On the jump set of solutions of the total variation flow

On the Jump set of Solutions of the Total Variation Flow

V. CASELLES(*) - K. JALALZAI(**) - M. NOVAGA(***)

ABSTRACT - We show that the jump set of the solution of the minimizing Total Variation flow decreases with time for any initial condition inBV(V)\L^N(V). We prove that the size of the jump also decreases with time.

MATHEMATICSSUBJECTCLASSIFICATION(2010). 35J20, 94A08, 26B30, 68U10 KEYWORDS. Total variation flow, demonising model, nonlinear parabolic equations,

functions of bounded variation

1. Introduction

The use of total variation as a regularization tool for image denoising and restoration was introduced by L. Rudin, S. Osher and E. Fatemi in [19]. IfVdenotes the image domain, the Total Variation denoising problem corresponds to solving the minumum problem

u2BV(V)min Z

V

jDuj ‡ 1 2l

Z

V

(uÿf)²dx; l>0 8<

:

9=

;: …1†

One of the main features of (1), confirmed by numerical simulations, is its ability to restore the discontinuities of the image [19, 14, 16]. The a

(*) Indirizzo dell'A.: Departament de Tecnologia, Universitat Pompeu-Fabra, Barcelona, Spain.

E-mail: vicent.caselles@upf.edu

(**) Indirizzo dell'A.: CMAP, CNRS UMR 7641, Ecole Polytechnique, 91128 Palaiseau, France.

E-mail: khalid.jalalzai@polytechnique.edu

(***) Indirizzo dell'A.: Dipartimento di Matematica, UniversitaÁ di Padova, Via Trieste 63, 35121 Padova, Italy.

E-mail: novaga@math.unipd.it

priori assumption is that functions of bounded variation (the BV model [3]) are a reasonable functional setting for many problems in image processing and, in particular, for denoising and restoration. Typically, functions of bounded variation admit a set of discontinuities which is countably rectifiable [3], being continuous (in the measure theoretic sense) away from discontinuities. The discontinuity set corresponds to the edges in the image, and the ability of Total Variation regularization to recover edges is one of the main features which advocates for the use of this model in image processing (its ability to describe textures is less clear, even if some textures can be recovered up to a certain scale of oscillation).

As a support to this idea, in [10] (see also [13, 12]) it has been proved that the jump set of the solutionuof the TV denoising problem (1) is contained in the jump set of the datumf, assuming thatf 2BV(V)\L¹(V). Moreover the size of the jump ofuat any pointxin its jump set is bounded by the size of the jump of f. This result was explicitly stated in [18] in a more general context, even if for the Total Variation it is essentially contained in [10] (see also [12]).

The purpose of this paper is to prove the corresponding result for so- lutions of the the minimizing Total Variation flow, with Neumann or Di- richlet boundary conditions, and for the Cauchy problem inR^N. That is, if u(t) is the solution of the TV flow andJu(t)denotes the jump set ofu(t), we prove that

Ju(t)Ju(s)Ju(0) H^Nÿ1 a.e. for any t>s>0:

…2†

Moreover, letting [u(t)] be the jump size ofu(t), we also show that [u(t)][u(s)] H^Nÿ1 a.e. onJ_u(t) for any ts0:

…3†

The inclusions in (2) have already been proved in [10], under the as- sumption that u₀ belongs to the domain of the operatorÿdiv Du

jDuj

in L¹(V). Some sufficient conditions for this to happen were given in [10]. By exploting (3), in the present paper we get rid of this condition onu(0).

Let us finally describe the plan of the paper. In Section 2 we recall some basic facts about functions of bounded variation that will be used in the sequel. In Section 3 we review the result of [10] on the jump of the solutions of the denoising problem (1), explicitly proving that the size of the jump of the solution is bounded by the size of the jump of the initial datum. Finally, in Section 4 we extend this result to the Total Variation flow.

2. Notation and preliminaries onBVfunctions

LetVbe an open subset ofR^N. A functionu2L¹(V) whose gradientDu in the sense of distributions is a (vector valued) Radon measure with finite total variation inV is called a function of bounded variation. The class of such functions will be denoted byBV(V). The total variation ofDuonV turns out to be

sup Z

V

udivz dx:z2C¹₀ (V;R^N);jz(x)j 18x2V 8<

:

9=

;; …4†

(where for a vectorvˆ(v₁;. . .;v_N)2R^N we setjvj²:ˆP^N

iˆ1v²_i) and will be denoted by jDuj(V) or by R

VjDuj. The map u! jDuj(V) is L¹_loc(V)-lower semicontinuous. BV(V) is a Banach space when endowed with the norm kuk:ˆR

V jujdx‡ jDuj(V).

A measurable setEV is said to be of finite perimeter inV if (4) is finite when uis substituted with the characteristic functionx_EofE. The perimeter of EinV is defined as P(E;V):ˆ jDx_Ej(V). We denote by L^N and H^Nÿ1, respectively, the N-dimensional Lebesgue measure and the (Nÿ1)-dimensional Hausdorff measure inR^N.

Let u2[L¹_loc(V)]^m. We say that u has approximate limit at x2V if there existsz2R^m such that

limr#0

1 jB(x;r)j

Z

B(x;r)

ju(y)ÿzjdyˆ0:

…5†

The set of points where this does not hold is called the approximate dis- continuity set ofu, and is denoted byS_u. Using Lebesgue's differentiation theorem, one can show that the approximate limitzexists atL^N-a.e.x2V, and is equal tou(x): in particular,jSuj ˆ0.

Ifx2VnSu, the vectorzis uniquely determined by (5) and we denote it by ~u(x). We say thatu is approximately continuous atxif x62S_u and u(x)~ ˆu(x), that is ifxis a Lebesgue point of uwith respect to the Le- besgue measure.

Foru2BV(V), the gradientDuis a Radon measure that decomposes into its absolutely continuous and singular parts DuˆD^au‡D^su. Then D^auˆ ru dxwhereruis the Radon-Nikodym derivative of the measure Du with respect to the Lebesgue measure inR^N. The function u is ap- proximately differentiableL^N a.e. inV and the approximate differential

coincides withru(x)L^Na.e. The singular partD^sucan be also split in two parts: thejumppartD^juand theCantorpartD^cu. We say thatx2Vis an approximate jump point of u if there exist u^‡(x)6ˆu^ÿ(x)2R and jn_u(x)j ˆ1 such that

limr#0

1 jB^‡_r(x;n_u(x))j

Z

B^‡r(x;nu(x))

ju(y)ÿu^‡(x)jdyˆ0

limr#0

1 jB^ÿ_r(x;nu(x))j

Z

B^ÿ_r(x;nu(x))

ju(y)ÿu^ÿ(x)jdyˆ0;

where

B^‡_r(x;nu(x))ˆ fy2B(x;r) : hyÿx;nu(x)i>0g B^ÿ_r(x;nu(x))ˆ fy2B(x;r) : hyÿx;nu(x)i50g:

We denote byJ_uthe set of approximate jump points ofu. Ifu2BV(V), the set S_u is countably H^Nÿ1 rectifiable, J_u is a Borel subset of S_u and H^Nÿ1(SunJu)ˆ0 (see [3]). In particular, we have thatH^Nÿ1-a.e.x2Vis either a point of approximate continuity ofu, or a jump point with two limits~ in the above sense. Eventually, we have

D^juˆD^su Juˆ(u^‡ÿu^ÿ)nuH^Nÿ1 Ju and D^cuˆD^su _(VnSu): Forx2Juwe set [u(x)]ˆu^‡(x)ÿu^ÿ(x) .

For a comprehensive treatment of functions of bounded variation we refer to [3].

3. The jump set of solutions of the TV denoising problem

Given a functionf 2L²(V) andl>0 we consider the minimum problem

u2BV(V)min Z

V

jDuj ‡ 1 2l

Z

V

(uÿf)²dxˆ:Fl(u): …6†

Notice that problem (6) always admits a unique solution u_l, since the functionalF_l is strictly convex.

As we mentioned in the Introduction, one of the main reasons to in- troduce the Total Variation as a regularization term in imaging problems is its ability to recover the discontinuities of the functionf. In this section we recall a result proved in [10] showing that the jump set ofu_lis always

contained in the jump set of f, that is, the model (6) does not create any new discontinuity besides the existing ones. We refine the proof given in [10] proving that the size of the jump decreases. This fact was observed in [18] where it was proved in the more general context of weighted total variation.

Let us recall the following Proposition, which is proved in [15, 1].

PROPOSITION3.1. Let u_lbe the (unique) solution of(6).Then, for any t2R,fu_l>tg (respectively, fu_ltg) is the minimal (resp., maximal) solution of the minimal surface problem

minEV P(E;V)‡1 l Z

E

(tÿf(x))dx:

…7†

In particular, for all t2Rbut a countable set,fu_lˆtghas zero measure and the solution of(7)is unique (up to a negligible set).

A proof that ful>tg and fultg both solve (7) is found in [15, Prop. 2.2]. The proof of Proposition 3.1 then follows from the co-area formula and from the following comparison result for solutions of (7) (see [1, Lemma 4]):

LEMMA3.2. Let f;g2L¹(V)and E and F be respectively minimizers of minE P(E;V)ÿ

Z

E

f(x)dx and min

F P(F;V)ÿ Z

F

g(x)dx:

Then, if f5g a.e.,jEnFj ˆ0(in other words, EF up to a negligible set).

From Proposition 3.1 and the regularity theory for surfaces of pre- scribed curvature (see for instance [2]), one has the following regularity result.

COROLLARY 3.3. Let f 2L^p(V), with p>N. Then, for all t2R the super-level set E_t:ˆ fu_l>tg (respectively, fu_ltg) has boundary of class C^1;a, for alla5(pÿN)=p, out of a closed singular setSof Hausdorff dimension at most Nÿ8. Moreover, if pˆ 1, the boundary of Et is of class W^2;qout ofS, for all q51, and is of class C^1;1if Nˆ2.

Before stating the main result of this section, we recall two simple Lemmata (see [11]).

LEMMA 3.4. Let U be an open set in R^Nand v2W^2;p(U), p1. We have that

div rv 1‡ jrvj² q

0 B@

1

CAˆtraceÿA(rv)D²v

a.e. in U;

where

A(j)_ij:ˆ 1

(1‡ jjj²)¹² d_ijÿ j_ij_j (1‡ jjj²)

!

j2R^N:

LEMMA3.5. Let U be an open set inR^Nand v2W^2;1(U). Assume that u has a minimum at y02U and

…8† lim

r!0‡

1 jB(y0;r)j Z

B(y0;r)

u(y)ÿu(y0)ÿ ru(y0)(yÿy0)ÿ1

2hD²v(y0)(yÿy0);yÿy0i

r² dyˆ0;

then D²v(y0)0.

Recall that, ifv2W^2;1(U), then (8) holds a.e. inU[20, Th. 3.4.2].

THEOREM1. Let f 2BV(V)\L¹(V). Then, for alll>0, J_ulJ_f

…9†

up to a set of zeroH^Nÿ1-measure. Moreover

[u_l(x)][f(x)] H^Nÿ1-a.e. onJ_ul: …10†

PROOF. Notice that (10) implies (9), so that it is enough to prove (10).

Let us first recall some consequences of Corollary 3.3. Let Et:ˆ ful >tg,t2R, and letStbe its singular set given by Corollary 3.3.

Sincef 2L¹(V), around each pointx2@EtnSt,t2R,@Et is locally the graph of a function in W^2;p for all p2[1;1) (hence in C^1;a for any a2(0;1)). LetQbe a countable dense set inRsuch thatfu_l>tgis a set of finite perimeter for any t2Q. If we let N :ˆ S

t2QS_t, we then have H^Nÿ1(N)ˆ0.

Since we can write (as in [3]) Julˆ [

t1;t22Q;t15t2

@Et1\@Et2;

in order to prove (10) it suffices to show

t₂ÿt₁[f(x)] H^Nÿ1-a.e. on@E_t1\@E_t2; …11†

for allt1;t22Qwitht15t2.

Fixt1;t22Qwitht15t2, and letB_R⁰ be the ball of radiusR>0 inR^Nÿ1 centered at 0. Let C_R:ˆB_R⁰ (ÿR;R), and denote x2C_R as xˆ(y;z)2B_R⁰ (ÿR;R). Given x2@E_t1\@E_t2n N, by Corollary 3.3 there is R>0 such that, after a suitable change of coordinates, we can write the set@Eti\CR,iˆ1;2, as the graph of a function v_i2W^2;p(B_R⁰), p2[1;1), such that xˆ(0;v_i(0))2C_RV andrv_i(0)ˆ0. Without loss of generality, we assume thatv_i>0 inB_R⁰, and thatE_tiis the supergraph of v_i. Sincet₁5t₂ we have E_t2E_t1, which gives in turnv₂v₁ in B_R⁰. We may also assume that

H^Nÿ1ÿfy2B_R⁰ :v₁(y)ˆv₂(y)g

>0:

…12†

Notice that, since @Eti is of finite H^Nÿ1-measure, we may cover

@E_t1\@E_t2n N by a countable set of such cylinders. By [3, Th. 3.108], for H^Nÿ1-a.e. y2B_R⁰ the function f(y;) belongs to BV((ÿR;R)), and the jumps off(y;) are the pointszsuch that (y;z)2Jf. Recalling thatviis a local minimizer of

minv

Z

B_R⁰



1‡ jrvj² q

dyÿ1 l

Z

B_R⁰

Z^v(y)

0

(t_iÿf(y;z))dz dyˆ:E_i(v);

by taking a positive smooth test functionc(y) witn compact support inB_R⁰, and computing

e!0lim^‡

E_i(v‡ec)ÿ E_i(v)

e 0;

we deduce that

div rv_i(y) 1‡jrv_i(y)j² q

0 B@

1 CA‡1

l…t_iÿf^‡(y;v_i(y)† 0; H^Nÿ1-a.e. inB_R⁰: …13†

In a similar way, we get

div rvi(y) 1‡jrv_i(y)j² q

0 B@

1 CA‡1

l…t_iÿf^ÿ(y;v_i(y)† 0; H^Nÿ1-a.e. inB_R⁰: …14†

Finally we observe that, sincev1;v22W^2;p(B_R⁰) for anyp2[1;1) and v2v1 in B_R⁰, by Lemma 3.5 we have that rv1(y)ˆ rv2(y) and D²(v₁ÿv₂)(y)0 H^Nÿ1-a.e. on fy2B_R⁰ :v₁(y)ˆv₂(y)g. Using both in- equalities (13) and (14) and Lemma 3.4, it follows that

05t2ÿt1lÿtrace(A(rv1(y))D²v1(y))ÿtrace(A(rv2(y))D²v2(y))

‡f^‡(y;v2(y))ÿf^ÿ(y;v1(y)) f^‡(y;v₂(y))ÿf^ÿ(y;v₁(y));

H^Nÿ1-a.e. onfy2B_R⁰ :v₁(y)ˆv₂(y)g, which gives (11) and concludes the

proof. p

4. The jump set of solutions of the Total Variation flow

Let V be an open bounded set with Lipschitz boundary. We consider the Total Variation flow

@u

@t ˆdiv Du

jDuj

in QT ˆ(0;T)V;

Du

jDujn^Vˆ0 in Q_T ˆ(0;T)@V;

…15†

with the initial condition

u(0;x)ˆf(x); x2V:

…16†

Let us recall that (15) it is theL²-gradient flow of the total variation as defined in [9]. In the general case we shall follow [4, 8]. The purpose of this Section is to prove the following result.

THEOREM 2. Let f 2L^N(V)\BV(V). Let u(t) be the solution of (15) with initial condition(16).Then u(t)2L¹(V)\BV(V)for any t>0, and

J_u(t) J_u(s)J_f 8t>s>0:

…17†

Moreover

[u(t)][u(s)][f] H^Nÿ1 a.e.8t>s>0:

…18†

The proof of Theorem 2 is based on the approach in [10] and uses the estimate (10). Let us first recall some basic facts about the operator ÿdiv Du

jDuj

inL^pspaces. Since it suffices for our purposes, we shall only

consider the case p2 N Nÿ1;1

. For any p2[1;1], let us define the space

X(V)p:ˆnz2L¹(V;R^N) :fdiv(z)2L^p(V)o :

If z2X(V)_p and w2BV(V)\L^q(V), p^ÿ1‡q^ÿ1ˆ1, we define the functional (zDw):C₀¹(V)!Rby the formula

h(zDw);Wi:ˆ ÿ Z

V

wWdivz dxÿ Z

V

w z rWdx:

Then (zDw) is a Radon measure in V, and (zDw)ˆz rw if w2W^1;1(V)\L^q(V).

Following [7], we observe that for anyz2X(V)_pthere exists a function [zn^V]2L¹(@V) satisfyingk[zn^V]k_L1(@V) kzk_L1(V;R^N), and such that for anyu2BV(V)\L^q(V) we have

Z

V

udivz dx‡ Z

V

(zDu)ˆ Z

@V

[zn^V]u dH^Nÿ1:

DEFINITION 4.1. We define the operator A_pL^p(V)L^p(V), with N

Nÿ1p 1, by:

(u;v)2 Ap if and only if u;v2L^p(V), u2BV(V) and there exists z2X(V)_pwithkzk₁1, such that(zDu)ˆ jDuj,[zn^V]ˆ0and

vˆ ÿdiv(z) inD⁰(V):

Byv2 A_puwe mean that (u;v)2 A_p. ByL¹_w((0;T);BV(V)) we denote the space of weakly measurable functionsw:[0;T]!BV(V) (i.e., the map t2[0;T]! hw(t);fiis measurable for anyf2BV(V) whereBV(V) de- notes the dual ofBV(V)) such thatR^T

0 kw(t)kdt51.

DEFINITION4.2. A function u2C([0;T];L^p(V)) is a strong solution of (15) if u2W_loc^1;1(0;T;L^p(V))\L¹_w((0;T);BV(V)) and there exists z2L¹(0;T)V;R^N

with kzk₁1 such that Z

V

(z(t)Du(t))ˆ Z

V

jDu(t)j for a.e.t>0:

…19†

[z(t)n^V]ˆ0 in@V for a.e.t>0:

…20†

and utˆdivz in D⁰…(0;T)V†:

PROPOSITION4.3. The operatorA_pis m-accretive in L^p(V), that is for any f 2L^p(V)and anyl>0there is a unique solution u2L^p(V)of the problem

u‡lApu3f: …21†

Moreover, if u1;u22L^p(V)are the solutions of(21)corresponding to the right hand sides f1;f22L^p(V), then

ku₁ÿu₂k_p kf₁ÿf₂k_p:

Moreover the domain ofA_pis dense in L^p(V)when p51.

We denote byR_lf the solution of (21) and byR^k_lf itsk-iterate, for any k1. We also recall the notion of strong solution for nonlinear semigroups generated by accretive operators.

DEFINITION4.4. A function u is called a strong solution in the sense of semigroups of

du

dt‡ A_pu30 with u(0)ˆf if

u2C([0;T];L^p(V))\W_loc^1;1((0;T);L^p(V))

u(t)2Dom(Ap) a.e. in t>0 andu⁰‡ Apu(t)30 a.e.t2(0;T) u(0)ˆf:

8>

<

>: …22†

By Crandall-Ligget's semigroup generation theorem [17], using Pro- position 4.3 as in [6, 10], one obtains the following result:

THEOREM 3. Let f 2L^p(V) if N

Nÿ1p51, or f 2Dom(A₁) if pˆ 1. Then there is a unique strong solution in the sense of semigroups u(t)ˆS(t)f :ˆ lim

l#0;kl!tR^k_lf 2C([0;T];L^p(V))of the problem du

dt ‡ A_pu30; u(0)ˆf: …23†

Moreover, the semigroup solution is a strong solution of (15) and con- versely, any strong solution of (15) is a strong solution in the sense of semigroups of(23).

REMARK4.5. Notice that givenp2[ N

Nÿ1;1] the limit lim

l#0;kl!tR^k_lfis taken inL^p(V).

To prove Theorem 2, we need the following Lemma.

LEMMA4.6. Let(u_n)_n2Nbe a sequence of functions in BV(V)\L¹(V).

Assume that J_unJ_u0, for all n2N, and u_n!u strongly in L¹(V). Then, H^Nÿ1-almost every point ofVnJu0is a Lebesgue point for u. In particular, if u2BV(V), then Ju Ju0. Moreover, if all the functions un are con- tinuous at x2V, then also u is continuous at x.

PROOF. The thesis follows observing that ifx2Vis a Lebesgue point for all the functionsun, then it is also a Lebesgue point foru, and the same is

true for a continuity point. p

PROOF OFTHEOREM2. We divide the proof into three steps.

Step 1. Assume thatf 2Dom(A₁)\BV(V). Recall thatR_lf denotes the solution of (21), and R^k_lf itsk-iterate, for anyk1. In this case, as mentioned in Remark 4.5, we know from [17] that R^k_lf !u(t) in L¹(V) when l!0^‡ and kl!t. Then the result follows from Theorem 1 and Lemma 4.6.

Step 2. Let f 2L^N(V). By Theorem 3, u2C([0;T];L^N(V)). We also know from [6] thatu(t)ˆS(t)f2C((0;T];L¹(V)) andu(t)2BV(V) for any t>0. Moreover, the following estimate is a consequence of the 0-homo- geneity of the operatorA₁ [4, 6]

d dtS(t)f

12kfk₁

t for anyt>0:

…24†

This implies thatu(t)2Dom(A1) for anyt>0. Notice that by Step 1 and Theorem 1, we know that

J_u(t) J_u(s) for allt>s>0:

…25†

Moreover, by (10) we have

[u(t)][u(s)] H^Nÿ1-a.e. onJ_u(t); for all t>s>0:

…26†

Step 3. Let f 2L^N(V)\BV(V). We shall prove that J_u(t)J_f (modulo an H^Nÿ1 null set) for anyt>0.

LetZJu(t) be such that [u(t;)]e>0 onZand letx0 be a point of H^Nÿ1-density 1 inZ.

Let us consider tn#0^‡, tn5t. By (25) we have J_u(tn)J_u(tnÿ1) and ZJ_u(tn)for alln2N(modulo aH^Nÿ1-null set). By (26) we also have

[u(t_n;x)][u(t;x)]e H^Nÿ1-a.e. onZ:

Now, observe that

Z

V

jDu(t)j Z

V

jDfj:

…27†

Since

lim sup

t!0^‡

Z

V

jDu(t)j Z

V

jDfj lim inf

t!0^‡

Z

V

jDu(t)j:

we have that

t!0lim^‡ Z

V

jDu(t)j ˆ Z

V

jDfj:

…28†

By Step 2 we know that u(t)!f inL^N(V). ThenDu(t)!Df ast!0‡ weakly* as vector measures in V. Since (28) holds, we also have that jDu(t)j ! jDfjast!0‡weakly* as measures inV([3], Proposition 1.62).

Letr>0 be such thatjDfj(@B(x₀;r))ˆ0. By [3, Proposition 1.62] we get

jDfj(B(x0;r))ˆ lim

n!1jDu(tn)j(B(x0;r))eH^Nÿ1(Z\B(x0;r)):

It follows that, for all pointsx₀ofH^Nÿ1density 1 inZ, we have lim inf

r!0^‡

jDfj(B(x₀;r)) vNÿ1r^Nÿ1 e;

wherev_Nÿ1 is the area of the (Nÿ1)-dimensional ball. This implies that ZJ_f (modulo anH^Nÿ1null set). Thus,J_u(t) J_f(modulo anH^Nÿ1null set)

for anyt>0. p

REMARK4.7. Theorem 2 still holds, with analogous proof, in the case of zero Dirichlet boundary conditions or inR^N [5, 8, 6]. Moreover, as in [10]

(see also [18]), it also holds for the anisotropic Total Variation flow, when the anisotropy is smooth and elliptic.

Acknowledgments. V. C. acknowledges partial support by MICINN project reference MTM2009-08171, by GRC reference 2009 SGR 773, and by ``ICREA AcadeÁmia'' prize for excellence in research funded both by the Generalitat de Catalunya. M. N. acknowledges partial support by the Fondazione CaRiPaRo Project ``Nonlinear Partial Differential Equations:

models, analysis, and control-theoretic problems''.

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Manoscritto pervenuto in redazione il 28 Aprile 2012.