Abstract

The moment approach to solve the Cauchy problems is investigated. First, we consider the Cauchy problem for the Laplace equation, and we present a moment method for solving it in the case of a flat boundary. Second, we consider the reciprocity gap concept used to solve the problem of crack detection, as a stopping criterion and we study the case of noise data. Finally, we propose an application to the Cauchy problem for the Laplace equation, for the inpainting problem. Some numerical results showing the efficiency of the method proposed are also given.

1. Introduction

This paper deals with the inverse problem of crack detection defined by overdetermined boundary data, in the framework of the Laplace equation. The Cauchy problem for the Laplace equation is well known to be highly ill posed since Hadamard. That means the solution does not depend continuously on the given Cauchy data and any small change in the given data may cause large change to the solution [1–3]. In [4], the authors propose a method for data matching via a moment problem and give an application by using the Legendre polynomials. The idea to transform the problem into a moment problem has been initiated by Cheng et al. in the work [5]; then, several numerical methods have been proposed to solve this problem, such as Gram-Schmidt orthonormalization technique [6], the Backus-Gilbert algorithm [7], the conjugate gradient method [8], the fourth-order derivative method [9], the Fourier method [10], and the wavelets method [11], and we refer the reader to [12] for theoretical aspects of moment problems. Let us note here that the Talenti method is efficient but is restricted to the Hausdorff moment problem. In [4], a general character of the orthonormal basis chosen is presented: for any orthonormal basis, the authors can calculate the moments associated. Our motivation in this paper is based on three main features that make it according to our knowledge an efficient method. The first feature concerns its robustness: it compares very well with [5] at recovering processes in the sense that it allows the reconstruction of highly oscillating data. The second feature we would like to point out concerns the cost of the present method: since we have converted the data-matching problem into a moment one, then the recovering process turns out to be quasi-explicit; that is, it does not require any resolution of the forward problem. Finally, the third feature is shown in the general character of possible applications. In fact, for a potential application of present boundary data completion algorithm, one can consider the image processing area and more precisely restoration and inpainting problems. Our motivation is the following: image inpainting refers to image restoration techniques which can be based on crack detection. So, as a crack can be detected as it was presented in [4], then it will be easy to reconstruct a particularly damaged part of an image using the same idea. On the other part, as the result of the reconstruction depends on the number of projection used, so we propose to use the gap reciprocity function introduced by Andrieux and Ben Abda [13, 14], as a stopping criterion and study the error estimate.

This paper is organized as follows. The opening section is devoted to the Cauchy problem of the Laplace equation and its reformulation as a moment problem. In Section 3, the gap reciprocity function is presented in the framework of crack detection. We propose then to adapt the reciprocity gap concept, as a stopping criterion. In Section 4, we treat the inpainting problem using the Cauchy problem for the Laplace equation. The concluding section is devoted to some discussions and open problems.

2. Formulation of the Problem and Transformation to a Moment Problem

Let us consider a domain . We suppose that is a regular Lipschitz bounded domain. Let us denote by the outward unit normal to the boundary . We assume that is partitioned into two parts and . Given both the Dirichlet and Neumann conditions on , one wants to reconstruct the corresponding conditions on the remaining part of the boundary . This problem arises in electrostatic. One can suppose that represents the electrostatic potential in a body of which only the part is accessible to measurements. Hence, in this application, the corresponding inverse problem can be seen as the reconstruction of the boundary by determining the inaccessible measurements from those imposed on . For different applications of this problem, we can refer the reader to [15–17]. This problem can be expressed as a Cauchy problem for the Laplace equation and therefore termed mathematically as follows.

Find and on the remaining part of the boundary such that there exists a solution satisfying

In [5], Cheng et al. give a numerical method to approximate the solution of (2.1) using Green’s formula. In fact, by considering the system the authors transform the original problem to a moment problem whose numerical solution can be obtained accurately and efficiently. We refer the reader to [5] for the details of the proof of the following proposition.

Proposition 2.1. If the Cauchy problem (2.1) has a solution such that , then satisfies the following moment problem: where (2.2).
Conversely if satisfies (2.3), then there exists a solution of the Cauchy problem (2.1) such that .

To approach the solution of (2.3), the authors in [5] proved that the Cauchy problem for the Laplace equation (2.1) is equivalent to the Hausdorff moment problem, which has been studied by many researchers. More precisely, to obtain the desired result, they used the method of Talenti [12], which is based on the Gram matrix of a given set of vectors, and discussed the error estimation. We note here that Hilbert matrices are well known to be extremely ill conditioned, which will disturb the convergence of the solution.

Let us first recall the moment problem as given by Talenti [12]. We consider a Hilbert space , and we consider given basis functions and real numbers .

The moment problem consists in determining an unknown function such that As a trivial example of such a basis, we can take the polynomial basis , defined in a one-dimensional space. Working in a two-dimensional space, one can take and choose .

We have assuring the density of in .

In [4], the authors generalize the above result obtained in [5]. In fact, the authors propose to recover and on from their moments on any orthonormal basis of . The idea is to make the components and appear directly according to a given orthonormal basis of .

For a numerical convenience, let us suppose that is a bounded simply Lipschitz connected domain of given by where and is the Lipschitz curve in which connects the two points and such that .

We consider and as follows:

Let be an orthonormal basis of , where represents . To determine the moments of the lacking data directly on , we give the following results (see [4] for the proofs).

Theorem 2.2. There exists a set such that where the moments are defined by

Theorem 2.3. There exists a set such that where the moments are defined by

We note that an approximation of the solution on is given by the function such that , where are the moments of the function and are given by relation (2.9), and an approximation of is given by the function such that , where are the moments of the function and are determined by relation (2.11).

We present in Table 1 the error reconstruction with respect to the number of moment calculated for both the solution and its derivative . We remark that this error is decreasing when the number of moments increases. The reconstruction of both the solution and its derivative is excellent when the number of moments calculated varies between 9 and 13. Then, reconstruction error increases when the number of moments increases too, characterizing the fact that the problem (2.1) is ill posed and then implying a well-known property of a numerical instability.

Figure 1 shows the numerical results of the error reconstruction for both and .

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(b)

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Figure 1 

reconstruction error for both (a) and (b).

In the rest of this work, we propose to use the reciprocity gap concept as a stopping criterion.

3. The Reciprocity Gap Concept

The reciprocity gap principle has been initially introduced in solid mechanics throughout the Maxwell-Betti theorem in 1872, relating the responses to different external and internal forcing terms. More precisely, let us consider given overdetermined boundary data (a flux imposed on and a temperature measured on ). The Maxwell-Betti theorem establishes that the effect work of the flux in any field in equilibrium in with zero source term, is equal to the effect work of the temperature in the flux , implying that where . Let us note here that (3.1) is fundamentally derived from the symmetry of the operator involved in the modelisation of the phenomenon.

Using (3.1), we define the reciprocity gap functional as follows:

On the other part, from divergence theorem, Green’s second identity, and working with an unperturbed domain , we deduce that the reciprocity gap functional is identically zero and then

The reciprocity gap functional has been applied in different theories. For example, we can find its application to electromagnetic inverse scattering problems [18], or to groundwater flow problems [19]. Particularly, the nonidentical nullity of the reciprocity gap functional implies the presence of cracks in the domain, and this functional has been used to localize and determine the shape of one or several cracks in the initial domain [13, 14]. We propose in this work to apply this functional in order to recover missing boundary data on a part of the boundary domain from overdetermined ones existing on the remaining part, by applying the moment method. More precisely, since the left-hand side of (3.3) is not exact, then the idea is to use its approximation by the moment method applied in the previous section and then to apply the reciprocity condition as a stopping criterion.

Let us consider the following equations: Since the exact solution on the remaining part is not available, then we consider an approximation of as follows: where and are determined thanks to Theorems 2.2 and 2.3.

Let

Numerically, if the approximated solution converges to the exact one and the approximated flux converges to the exact one , then (3.6) converges to zero.

On the other part, from (3.6), we deduce that the condition yields which gives an error estimate of the recovering missing boundary data and then characterizes a nice stopping criteria. By considering the scalar product defined on the Hilbert space , numerically, the stopping criterion is such that the quantity is small enough.

4. Application to Image Inpainting

As a potential application of present boundary data completion algorithm, one can think of the image processing area. We consider here the inpainting problem. The goal of inpainting is to restore a damaged or corrupted image where one or several pieces of information have been lost, see [20–25]. Inpainting may also be an interesting tool for those who need to remove some parts of an image artificially such as scratches. The third application we are dealing with in this work is exactly this kind of problem, which is an inverse problem: the reconstruction of the image processed from its damaged version. Our motivation is the following: image inpainting refers to image restoration techniques which can be based on cracks detection. So, as a crack can be detected as it was presented in [4], then it will be easy to reconstruct a particularly damaged part of an image using the same idea. More precisely, let us consider the image processed, the whole external boundary, and the missing part of the image which represents in this work a given scratch. is determined by the intensity of its grey level and it splits the domain into two subdomains and . will represent the value of the image on and its corresponding flux. By considering the following Cauchy problem: we give the inpainting algorithm as follows.

Reconstruction Algorithm
(i)Calculation of and solutions of (4.1). (ii) detection of the crack by minimizing the following criterion:(iii) calculation of solution of the Neumann problem with this crack

Figure 2 illustrates this technique. We have considered the inpainting of a black line on the Lena image. This line divides the original domain into two subdomains. The black line represents : the missing part of the image. The corresponding reconstruction is shown in Figure 2(d). Then we added a Gaussian noise such that . The restored image with the missing part is shown in Figure 2(f). One can see that the image is still very well inpainted. However, it could be more interesting to generalize the inpainting problem for general missing parts and more general cracks in the image processed and not only parts determined by cracks in only one direction. This will be part of a forthcoming paper.

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(b)

(c)

(d)

(e)

(f)

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(b)
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Figure 2 

Inpainting of a black line on Lena image: original image (a), occluded image (b), detection of the missing part (c), inpainted image (d), occluded noisy image with (e), and inpainted noisy image (f).

5. Conclusion

We have presented in this work an application of the moment method used for crack detection to a particular problem in image processing related to image inpainting. A stopping criterion based on the reciprocity gap functional has been also introduced in this work. Our data recovering has a major advantage: it concerns the cost of the present method. In fact, since the initial Cauchy problem for the Laplace equation has been converted to a moment one, then the recovering process turns out to be quasi-explicit; it does not require any resolution of the forward problem. However, one can consider the very particular character of our method concerning the application to the inpainting problem, which represents here a major drawback, in the sense that this recovering is strictly related to a flat boundary. This drawback will be considered in a forthcoming paper by considering the reciprocity gap functional related to geometrical moments. Finally, some attention should be devoted to noisy data. We propose to study the robustness of our model. In fact, we propose to define a stopping criterion based on the reciprocity gap functional in case of Cauchy data noise, for a better comparison to real-life applications, and particularly for image processing applications as restoration.

Acknowledgment

This paper has been supported by the Deanship of Scientific Research at the University of Dammam under the Reference 201036.