ofC¹increasing functions h_n(t) :=h ρ1

n(t) +_n^t, 0≤ρ1

n being a symmetric mollifier. Therefore, it is easy to check that h_n(u) strictly converges toh(u) and we finish the proof

using (A.5).

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Salvador Moll and Vicent Pallard´o Departament d’An`alisi Matem`atica Universitat de Val`encia

C/ Dr. Moliner, 50 46100 Burjassot Spain

e-mail:j.salvador.moll@uv.es

Vicent Pallard´o

e-mail:vicentpallardojulia@gmail.com

Received: May 13, 2021.

Revised: August 29, 2021.

Accepted: April 5, 2022.