Theoretical Foundation of the Weighted Laplace Inpainting Problem
Laurent Hoeltgen Andreas Kleefeld Isaac Harris Michael Breuß
math.AP 35J15, 35J70, 46E35, 94A08






















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    remarkRemarkhypothesisHypothesishypothesisHypothesisHypothesesclaimClaim
    
      Theoretical Foundation of the Weighted Laplace Inpainting Problem
      L. Hoeltgen and A. Kleefeld and I. Harris and M. Breuß
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    PDEPDEpartial differential equation
    Theoretical Foundation of the Weighted Laplace Inpainting Problem
        
      Laurent HoeltgenInstitute for Mathematics, Brandenburg Technical University, Platz der
        Deutschen Einheit 1, 03046 Cottbus, Germany,
        (hoeltgen@b-tu.de, breuss@b-tu.de).
      Andreas KleefeldInstitute for Advanced Simulation, Forschungszentrum Jülich GmbH, Jülich
        Supercomputing Centre, Wilhelm-Johnen-Straße, 52425 Jülich, Germany
        (a.kleefeld@fz-juelich.de).
      Isaac HarrisDepartment of Mathematics, Texas A&M University, 621A Blocker Building,
        3368 TAMU College Station TX 77843, USA
        (iharris@math.tamu.edu).
      Michael Breuß[1]
    
        
    ============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================
    
    
    
          Laplace interpolation is a popular approach in image inpainting using partial differential
          equations. The classic approach considers the Laplace equation with mixed boundary conditions.
          Recently a more general formulation has been proposed where the differential operator consists
          of a point-wise convex combination of the Laplacian and the known image data. We provide the
          first detailed analysis on existence and uniqueness of solutions for the arising mixed
          boundary value problem. Our approach considers the corresponding weak formulation and aims at
          using the Theorem of Lax-Milgram to assert the existence of a solution. To this end we have to
          resort to weighted Sobolev spaces. Our analysis shows that solutions do not exist
          unconditionally. The weights need some regularity and fulfil certain growth conditions. The
          results from this work complement findings which were previously only available for a discrete
          setup.
    
    
            Image Inpainting,
            Image Reconstruction,
            Laplace Equation,
            Laplace Interpolation,
            Mixed Boundary Conditions,
            Partial Differential Equations,
            Weighted Sobolev Space
    
    
          35J15, 35J70, 46E35, 94A08
    
    
    
    § INTRODUCTION
    
    Image inpainting deals with recovering lost image regions or structures by means of interpolation.
    It is an ill-posed process; as soon as a part of the image is lost it cannot be recovered correctly
    with absolute certainty unless the original image is completely known. The inpainting problem goes
    back to the works of Masnou and Morel as well as Bertalmío and colleagues <cit.>,
    although similar problems have been considered in other fields already before. There exist many
    inpainting techniques, often based on interpolation algorithms, but PDE[-based] approaches are
    among the most successful ones, see e.g. <cit.>. Among these, strategies based on the
    Laplacian stand out <cit.>. In that context, the elliptic mixed
    boundary value problem
    
    
        {3
                          -Δ u    = 0
                               in           Ω∖Ω_K
        
                          u    = f
                             in         ∂Ω_K
        ∂_n u    = 0
                             in         ∂Ω∖∂Ω_K.
    
    is very popular. Here, f represents known image data in a region Ω_K⊂Ω
    (resp. on the boundary ∂Ω_K) of the whole image domain Ω. Further,
    ∂_n u denotes the derivative in outer normal direction. An exemplary sketch of the layout
    of the problem is given in <ref>. Equations like <ref>, that involve different kinds
    of boundary conditions, are commonly referred to as mixed boundary value problems and in rare cases
    also as Zaremba's problem <cit.>. Image inpainting based on <ref> appears under various
    names in the literature: Laplace interpolation <cit.>, harmonic
    interpolation <cit.>, or homogeneous diffusion inpainting <cit.>. The latter name is
    often used in combination with the steady state solution of the parabolic counterpart
    of <ref>.
    
    Applications of image inpainting are manifold and range from art restoration to image compression.
    The earliest uses of <ref> go back to Noma and Misulia (1959) <cit.> and Crain
    (1970) <cit.> for generating topographic maps. Further applications include the works of
    Bloor and Wilson (1989) <cit.>, who studied partial differential equations for generating
    blend surfaces. Finally, refer to <cit.> for a broad overview on PDE[-based]
    inpainting and the closely related problem of PDE[-based] image compression.
    In the context of image reconstructions, (<ref>) is often favoured over other more complex
    models due to its mathematically sound theory, even though the strong smoothing properties may yield
    undesirable blurry reconstructions. Models based on anisotropic diffusion <cit.> or
    total variation <cit.> may be more powerful, but are much harder to grasp from a
    mathematical point of view. In the context of image compression, the data Ω_K used for the
    reconstruction can be freely chosen, since the complete image is known beforehand. The difficulty in
    compressing an image with a PDE lies in the fact that one has to optimise two contradicting
    constraints. On the one hand, the size of the data Ω_K should be small to allow an
    efficient coding, but on the other hand one wishes to have an accurate reconstruction from this
    sparse amount of information, too. The optimal data depends on the choice of the differential
    operator and the simplicity of the Laplacian offers many design choices for optimisation strategies
    to find the best Ω_K. Some of these approaches belong to the state-of-the-art methods in
    PDE[-based] image compression. We refer to <cit.> for a comparison of different
    PDE[-based] models and to <cit.> for data optimisation strategies in
    the compression context. <Ref> demonstrates the potential of such a careful data
    optimisation. <Ref> and <ref> show the reconstruction of an
    arbitrary rectangle. The reconstruction is severely blurred and the texture of the scarf is almost
    completely lost. On the other hand, <ref> represents an optimised set of 5% of the
    data points with the corresponding colour values. <Ref> depicts the corresponding
    reconstruction. Although the reconstruction has a few artefacts, its overall quality is very
    convincing.
    
    
    As already mentioned, finding the best pixel-data is a very challenging task. Mainberger et
    al. <cit.> consider the combinatorial point of view of this task while Belhachmi and
    colleagues <cit.> approach the topic from the analytic side. Recently <cit.>, the
    “hard” boundary conditions in <ref> have been replaced by softer weighting schemes that lead
    to models such as
    
    
        {
                          c( u-f ) + ( 1-c ) ( -Δ) u    = 0
                             in    Ω∖Ω_K
        ∂_n u    = 0
                             in    ∂Ω
        .
    
    with a weighting function c. Optimising such a weighting function is notably simpler, at least in
    discrete setups.
    <Ref> is well understood and there exist many results on existence, uniqueness, and regularity
    of solutions, see <cit.> for a generic analysis and <cit.> for a more
    specific analysis in the inpainting context with Dirichlet boundary conditions only. Finite
    difference discretisations of <ref> and <ref> have also been subject of several
    investigations in the past. One can show that the discrete counterpart of <ref> admits a
    unique solution as soon as the Dirichlet boundary set is non-empty <cit.>. Similarly, the
    discrete finite difference formulation of <ref> admits a unique solution if c is positive in
    at least one position <cit.>.
    An important question that arises in this context is what these discrete requirements relate to in
    the continuous setting. If we consider for example the following model problem that one may extract
    from the formulation of <ref>,
    
    
        {3
                          -Δ u    = 0
                               in           B_1∖ B_ε
        
                          u    = 0
                             in         ∂ B_1
        
                          u    = 1
                             in         ∂ B_ε.
    
    where B_r is a ball or radius r with centre at the origin, then one can show that a smooth
    solution exists for every ε > 0, but that no solution exists in the limiting case
    ε→ 0. Indeed, the solution is given by
    
    
        u(x,y) = ln( x^2+y^2)/2ln(ε)
    
    Yet, the discrete formulation will admit a unique solution independently of the choice
    of ε. It suffices that the corresponding matrix is block irreducible. We refer
    to <cit.> for a detailed discussion on the existence of solutions. To remedy the
    situation for the continuous formulation in <ref>, the authors of <cit.> have
    required that the set Ω_K should have positive α-capacity. The α-capacity
    (α > 0) of a subset E⊂ D of a smooth, bounded, and open set D is given by
    
    
        inf *∫_D∇ u^2 + α*u^2xu∈ U_E
    
    where U_E is the set of all functions u of the Sobolev space H^1_0(D) such that u
    ⩾ 1 almost everywhere in a neighbourhood of E. If Ω_K has positive
    α-capacity, then a solution of <ref> exists in the Sobolev space
    H^1(Ω) <cit.>. This requirement, that Ω_K must have positive capacity,
    can be understood as requiring that image pixels are “fat enough” to allow a reconstruction. It
    reconciles the continuous and discrete worlds and leads to a consistent theory on both sides. A
    higher regularity than H^1(Ω) can be achieved for specific constellations of the boundary
    data. A rather general theory is given in <cit.>. The author of <cit.>
    shows that a Hölder continuous solution exists if the data is regular enough. Finally, <cit.>
    discusses the regularity of solutions on Lipschitz domains. Let us mention that the authors
    of <cit.> have also discussed this inability of the Laplacian to recover images from
    isolated points and that they suggested absolutely minimising Lipschitz extensions as an
    alternative.
    The authors of this manuscript are not aware of any similar theory that would remedy the apparent
    discrepancy between <ref> and its discrete counterpart. The discrete setup is almost always
    solvable. On the other hand, solutions for the continuous model are only known for some special
    cases such as c being bounded between two positive constants in the interval (0,1), or c being
    itself a constant <cit.>. For inpainting purposes it is important that c may map to
    the whole unit interval and even beyond. Regions with c≡ 1 keep the data fixed and if c
    exceeds the value 1, then contrast enhancing in the reconstruction is possible,
    see <cit.>.
    Here, we attempt to bridge that gap between the discrete setup and the continuous model for the case
    when c maps to [0,1]. We show that a weak solution exists if certain assumptions on the weight
    functions are met. Special interest will be paid to occurring requirements on c and whether they
    correspond to discrete counterparts. We aim at applying the theorem of Lax-Milgram in purpose-built
    weighted Sobolev spaces. As a such, the contributed novelties of this manuscript are twofold. First
    we complement the well-posedness study of <ref> and c > 1, which has recently been discussed
    in <cit.> with the missing case where c maps to [0,1] and secondly, we introduce
    weighted Sobolev spaces to the image processing community. These spaces bear a certain number of
    interesting properties that can also be useful for other image analysis tasks.
    In the next section we first derive the weak formulation corresponding to <ref> and introduce
    the weighted Sobolev spaces where the solution is sought. Then we will state the necessary
    conditions on the weight function c that must be fulfilled to assert the existence of a solution.
    Finally we show that a unique solution exists.
    
    
    
    
    § INPAINTING WITH THE WEIGHTED LAPLACIAN
    
    
    As already mentioned in the previous section, the classic formulation for PDE[-based]
    inpainting with the Laplacian reads
    
    
        {3
                          -Δ u    = 0
                               in           Ω∖Ω_K
        
                          u    = f
                             in         ∂Ω_K
        ∂_n u    = 0
                             in         ∂Ω∖∂Ω_K.
    
    Using the findings from <cit.>, it is easy to show that <ref> is well-posed
    and that a unique weak solution exists in a subspace of H^1(Ω). With a weight function c
    that maps from Ω to 0,1, <ref> can also be rewritten as
    
    
        {3
                          c ( u-f ) + ( 1-c )( -Δ) u     = 0
                               in           Ω
        ∂_n u     = 0
                             in         ∂Ω
        .
    
    Interestingly, the latter formulation also makes sense if c:Ω→, a fact which was first
    exploited in <cit.>. If c has binary values in the set 0,1, then <ref> is
    equivalent to <ref> with the Dirichlet boundary conditions specified by f at those regions
    where c equals 1. <Ref> can also be interpreted from a physical or chemical point of view.
    We are in the presence of a stationary reaction-diffusion equation. The diffusive term
    (c-1)Δ u is responsible for spreading the information generated by the reactive term
    c  (u-f). The weight c is responsible for the speed at which information is generated and
    spread.
    If c is bounded between two non-negative numbers strictly smaller than one, then it follows
    from <cit.> that a solution exists in C^2,α( Ω). For inpainting purposes it is however important to allow c(x)=1 or even c(x) > 1. In
    order to derive the weak formulation of <ref> we follow the presentation in <cit.>,
    where the setup in <ref> with c > 1 was discussed by outlining its relationship to the
    Helmholtz equation.
    Let us now rewrite (<ref>) in a more suitable form. In a first step we explicitly set the
    regions where c≡ 1 apart.
    
    
        {
                          c( u-f ) + ( 1-c ) ( -Δ) u    = 0
                             in    Ω∖Ω_K
        
                          u    = f
                             in    ∂Ω_K
        ∂_n u    = 0
                             in    ∂Ω∖∂Ω_K
        .
    
    The previous reformulation implies that c < 1 almost everywhere in Ω∖Ω_K. A
    small detail that will become important in the forthcoming discussions. If we further assume that
    c∈ H^1(Ω,[0,1]), then we can apply the product rule and rewrite <ref> as
    
    
        {3
                          -( (1-c) ∇ u ) - ∇ c ·∇ u + c (u -f)    = 0
                               in           Ω∖Ω_K
        
                          u    = f
                             in         ∂Ω_K
        ∂_n u     = 0
                             in         ∂Ω∖∂Ω_K
        .
    
    In order to derive the weak formulation of <ref> within weighed Sobolev spaces let us first
    remark that if u solves <ref>, then v  u - f solves
    
    
        {3
                          -( (1-c) ∇ v ) - ∇ c ·∇ v + c v    = g
                               in          Ω∖Ω_K
        
                          v    = 0
                             in         ∂Ω_K
        ∂_n v    = h   in         ∂Ω∖∂Ω_K.
    
    with g  (1-c) Δ f and h  - ∂_n f. For convenience of writing, we
    will continue calling the sought solution of <ref>u and not v. Being able to solve
    <ref> is equivalent to being able to solve <ref>. Yet, this change lets us reduce the
    problem to the case with homogeneous Dirichlet boundary conditions. Deriving the associated weak
    formulation is now straightforward. Multiplying with a test function φ and integrating
    <ref> by parts implies that we must seek a function u∈ V, which solves
    
    
        ∫_Ω∖Ω_Kx
                (1-c) ∇ u ·∇φ -
                (∇ c ·∇ u) φ +
                c u φx_ B^c(u,φ)
              =
              ∫_Ω∖Ω_Kx g φx +
                ∫_∂Ω∖∂Ω_Kx h φx_ F(φ)  ∀φ∈ V
    
    Since c maps to the unit interval, we are in the presence of a so called degenerate elliptic
    equation <cit.> or sometimes also referred to as a PDE with non-negative
    characteristic form <cit.>. Such PDE are characterised by the fact, that their
    highest order term is allowed to vanish. This fact that the second order differential operator may
    vanish locally requires a more sophisticated analysis. The key issue to approach this kind of
    problems is to select the correct function space V and to place certain necessary restrictions
    onto c.
    Let us briefly explain why additional restrictions on c, resp. the solution space V, are vital
    to solve <ref>. The standard approach to show existence and uniqueness of a weak solution
    consists in applying the Lax-Milgram Theorem <cit.>. The crucial part will be the coercivity
    of the bilinear form B^c and the boundedness of B^c and F. Obviously the boundedness of
    B^c and F depends a lot on the choice of the space V and c. To show coercivity of the
    bilinear form, we must study the behaviour of
    
    
        ∫_Ω∖Ω_Kx (1-c) *∇ u^2 -
              (∇ c ·∇ u) u + c u^2x
    
    If c is, for example, piecewise constant in <ref>, then the middle term vanishes almost
    everywhere. Any function u which is equal to 0 whenever c is positive and equal to an arbitrary
    constant when c is 0 will force the bilinear form to be 0. Yet the norm of u can be arbitrarily
    large, which hinders us from showing coercivity. In order to prevent this situation, the following
    assumptions seem reasonable:
    
    
      *  The function c should have a certain regularity, e.g. being continuous. Then arbitrary
          switching between regions where c takes different constants is not possible anymore.
    
      *  The space V in which we seek solutions should fix its elements at certain boundaries. This
          would prevent solutions u“slipping” away by adding constants that are invisible to the
          bilinear form.
    
    From these observations it becomes apparent that the function c must go, at least partially, into
    the definition of the space V. We consider such an approach in the following section by using
    weighted Sobolev spaces and provide precise requirements that assert the well-posedness
    of <ref>.
    
    
    
    
     §.§ Weighted Sobolev Spaces
    
    
    Weighted Sobolev spaces have been studied intensively in the past. Their uses are manifold, but they
    are most often found in the analysis of PDE with vanishing or singular diffusive term. The
    works <cit.> give an almost complete overview of their usefulness.
    For the sake of completeness, we shortly summarise how these spaces are set up.
    In the following we denote by W_Ω the set of weight functions ω, i.e. ω is a
    measurable and almost everywhere positive function in some domain Ω. For 1 ⩽ p <
    ∞ and ω∈ W_Ω we define the corresponding weighted L^p space as
    
    
        L^p(Ω;ω) *uΩ→*u_L^p(Ω;ω)( ∫_Ω*u(x)^pω(x)x)^1/p < ∞
    
    In a similar way as Sobolev spaces refine the Lebesgue spaces we can also refine our weighted
    L^p spaces by including the weak derivatives (defined in the usual sense) into the norm. In such
    cases, different weights for different derivatives are also possible. For a given collection
    S_kω_α∈ W_Ω α⩽ k of weight
    functions, we denote by W^k,p(Ω; S_k) the set of all functions u defined on Ω
    whose (weak) derivatives D^αu of order *α⩽ k (α being a
    multi-index) belong to L^p( Ω;ω_α). We can equip this vector space
    with the norm
    
    
        *u_W^k,p(Ω;S_k)(
                          ∑_*α⩽k∫_Ω*D^αu(x)^pω_α(x)x)^1/p
                    =
                    (
                          ∑_*α⩽k*D^αu^p_L^p(Ω;ω_α))^1/p
    
    One can show that the space W^k,p(Ω;S_k) is a Banach space if ω_α∈
    L^1_loc(Ω) and ω_α^-1/p-1∈ L^1_loc(Ω) for
    all α⩽ k, see <cit.>. Note that this requires that all
    derivatives up to the order k must be attributed such a weight ω_α. However, one can
    also show that W^k,p(Ω;S̃_k) is still complete if S̃_k⫋
    S_k contains at least one weight ω_α with α = k and a weight for
    α = 0, see <cit.>.
    We remark that for p=2 there is a canonical choice for a scalar product:
    
    
        *uv_W^k,2(Ω;S_k)∑_*α⩽k∫_Ω D^αu(x)D^αv(x)ω_α(x)x
    
    Thus, with a suitable choice of weights we obtain a Hilbert space. If all the weight functions are
    constant and equal to one, then our weighted spaces coincide with the usual definition of Sobolev
    spaces. We refer to <cit.> for a more complete listing of possible weighted Sobolev
    space constructions. Finally, we remark that an alternative description of reasonable weight
    functions can be given in terms of so called Muckenhoupt A_p weights. We refer to <cit.>
    for more information.
    By looking at <ref> it becomes apparent why these weighted Sobolev spaces are useful. The
    function c (resp. 1-c) can be considered as a weight function and simply be integrated into the
    space definition. This simplifies the proofs to show existence and uniqueness, since boundedness and
    coercivity are easier to show and theorems such as Lax-Milgram can by applied in any Hilbert
    space.
    Our goal now will be to consider the corresponding weak formulation of <ref> in a suitable
    weighted Sobolev space V. By applying the Theorem of Lax-Milgram in these spaces we will show the
    existence and uniqueness of a weak solution of <ref>.
    We make the following assumptions on our setup. These assumptions will hold throughout the whole
    paper, unless mentioned otherwise. We remark that some of these assumptions can probably be
    weakened, nevertheless they are not uncommon for image processing purposes and ease the discussion
    on a few occasions.
    
    
      * Ω is an open, connected and bounded subset of ^2 with 𝒞^∞
          boundary ∂Ω.
      * Ω_K⫋Ω is a closed subset of Ω with positive Lebesgue measure.
          It represents the known data locations used to recover the missing information on
          Ω∖Ω_K. The interpolation data is given by f(Ω_K). The boundary
          ∂Ω_K, is assumed to be 𝒞^∞, too. This set Ω_K is
          characterised by c(x)≡ 1.
      * fΩ→ is a 𝒞^∞ function representing the given
          image data to be interpolated by the underlying PDE.
      *  The boundaries ∂Ω and ∂Ω_K do not intersect and neither of the
          boundaries ∂Ω or ∂Ω_K are empty.
      *  The function c maps from Ω to the interval [0,1], admits weak first order
          derivatives, and is an element of
          L^1_loc( Ω∖Ω_K).
    Let us briefly comment on these requirements. The first part of <ref> is trivially fulfilled
    by images. Its second part is more restrictive. Assuming the boundary of Ω to be piecewise
    𝒞^∞ would be more realistic, but this would also reduce the regularity of the
    solution. <Ref> and <ref> do not impose any severe restrictions for image processing
    tasks. Images can always be rendered 𝒞^∞ by convolving them with a Gaussian.
    <Ref> is necessary for technical reasons. If the Neumann and Dirichlet boundary conditions
    meet each other, it is possible to generate setups that lead to contradicting requirements. Finally,
    <ref> is necessary to assert the existence of our weighted Sobolev spaces.
    The weights for our space definition should be chosen such that the bilinear form is equivalent to
    the norm of our space. Often, the multiplicative factors of the individual derivatives in the
    bilinear form offer themselves as viable choices for this task. In our case however, the function
    c may vanish locally. This prevents us from using 1-c and c as weights to define a norm. They
    only give us a seminorm structure. Such a situation is briefly described in <cit.>. We mostly
    follow that presentation and we propose the following correspondence between multi-indices
    α∈_0^2 and weights ω_α
    
        ω_00 1,  ω_10 1-c,  ω_01 1-c
    
    This yields the scalar product and norm
    
    
        uv_V   ∫_Ω∖Ω_Kx
                                    (1-c) ∇ u ·∇ v + u v x
        *u_V   ( ∫_Ω∖Ω_Kx
                                (1-c) *∇ u^2 + u^2x)^1/2
    
    as well as the following definition for our space V:
    
    
        V ϕ∈ W^1,2(Ω∖Ω_K;S_c)
              ϕ≡ 0 on ∂Ω_K
    
    In addition, we define the following seminorm
    
    
        u_V( ∫_Ω∖Ω_Kx
                          (1-c) *∇ u^2x)^1/2
    
    Finally, following the presentation in <cit.>, we note that the bilinear form B^c in
    <ref> can be written compactly as a ternary quadratic form
    
    
        B^c(u,φ) = ∑_α,β⩽ 1∫ _Ω∖Ω_Kx
              a_α,β D^β u D^αφx
    
    where α, β are multi-indices in _0^2. The weights a_α,β must
    be set as follows to yield our model:
    
    
        a_10,10 = a_01,01 = 1 - c(x),  
                    a_00,00 = c(x) 
        
                    a_00,10 = - ∂_x c(x),  
                    a_00,01 = ∂_y c(x)
    
    and a_α,β = 0 for any other combination of multi-indices. In addition to the
    previous assumptions, we assume further:
    
    
      *  There exists a constant κ > 0, such that for all α,
          β⩽ 1, α≠β,
          
        *a_α,β⩽κ√(a_α,αa_β,β)
    
          almost everywhere in Ω∖Ω_K. For our choice in <ref>, this reduces to
          
          
        ∂_x c⩽κ√(c(1-c)),  ∂_y c⩽κ√(c(1-c))
    
          almost everywhere in Ω∖Ω_K.
    
      *  There exists a constant κ' > 0, such that for all real vectors ξ∈^3 with
          entries ξ_γ (γ being a multi-index in ℕ_0^2 such that
          γ⩽ 1) we have
          
          
        ∑_α,β⩽1 a_α,βξ_αξ_β⩾κ'∑_γ⩽1 a_γ,γξ_γ^2
    
          almost everywhere in Ω∖Ω_K. For our choice in <ref>, this reduces to
          
          
        [][0.8]
                                c ξ_1^2 +
                                (1-c) ξ_2^2 +
                                (1-c) ξ_3^2
                                - ∂_x c  ξ_1ξ_3 + ∂_y c ξ_1ξ_2
        ⩾κ'
                                ( (1-c) ξ_3^2 + (1-c) ξ_2^2 + c ξ_1^2)
                          
        ⇔     (∂_y c) ξ_1ξ_2 - (∂_x c) ξ_1ξ_3⩾
                          (κ'-1)
                          ( (1-c) ξ_3^2 + (1-c) ξ_2^2 + c ξ_1^2)
    
          almost everywhere in Ω∖Ω_K.
    <Ref> and <ref> are technical requirements that are necessary for the coercivity and
    the boundedness of B^c. They cannot be avoided without substantial changes to the forthcoming
    proofs. Let us remark, that <ref> can be deduced from <ref>, provided that
    κ < 1/2 holds. We refer to <cit.> for a detailed proof. Equations <ref>
    and <ref> enforce a certain well-behaviour on c, by restricting for example the growth
    speed.
    The following findings are a direct consequence of the foregoing results.
    
    
          The bilinear form B^c from <ref> is continuous.
          
          
                By using <ref> and the Hölder inequality we obtain.
                
                
        *B^c(u,φ)   ⩽∑_α,β⩽ 1∫ _Ω∖Ω_Kxa_α,βD^β uD^αφx
           ⩽maxκ, 1∑_α,β⩽ 1∫ _Ω∖Ω_KxD^β u√(a_β,β)D^αφ√(a_α,α)x
           ⩽ K D^β u_VD^αφ_V
    
                where K is some positive constant. We emphasise that the last estimate requires
                c ⩽ 1 almost everywhere to be valid.
          
    
          There exists a constant κ' >0 such that the bilinear form B^c from <ref>
          satisfies the estimate B^c(u,u) ⩾κ' u_V^2.
          
          
                We replace ξ_α by D^α u and ξ_β by D^β u in
                <ref>. Integrating the resulting inequality over Ω∖Ω_K
                yields
                
                
        B^c(u,u)
                          =
                          ∑_α,β⩽1∫_Ω∖Ω_Kx
                          a_α,βD^α u D^β u
                          x⩾κ'∑_γ⩽1∫_Ω∖Ω_Kx
                          a_γ,γ( D^γ u )^2x⩾κ' u_V^2
    
    To complete the proof of the coercivity of the bilinear form B^c we need a Friedrichs-like
    estimate of the form u_V ⩽ K  u_V with some positive
    constant K. The particular formulation and preliminaries that we need can be found
    in <cit.> as Theorem 2.3. We repeat it here verbatim for the sake of completeness but refer
    to its source for a detailed proof.
    In the following theorem we denote by W_c(X) the subset of weights on the space X which are
    bounded from above and below by positive constants on each compact subset Q⊂ X, i.e. we only
    allow our weights to degenerate at the boundary of the domain. The next theorem also considers a
    constant A which is defined as follows. For an arbitrary domain X we assume that we can
    write
    
    
        X = ⋃_k=1^∞ X_k
    
    where (X_k)_k is a sequence of bounded domains whose boundary can be locally described by
    functions satisfying a Lipschitz condition and where X_k⊂X_k⊂ X_k+1
    holds for each k. Finally, let X^k X∖X_k and define
    
    
        A_k = sup_u_W^k,p(X; S_k)⩽ 1u_L^p(X^k; w_0)
    
    where w_0∈ S_k is the weight that corresponds to *α=0. We define additionally
    A lim_k→∞ A_k. Obviously A∈ [0,1] always holds. This number A is also
    the ball measure of non-compactness of the embedding W^k,p(X; S_k) → L^p(X; w_0),
    see <cit.>. One can interpret the number A as the distance from the embedding
    operator to the next closest compact operator from W^k,p(X; S_k) into L^p(X; w_0). Also,
    the numbers A_k can be understood as indicators on how much “weight” is put onto the function
    along the boundary. A_k < 1 means that there is at least some weight on the derivatives or
    inside the domain. Note that in our setup <ref> simplifies to
    
    
        A_k = sup_u_W^1,2(Ω∖Ω_K;S_k)⩽ 1u_L^2(X^k)
    
    where X^k is the complement of a set X_k⊂Ω∖Ω_K and where
    S_k is the set of weights from <ref>.
    For the following theorem it is important that A < 1, i.e. the weight is not completely
    concentrated on the boundary. Let us remark that this requirement is in accordance with the discrete
    theory established in <cit.>. In the discrete setting, there should be at least one
    position with positive weight in the interior of the domain.
    Let us emphasise that for our task at hand, such a construction with the requirement that A < 1 is
    an additional regularity assumption on our image data f and the mask function c. Indeed, part of
    the boundary of the domain that we consider is fixed where c≡ 1. Since the Ω_k need
    boundaries that can be described locally by functions that fulfil a Lipschitz condition, this
    requirement carries over to the function c.
    As already mentioned, the next theorem is a almost verbatim copy of Theorem 2.3 in <cit.>.
    
    
          Suppose 1 ⩽ p < ∞ and S_k⊂ W_c(X). Let ℓ be a functional on
          W^k,p(X; S_k) with the following properties.
          
          
      * ℓ is continuous on W^k,p( X; S_k)
      * ℓ(λ u) = λℓ(u) for all λ > 0 and all
                u∈ W^k,p(X, S_k).
          
      *  If u∈ P_k-1∩ W^k,p(X; S_k) (P_k-1 being the set of all
                polynomials on ^n of degree less than k) and ℓ(u) = 0, then u = 0.
          
          Let A < 1. Then there is a constant κ_0 such that
          
          
        ∫_X*u^p w_0x⩽κ_0(
                          *ℓ(u)^p + ∑_α=k*D^α u_L^p(X;w_α)^p)
    
          Here, w_0 is the weight that corresponds to α=0.
    
    The previous theorem can be seen as a generalisation to weighted spaces of a well-known theorem for
    constructing equivalent norms out of seminorms in regular Sobolev spaces. See Theorem 7.3.12
    in <cit.>. <Ref> can also be considered as a higher dimensional generalisation of the
    Hardy inequality. We refer to <cit.> for an extensive treatise on this inequality.
    We now use <ref> with p=2, k=1, n=2, w_0≡ 1, w_α = 1 - c for all
    α and
    
    
        ℓ(u) = ∫_∂Ω_Kx u x
    
    With these choices we obtain the Friedrichs' inequality in our space V:
    
    
        u^2_L^2(Ω∖Ω_K)⩽κ_0 u_V^2
    
    <Ref> is the final key building block in showing the existence and uniqueness of a solution
    of our PDE. It allows us to show the coercivity of our bilinear form.
    
    
          If <ref> holds, i.e. the requirements of <ref> are fulfilled for the choice of
          ℓ from <ref> and for our selection of weights for our space V, then the bilinear
          form B^c from <ref> is coercive.
          
          
                Equation <ref> immediately implies that
                u_V^2⩽ (1+κ_0) u_V^2. In
                combination with <ref> it follows that
                
                
        B^c(u,u) ⩾κ' u_V^2⩾κ'/1+κ_0u_V^2
    
          
    <Ref> completes the analysis of our bilinear form B^c. It remains to show that the
    right-hand side of our weak formulation is continuous if we want to apply the Theorem of Lax-Milgram.
    This final step is done in the following proposition.
    
    
          The linear operator F from <ref> is continuous, provided that g, Δ f, and
    ∇ f/√(1-c) are in L^2(Ω∖Ω_K).
          
          
                We remark that φ∈ V is 0 along ∂Ω_K, and thus we can extend
                the boundary integral over that part. Using the Hölder inequality and Green's first
                identity, we obtain
                
                
        F(φ)   ⩽∫_Ω∖Ω_Kxgφx +
                          *∫_∂Ω∖∂Ω_Kx hφx
           ⩽g_L^2(Ω∖Ω_K)φ_L^2(Ω∖Ω_K) +
                          *∫_Ω∖Ω_KxΔ f φ + ∇ f ·∇φx
           ⩽g_L^2(Ω∖Ω_K)φ_V +
                          Δ f_L^2(Ω∖Ω_K)φ_V +
                          *∫_Ω∖Ω_Kx∇ f ·∇φx
    
          The last integral can be estimated as follows
          
          
        *∫_Ω∖Ω_Kx∇ f ·∇φx   =
                          *∫_Ω∖Ω_K∇ f/√(1-c)√(1-c) ∇φx
           ⩽*∇ f/√(1-c)_L^2(Ω∖Ω_K)*∇φ_L^2(Ω∖Ω_K;1-c)
           ⩽*∇ f/√(1-c)_L^2(Ω∖Ω_K)*φ_V
    
          Therefore, it follows that
          
          
        F(φ)⩽( g_L^2(Ω∖Ω_K) +
                    Δ f_L^2(Ω∖Ω_K) +
                    *∇ f/√(1-c)_L^2(Ω∖Ω_K))
                    φ_V
    
          Thus, F is a bounded linear functional.
          
    
    We can now combine our results to prove our main result.
    
    
          The weak formulation <ref> of the mixed boundary value problem <ref> has a
          unique solution in the space V. In addition, we know that
          
          
        *u_V⩽1 + κ_0/κ'*F_V^*
    
          where κ'/1 + κ_0 is the constant from <ref>. Here, V^*
          denotes the dual space of V.
          
          
                From <ref> and <ref> it follows that our bilinear form B^c is bounded
                and coercive. <Ref> shows that the corresponding right-hand side F is bounded,
                too. Therefore, from the Theorem of Lax-Milgram (see <cit.>) it follows that
                there exists a unique u∈ V such that B^c(u,φ) = F(φ) holds for all
                φ∈ V. In addition, this u fulfils
                u⩽1 + κ_0/κ'*F_V^*
    
    Our weighted Sobolev space V might be unsuited for other applications as it is very problem
    specific. Having an embedding from V to some standard Sobolev space W^k,2 would be very useful
    in view of the many embedding theorems for these latter spaces which can be used to show a higher
    regularity of the solution. It would also help in comparing solutions obtained when c maps to the
    unit interval but does not reach 0 or 1. In that case, the solutions live in W^1,2. By
    construction of our space V, we immediately obtain u∈ L^2(Ω∖Ω_K).
    However, this result does not even acknowledge the existence of the weak derivatives. Any claims
    beyond that are difficult to do. There exist a certain number of results concerning the embedding of
    weighted Sobolev spaces into other spaces, however, their assumptions are often very abstract or
    quite restrictive, e.g. all weights must be identical. We refer the reader to the
    works <cit.> for a detailed analysis on this topic.
    
    
    
    
     §.§ What happens if c greater equal 1?
    
    
    Let us shortly discuss the consequences of c exceeding its upper limit 1. Similar conclusions can
    also be drawn for the case c ⩽ 0, however, this latter situation usually does not occur
    in practice.
    There are no restrictions on c when establishing the weak formulation. Applying c ⩾ 1,
    the main difference would be that 1-c and c would have different signs. In order to follow the
    same strategy as in this paper one would have to find suitable weights for the space definition. In
    <cit.> the authors discuss the situation when one of the weights in the weak formulation is
    negative and they suggest to multiply the negative weight with another negative constant to render
    it positive. Afterwards, a similar approach as in this paper could be possible.
    In our situation there exists a second issue that may be harder to resolve. We required certain
    restrictions on the growth of the function c, which were of the form
    
    
        *∂_z c⩽κ√(c(1-c))
    
    for z being either x or y. The left-hand side of this inequality will always be a non-negative
    real number. However, the right-hand side becomes complex-valued once c exceeds 1. These growth
    restrictions were important to show the coercivity of the bilinear form.
    To conclude this section we remark that an alternative approach by means of the Helmholtz equation
    already exists for the case c>1, see <cit.>. However, this approach uses different
    assumptions and yields a well-posedness theory in different spaces.
    
    
    
    
     §.§ Summary: What is needed to assert the existence of a solution?
    
    
    In this section, we summarise the necessary conditions that we had to impose on our data throughout
    the paper.
    
    
      *  The function c must have weak derivatives of first order and map the domain Ω to the
          interval [0,1].
    
      *  The function c must fulfil <ref>.
    
      *  The function c must fulfil <ref>.
    
      *  The sequence (A_k)_k defined by <ref> must fulfil
          lim_k→∞ A_k < 1.
    
    The first three requirements can easily be verified if a concrete instance of c is given. However,
    the last requirement can probably only be checked in particular cases.
    
    
    
    
    § CONCLUSION
    
    We have shown that a solution to the inpainting problem with the weighted Laplacian exists if the
    weight is a function that maps into the interval [0,1]. The effort to assert the existence and
    uniqueness of such a solution was significant. The well-posedness of the task can be asserted if
    certain regularity conditions on the weight function c are met. These requirements are similar to
    what is needed to show existence and uniqueness of a solution in a discrete setting. The results in
    this manuscript complete the analysis of the inpainting problem with the Laplacian. While the theory
    for the discrete setup was complete for any choice of c ⩾ 0, the continuous theory only
    covered the setup where c > 1. This work complements the setup where c maps to [0,1].
    
    siamplain
