On the Taut String Interpretation of the One-dimensional

Rudin–Osher–Fatemi Model

Niels Chr Overgaard

Centre for Mathematical Sciences, Lund University, S

olvegatan 18A, 221 00 Lund, Sweden

Keywords:

Total Variation Minimization, Taut String, Denoising, Regression Splines, Lewy–Stampacchia Inequality,

Sub-modularity, Moreau-Yosida Approximation, Isotonic Regression.

Abstract:

A new proof of the equivalence of the Taut String Algorithm and the one-dimensional Rudin–Osher–Fatemi

model is presented. Based on duality and the projection theorem in Hilbert space, the proof is strictly ele-

mentary. Existence and uniqueness of solutions (in the continuous case) to both denoising models follow as

by-products. The standard convergence properties of the denoised signal, as the regularizing parameter tends

to zero, are recalled and efﬁcient proofs provided. Moreover, a new and fundamental estimate on the denoised

signal is derived. It implies, among other things, the strong convergence (in the space of functions of bounded

variation) of the denoised signal to the in-signal as the regularization parameter vanishes.

1 INTRODUCTION

In 2017 it was 25 years ago Leonid Rudin, Stanley

Osher and Emad Fatemi proposed their now classical

model for edge-preserving denoising of images (Ru-

din et al., 1992). The present paper will investi-

gate the properties of the one-dimensional version of

the Rudin-Osher-Fatemi (ROF) model: To a given

(noisy) signal f ∈ L

(I), deﬁned on a bounded inter-

val I = (a, b), associate the (ROF) functional

(u) = λ

(x)|dx +

( f (x) − u(x))

dx ,

where λ > 0 is a parameter. Deﬁne the denoised sig-

nal as the function u

∈ BV (I) which minimizes this

energy, i.e.,

:= arg min

u∈BV (I)

(u) . (1)

The ﬁrst term in the ROF-functional is λ times the

total variation

|dx of the function u and BV (I)

denotes the set of functions on I with ﬁnite total varia-

tion. Precise deﬁnitions will be given below, in Secti-

ons 2 and 3.

The one-dimensional ROF model is compared to

the Taut string algorithm—an alternative method for

denoising of signals with applications in statistics,

non-parametric estimation, real-time communication

systems and stochastic analysis. In the continuous

setting, for analogue signals, the Taut string algorithm

can be stated in the following manner (cf. Figure 1):

Algorithm 1: The Taut String Algorithm.

INPUT: A bounded interval I = (a,b), a (noisy) signal

f ∈ L

(I) and a parameter λ > 0.

OUTPUT: The denoised signal f

∈ L

(I).

STEP 1. Compute the cumulative signal,

F(x) =

f (t)dt , x ∈

I = [a, b].

STEP 2. Set

W ∈ H

(I) : W (a) = F(a), W (b) = F(b),

and F − λ ≤ W ≤ F +λ

(The set of L

-functions with weak derivatives in L

and graphs lying within a tube around F of width λ.)

STEP 3. Compute the unique minimizer W

∈ T

(the

‘Taut string’) of the energy

min

W ∈T

E(W ) :=

(x)

dx . (2)

STEP 4. Set f

= W

(distributional derivative.)

END.

The taut string algorithm has been extensively stu-

died in the discrete setting by (Mammen and van de

Geer, 1997; Davies and Kovac, 2001) and (D

umbgen

and Kovac, 2009). Very recently, using methods from

Overgaard, N.

On the Taut String Interpretation of the One-dimensional Rudin–Osher–Fatemi Model.

DOI: 10.5220/0006720402330244

In Proceedings of the 7th International Conference on Pattern Recognition Applications and Methods (ICPRAM 2018), pages 233-244

ISBN: 978-989-758-276-9

233

interpolation theory (Peetre’s K-functional and the

notion of invariant K-minimal sets), Setterqvist has

investigated the limits to which taut string methods

may be extended (Setterqvist, 2016).

In its original formulation, the Taut string algo-

rithm instruct us to ﬁnd the solution of the shortest

path problem

min

W ∈T

L(W ) :=

1 +W

(x)

dx , (3)

hence the epithet ‘taut string’. However, the ‘stret-

ched rubber band’-energy E in step 3 of the algorithm

is not only easier to handle analytically, it also has

precisely the same solution as (3). While this is intui-

tively clear from our everyday experience with rub-

ber bands and strings, the assertion is, mathematically

speaking, not equally self-evident so a proof is offered

in Appendix A.

The main purpose of this paper, the ﬁrst of two,

is to present a new, elementary proof of the following

remarkable result:

Theorem 1. The Taut string algorithm and the ROF

model yield the same solution; f

= u

This is not new; a discrete version of this theorem

was proved in (Mammen and van de Geer, 1997) and

in (Davies and Kovac, 2001). In the continuum set-

ting, the equivalence result was explicitly stated and

proved in (Grassmair, 2007). There is also an exten-

sive treatment in (Scherzer et al., 2009, Ch. 4). In-

deed, a few years earlier (Hinterm

uller and Kunisch,

2004, p.7), in a brief (but inconclusive) remark, refer

to the close relation between the ROF model and the

Taut string algorithm.

The second main result of the paper, whose proof

we give in Section 6, is the following “fundamental”

estimate on the denoised signal:

Theorem 2. If the signal f belongs to BV (I) then, for

any λ > 0, the denoised signal u

satisﬁes the inequa-

lity

−( f

)

−

≤ u

≤ ( f

)

, (4)

where ( f

)

and ( f

)

−

denote the positive and the

negative variations, respectively, of f

(distributional

derivative).

Just like f

, the derivative u

is computed in the

distributional sense and is, in general, a signed mea-

sure. Recall that ( f

)

and ( f

)

−

are ﬁnite positive

measures satisfying f

= ( f

)

− ( f

)

−

, see e.g. (Ru-

din, 1986, Sec. 6.6). As an example, compute the

derivatives of f and u

= f

as shown in Figure 1.

The proof of the theorem is based on (an extension of)

the Lewi–Stampacchia-inequality (Lewy and Stam-

pacchia, 1970) and uses the Taut String-interpretation

of the ROF model (Theorem 1) in an essential way.

A signiﬁcant consequence of Theorem 2 is that

for an in-signal f belonging to BV (I) we get u

→ f

strongly in BV (I) as λ → 0+ . The usual Moreau–

Yosida approximation result, see e.g. (Ambrosio

et al., 2000, Ch. 17), only gives the weaker u

→ f in

(I) and

|dx →

| f

|dx as λ tends to zero.

Further contributions of the paper are: i) The re-

derivation some known properties of the ROF model

(Propositions 2) and ii) proof of some precise results

on the rate of convergence of u

→ f as λ tends to

zero Propositions 3 and 4—collecting all such result

in one place! iii) A new and slick proof of the (known)

fact that u

is a semi-group with respect to λ (Propo-

sition 8). iv) Finally we indicated how our method of

proof can be modiﬁed and applied to the problem of

isotonic regression.

The paper is based on the author’s preprint (Over-

gaard, 2017) and is completely theoretical. All exam-

ples, including the ones in the ﬁgures, are computed

by hand using the taut string interpretation. Howe-

ver, we predict that the theory developed can be used

to construct new fast non-iterative algorithms for de-

noising using the ROF-model or, at least, can be used

to shed new light on existing such algorithms such

as (Condat, 2013).

2 OUR ANALYSIS TOOLBOX

Throughout this paper I denotes an open, bounded

interval (a, b), where a < b are real numbers, and

I = [a, b] is the corresponding closed interval.

(I) denotes the space of continuously differen-

tiable (test-)functions ξ : I → R with compact support

in I, and C(

I) is the space of continuous functions on

the closure of I.

For 1 ≤ p ≤ ∞, L

(I) denotes the Lebesgue space

of measurable functions f : I → R with ﬁnite p-norm;

k f k



| f (x)|



1/p

< ∞, when p is ﬁnite,

and k f k

∞

= esssup

x∈I

| f (x)| < ∞ when p = ∞. The

space L

(I) is a Hilbert space with the inner product

h f , gi = h f ,gi

(I)

f (x)g(x) dx and the corre-

sponding norm k f k :=

h f , f i

(I)

= k f k

We are going to need the Sobolev spaces over L

(I) =



u ∈ L

(I) : u

∈ L

(I)



were u

denotes the distributional derivative of u.

This is a Hilbert space with inner product hu, vi

hu,vi + hu

i and norm kuk

= (ku

+ kuk

)

1/2

Any u ∈ H

(I) can, after correction on a set of mea-

sure zero, be identiﬁed with a unique function in C(

I).

In particular, a unique value u(x) can be assigned to u

for every x ∈

ICPRAM 2018 - 7th International Conference on Pattern Recognition Applications and Methods

234

−1

(a) The input signal f .

−λ

F +λ

F −λ

(b) The cumulative signal F and the tube T

−λ

−1

f = F

= W

(d) The denoised signal f

together with the input sig-

nal.

Figure 1: A graphical illustrations of the steps in the Taut string algorithm applied to a piecewise constant signal.

The following subspace of H

(I) plays an impor-

tant role in our analysis:

(I) =



u ∈ H

(I) : u(a) = 0 and u(b) = 0



Here hu,vi

(I)

(x)v

(x)dx deﬁnes an inner

product on H

(I) whose induced norm kuk

(I)

is equivalent to the norm inherited from H

(I).

Finally, let H be a (general) real Hilbert space with

inner product between u, v ∈ H denoted by hu,vi and

the corresponding norm kuk =

hu,ui. The follo-

wing result is standard (Br

ezis, 1999, Th

eor

eme V.2):

Proposition 1 (Projection Theorem). Let K ⊂ H be

a non-empty closed convex set. Then for every ϕ ∈ H

there exists a unique point u ∈ K such that

kϕ − uk = min

v∈K

kϕ − vk.

Moreover, the minimizer u is characterized by the fol-

lowing property:

u ∈ K and hϕ − u,v − ui ≤ 0, for all v ∈ K.

The point u is called the projection of ϕ onto K, and

is denoted u = P

(ϕ).

3 PRECISE DEFINITION OF THE

ROF MODEL

The expression

|dx for the total variation, makes

sense for u ∈ H

(I) but is otherwise merely a conve-

nient symbol. A more general and precise deﬁnition

is needed; one which works when u

does not exist

in the classical sense. The standard way to deﬁne the

total variation is via duality: For u ∈ L

(I) set

J(u) = sup

u(x)ξ

(x)dx :

ξ ∈ C

(I), kξk

∞

≤ 1

. (5)

If J(u) < ∞, u is said to be a function of bounded

variation on I, and J(u) is called the total variation

of u (using the same notation as (Chambolle, 2004)).

The set of all integrable functions on I of bounded

variation is denoted BV (I), that is, BV (I) =



u ∈

(I) : J(u) < ∞



. This becomes a Banach space

when equipped with the norm kuk

:= J(u)+kuk

Notice that, as already mentioned, if u ∈ H

(I) then

J(u) =

|dx < ∞, so u ∈ BV (I).

Let us illustrate how the deﬁnition works for a

function with a jump discontinuity:

On the Taut String Interpretation of the One-dimensional Rudin–Osher–Fatemi Model

235

Example 1. Let u(x) = sign(x) on the interval I =

(−1,1). For any ξ ∈ C

(I), satisfying |ξ(x)| ≤ 1 for

all x ∈ I, we have

−1

u(x)ξ

(x)dx

(x)dx −

−1

(x)dx = −2ξ(0) ≤ 2,

where equality holds for any admissible ξ which

satisﬁes ξ(0) = −1. So J(u) = 2 and u ∈ BV (I), as

predicted by intuition.

In this example the supremum is attained by many

choices of ξ. This is not always the case; if u(x) = x

on I = (0, 1) then J(u) = 1, but the supremum is not

attained by any admissible test function.

The following lemma shows that the deﬁnition of

the total variation J and the space BV (I) can be moved

to a Hilbert space-setting involving L

and H

Lemma 3. Every u ∈ BV (I) belongs to L

(I) and

J(u) = sup

ξ∈K

hu,ξ

(I)

, (6)

where K = {ξ ∈ H

(I) : kξk

∞

≤ 1 }, which is a closed

and convex set in H

(I).

Proof. If u ∈ BV (I) then Sobolev’s lemma for functi-

ons of bounded variation, see (Ambrosio et al., 2000,

p. 152), ensures that u ∈ L

∞

(I). This in turn implies

u ∈ L

(I) because I is bounded. The (ordinary) So-

bolev’s lemma asserts that H

(I) is continuously em-

bedded in L

∞

(I). Since K is the inverse image under

the embedding map of the unit ball in L

∞

(I), which is

both closed and convex, we draw the conclusion that

K is closed and convex in H

It only remains to prove (6). Clearly J(u) cannot

exceed the right hand side because the set {ξ ∈ C

(I) :

kξk

∞

≤ 1 } is contained in K. To verify that equality

holds it is enough to prove the inequality

hu,ξ

(I)

≤ J(u)kξk

∞

, for all ξ ∈ H

(I), (7)

as it implies that the right hand side of (6) cannot ex-

ceed J(u). To do this, we ﬁrst notice that the ine-

quality in holds for all ζ ∈ C

(I). This follows by

applying homogeneity to the deﬁnition of J(u). Se-

condly, if ξ ∈ H

(I) we can use that C

(I) is dense in

(I) and ﬁnd functions ζ

∈ C

(I) such that ζ

→ ξ

in H

(I) (and in L

∞

(I) by the continuous embedding).

It follows that

hu,ξ

(I)

= lim

n→∞

hu,ζ

(I)

≤ J(u) lim

n→∞

kζ

∞

= J(u)kξk

∞

which establishes (7) and the proof is complete.

The inequality (7) combined with the Riesz repre-

sentation theorem (cf. e.g. (Ambrosio et al., 2000,

Thm. 1.54)) implies that the distributional derivative

of u ∈ BV (I) is a signed (Radon) measure µ on I,

and that we may write hu,ξ

(I)

ξdµ. This will

be useful later on.

We can now give the precise deﬁnition of the ROF

model: For any f ∈ L

(I) and any real number λ > 0

the ROF functional is the function E

: BV (I) → R

given by

(u) = λJ(u) +

k f − uk

(I)

. (8)

Denoising according to the ROF model is the map

(I) 3 f 7→ u

∈ BV (I) deﬁned by (1). To empha-

sise the role of the in-signal f we sometimes write

( f ;u) instead of E

(u). Well-posedness of the ROF

model is demonstrated in the next section.

4 EXISTENCE THEORY FOR

THE ROF MODEL

We begin with a simple observation: if u ∈ BV (I)

then J(u + c) = J(u) for any real constant c. This

property of the total variation has two important con-

sequences. First of all, E

( f ;u) = E

( f − c; u − c)

for any constant c. Taking c to be the mean value

of f shows that we may assume, as we do throug-

hout this paper, that the in-signal satisﬁes

f dx = 0.

This assumption implies that the cumulative signal

F(x) satisﬁes F(a) = F(b) = 0, hence F ∈ H

(I).

This plays an important role in our analysis. Se-

condly, since f has mean value zero, it is enough to

minimize E

over the subspace of BV (I) consisting

of functions with mean value zero. To see this, let

P be the orthogonal projection (in L

(I)) onto this

subspace. An easy computation yields the identity

(Pu) = E

(u) −

ku − Puk

, which shows that u

can be a minimizer of E

only if it belongs to the

range of P.

The following theorem contains the key result of

our paper.

Theorem 4. We have the equality

min

u∈BV (I)

(u) = max

ξ∈K

k f k

(I)

− k f − λξ

(I)

(9)

with the minimum achieved by a unique u

∈ BV (I)

and the maximum by a unique ξ

∈ K. The two functi-

ons are related by the identity

= f − λξ

, (10a)

and satisfy

J(u

) = hu

,ξ

(I)

. (10b)

ICPRAM 2018 - 7th International Conference on Pattern Recognition Applications and Methods

236

Moreover, if u

6= 0, then kξ

∞

= 1. Conversely, the

conditions (10a) and (10b) characterizes the solution;

if a pair of functions ¯u ∈ BV (I) and

ξ ∈ K satisfy ¯u =

f −λ

and J( ¯u) = h ¯u,

(I)

, then ¯u = u

and

ξ = ξ

This result is a special instance of the Fenchel–

Rockafellar theorem, see e.g. (Br

ezis, 1999, p. 11). It

is tailored with our speciﬁc needs in mind and will be

proved with our bare hands using the projection theo-

rem. The general version is used in (Hinterm

uller and

Kunisch, 2004) in their analysis of the multidimensi-

onal ROF model (with the ‘Manhattan metric’). The

equality (9) has played an important role in the de-

velopment of numerical algorithms for total variation

minimization, both directly, as for instance in (Zhu

et al., 2007) or, indirectly, as in (Chambolle, 2004).

Before the proof starts, let us remind the reader of

the following general fact: If M and N are arbitrary

non-empty sets and Φ : M × N → R is any real valued

function, then it is easy to check that

inf

x∈M

sup

y∈N

Φ(x,y) ≥ sup

y∈N

inf

x∈M

Φ(x,y) , (11)

is always true. The use of inf’s and sup’s are impor-

tant, as neither the greatest lower bounds nor the least

upper bounds are necessarily attained.

Proof. Since E

(u) = sup

ξ∈K

λhu,ξ

i +

k f − uk

follows from (11) that

inf

u∈BV (I)

(u) ≥ sup

ξ∈K

inf

u∈BV (I)

λhu,ξ

i +

ku − f k

We ﬁrst solve, for ξ ∈ K ﬁxed, the minimization pro-

blem on the right hand-side. Expanding k f − uk

and

completing squares with respect to u yields:

λhu,ξ

i +

ku − f k

ku − ( f − λξ

− k f − λξ

+ k f k

The right hand-side is clearly minimized by the L

(I)-

function u = f − λξ

and

inf

u∈BV (I)

(u) ≥ sup

ξ∈K

k f k

− k f − λξ

(12)

holds. The maximization problem on the right hand

side is equivalent to

inf

ξ∈K

k f − λξ

k = inf

ξ∈K

− λξ

= λ inf

ξ∈K

kλ

−1

F − ξk

(I)

. (13)

By Proposition 1, this problem has the unique solution

= P

(λ

−1

F) ∈ K, so the supremum is attained in

(12). Now, let the function u

be deﬁned by (10a) in

the theorem. A priori, u

belongs to L

(I), but we are

going to show that u

∈ BV (I): The characterization

of ξ

according in the projection theorem states that

∈ K and h f − λξ

,λξ

− λξ

i ≤ 0 for all ξ ∈ K. If

we use the deﬁnition of u

and divide by λ > 0 this

characterization becomes

,ξ

i ≤ hu

,ξ

i for all ξ ∈ K,

where the right hand-side is ﬁnite. It follows from the

deﬁnition of the total variation that u

∈ BV (I) with

J(u

) = hu

,ξ

i, as asserted in the theorem. (This

reasoning can be reversed; if (10b) is true then ξ

the minimizer in (13).) Also, if u

6= 0 then kξ

∞

< 1

is not consistent with the maximizing property (10b),

hence kξ

∞

= 1, as claimed.

It remains to be veriﬁed that u

minimizes E

and

that equality holds in (12). This follows from a direct

calculation:

inf

u∈BV (I)

(u) ≥ max

ξ∈K

k f k

− k f − λξ

k f k

−

k f k

− ku

k f k

− hu

, f − λξ

k f − u

+ hu

,λξ

k f − u

+ λJ(u

)

= E

) .

So infE

(u) = E

), the inﬁmum is attained, and

equality holds in (12). The inequality E

(u) −

) ≥

ku − u

implies the uniqueness of u

The converse statement is proved by back-tracking

the steps of the above proof.

Denoising is a non-expansive mapping:

Corollary. If f and

f are signals in L

(I) and the

corresponding denoised signals are denoted u

and

˜u

, respectively, then k ˜u

− u

(I)

≤ k

f − f k

(I)

This is a special instance of a more general result

about Moreau–Yosida approximation (or of the proxi-

mal map), see (Attouch et al., 2015, Theorem 17.2.1).

However, the result is easily veriﬁed by the reader

using the characterization of the ROF-minimzer given

in the theorem.

The equivalence of the two denoising models can

now be established:

On the Taut String Interpretation of the One-dimensional Rudin–Osher–Fatemi Model

237

Proof of Theorem 1. It follows from Theorem 4 that

the minimizer u

of the ROF functional is given by

= f − λξ

where ξ

is the unique solution of

min

ξ∈K

k f − λξ

(I)

. (14)

If we introduce the new variable W := F − λξ, where

F ∈ H

(I) is the cumulative signal, then W ∈ H

(I)

and the condition kξk

∞

≤ 1 implies that W satis-

ﬁes F(x) − λ ≤ W (x) ≤ F(x) + λ on I. Therefore

(14) is equivalent to min

W ∈T

(1/2)kW

(I)

, which

is the minimization problem in step 3 of the Taut

string algorithm whose solution we denoted W

. It

follows that W

= F − λξ

and differentiation yields

= W

= f − λξ

= u

, the desired result.

It is interesting to note that Theorem 4 associates

a unique test function (or ‘dual variable’) ξ

∈ K

with the solution u

of the ROF model such that

J(u

) = hu

,ξ

, in particular if we compare to

the situation in Example 1. A concrete case looks as

follows:

Example 2. Let f (x) = sign(x) be the step function

deﬁned on I = (−1, 1). An easy calculation, based on

the Taut string interpretation, shows that if 0 < λ < 1

then u

= (1 − λ) sign(x) and ξ

= |x| − 1 ∈ H

(I).

Here ξ

is not in C

(I), so the extension of the space

of test functions from C

to H

is essential to our the-

ory. For λ ≥ 1 we ﬁnd u

= 0 and ξ

= λ

−1

(|x| − 1).

Notice that kξ

∞

= 1 when u

6= 0.

Our proof of Theorem 1 is essentially a change of

variables and, as such, becomes almost a ‘derivation’

of the taut string interpretation. We also get the ex-

istence and uniqueness of solutions to both models in

one stroke. The proof given in (Grassmair, 2007) ﬁrst

shows that u

and W

satisfy the same set of three ne-

cessary conditions, and that these conditions admit at

most one solution. Then it proceeds to drive home

the point by establishing existence separately for both

models. The argument assumes f ∈ L

∞

and involves a

fair amount of measure theoretic considerations. The

proof of equivalence given in (Scherzer et al., 2009)

is based on a thorough functional analytic study of

Meyer’s G-norm and is not elementary.

5 CONSEQUENCES OF THE

EQUIVALENCE RESULT

We now prove some known, and some new, properties

of the ROF model.

The Taut string algorithm suggests that W

= 0,

and therefore u

= 0, when λ is sufﬁciently large, and

that W

must touch the sides F ±λ of the tube T

when

λ is small. These assertions can be made precise:

Proposition 2. (a) The denoised signal u

= 0 if and

only if λ ≥ kFk

∞

, and

(b) if 0 < λ < kFk

∞

then kF −W

∞

= λ.

∞

= max(0, kFk

∞

− λ).

The results (a) and (b) are well-known and proofs,

valid in the multi-dimensional case, can be found in

Meyer’s treatise (Meyer, 2000). The natural estimate

in (c) seems to be stated here for the ﬁrst time. Notice

that the maximum norm kFk

∞

of the cumulative sig-

nal F coincides, in one dimension, with the Meyer’s

G-norm k f k

∗

of the signal f . Theorem 4 and the

taut string interpretation of the ROF model allow us

to give very short and direct proofs of all three pro-

perties.

Proof. (a) By Theorem 1, the denoised signal u

zero if and only if the taut string W

is zero. We know

that W

= F −λξ

where, as seen from (13), ξ

is the

projection in H

(I) of λ

−1

F onto the closed convex

set K. Therefore u

= 0 if and only if λ

−1

F ∈ K, that

is, if and only if kFk

∞

≤ λ, as claimed.

(b) If 0 < λ < kFk

∞

then u

6= 0 hence kξ

∞

= 1,

by Theorem 4. The assertion now follows by taking

norms in the identity λξ

= F −W

∞

be-

cause W

= 0 by (a). When c := kFk

∞

− λ > 0 we

use a truncation argument: If W belongs to T

then

so does

W := min(c,W ), in particular c > 0 ensu-

res that

W (a) =

W (b) = 0. Since E(

W ) ≤ E(W ),

and W

is the (unique) minimizer of E over T

, we

conclude that max

≤ c. A similar argument gi-

ves −min

≤ c. Thus kW

∞

≤ max(0, kFk

∞

−λ).

The reverse inequality follows from (b).

Now deﬁne, for λ > 0, the value function

e(λ) := inf

u∈BV (I)

(u),

that is, e(λ) = E

). The next two theorems con-

tains essentially well-known results.

Proposition 3. The function e : (0,+∞) → (0,+∞)

is nondecreasing and concave, hence continuous, and

satisﬁes e(λ) = k f k

/2 for λ ≥ kFk

∞

. Moreover, for

f ∈ L

(I)

lim

λ→0+

e(λ) = 0.

and if f ∈ BV (I) then e(λ) = O(λ) as λ → 0+.

Proof. If λ

≥ λ

> 0 then the inequality E

(u) ≥

(u) holds trivially for all u. Taking inﬁmum over

ICPRAM 2018 - 7th International Conference on Pattern Recognition Applications and Methods

238

the functions in BV (I) yields e(λ

) ≥ e(λ

), so e is

nondecreasing.

For any u the right hand side of the inequality

e(λ) ≤ E

(u) = λJ(u) +

ku − f k

is an afﬁne, and therefore a concave, function of

λ. Because the inﬁmum of any family of concave

functions is again concave, it follows that e(λ) =

inf

u∈BV (I)

(u) is concave.

For λ ≥ kFk

∞

we know from the previous theorem

that u

= 0, so e(λ) = E

(0) = k f k

/2.

To prove the assertion about e(λ) as λ tends to

zero from the right, we ﬁrst assume that f ∈ BV (I), in

which case it follows that 0 < e(λ) ≤ E

( f ) = λJ( f ),

so e(λ) = O(λ) because J( f ) < ∞.

If we merely have f ∈ L

(I) an approximation ar-

gument is needed: For any ε > 0 take a function f

∈

(I) such that k f − f

/2 < ε. Then f

∈ BV (I)

and 0 ≤ e(λ) ≤ E

( f

) < λJ( f

) + ε. It follows that

0 ≤ lim sup

λ→0+

e(λ) < ε. Since ε is arbitrary, we get

lim

λ→0+

e(λ) = 0.

The ﬁrst part of next the proposition is a special

instance of a much more general result, see (Attouch

et al., 2015, Theorem 17.2.1). The second part con-

tains a quantiﬁcation of the rate of convergence which

is not easily located in the literature.

Proposition 4. For any f ∈ L

(I) we have u

→ f in

as λ → 0+. Moreover, if f ∈ BV (I) then ku

−

f k

(I)

= o(λ

1/2

) and J(u

) → J( f ) as λ → 0+.

Proof. The obvious inequality k f − u

/2 ≤ e(λ)

and the fact lim

λ→0+

e(λ) = 0, proved above, implies

the ﬁrst assertion. When f ∈ BV (I) it follows from the

inequality λJ(u

− f k

(I)

= e(λ) ≤ E

( f ) =

λJ( f ) that

− f k

(I)

≤ 2λ(J( f ) − J(u

)) . (15)

Consequently ku

− f k

(I)

= O(λ) and J(u

) ≤ J( f )

for all λ > 0. But we can do slightly better than

that. Since u

→ f in L

as λ → 0+, we get J( f ) ≤

liminf

λ→0+

J(u

), by the lower semi-continuity of the

total variation J, cf. (Ambrosio et al., 2000). Since

J(u

) ≤ J( f ) we also obtain an estimate from be-

low: lim sup

λ→0+

J(u

) ≤ J( f ). We conclude that

lim

λ→0+

J(u

) = J( f ). If this is used in (15) we ﬁnd

that ku − f k

(I)

= o(λ) as λ → 0+.

6 PROOF AND APPLICATIONS

OF THEOREM 2

We begin with the proof of the fundemental estimate

on the derivative of the denoised signal:

Proof of Theorem 2. The estimate (4) is a conse-

quence of the extension to bilateral obstacle pro-

blems of the original Lewy–Stampacchia inequa-

lity (Lewy and Stampacchia, 1970) which we ex-

plain here. The bilateral obstacle problem, in the

one-dimensional setting, is to minimize the energy

E(u) :=

(x)

dx in (2) over the closed convex

set C = {u ∈ H

(I) : φ(x) ≤ u(x) ≤ ψ(x) a.e. I}. The

obstacles are functions φ,ψ ∈ H

(I) which satisfy the

conditions φ < ψ on I, and φ < 0 < ψ on ∂I = {a,b}.

This ensures that C is nonempty.

Suppose φ

and ψ

are in BV (I), such that φ

and ψ

are signed measures, then the solution u

min

u∈C

E(u) satisﬁes the following inequality (as me-

asures)

−(φ

)

−

≤ u

≤ (ψ

)

. (16)

Here the notation µ

and µ

−

is used to denote the po-

sitive and negative variation, respectively, of a signed

measure µ. This is the generalization of the Lewy–

Stampacchia inequality, proof of which can be found

in Appendix B. This proof is based on the abstract

proof, valid in a much more general setting, given

in (Gigli and Mosconi, 2015). The assumption of our

theorem, that f ∈ BV (I), implies that F

= f

is a

signed measure. If we apply (16) with φ = F − λ and

ψ = F +λ then we ﬁnd that the taut string W

satisﬁes

−(F

)

−

≤ W

≤ (F

)

The estimate (4) follows if we use the identities F

f and W

= u

into the above inequality.

Having established Theorem 2 we are able to

prove the following result about the strong conver-

gence in BV (I) of the ROF-minimizer as the regulari-

zation weight approaches zero.

Proposition 5. If f ∈ BV (I) then

J( f − u

) = J( f ) − J(u

). (17)

In particular, both J( f − u

) and k f − u

tend to

zero as λ → 0+.

Proof. The measures ( f

)

and ( f

)

−

are concentra-

ted on disjoint measurable sets (Hahn decomposition,

see (Rudin, 1986, Sec. 6.14)), so Proposition 2 im-

plies the pair of inequalities, 0 ≤ (u

)

≤ ( f

)

and

0 ≤ (u

)

−

≤ ( f

)

−

. A direct calculation, using the

fact that J(v) = (v

)

(I) + (v

)

−

(I) for any function

v ∈ BV (I), yields

J( f −u

) = ( f

− u

)

(I) + ( f

− u

)

−

(I)

= ( f

)

(I) − (u

)

(I) + ( f

)

−

(I) − (u

)

−

(I)

= J( f ) − J(u

where the right hand-side tends to zero as λ → 0+, by

Proposition 4.

On the Taut String Interpretation of the One-dimensional Rudin–Osher–Fatemi Model

239

Theorem 2 also implies the ﬁrst part of

Proposition 6. If f is piecewise constant function on

I, then so is u

for all λ > 0. Moreover, there exists a

number

λ > 0 and a piecewise linear function

ξ ∈ K

such that ξ

ξ for all λ, 0 < λ ≤

λ.

We only give the proof of the ﬁrst part of this theo-

rem, which is simple, and omit the proof of the second

part, which is rather lengthy.

Proof. If f is piecewise constant then there exists

nodes a = x

< x

< . .. < x

N−1

< x

= b which

partitions the interval I = (a,b] into N subintervals

= (x

i−1

] such that f equals the constant value

∈ R on I

for i = 1, ...,N. That is,

f =

∑

i=1

where, as usual, χ

denotes the characteristic function

of the set A. The derivative of the signal becomes

∑

N−1

i=1

( f

i+1

− f

)δ

, with δ

denoting the Di-

rac measure supported at x, and therefore J( f ) =

∑

N−1

i=1

| f

i+1

− f

| < ∞. Therefore f belongs to BV (I)

and Theorem 2 may be applied:

N−1

∑

i=1

min{0, f

i+1

− f

}δ

= −( f

)

−

≤u

≤ ( f

)

N−1

∑

i=1

max{0, f

i+1

− f

}δ

This estimate shows that u

∑

N−1

i=1

(λ)δ

where

the real numbers c

(λ) satisfy 0 ≤ c

(λ) · ( f

i+1

−

)

−1

≤ 1 for i = 1,..., N − 1. Since the deriva-

tive is zero except at a ﬁnite set of points we draw

the conclusion that u

is piecewise constant u

∑

i=1

)

with nodes contained in the node set of

f . (The latter is the “edge-preserving” property of the

one-dimensional ROF model.) Once the c

(λ)’s are

known the N real numbers (u

)

may be determined

from the N linear equations (u

)

i+1

− (u

)

= c

(λ),

i = 1,... ,N −1, and

∑

i=1

)

−x

i−1

) =

dx =

f dx = 0.

The latter half of the above proposition can be pro-

ved by “guessing” the the dual variable ξ

—it must be

a continuous piecewise linear function with the same

nodes as f —and then use the characterization of so-

lutions from Theorem 4. This result is mentioned be-

cause it implies what is possibly the strongest imagi-

nable approximation result:

Proposition 7. If f is piecewise constant function,

then k f − u

(I)

= O(λ), λ → 0+.

Proof. We know from (15) that (1/2)k f − u

(I)

≤

λ(J( f ) − J(u

)) so an estimate of the difference

J( f ) − J(u

) is needed. By Theorem 4, J(u

) =

,ξ

(I)

. Since ξ

ξ when λ is close to zero

it follows that

J( f ) = lim

λ→0+

J(u

) = lim

λ→0+

(I)

= h f ,

(I)

Moreover, the scalar product of u

= f −λξ

and ξ

is J(u

) = hu

i = h f − λ

i = J( f ) − λk

Hence J( f ) − J(u

) = O(λ), λ → 0+.

Our interest in the various limits as λ → 0+ is mo-

tivated by the fact that λ 7→ u

is a semi-group; state-

ments about limits at λ = 0 can be translated to limits

at any λ > 0.

Proposition 8 (Semi-group property). Let f ∈ L

(I).

With the convention (mentioned above) that u

= f

the formula

)

= u

λ+µ

holds for all λ,µ ≥ 0.

Here we have tweaked the notation slightly to

make the statement more compact: By using the let-

ter u in place of f for the in-signal, the operation of

denoising, for some λ > 0, is indicated by adding the

subscript ‘λ’ to u, thus obtaining u

. This makes sense

even for λ = 0 if we agree to set u

= u.

A proof of the semi-group property can be found

in (Scherzer et al., 2009). However, the fundamen-

tal estimate in Theorem 2 and the characterization of

the ROF-minimizer in Theorem 4 allow us to present

short and very direct proof of this result:

Proof. The assertion holds trivially if either λ or µ

equals zero, so we may assume that λ,µ > 0. The

idea of the proof is then to set ¯u = (u

)

and show

that there exists a function

ξ ∈ K such that

(

¯u = f − (λ + µ)

and

J( ¯u) = h ¯u,

The characterization of solutions to the ROF model

in Theorem 4 then implies that ¯u equals u

λ+µ

. Since

and ¯u are the ROF-minimizers of E

( f ;·) and

;·), respectively, they both satisfy the conditi-

ons (10a) and (10b), that is

(

= f − λξ

J(u

) = hu

,ξ

and

(

¯u = u

− µ

J( ¯u) = h ¯u,

for a uniquely determined pair of functions ξ

and

in K. Now, if we set

ξ =

λξ

+ µ

λ + µ

ICPRAM 2018 - 7th International Conference on Pattern Recognition Applications and Methods

240

then

ξ ∈ K because it is the convex combination of

two elements of K. Using what is known about u

and ¯u, the following calculation reveals why we make

this deﬁnition of

ξ, in fact f − (λ + µ)

= f − λξ

−

= u

− µ

= ¯u, hence ¯u and

ξ fulﬁl the condition

(10a) by construction. It remains to verify that (10b)

is fulﬁlled as well. Since

h ¯u,

i =

λ + µ

h ¯u,ξ

i +

λ + µ

h ¯u,

λ + µ

h ¯u,ξ

i +

λ + µ

J( ¯u) .

we see that the second condition follows if it can

show that h ¯u,ξ

i = J( ¯u). This essentially follows

from the identity in Proposition 5 which states that

J( ¯u) = J(u

) − J(u

− ¯u). In fact, using this identity

we get the inequality

J( ¯u) ≤ J(u

) − hu

− ¯u,ξ

= J(u

) − J(u

) + h ¯u,ξ

i = h ¯u,ξ

But J( ¯u) ≥ h ¯u,ξ

i for all ξ ∈ K, so J( ¯u) = h ¯u,ξ

i, and

the proof is complete.

The last part of the proof yields

Corollary. If λ > 0 then J(u

) = hu

,ξ

(I)

for all

µ, 0 < µ ≤ λ.

That is, the total variation of u

can be computed

by taking inner product with any of the previous ξ

’s.

7 APPLICATION TO ISOTONIC

REGRESSION

We illustrates the usefulness of our approach by

brieﬂy outlining (without proofs) how the theory de-

veloped earlier can be modiﬁed in order to derive the

so-called “lower convex envelope” interpretation of

the solution to the problem of isotonic regression. Iso-

tonic regression is a method from mathematical statis-

tics used for non-parametric estimation of probability

distributions, see for instance (Anevski and Soulier,

2011). It is a least-squares problem with a mono-

tonicity constraints: given f ∈ L

(I), ﬁnd the non-

decreasing function u

↑

∈ L

(I) which solves the mi-

nimization problem,

min

u∈L

↑

(I)

ku − f k

(I)

, (18)

where L

↑

(I) denotes the set of all non-decreasing

functions in L

(I). The “lover convex envelope” in-

terpretation is shown for a piecewise constant signal

f in Fig. 2.

The idea is to re-formulate (18) as an unconstrai-

ned optimization problem by replacing the total va-

riation term J of the ROF functional by regulariza-

tion term J

↑

which can distinguish between functions

that are non-decreasing or not. To achieve this we set



ξ ∈ H

(I) : ξ (x) ≥ 0 for all x ∈ I



and deﬁne

↑

(u) = sup

ξ∈K

hu,ξ

(I)

. It can be shown that

↑

(u) =

(

0 if u ∈ L

↑

(I) ,

+∞ otherwise.

The isotonic regression problem (18) now becomes

equivalent to ﬁnding the minimizer u

↑

in L

(I) of the

functional

↑

(u) := J

↑

(u) +

ku − f k

(I)

. (19)

Notice that there is no need for a positive weight in

this functional because the regularizer assumes only

the values zero and inﬁnity.

Again we may assume the mean value f to be zero

so that the cumulative function F belongs to H

(I).

Mimicking the proof of Theorem 4 we get:

min

u∈L

(I)

↑

(u) = max

W ∈T

k f k

−

(I)

where W = F − ξ, ξ ∈ K

, and T = {W ∈

(I) : W (x) ≤ F(x), x ∈ I}. The minimiza-

tion of (19) is equivalent to the obstacle pro-

blem min

W ∈T

(I)

which admits a unique

solution W

↑

by the Projection theorem. It fol-

lows that (19) also has the unique solution u

↑

(distributional derivative) which belongs to

↑

(I) because E

↑

) is ﬁnite.

The solution W

↑

of the obstacle problem satisﬁes

↑

≥ 0 (this is the ‘easy’ part of the original Lewy-

Stampacchia inequality, 0 ≤ W

↑

≤ (F

)

) and is the-

refore automatically a convex function. In fact, by

optimality, W

↑

is the maximal convex function lying

below F, i.e., it is the lower convex envelope of F.

Similar problems are considered in the multidimen-

sional case, using higer-order methods (the space of

functions with bounded Hessians), in Hinterberger

and Scherzer (Hinterberger and Scherzer, 2006).

8 CONCLUDING REMARKS

We have developed the theory for the one-

dimensional ROF model in the continuous setting for

a quite general class of signals and proved several

properties of solution of the model including a use-

ful fundamental estimate, Theorem 2, on the denoi-

sed signal. The theory may ﬁnd practical applicati-

ons in signal processing and image analysis alike. In-

deed, by using the fundamental estimate we saw in

On the Taut String Interpretation of the One-dimensional Rudin–Osher–Fatemi Model

241

↑

= W

↑

(a) The piecewise constant input signal f and the mono-

tonic solution u

↑

to the isotonic regression problem.

a b

↑

(b) The cumulative signal F and the corresponding

lower convex envelope (or taut string) W

↑

Figure 2: A graphical illustrations of the taut string interpretation of isotonic regression.

Proposition 6 how application of the ROF model to

a piecewise constant signal leads to a piecewise con-

stant denoised signal with the same nodes (i.e. the

model is “edge-preserving”). This observation, toget-

her with the semi-group property, immediately sug-

gests a (perhaps not so efﬁcient) non-iterative algo-

rithm for the computation of the denoising of a piece-

wise constant signal which is different from the one

in (Condat, 2013). If very fast non-iterative schemes

for ﬁnding the solution to the one-dimensional ROF

model can be devised then, as already indicated by L.

Condat, efﬁcient iterative algorithms for image denoi-

sing using the two-dimensional ROF model (with the

‘Manhattan metric’ as measure of the image gradient

magnitude) may be constructed as well. We hope to

return to this topic in the future.

ACKNOWLEDGEMENTS

The author whishes to thank Viktor Larsson at the

Centre for Mathematical Sciences, Lund University,

for reading and commenting on several drafts of this

paper.

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APPENDIX A

As promised in the introduction, we are going to

prove that the solution of the minimization problem

(2) in STEP 3 of the Taut string algorithm coincides

with the solution of the shortest path problem (3). In

fact we prove the slightly more general statement:

Proposition. Let H denote any strictly convex C

function deﬁned on R and set

(W ) =

H(W

(x))dx .

Then the problem min

W ∈T

(W ) has precisely

the same solution as the minimization problem

min

W ∈T

E(W ) in (2).

The case we need in our analysis follows by taking

H(s) = (1 + s

)

1/2

Proof. The idea of the proof is to verify that W

argmin

W ∈T

E(W ) solves the variational inequality:

h(W

(x))(W

(x) −W

(x)

)dx ≥ 0 , for all W ∈ T

(20)

where h = H

. This condition is both necessary and

sufﬁcient for W

to be a minimizer of L

over T

and since L

is a strictly convex functional, there is

at most one such minimizer.

Being the minimizer of E over T

, W

∈ T

satis-

ﬁes the variational inequality (which is a special case

of (20) if we take H(s) = s):

−W

)dx ≥ 0, for all W ∈ T

. (21)

Set C

= {x ∈ I : W

(x) = F(x) + λ} and C

−

= {x ∈

I : W

(x) = F(x) − λ}. These are the sets where the

solution touches the upper and the lower obstacles,

respectively. Since F and W

are continuous, both sets

are closed. In fact, C

and C

−

are compact because

λ > 0 implies that they do not reach the boundary of

I. They are disjoint, C

∩ C

−

0, and their union,

C = C

∪C

−

, is the contact set for W

For any non-negative ξ ∈ C

(I\C

) there exists an

ε > 0 such that W := W

+ εξ belongs to T

. If this W

is substituted into (21) we ﬁnd that

dx ≥ 0 for all ξ ∈ C

(I\C

) with ξ ≥ 0.

It follows that −W

is a positive measure on I\C

hence −W

is non-decreasing on each connected

component of I\C

. Similarly one proves that −W

is non-increasing on each connected component of

I\C

−

. This means, in particular, that W

constant on

each connected component of I\C.

Since h is non-decreasing, the composite function

−h(W

) has the same monotonicity properties as

−W

. Therefore the distributional derivative −h(W

)

is a positive measure µ

on I\C

and minus a posi-

tive measure −µ

−

on I\C

−

. Clearly supp µ

⊂ C

−

and supp µ

−

⊂ C

, so −h(W

)

is a signed measure µ

with the Jordan decomposition µ = µ

− µ

−

. The fol-

lowing calculation now veriﬁes (20): For any W ∈ T

we have

h(W

)(W

−W

)dx = −

W −W

dµ

= −

W −W

dµ

W −W

dµ

−

≥ 0

which holds because W − W

≥ 0 on C

−

and W −

≤ 0 on C

APPENDIX B

We prove the inequality (16) used in the proof of The-

orem 2. With the notation introduced in this proof we

can formulate this result as follows:

Proposition (Lewy–Stampacchia inequality). Sup-

pose φ

and ψ

are signed measures. Then the mi-

nimizer u

of E over C = {u : φ ≤ u ≤ ψ} satisﬁes

−(φ

)

−

≤ u

≤ (ψ

)

where µ

and µ

−

denote the positive and negative va-

riations, respectively, of the signed measure µ.

Note that the functional E is deﬁned, convex and

differentiable on H

(I) and therefore satisﬁes the in-

equality

E(v) − E(u) ≥

(x)(v

(x) − u

(x))dx, (22)

for all u,v ∈ H

(I), as is easily checked. We also

know that min

E has a unique solution u

∈ C (use

the projection theorem) which satisﬁes the necessary

and sufﬁcien condition:

− u

)dx ≥ 0 for all v ∈ C.

On the Taut String Interpretation of the One-dimensional Rudin–Osher–Fatemi Model

243

Proposition 8 was ﬁrst proved for the unilateral

obstacle problem (in multiple dimensions) in (Lewy

and Stampacchia, 1970) and since extended to bilate-

ral obstacle problems. Here we present a proof based

on the sub-modularity of the functional E, that is, for

all u,v ∈ H

(I),

E(u ∧ v) + E(u ∨ v) ≤ E(u)+ E(v), (23)

where u ∧ v := max(u, v) and u ∨ v := min(u, v) both

belong to H

(I). In fact, the functional E is so sim-

ple that equality holds for all u,v. This method of

proof was invented recently by (Gigli and Mosconi,

2015) and used to prove a very general version of the

bilateral Lewy–Stampacchia inequality. We use their

approach.

Proof. We prove the rightmost inequality, u

≤

(ψ

)

, the leftmost one then follows by a symme-

try argument: −u

minimizes E over the −C = {u ∈

(I) : −ψ ≤ u ≤ −φ}.

To simplify notation we set ψ

= µ. Deﬁne the

new functional

E(u) = E(u) + hµ

,ui and consider

the minimization problem

min

u∈H

(I):u≥u

E(u). (24)

The goal is to prove that u

itself solves this problem.

This will imply the desired inequality, as we shall see

below.

First the following claim is proved: For any u ∈

(I) satisfying u ≥ u

we have

E(u ∧ ψ) ≤

E(u). (25)

Since ψ > 0 on ∂I and u ∈ H

(I) we get u∧ψ ∈ H

(I)

which is therefore admissible for the min-problem

above. The claim is proved using the sub-modularity

(23) of E and the identity u ∧ v + u ∨ v = u + v with v

replaced by ψ:

E(u) −

E(u ∧ ψ)

= E(u) − E(u ∧ ψ) + hµ

,u − u ∧ ψi

≥ E(u ∨ ψ) − E(ψ) + hµ

,u ∨ ψ − ψi

≥ E(u ∨ ψ) − E(ψ) + hψ

,u ∨ ψ − ψi

≥ E(u ∨ ψ) − E(ψ) − hψ

,(u ∨ ψ)

− ψ

i ≥ 0,

where the last inequality follows from (22).

The claim (25) shows that the minimum of

E over

{u ≥ u

} coincides with the minimum over {ψ ≥ u ≥

}. Therefore, since u − u

≥ 0, we ﬁnd

E(u) −

E(u

) = E(u) − E(u

) + hµ

,u − u

≥ E(u) − E(u

) ≥ 0,

where the last estimate follows from the observation

that {u : u

≤ u ≤ ψ} ⊂ C and that u

minimizes E

over C. That is, u

is the solution of (24).

The necessary condition for u

to be a minimizer

for

E reads:

dα

E((1 − α)u

+ αu)



α=0

≥ 0

for all u ∈ H

(1) satisfying u ≥ u

. That is,

− u

)dx + hµ

,u − u

i ≥ 0

for all u ∈ H

(I) satisfying u ≥ u

. This implies

h−u

+ µ

,ϕi ≥ 0 for all ϕ ∈ C

(I) such that ϕ ≥ 0

on I. Therefore −u

+µ

is a positive measure, hence

≤ µ

, which is the desired result.

ICPRAM 2018 - 7th International Conference on Pattern Recognition Applications and Methods

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