Regularization with Metric Double Integrals of Functions with Values in a Set of Vectors
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Abstract
We present an approach for variational regularization of inverse and imaging problems for recovering functions with values in a set of vectors. We introduce regularization functionals, which are derivativefree double integrals of such functions. These regularization functionals are motivated from double integrals, which approximate Sobolev seminorms of intensity functions. These were introduced in Bourgain et al. (Another look at Sobolev spaces. In: Menaldi, Rofman, Sulem (eds) Optimal control and partial differential equationsinnovations and applications: in honor of professor Alain Bensoussan’s 60th anniversary, IOS Press, Amsterdam, pp 439–455, 2001). For the proposed regularization functionals, we prove existence of minimizers as well as a stability and convergence result for functions with values in a set of vectors.
Keywords
Regularization Manifoldvalued data Nonconvex Metric Double integral Fractional Sobolev space Bounded variation1 Introduction

Interferometric Synthetic Aperture Radar (InSAR) is a technique used in remote sensing and geodesy to generate, for example, digital elevation maps of the earth’s surface. InSAR images represent phase differences of waves between two or more SAR images, cf. [44, 53]. Therefore, InSAR data are functions \(f:\Omega \rightarrow {\mathbb {S}}^1\subseteq {\mathbb {R}}^2\). The pointwise function values are on the \({\mathbb {S}}^1\), which is considered embedded into \({\mathbb {R}}^2\).

A color image can be represented as a function in HSV space (hue, saturation, value) (see, e.g., [48]). Color images are then described as functions \(f:\Omega \rightarrow K \subseteq {\mathbb {R}}^3\). Here \(\Omega \) is a plane in \({\mathbb {R}}^2\), the image domain, and K (representing the HSV space) is a cone in threedimensional space \({\mathbb {R}}^3\).

Estimation of the foliage angle distribution has been considered, for instance, in [39, 51]. Therefore, the imaging function is from \(\Omega \subset {\mathbb {R}}^2\), a part of the Earth’s surface, into \(\mathbb {S}^2 \subseteq {\mathbb {R}}^3\), representing foliage angle orientation.

Estimation of functions with values in \(SO(3) \subseteq {\mathbb {R}}^{3 \times 3}\). Such problems appear in CryoElectron Microscopy (see, for instance, [38, 58, 61]).

w is an element of the set of admissible functions.

Open image in new window is an operator modeling the image formation process (except the noise).

\(\mathcal {D}\) is called the data or fidelity term, which is used to compare a pair of data in the image domain, that is to quantify the difference of the two data sets.

\(\mathcal {R}\) is called regularization functional, which is used to impose certain properties onto a minimizer of the regularization functional \(\mathcal {F}\).

\(\alpha > 0\) is called regularization parameter and provides a trade off between stability and approximation properties of the minimizer of the regularization functional \(\mathcal {F}\).

\(v^\delta \) denotes measurement data, which we consider noisy.

\(v^0\) denotes the exact data, which we assume to be not necessarily available.
1.1 Variational Regularization for Reconstruction of Intensity Data
Opposite to what we consider in the present paper, most commonly, imaging data v and admissible functions w, respectively, are considered to be representable as intensity functions. That is, they are functions from some subset \(\Omega \) of an Euclidean space with real values.
In such a situation, the most widely used regularization functionals use regularization terms consisting of powers of Sobolev (see [12, 15, 16]) or total variation seminorms [54]. It is common to speak about Tikhonov regularization (see, for instance, [59]) when the data term and the regularization functional are squared Hilbert space norms, respectively. For the Rudin, Osher, Fatemi (ROF) regularization [54], also known as total variation regularization, the data term is the squared \(L^2\)norm and \(\mathcal {R}(w) = w_{TV}\) is the total variation seminorm. Nonlocal regularization operators based on the generalized nonlocal gradient are used in [35].
Other widely used regularization functionals are sparsity promoting [22, 41], Besov space norms [42, 46] and anisotropic regularization norms [47, 56]. Aside from various regularization terms, there also have been proposed different fidelity terms other than quadratic norm fidelities, like the pth powers of \(\ell ^p\) and \(L^p\)norms of the differences of F(w) and v , [55, 57], maximum entropy [26, 28] and Kullback–Leibler divergence [52] (see [50] for some reference work).
1.2 Regularization of Functions with Values in a Set of Vectors
In this paper we generalize the derivativefree characterization of Sobolev spaces and functions of bounded variation to functions \(u:\Omega \rightarrow K\), where K is some set of vectors, and use these functionals for variational regularization. The applications we have in mind contain that K is a closed subset of \({\mathbb {R}}^M\) (for instance, HSV data) with nonzero measure, or that K is a submanifold (for instance, InSAR data).
The reconstruction of manifoldvalued data with variational regularization methods has already been subject to intensive research (see, for instance, [4, 17, 18, 19, 40, 62]). The variational approaches mentioned above use regularization and fidelity functionals based on Sobolev and TV seminorms: a total variation regularizer for cyclic data on \({\mathbb {S}}^1\) was introduced in [18, 19], see also [7, 9, 10]. In [4, 6] combined first and secondorder differences and derivatives were used for regularization to restore manifoldvalued data. The later mentioned papers, however, are formulated in a finitedimensional setting, opposed to ours, which is considered in an infinitedimensional setting. Algorithms for total variation minimization problems, including halfquadratic minimization and nonlocal patchbased methods, are given, for example, in [4, 5, 8] as well as in [37, 43]. On the theoretical side the total variation of functions with values in a manifold was investigated by Giaquinta and Mucci using the theory of Cartesian currents in [33, 34], and earlier [32] if the manifold is \({\mathbb {S}}^1\).
1.3 Content and Particular Achievements of the Paper
The contribution of this paper is to introduce and analytically analyze double integral regularization functionals for reconstructing functions with values in a set of vectors, generalizing functionals of the form Eq. 1.3. Moreover, we develop and analyze fidelity terms for comparing manifoldvalued data. Summing these two terms provides a new class of regularization functionals of the form Eq. 1.2 for reconstructing manifoldvalued data.
 (i)
The admissible functions, where we minimize the regularization functional on, do form only a set but not a linear space. As a consequence, wellposedness of the variational method (that is, existence of a minimizer of the energy functional) cannot directly be proven by applying standard direct methods in the Calculus of Variations [20, 21].
 (ii)
The regularization functionals are defined via metrics and not norms, see Sect. 3.
 (iii)
In general, the fidelity terms are nonconvex. Stability and convergence results are proven in Sect. 4.
2 Setting
In the following we introduce the basic notation and the set of admissible functions which we are regularizing on.
Assumption 2.1

\(p_1, p_2 \in [1, +\infty )\), \(s \in (0,1]\),

\(\Omega _1, \Omega _2 \subseteq {\mathbb {R}}^N\) are nonempty, bounded and connected open sets with Lipschitz boundary, respectively,

\(k \in [0,N]\),

\(K_1 \subseteq {\mathbb {R}}^{M_1}, K_2 \subseteq {\mathbb {R}}^{M_2}\) are nonempty and closed subsets of \({\mathbb {R}}^{M_1}\) and \({\mathbb {R}}^{M_2}\), respectively.

Open image in new window and \(\Vert \cdot \Vert _{{\mathbb {R}}^{M_i}}, \ i=1,2,\) are the Euclidean norms on \({\mathbb {R}}^N\) and \({\mathbb {R}}^{M_i}\), respectively.

Open image in new window denotes the Euclidean distance on \({\mathbb {R}}^{M_i}\) for \(i=1,2\) and
 Open image in new window denote arbitrary metrics on \(K_i\), which fulfill for \(i=1\) and \(i=2\)In particular, this assumption is valid if the metric \(d_i\) is equivalent to Open image in new window . When the set \(K_i, \ i=1,2\), is a suitable complete submanifold of \({\mathbb {R}}^{M_i}\), it seems natural to choose \(d_i\) as the geodesic distance on the respective submanifolds.

\(\,{\mathrm {d}}_i\) is continuous with respect to Open image in new window , meaning that for a sequence Open image in new window in \(K_i \subseteq {\mathbb {R}}^{M_i}\) converging to some \(a \in K_i\) we also have Open image in new window .
 \((\rho _{\varepsilon })_{\varepsilon > 0}\) is a Dirac family of nonnegative, radially symmetric mollifiers, i.e., for every \(\varepsilon > 0\) we haveWe demand further that, for every \(\varepsilon > 0\),
 (i)
\(\rho _\varepsilon \in \mathcal {C}^{\infty }_{c}({\mathbb {R}}^N, {\mathbb {R}})\) is radially symmetric,
 (ii)
\(\rho _\varepsilon \ge 0\),
 (iii)
\(\int \limits _{{\mathbb {R}}^N} \rho _\varepsilon (x) \,{\mathrm {d}}x= 1\), and
 (iv)
for all \(\delta > 0\), Open image in new window .
This condition holds, e.g., if \(\rho _{\varepsilon }\) is a radially decreasing continuous function with \(\rho _{\varepsilon }(0) > 0\). (v)
there exists a \(\tau > 0\) and \(\eta _{\tau }> 0\) such that Open image in new window .
 (i)

When we write p, \(\Omega \), K, M, then we mean \(p_i\), \(\Omega _i\), \(K_i\), \(M_i\), for either \(i=1,2\). In the following we will often omit the subscript indices whenever possible.
Example 2.2
 by substitution \(x = t \theta \) with \(t > 0, \theta \in \mathbb {S}^{N1}\) and \(\hat{t}=\frac{t}{\varepsilon }\),(2.1)
 Again by the same substitutions, taking into account that \(\hat{\rho }\) has compact support, it follows for \(\varepsilon > 0\) sufficiently small that(2.2)
In the following we write down the basic spaces and sets, which will be used in the course of the paper.
Definition 2.3
 The Lebesgue–Bochner space of \({\mathbb {R}}^M\)valued functions on \(\Omega \) consists of the set which is associated with the norm Open image in new window , given by
 Let \(0< s < 1\). Then the fractional Sobolev space of order s can be defined (cf. [1]) as the set equipped with the norm(2.3)(2.4)
 For \(s = 1\) the Sobolev space \(W^{1,p}(\Omega , {\mathbb {R}}^M)\) consists of all weakly differentiable functions in \(L^1(\Omega ,{\mathbb {R}}^M)\) for which where \(\nabla w\) is the weak Jacobian of w.
 Moreover, we recall one possible definition of the space Open image in new window from [2], which consists of all Lebesgue–Borel measurable functions \(w:\Omega \rightarrow {\mathbb {R}}^M\) for which where where \(\left\ \varphi (x)\right\ _F\) is the Frobeniusnorm of the matrix \(\varphi (x)\) and \(\text {Div}\varphi = (\text {div} \varphi _1, \dots , \text {div} \varphi _M)^\text {T}\) denotes the row–wise formed divergence of \(\varphi \).
Lemma 2.4
Proof
The first result can be found in [24] for \(0< s < 1\) and in [29] for \(s = 1\). The second assertion is stated in [2]. \(\square \)
Remark 2.5

Let \(p > 1\), \(s\in (0,1]\) and assume that \((w_n)_{n \in {\mathbb {N}}}\) is bounded in Open image in new window . Then there exists a subsequence \((w_{n_k})_{k \in {\mathbb {N}}}\) which converges weakly in Open image in new window .

Assume that \((w_n)_{n \in {\mathbb {N}}}\) is bounded in Open image in new window . Then there exists a subsequence \((w_{n_k})_{k \in {\mathbb {N}}}\) which converges weakly* in Open image in new window .
Before introducing the regularization functional, which we investigate theoretically and numerically, we give the definition of some sets of (equivalence classes of) admissible functions.
Definition 2.6

Open image in new window is associated with the strong Open image in new window topology,

Open image in new window is associated with the weak Open image in new window topology, and

Open image in new window is associated with the weak* Open image in new window topology.

is associated with the weak Open image in new window topology in the case \(p \in (1, \infty )\) and \(s \in (0,1]\) and

with the weak* Open image in new window topology when \(p=1\) and \(s=1\).
Remark 2.7

In general Open image in new window and Open image in new window are sets which do not form a linear space.

If \(K = {\mathbb {S}}^1\), then Open image in new window as occurred in [13].

For an embedded manifold K, the dimension of the manifold is not necessarily identical with the space dimension of \({\mathbb {R}}^M\). For instance, if \(K = {\mathbb {S}}^1\subseteq {\mathbb {R}}^2\), then the dimension of \({\mathbb {S}}^1\) is 1 and \(M=2\).
The following lemma shows that \(W(\Omega ,K)\) is a sequentially closed subset of Open image in new window .
Lemma 2.8
 (i)
Let Open image in new window and \((w_n)_{n\in {\mathbb {N}}}\) be a sequence in Open image in new window with \(w_n \overset{W(\Omega , {\mathbb {R}}^M)}{\longrightarrow } w_*\) as \(n \rightarrow \infty \). Then Open image in new window and Open image in new window in Open image in new window .
 (ii)
Let Open image in new window and \((v_n)_{n \in {\mathbb {N}}}\) be a sequence in Open image in new window with \(v_n \rightarrow v_*\) in Open image in new window as \(n \rightarrow \infty \). Then Open image in new window and there is some subsequence \((v_{n_k})_{k \in {\mathbb {N}}}\) which converges to \(v_*\) pointwise almost everywhere, i.e., \(v_{n_k}(x) \rightarrow v_*(x)\) as \(k \rightarrow \infty \) for almost every \(x \in \Omega \).
Proof
For the proof of the second part, cf. [27], Chapter VI, Corollary 2.7, take into account the closedness of \(K \subseteq {\mathbb {R}}^M\). The proof of the first part follows from standard convergence arguments in Open image in new window , Open image in new window and Open image in new window , respectively, using the embeddings from Lemma 2.4, an argument on subsequences and part two. \(\square \)
Remark 2.9
Lemma 2.4 along with Lemma 2.8 imply that Open image in new window is compactly embedded in Open image in new window , where these sets are equipped with the bornology inherited from Open image in new window and the topology inherited from Open image in new window , respectively.
In the following we postulate the assumptions on the operator Open image in new window which will be used throughout the paper:
Assumption 2.10
Let Open image in new window be as in Eq. 2.5 and assume that Open image in new window is an operator from Open image in new window to Open image in new window .
We continue with the definition of our regularization functionals:
Definition 2.11
Let Assumptions 2.1 and 2.10 hold. Moreover, let \(\varepsilon > 0\) be fixed and let \(\rho :=\rho _\varepsilon \) be a mollifier.
 (i)
 (ii)
\(s \in (0,1]\),
 (iii)
\(\alpha \in (0, +\infty )\) is the regularization parameter,
 (iv)
\(l \in \left\{ 0, 1\right\} \) is an indicator and
 (v)
\({\left\{ \begin{array}{ll} k \le N &{}\text{ if } W (\Omega _1, K_1) = W^{s,p_1}(\Omega _1, K_1), \ 0{<}s{<}1, \\ k=0 &{} \text{ if } W (\Omega _1, K_1) = W^{1,p_1}(\Omega _1, K_1)\text { or if }\\ &{}\quad W (\Omega _1, K_1) = BV(\Omega _1, K_1), \text { respectively.} \end{array}\right. }\)
Remark 2.12
 (i)\(l = \left\{ 0,1\right\} \) is an indicator which allows to consider approximations of Sobolev seminorms and double integral representations of the type of Bourgain et al. [14] in a uniform manner.We expect a relation between the two classes of functionals for \(l=0\) and \(l=1\) as stated in Sect. 5.2.
 (ii)
When \(d_1\) is the Euclidean distance then the second term in Eq. 2.6 is similar to the ones used in [3, 11, 14, 23, 49].
In the following we state basic properties of Open image in new window and the functional Open image in new window .
Proposition 2.13
 (i)
Then the mapping Open image in new window Open image in new window satisfies the metric axioms.
 (ii)
Let, in addition, Assumption 2.10 hold, assume that Open image in new window and that both metrics \(d_i\), \(i=1,2\), are equivalent to Open image in new window , respectively. Then the functional Open image in new window does not attain the value \(+\infty \) on its domain Open image in new window .
Proof
 (i)The axioms of nonnegativity, identity of indiscernibles and symmetry are fulfilled by Open image in new window since Open image in new window is a metric. To prove the triangle inequality, let \(\phi ,\xi ,\nu \in L^{p_2}(\Omega _2, K_2)\). In the main case Open image in new window Hölder’s inequality yields meaning If Open image in new window , the triangle inequality is trivially fulfilled.In the remaining case Open image in new window applying the estimate \((a+b)^p \le 2^{p1} (a^p + b^p)\), see, e.g., [55, Lemma 3.20], to Open image in new window and Open image in new window yields implying the desired result.
 (ii)We emphasize that Open image in new window because every constant function \(w(\cdot ) = a \in K_1\) belongs to Open image in new window for \(p_1 \in (1, \infty )\) and \(s \in (0,1]\) as well as to Open image in new window for \(p_1 = 1\) and \(s = 1\). Assume now that the metrics \(d_i\) are equivalent to Open image in new window for \(i=1\) and \(i=2\), respectively, so that we have an upper bound Open image in new window . We need to prove that Open image in new window for every Open image in new window . Due to Open image in new window for all Open image in new window it is sufficient to show Open image in new window for all Open image in new window .

For Open image in new window this is guaranteed by [49, Theorem 1.2].

For Open image in new window by [14, Theorem 1].

For Open image in new window , \(s \in (0,1)\), we distinguish between two cases.
If Open image in new window , we have that Open image in new window for \(k \le N\) and hence If Open image in new window , we can estimate In summary adding yields Open image in new window . \(\square \)

3 Existence
In order to prove existence of a minimizer of the functional Open image in new window , we apply the direct method in the Calculus of Variations (see, e.g., [20, 21]). To this end we verify continuity properties of Open image in new window and Open image in new window , resp. Open image in new window and apply them along with the sequential closedness of Open image in new window , already proven in Lemma 2.8.
In this context we point out some setting assumptions and their consequences on Open image in new window , resp. Open image in new window and \({\mathcal {R}}\) in the following remark. For simplicity we assume \(p :=p_1 = p_2 \in (1, \infty )\), \(\Omega :=\Omega _1 = \Omega _2\) and Open image in new window .
Remark 3.1

The continuity of Open image in new window with respect to Open image in new window guarantees lower semicontinuity of Open image in new window and Open image in new window .

The inequality Open image in new window carries over to the inequalities Open image in new window for all Open image in new window , and Open image in new window for all Open image in new window , allowing to transfer properties like coercivity from Open image in new window to Open image in new window . Moreover, the extended realvalued metric space Open image in new window stays related to the linear space Open image in new window in terms of the topology and bornology induced by Open image in new window , resp. those inherited by Open image in new window .

The closedness of \(K \subseteq {\mathbb {R}}^M\) is crucial in showing that Open image in new window is a sequentially closed subset of the linear space Open image in new window . This closedness property acts as a kind of replacement for the, a priori not available, notion of completeness with respect to the “space” Open image in new window .
We will use the following assumption:
Assumption 3.2
Let Assumption 2.1 hold, Open image in new window and let Open image in new window and the associated topology be as defined in Eq. 2.5.

Open image in new window is well defined and sequentially continuous with respect to the specified topology on Open image in new window and
 For every \(t > 0\) and \(\alpha > 0\), the level sets(3.1)

There exists a \(\bar{t} > 0\) such that Open image in new window is nonempty.

Only those Open image in new window are considered which additionally fulfill Open image in new window .
Remark 3.3
The third condition is sufficient to guarantee Open image in new window . In contrast, the condition Open image in new window , cf. Definition 2.11, might not be sufficient if \(d_2\) is not equivalent to Open image in new window .
Lemma 3.4
 (i)
The mapping Open image in new window is sequentially lower semicontinuous, i.e., whenever sequences Open image in new window , Open image in new window in Open image in new window converge to Open image in new window and Open image in new window , respectively, we have Open image in new window .
 (ii)The functional Open image in new window is sequentially lower semicontinuous, i.e., whenever a sequence \((w_n)_{n \in {\mathbb {N}}}\) in Open image in new window converges to some Open image in new window we have
 (iii)
The functional Open image in new window is sequentially lower semicontinuous.
Proof
 (i)It is sufficient to show that for every pair of sequences Open image in new window , Open image in new window in Open image in new window which converge to previously fixed elements Open image in new window and Open image in new window , respectively, we can extract subsequences \((\phi _{n_j})_{j \in {\mathbb {N}}}\) and \((\nu _{n_j})_{j \in {\mathbb {N}}}\), respectively, with To this end let \((\phi _n)_{n \in {\mathbb {N}}},(\nu _n)_{n \in {\mathbb {N}}}\) be some sequences in Open image in new window with Open image in new window and Open image in new window in Open image in new window . Lemma 2.8 ensures that there exist subsequences \((\phi _{n_j})_{j \in {\mathbb {N}}}, (\nu _{n_j})_{j \in {\mathbb {N}}}\) converging to \(\phi _*\) and \(\nu _*\) pointwise almost everywhere, which in turn implies \(\big (\phi _{n_j}(\cdot ), \nu _{n_j}(\cdot ) \big ) \rightarrow \big ( \phi _*(\cdot ), \nu _*(\cdot ) \big )\) pointwise almost everywhere. Therefrom, together with the continuity of Open image in new window with respect to Open image in new window , cf. Sect. 2, we obtain by using the quadrangle inequality that and hence for almost every \(x \in \Omega _2\). Applying Fatou’s lemma, we obtain
 (ii)Let \((w_n)_{n \in {\mathbb {N}}}\) be a sequence in Open image in new window with Open image in new window as Open image in new window . By Lemma 2.8 there is a subsequence \((w_{n_j})_{j \in {\mathbb {N}}}\) which converges to \(w_*\) both in Open image in new window and pointwise almost everywhere. This further implies that for almost everyDefining for all \(j \in {\mathbb {N}}\) and we thus have Open image in new window for almost every \((x,y) \in \Omega _1 \times \Omega _1\). Applying Fatou’s lemma to the functions \(f_j\) yields the assertion, due to the same reduction as in the proof of the first part.$$\begin{aligned} (x,y) \in \Omega _1 \times \Omega _1 \supseteq \{(x,y) \in \Omega _1 \times \Omega _1 : x \ne y \} =:A.\nonumber \\ \end{aligned}$$(3.2)
 (iii)
It is sufficient to prove that the components Open image in new window and Open image in new window of Open image in new window are sequentially lower semicontinuous. To prove that \(\mathcal {G}\) is sequentially lower semicontinuous in every \(w_* \in W(\Omega _1, K_1)\), let \((w_n)_{n \in {\mathbb {N}}}\) be a sequence in \(W(\Omega _1, K_1)\) with Open image in new window as Open image in new window . Assumption 3.2, ensuring the sequential continuity of Open image in new window , implies hence Open image in new window in Open image in new window as Open image in new window . By item (i) we thus obtain Open image in new window .
\({\mathcal {R}}\) is sequentially lower semicontinuous by item (ii).
3.1 Existence of Minimizers
The proof of the existence of a minimizer of Open image in new window is along the lines of the proof in [55], taking into account Remark 3.1. We will need the following useful lemma, cf. [55], which links Open image in new window and Open image in new window for Open image in new window .
Lemma 3.5
Proof
Theorem 3.6
Let Assumption 3.2 hold. Then the functional Open image in new window attains a minimizer.
Proof
In the following we investigate two examples, which are relevant for the numerical examples in Sect. 6.
Example 3.7
We consider that \(W(\Omega _1,K_1) = W^{s, p_1}(\Omega _1, K_1)\) with \(p_1>1, \ 0< s < 1\) and fix \(k = N\).
Case 1 Open image in new window .
Case 2 Open image in new window .
If \(l=0\), then Open image in new window is exactly the \(W^{s,p_1}(\Omega _1, {\mathbb {R}}^{M_1})\)seminorm \(w_{W^{s,p_1}(\Omega _1, {\mathbb {R}}^{M_1})}\) and we trivially get the desired result.
Splitting \(\Omega _1 \times \Omega _1\) into \({\mathcal {S}}_{\tau }=:{\mathcal {S}}\) and its complement \((\Omega _1 \times \Omega _1) \setminus {\mathcal {S}}_{\tau }=:{\mathcal {S}}^{c}\), we accordingly split the integrals Open image in new window and consider again two cases Open image in new window and Open image in new window , respectively.
Case 2.1 Open image in new window .
The second example concerns the coercivity of Open image in new window , defined in Eq. 2.9, when Open image in new window denotes the masking operator occurring in image inpainting. To prove this result, we require the following auxiliary lemma:
Lemma 3.8
Proof
The proof is inspired by the proof of Poincaré’s inequality in [29]. It is included here for the sake of completeness.
 (i)
\(\left\ w_n\right\ _{L^{p_1}\left( D, {\mathbb {R}}^{M_1}\right) }^{p_1} = 1\).
 (ii)
\(\left\ w_n\right\ _{L^{p_1}(\Omega _1 \setminus D, {\mathbb {R}}^{M_1})}^{p_1} < \frac{1}{n}\),
 (iii)
 (i)
\(\left\ w^*\right\ _{L^{p_1}\left( D, {\mathbb {R}}^{M_1}\right) }^{p_1} = 1\), in particular, \(w^*\) is not the null function on D,
 (ii)
\(\left\ w^*\right\ _{L^{p_1}(\Omega _1 \setminus D, {\mathbb {R}}^{M_1})}^{p_1} = 0\) since \(n \in {\mathbb {N}}\) is arbitrary and hence \(w^* \equiv 0\) on \(\Omega _1 \setminus D\).
 (iii)i.e., \(w^*(x) = w^*(y) \) for \((x,y) \in {\mathcal {S}}\) yielding that \(w^*\) locally constant and hence even constant since \(\Omega _1\) is connected,
In the case \(l=0\) we use similar arguments, where the distance Open image in new window in the last inequality can be estimated by \(\text {diam}\Omega _1\) (instead of \(\eta \)) since \(\Omega _1\) is bounded. \(\square \)
Remark 3.9
Example 3.10
As in Example 3.7, we consider that \(W(\Omega _1,K_1) = W^{s, p_1}(\Omega _1, K_1)\) with \(p_1>1, \ 0< s < 1\) and fix \(k = N\).
 Case 1.1
 Case 1.2
4 Stability and Convergence
In this section we will first show a stability and afterwards a convergence result. We use the notation introduced in Sect. 2. In particular, \(W(\Omega _1, K_1)\) is as defined in Eq. 2.5. We also stress that we use notationally simplified versions Open image in new window of Open image in new window and \({\mathcal {R}}\) of Open image in new window whenever possible. See Eqs. 2.6, 2.7 and 2.8.
Theorem 4.1
The subsequent proof of Theorem 4.1 is similar to the proof of [55, Theorem 3.23].
Proof
For the ease of notation, we simply write Open image in new window instead of Open image in new window and Open image in new window .
Before proving the next theorem, we need the following definition, cf. [55].
Definition 4.2
The following theorem and its proof are inspired by [55, Theorem 3.26].
Theorem 4.3
Moreover, if \(w^\dagger \) is unique, it follows that Open image in new window and Open image in new window .
Proof
Now assume that the solution fulfilling Eq. 4.3 is unique; we call it \(w^\dagger \). In order to prove that Open image in new window , it is sufficient to show that any subsequence has a further subsequence converging to \(w^\dagger \), cf. [55, Lemma 8.2]. Hence, denote by \((w_{n_k})_{k \in {\mathbb {N}}}\) an arbitrary subsequence of \((w_n)\), the sequence of minimizers. Like before we can show that Open image in new window is bounded and we can extract a converging subsequence \((w_{n_{k_l}})_{l \in {\mathbb {N}}}\). The limit of this subsequence is \(w^\dagger \) since it is the unique solution fulfilling Eq. 4.3, showing that Open image in new window . Moreover, \(w^\dagger \in W(\Omega _1, K_1)\). Following the arguments above, we obtain as well Open image in new window \(\square \)
Remark 4.4
Theorem 4.1 guarantees that the minimizers of Open image in new window depend continuously on \(v^\delta \), while Theorem 4.3 ensures that they converge to a solution of Open image in new window , \(v^0\) the exact data, while \(\alpha \) tends to zero.
5 Discussion of the Results and Conjectures
In this section we summarize some open problems related to double integral expressions of functions with values on manifolds.
5.1 Relation to Single Integral Representations
In the following we show for one particular case of functions that have values in a manifold, that the double integral formulation Open image in new window , defined in Eq. 2.8, approximates a single energy integral. The basic ingredient for this derivation is the exponential map related to the metric \(d_1\) on the manifold. In the following we investigate manifoldvalued functions \(w \in W^{1,2}(\Omega , \mathcal {M})\), where we consider \(\mathcal {M} \subseteq {\mathbb {R}}^{M \times 1}\) to be a connected, complete Riemannian manifold. In this case some of the regularization functionals Open image in new window , defined in Eq. 2.8, can be considered as approximations of single integrals. In particular, we aim to generalize Eq. 1.3 in the case \(p=2\).
Example 5.1
From these considerations we can view Open image in new window as functionals, which generalize Sobolev and \(\text {BV}\) seminorms to functions with values on manifolds.
5.2 A Conjecture on Sobolev Seminorms
 In the case \(l=0\), \(k=N\), \(0<s<1\) and Open image in new window the functional Open image in new window from Eq. 2.8 simplifies to the pth power of the Sobolev seminorm and reads(5.9)
 On the other hand, when we choose \(k=0\), \(l=1\) and Open image in new window , then Open image in new window from Eq. 2.8 reads (note \(\rho =\rho _\varepsilon \) by simplification of notation):(5.10)
6 Numerical Examples
In this section we present some numerical examples for denoising and inpainting of functions with values on the circle \({\mathbb {S}}^1\). Functions with values on a sphere have already been investigated very diligently (see, for instance, [13] out of series of publications of these authors). Therefore, we review some of their results first.
6.1 \({\mathbb {S}}^1\)Valued Data
Lemma 6.1

Let \(\Omega \subset {\mathbb {R}}\), \(0< s < \infty \), \(1< p < \infty \). Then for all Open image in new window there exists Open image in new window satisfying Eq. 6.1.

Let \(\Omega \subset {\mathbb {R}}^N\), \(N \ge 2\), \(0< s < 1\), \(1< p < \infty \). Moreover, let \(sp < 1\) or \(sp \ge N\), then for all Open image in new window there exists Open image in new window satisfying Eq. 6.1.
If \(sp \in [1,N)\), then there exist functions Open image in new window such that Eq. 6.1 does not hold with any function Open image in new window .
Remark 6.2

We note that in the case \(k=0\), \(s=1\) and \(l=1\) these integrals correspond with the ones considered in Bourgain et al. [14] for functions with values on \({\mathbb {S}}^1\).

If we choose \(k=N\), \(s=1\) and \(l=0\), then this corresponds with Sobolev seminorms on manifolds.
 Let \(\varepsilon > 0\) fixed (that is, we consider neither a standard Sobolev regularization nor the limiting case \(\varepsilon \rightarrow 0\) as in [14]). In this case we have proven coercivity of the functional Open image in new window only with the following regularization functional, cf. Example 3.7 and Example 3.10:
We summarize a few results: The first lemma follows from elementary calculations:
Lemma 6.3
Open image in new window and \(\,{\mathrm {d}}_{{\mathbb {R}}^2}\big _{{\mathbb {S}}^1\times {\mathbb {S}}^1}\) are equivalent.
Lemma 6.4
Let Open image in new window . Then Open image in new window .
Proof
This follows directly from the inequality \(\Vert {\mathrm {e}}^{ia}{\mathrm {e}}^{ib}\Vert \le \Vert ab\Vert \) for all \(a,b \in {\mathbb {R}}\). \(\square \)
Below we show that Open image in new window is finite on Open image in new window .
Lemma 6.5
Open image in new window maps Open image in new window into \([0,\infty )\) (i.e., does not attain the value \(+\infty \)).
Proof
Let Open image in new window . Then by Lemma 6.4 we have that Open image in new window . Therefore, from Lemma 6.3 and Proposition 2.13 item (ii) it follows that Open image in new window . Hence, by definition, Open image in new window . \(\square \)
6.2 Setting of Numerical Examples
6.3 Regularization Functionals
Lemma 6.6
Let \(\emptyset \ne \Omega \subset {\mathbb {R}}\) or \({\mathbb {R}}^2\) be a bounded and simply connected open set with Lipschitz boundary. Let \(1< p < \infty \) and \(s \in (0,1)\). If \(N=2\) assume that \(sp < 1\) or \(sp \ge 2\). Moreover, let Assumption 3.2 and Assumption 2.10 be satisfied. Then the mapping Open image in new window attains a minimizer.
Proof
Let Open image in new window . Then by Lemma 6.4 we have that Open image in new window . As arguing as in the proof of Lemma 6.5, we see that Open image in new window .
Since we assume that Assumption 3.2 is satisfied, we get that Open image in new window attains a minimizer Open image in new window . It follows from Lemma 6.1 that there exists a function \(u^* \in W^{s,p}(\Omega , {\mathbb {R}})\) that can be lifted to \(w^*\), i.e., \(w^* = \Phi (u^*)\). Then \(u^*\) is a minimizer of (6.6) by definition of Open image in new window and \(\Phi \). \(\square \)
6.4 Numerical Minimization
In our concrete examples, we will consider two different operators Open image in new window . For numerical minimization we consider the functional from Eq. 6.6 in a discretized setting. For this purpose, we approximate the functions \(u \in W^{s, p}(\Omega ,{\mathbb {R}})\), \(0<s<1,1<p<\infty \) by quadratic Bspline functions and optimize with respect to the coefficients. We remark that this approximation is continuous and thus that sharp edges correspond to very steep slopes.
The noisy data \(u^\delta \) are obtained by adding Gaussian white noise with variance \(\sigma ^2\) to the approximation or the discretized approximation of u.
We apply a simple Gradient Descent scheme with fixed step length implemented in \(\text {MATLAB}\).
6.5 Denoising of \({\mathbb {S}}^1\)Valued Functions: The InSAR Problem
In the examples we will just consider the continuous approximation again denoted by u.
6.6 OneDimensional Test Case
Let \(\Omega = (0,1)\) and consider the signal Open image in new window representing the angle of a cyclic signal.
In this experiment we show the influence of the parameters s and p. In all cases the choice of the regularization parameter \(\alpha \) is 0.19 and \(\varepsilon = 0.01\).
The red signal in Fig. 2b is obtained by choosing \(s = 0.1\) and \(p = 1.1\). We see that the periodicity of the signal is handled correctly and that there is nearly no staircasing. In Fig. 2c the parameter s is changed from 0.1 to 0.6. The value of the parameter p stays fixed. Increasing of s leads the signal to be more smooth. We can observe an even stronger similar effect when increasing p (here from 1.1 to 2) and letting s fixed, see Fig. 2d. This fits the expectation since s only appears once in the denominator of the regularizer. At a jump, increasing of s leads thus to an increasing of the regularization term. The parameter p appears twice in the regularizer. Huge jumps are hence weighted even more.
In Fig. 3a we considered a simple signal with a single huge jump. Again it is described by the angular value. We proceeded as above to obtain the approximated discrete original data (black) and noisy signal with \(\sigma = 0.1\) (blue). We chose again \(\varepsilon = 0.01\).
Moreover, we have seen that increasing of p leads to an even more smooth signal. In Fig. 3c we choose a quite large value of p, \(p=2\) and a rather small value of s, \(s = 0.001\). Even for this very simple signal, it was not possible to get sharp edges. This is due to the fact that the parameter p (but not s) additionally weights the height of jumps in the regularizing term.
6.7 Denoising of a \({\mathbb {S}}^1\)Valued Image
Our next example concerned a twodimensional \({\mathbb {S}}^1\)valued image represented by the corresponding angular values. We remark that in this case where \(N=2\) the existence of such a representation is always guaranteed in the cases where \(sp < 1\) or \(sp \ge 2\), see Lemma 6.1.
This experiment shows the difference of our regularizer respecting the periodicity of the data in contrast to the classical total variation regularizer. The classical TV minimization is solved using a fixed point iteration ([45]); for the method see also [60].
In Fig. 5a the function u can be seen from the top, i.e., the axes correspond to the i resp. j axis in Fig. 4. The noisy data are obtained by adding white Gaussian noise with \(\sigma = \sqrt{0.001}\) using the builtin function \(\texttt {imnoise}\) in \(\text {MATLAB}\). It is shown in Fig. 5b. We choose as parameters \(s=0.9, \ p=1.1, \ \alpha = 1,\) and \(\varepsilon = 0.01\). We observe significant noise reduction in both cases. However, only in Fig. 5d the color transitions are handled correctly. This is due to the fact that our regularizer respects the periodicity, i.e., for the functional there is no jump in Fig. 4 since 0 and \(2\pi \) are identified. Using the classical TV regularizer, the values 0 and \(2\pi \) are not identified and have a distance of \(2\pi \). Hence, in the TVdenoised image there is a sharp edge in the middle of the image, see Fig. 5c.
6.8 Hue Denoising
The \(\text {HSV}\) color space is shorthand for Hue, Saturation, Value (of brightness). The hue value of a color image is \({\mathbb {S}}^1\)valued, while saturation and value of brightness are realvalued. Representing colors in this space better match the human perception than representing colors in the RGB space.
In Fig. 6a we see a part of size \(70 \times 70\) of the RGB image “fruits” (https://homepages.cae.wisc.edu/~ece533/images/).
The corresponding hue data are shown in Fig. 6b, where we used again the colormap HSV, cf. Fig. 4. Each pixel value lies, after transformation, in the interval \([0, 2\pi )\) and represents the angular value. Gaussian white noise with \(\sigma = \sqrt{0.001}\) is added to obtain a noisy image, see Fig. 6c.
To obtain the denoised image, in Fig. 6d we again used the same fixed point iteration, cf. [45], as before.
We see that the denoised image suffers from artifacts due to the nonconsideration of periodicity. The pixel values in the middle of the apple (the red object in the original image) are close to \(2\pi \) while those close to the border are nearly 0, meaning they have a distance of around \(2\pi \).
We use this TVdenoised image as starting image to perform the minimization of our energy functional. As parameters we choose \(s = 0.49, \ p = 2, \ \alpha = 2, \ \varepsilon = 0.006\).
Since the cyclic structure is respected, the disturbing artifacts in image in Fig. 6d are removed correctly. The edges are smoothed due to the high value of p, see Fig. 6e.
6.9 \({\mathbb {S}}^1\)Valued Image Inpainting
According to Example 3.10, the functional Open image in new window is coercive and Assumption 3.2 is satisfied. For \(\emptyset \ne \Omega \subset {\mathbb {R}}\) or \({\mathbb {R}}^2\) a bounded and simply connected open set, \(1< p < \infty \) and \(s \in (0,1)\) such that additionally \(sp < 1\) or \(sp \ge 2\) if \(N=2\) Lemma 6.6 applies which ensures that there exists a minimizer \(u \in W^{s, p}(\Omega ,{\mathbb {R}})\) of the lifted functional Open image in new window \(u \in W^{s, p}(\Omega ,{\mathbb {R}})\)
6.10 Inpainting of a \({\mathbb {S}}^1\)Valued Image
As a first inpainting test example, we consider two \({\mathbb {S}}^1\)valued images of size \(28 \times 28\), see Fig. 7, represented by its angular values. In both cases the ground truth can be seen in Fig. 7a, f. We added Gaussian white noise with \(\sigma = \sqrt{0.001}\) using the \(\text {MATLAB}\) buildin function \(\texttt {imnoise}\). The noisy images can be seen in Fig. 7b, g. The region D consists of the nine red squares in Fig. 7c, h.
The reconstructed data are shown in Fig. 7d, i.
For the twocolored image, we used as parameters \(\alpha = s = 0.3\), \(p = 1.01\) and \(\varepsilon = 0.05\). We see that the reconstructed edge appears sharp. The unknown squares, which are completely surrounded by one color, are inpainted perfectly. The blue and green color changed slightly.
As parameters for the threecolored image, we used \(\alpha = s = 0.4\), \(p=1.01\) and \(\varepsilon = 0.05\). Here again the unknown regions lying entirely in one color are inpainted perfectly. The edges are preserved. Just the corner in the middle of the image is slightly smoothed.
In Fig. 7e, j the TVreconstructed data are shown. The underlying algorithm ([31]) uses the split Bregman method (see [36]).
In Fig. 7e the edge is not completely sharp. There are some lighter parts on the blue side. This can be caused by the fact that the unknown domain in this area is not exactly symmetric with respect to the edge. This is also the case in Fig. 7j where we observe the same effect. Unknown squares lying entirely in one color are perfectly inpainted.
6.11 Hue Inpainting
As a last example, we consider again the hue component of the image “fruits”, see Fig. 8a. The unknown region D is the string \(\textit{01.01}\) which is shown in Fig. 8b. As parameters we choose \(p=1.1\), \(s=0.1\), \(\alpha = 2\) and \(\varepsilon = 0.006\). We get the reconstructed image shown in Fig. 8c. The edges are preserved and the unknown area is restored quite well. This can be also observed in the TVreconstructed image in Fig. 8d, using again the split Bregman method as before, cf. [31].
6.12 Conclusion
In this paper we developed a functional for regularization of functions with values in a set of vectors. The regularization functional is a derivativefree, nonlocal term, which is based on a characterization of Sobolev spaces of intensity data derived by Bourgain, Brézis, Mironescu and Dávila. Our objective has been to extend their double integral functionals in a natural way to functions with values in a set of vectors, in particular functions with values on an embedded manifold. These new integral representations are used for regularization on a subset of the (fractional) Sobolev space \(W^{s,p}(\Omega , {\mathbb {R}}^M)\) and the space \(BV(\Omega , {\mathbb {R}}^M)\), respectively. We presented numerical results for denoising of artificial InSAR data as well as an example of inpainting. Moreover, several conjectures are at hand on relations between double metric integral regularization functionals and single integral representations.
Notes
Acknowledgements
Open access funding provided by Austrian Science Fund. We thank Peter Elbau for very helpful discussions and comments. MH and OS acknowledge support from the Austrian Science Fund (FWF) within the national research network Geometry and Simulation, Project S11704 (Variational Methods for Imaging on Manifolds). Moreover, OS is supported by the Austrian Science Fund (FWF), with SFB F68, Project F6807N36 (Tomography with Uncertainties) and I3661N27 (Novel Error Measures and Source Conditions of Regularization Methods for Inverse Problems).
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