Exact solutions for the total variation denoising problem of piecewise constant images in dimension one

1 Introduction

When an image is acquired, it comes, unavoidably, with some distortion. Indeed, external conditions, other than defects or limitations of the instruments that are used to obtain them, affect the quality of the acquired data. Thus, in order to perform any task on the image, it is important to be able to recover the clean version in the best possible way, i.e., with optimal fidelity. If we denote it by u:Ω→ℝ and the acquired, corrupted image by f:Ω→ℝ, where Ω⊂ℝN is an open set, it is usually assumed that the two are related via

Here A is a bounded linear operator representing, for instance, the blurring effect and n is the instance of the random noise. One of the aims of image reconstruction is deblurring and denoising f in order to recover u (see [8, 24]).

Here we are interested in the denoising problem, i.e., when the operator A is the identity and we have to remove the noise. Problem (1.1) is, in general, ill-posed (in the sense of Hadamard) and thus we need to regularize it (see [1, 55]). A widely used variational technique for this purpose was introduced by Rudin, Osher and Fatemi in [52], where they proposed to recover u via the minimization problem

for some fixed σ>0, where f is assumed to be in L2⁢(Ω) and |D⁢u|⁢(Ω) denotes the total variation of the function u in Ω. There are some interesting cases though, where the real image is not represented by a function of bounded variation (see [35]). The constrained minimization problem (1.2) has been shown to be equivalent to the following penalized minimization problem (known as the total variation denoising model with L2 fidelity term):

for some Lagrange multiplier λ>0 (see [19]).

Today’s literature on the study of problem (1.3) is extensive, and here we limit ourselves to recall that properties of the solutions have been studied, for instance, in [2, 3, 4, 9, 10, 16, 17, 22, 25, 29, 31, 38, 37, 36, 42, 51, 56, 57], the analysis of variants of (1.3) that use the generalized total variation have been performed in [13, 45, 47, 50], anisotropic models are undertaken in [26, 30, 32, 41], while the effects of considering high-order models have been investigated in [21, 23, 27, 39, 46]. Finally, other variants of (1.2) have been addressed in [6, 7, 33, 34, 44], and algorithmic considerations may be found in [15, 18, 20, 28, 43].

The seeking for an analytical method to find exact solutions to the minimization problem (1.3) (and some variants of it, both for piecewise constant and general initial data f) has been the topic of many studies. For instance, the interested reader can consult [11, 12, 14, 45, 48, 49, 50, 53] and the references therein. The starting point of our investigation is the work of Strong and Chan (see [54]), where they consider the minimization problem (1.3) (essentially just in one dimension) where the initial data f is a piecewise constant function with noise. Under certain conditions on the amplitude of the noise, they are able to determine the solution of the minimization problem (1.3) in the case λ≫1, namely when the solution has jumps where f has.

The aim of this paper is to carry on the previous analysis and to determine the solution of the minimization problem (1.3) for f being a piecewise constant function (without noise) for every value of the parameter λ. To allow for more generality in the choice of the fidelity term, we generalize the L2 norm to an Lp one, with p∈(1,∞). Some results are also presented in the case p=1. To be precise, we consider the minimization problem

where Ω:=(a,b)⊂ℝ and

for a given initial piecewise constant data f. Our aim is to provide a method for finding the solution to the minimization problem (1.4) in the case p>1 for all values of λ>0.

The novelty of this paper relies on the fact that our approach is aimed at getting the solution uλ to the minimization problem (1.4) by considering the solution uλ for large values of λ and then obtaining the ones for small values of λ by decreasing the parameter λ, whereas all the above-mentioned papers used a fixed value of λ. In particular, the main result we obtain is Theorem 5.3 about the behavior of neighboring values of the solution uλ close to a jump point when the parameter λ is moving. Albeit some of the results could be proved by using the primal-dual optimality condition (see [14, 51]) and the semigroup property of the total variation flow in dimension one (see [53]), here we prefer to employ more elementary techniques to study the problem.

We next explain the main idea behind the strategy we propose. The rigid structure of the initial data forces the solution to be piecewise constant itself, with jump set contained in the one of f (see Corollary 3.2). Moreover, a simple truncation argument shows that the solution takes values between the minimum and the maximum of f. Hence, the minimization problem (1.4) with f of the form

is equivalent to the following minimization problem:

where Q:=[min⁡f,max⁡f]k and G:ℝk→ℝ is the function defined by

with v=(v1,…,vk) and Li:=xi-xi-1. The function G is convex, but it lacks differentiability on the hyperplanes where {vi-1=vi}. Thus, in principle, one should minimize the function G over several compact regions and then compare all the minimum values in order to find the global minimizer.

Our method aims at overcoming this difficulty. We will be able, for each λ, to predict a priori – that is, without knowing explicitly uλ (the minimizer of G corresponding to the parameter λ) – what the relative position of each uiλ with respect to ui-1λ and fi will be. Knowing that, it is possible to look for the minimizer uλ only in a specific region of ℝk, where the absolute values present in the expression of G can be written explicitly. Hence, uλ can be found by solving the appropriate Euler–Lagrange equation.

We give a more detailed description of our method: the function λ↦uλ is continuous and uλ→f as λ→∞ (see Lemma 5.1). Hence, for λ≫1, we have that uiλ is very close to fi, and this allows us to predict the relative position of uiλ with respect to ui-1λ. Moreover, thanks to the qualitative properties of the solutions we will prove in Lemma 5.2 and Proposition 5.5, we will also be able to tell the relative position of each ui with respect to fi. These information allows us to write explicitly the absolute values present in the expression of G, as well as to write explicitly the Euler–Lagrange equation, whose solution will give us the minimizer uλ. With this reasoning, we find the minimizers for λ large (how large λ has to be will be determined a posteriori).

The idea now is to let λ decrease. Since uλ is constant for small values of λ (see Lemma 3.6), by continuity of λ↦uλ eventually two neighboring values uiλ and ui-1λ will happen to be the same. The main technical result (Theorem 5.3) tells us that the same will be true for all smaller values of λ. As a result we now have to consider the function G restricted to the subspace {vi-1=vi}, thus reducing the number of variables. By continuity of λ↦uλ, it is then possible to predict the relative position of every ui with respect to ui-1, while the qualitative properties of the solutions will give us the relative position of ui with respect to fi. As a consequence, also in this case, we are able to write explicitly the Euler–Lagrange equation.

We observe that the price to pay for applying this method is that, in order to determine the solution of the minimization problem (1.5) for a certain value λ¯, we first need to know it for all λ>λ¯. This, in the end, boils down to solve some equations (linear if p=2), whose number can be roughly bounded above by k⁢(k+1)/2.

Finally, we would like to comment on the case p=1. The reason why the strategy described above fails for p=1 is because we cannot use the continuity of the map λ↦uλ. Indeed, even if for p=1 there is no uniqueness for the solution of the minimization problem (1.5) (see an example in Proposition 4.1), there is always a solution taking only the values that f takes (see Corollary 3.3). But this jumping behavior of the solution prevents us to use continuity arguments, which are at the core of the strategy sketched above. Although it could be possible to obtain a solution of the minimization problem (1.5) in the case p=1 by comparing the value of the functional G over all the vectors v∈ℝk of the form vi=fσ⁢(i), where σ:{1,…,k}→{1,…,k} (see Corollary 3.3), we are currently investigating the possibility to obtain a more efficient analytic method to fulfill the task.

The paper is organized as follows. After a brief recalling of the main properties of one-dimensional functions of bounded variation in Section 2, we devote Section 3 to stating and proving basic results we will need in the sequel concerning the solutions of our minimization problem. In Section 4, we illustrate with a simple case the different behaviors of the solution in the cases p=1 and p>1. Section 5 contains the main technical results needed to justify the strategy to determine the solution of the minimization problem (1.5) we describe. In Section 6, we conclude with an explicit example.

2 Preliminaries

In this section, we review basic definitions of one-dimensional functions of bounded variation. For more details, see [5, 40]. Here we assume a,b∈ℝ with a<b.

The relation between the total and the pointwise variation is given by the following result. In the following, ℒ1 will denote the one-dimensional Lebesgue measure on ℝ.

Theorem 2.4 allows us to single out some well behaving representative of a BV function.

3 The general structure of the solutions

This section is devoted to stating and proving some basic results concerning the solution of the minimization problem (1.4). Albeit some of these properties may be known, we present here the proofs for the reader’s convenience.

We start by stating a well-known result about the jump set of the solution (for a proof see, for instance, [14]).

In higher dimension, the inclusion Ju⊂Jf has been proved in [16, 56] in the case p>1, while it is known to be not always true if p=1 (see [22, 31]). The above result allows us to study an equivalent finite-dimensional minimization problem in the case in which f is a piecewise constant function.

Thus, we now concentrate on the study of the minimization problem (3.2).

The cases p=1 and p>1 turn out to be quite different. Heuristically, the difference lies in the fact that, in the first case, the two terms of the energy are of the same order while, for p>1, the fidelity term is of higher order than the total variation. This leads to very different behavior of the solutions in the two cases.

Because of the strict convexity of the functional G for p>1, the solution of the minimization problem (3.2) is unique, while for p=1 we have lack of uniqueness (see Proposition 4.1). Nevertheless, it is possible to identify a solution with a particular structure.

We conjecture that, in the case p=1, non-uniqueness of the solution of the minimization problem (1.4) happens only for a finite number of critical values of λ, where a continuum of solutions is present.

In the rest of this section, we seek to understand the behavior of the solution uλ in the limiting cases for λ, i.e., when λ≪1 and when λ≫1. In the former case the predominant term of the energy is given by the total variation, thus we expect uλ to minimizes it. We first treat the case p>1.

In the case p>1, the only thing we can say about the case λ≫1 is that

Indeed, from

we conclude that

We now consider the asymptotic behavior of the solution in the case p=1.

4 Explicit solutions in a simple case

Here we study the case where k=2. This analysis, albeit its simplicity, is important to underline some features that distinguish the behavior of the solution of the minimization problem (1.4) in the cases p=1 and p>1.

Figure 1

The case L1<L2.

In the case p>1 the landscape of the solutions is quite different.

5 The behavior of the solution for p>1

This section contains the main result of this paper, namely Theorem 5.3, that is derived from the qualitative properties of the solutions proved in the following two lemmas and in Proposition 5.5. Although the same result can be deduced by using the semigroup property of the total variation in dimension one [53], we prefer to get it from more elementary observations of qualitative nature on the behavior of the solution uλ when the parameter λ varies. These observations allow to predict how the solution behaves when the parameter λ varies.

We start by proving the continuity (with respect to the Euclidean topology of ℝk) of the solution uλ with respect to λ.

We now prove several qualitative properties regarding the behavior of the solution uλ as λ varies. Some of the following results could be stated in a more inclusive way, but since they can be used to deduce qualitative properties of the solutions when no direct analysis can be performed, for clarity of exposition we opt to present each of them separately.