## Abstract

This article investigates a Fourier-based algorithm for computing heterogeneous material parameter distributions from internal measurements of physical fields. Within the framework of the periodic scalar conductivity model, a pair of dual Lippmann–Schwinger integral equations is derived for the sought constitutive parameters based on full intensity or current density field measurements. A numerical method based on the fast Fourier transform and fixed-point iterations is proposed. Convergence, stability and approximation quality of the method are analysed. For materials with small contrast, a first-order Born-like approximation is also obtained. Overall, the proposed reconstruction approach enables a direct conversion of full-field measurement images, possibly noisy, into maps of material conductivity. A set of numerical results is presented to illustrate the performance of the method.

## 1. Introduction

This study focuses on the quantitative imaging problem for fields of constitutive parameters associated with a given material body from full-field images acquired when monitoring its behaviour under applied excitation. This inverse problem bears relevance to applications such as material characterization, non-destructive material testing or medical imaging. Available experimental methods bring a wealth of technical solutions for full-field measurements of internal physical fields that actually encode information about the unknown material parameters. State-of-the-art and non-invasive two-dimensional or three-dimensional imaging modalities include X-ray, neutron diffraction tomography, digital image correlation [1], ultrasound or magnetic resonance imaging [2], as well as multi-physics imaging approaches such as photoacoustic, thermoacoustic or electroacoustic tomography [3].

For a given physical model governed by a partial differential equation (PDE), which combines the constitutive and equilibrium equations, a typical scenario is that a high-resolution imaging modality provides internal measurements of solutions to that PDE for a set of prescribed excitations. Such full-field data then serve in a second step where the forward model is revisited as a redundant PDE system for the unknown parameter fields in which the available measurements constitute the spatially varying coefficients [4,5]. Such an inversion from full-field data allows the recovery of quantitative information at small scales. This inverse problem is mathematically mildly ill-posed [6] so that regularization is required to accommodate noisy measurements.

Inverse problems have witnessed the growth and flourishing of reconstruction methods from full-field measurements, in particular in the area of solid mechanics [7] for the identification of mechanical parameters from displacement or strain field data. Dedicated methods include finite-element model updating [8], the constitutive equation gap method [9], the virtual field method [10], the equilibrium gap method [11], as well as adjoint-weighted and gradient-based variational approaches [12,13]. The wealth of existing approaches to the inverse problem we consider here is somehow a consequence of the branching of forward solution methods themselves.

The full-field image conversion method proposed here stems from Fourier-based methods for simulating the behaviour of heterogeneous materials. Since the seminal paper [14], fast Fourier transform (FFT)-based methods have been successfully applied to a wide range of mechanical problems [15–18] as they allow fast and accurate computations of complex material responses through direct processing of two-dimensional or three-dimensional *images* of their constitutive parameters. At the core of these approaches is a FFT-based iteration method to compute the mechanical field solution to a Lippmann–Schwinger equation. In the field of inverse problems, analogue integral equation-based formulations and reconstruction methods have been developed in optical tomography [19], inverse scattering [20–22] and electrical impedance tomography [23] based on boundary measurements.

On this basis, this study aims at recasting the forward FFT-based approach to tackle the *inverse* problem with internal measurements by converting images of physical fields into constitutive parameter maps for heterogeneous materials. While our overarching goal is the treatment of full-field data pertaining to solid mechanics, the proposed method is discussed here in the context of the scalar conductivity model which has the same mathematical structure as the elasticity model yet avoids resorting to tensorial notations. Note that a Fourier-based *forward* solution method corresponding to this model is discussed in [24,25]. In [26], ch. 2, it is shown that the conductivity model arises in a number of physical problems such as electrical conductivity, magnetism, diffusion or thermal conduction. In particular, experimental infrared techniques are applicable to the latter case [1] and full-field thermal images constitute data suitable for this study.

The contribution of this work is twofold. First, in §2 it establishes a pair of dual Lippmann–Schwinger integral equations for the sought material conductivity and resistivity fields based on knowledge of a set of internal measurements of intensity or current density fields. Second, it investigates an associated iterative calculation method for the sought parameters in §3, and the overall reconstruction approach is analysed in §4. Finally, a set of numerical examples based on synthetic full-field measurements is presented in §5 to assess the performance of the proposed method.

### (a) Statement of the problem

Consider a periodic medium with representative volume element *d*=2 or 3, and the associated conductivity problem

where *u* is the associated scalar potential. Defining the intensity field as ** e**:=

**∇**

*u*+

**E**, the angular brackets 〈⋅〉 are used to denote the spatial average over

*u*to (1.1) that is unique up to an additive constant and that belongs to the functional space

*u*〉=0.

The framework adopted in this study relies on the assumption that a number *L* of experiments can be performed by varying the imposed mean intensities as **E**=**E**_{ℓ} with ℓ=1,…,*L* in problem (1.1), in association with the ability to measure internally the corresponding set of intensity field solutions, i.e. *e*_{ℓ}(** x**)=

**∇**

*u*

_{ℓ}(

**)+**

*x***E**

_{ℓ}, for all

*e*_{ℓ}(

**)} in order to reconstruct the material conductivity distribution**

*x***↦**

*x**γ*(

**) within**

*x**γ*〉.

The full-field data constitute a set of images whose frame is the unit cell *d*,
*d*-dimensional Fourier transform

## 2. Integral formulations

The first step to derive the sought numerical method consists in establishing a Lippmann–Schwinger equation for the unknown conductivity field. In this section, we show how to obtain such an integral equation, in terms of either the conductivity function *γ*(** x**) in a primal formulation or the resistivity function

*ρ*(

**):=**

*x**γ*

^{−1}(

**) in a dual one.**

*x*### (a) Primal formulation

#### (i) Auxiliary problems

As a starting point, given *k*=1,…,*d* together with right-hand side terms *γ*,
*b*_{k}} constitutes an orthogonal basis of *d* equations entails
** ξ** in equation (2.2), then the material conductivity function

*γ*featured in the set (2.1) of

*d*problems satisfies

To express this solution in real space, let the spatially convoluted product *f***g* at point ** x** be defined as

*f*,

*g*. The associated discrete convolution is such that

### Remark 2.1

In equation (2.6), the real-space counterpart ** Γ**(

**) of the Green tensor**

*x*The vectorial relation (2.2) in *γ*(** x**). It is associated with the following set of necessary compatibility conditions that have to be satisfied by the prescribed data terms

*τ*_{k}(

**) for**

*x**k*=1,…,

*d*:

Moreover, when *γ*:
*τ*_{k} for *k*=1,…,*d* are

Reciprocally, the solution (2.6) to equation (2.9) is also a solution to the initial set of auxiliary problems (2.1) provided that the compatibility condition (2.8) is satisfied. Indeed, equation (2.8) entails that there exists ** T**(

**)=−**

*x***∇**

*ϕ*. From (2.9), one deduces that Δ(

*γ*−

*ϕ*) = 0 and thus

**∇**

*γ*=−

**(**

*T***), which yields a solution to the initial problems according to the definition (2.7) of**

*x***(**

*T***).**

*x*#### (ii) Lippmann–Schwinger equation

Now the set of auxiliary problems and the associated solution (2.6) are readily used as follows. For all *k*=1,…,*d*, let ℓ_{k}∈{1,…,*L*} denote a companion label, to be determined, such that the original problem (1.1) with prescribed mean **E**_{ℓk} and associated solution *e*_{ℓk} is recast as an auxiliary problem given by equation (2.1) for the conductivity field *γ*(** x**) and where the right-hand-side term is defined according to:

### Definition 2.2

For all *k*=1,…,*d* and ℓ_{k}∈{1,…,*L*} let

Therefore, one readily obtains the following Fredholm integral equation of the second kind for the unknown conductivity field *γ*:
*γ*. Equation (2.10) constitutes the sought Lippmann–Schwinger integral equation, which is a linear equation for the unknown *γ*(** x**).

### (b) Dual formulation

A dual version of the integral equation (2.10) can be established by observing (e.g. [25]) that the forward problem (1.1) itself can be recast in dual form in terms of a vector potential ** v** as

*ρ*:=

*γ*

^{−1}while

**:=curl**

*j***+**

*v***denotes the current density with prescribed mean**

*J***, up to an additive constant, that belongs to the space**

*v**L*full-field measurement maps of current solutions

*j*_{ℓ}(

**)=curl**

*x*

*v*_{ℓ}(

**)+**

*x*

*J*_{ℓ}, for all

*L*that are associated with experiments with varying mean field

*J*_{ℓ}.

#### (i) Auxiliary problems and Lippmann–Schwinger equation

As we have done in §2a to establish the primal integral formulation, a set of auxiliary problems associated with equation (2.11) is now introduced. Given a prescribed mean resistivity *b*_{k}} of *k*=1,…,*d*, let us consider the following auxiliary problems for the unknown scalar field *ρ*:

In Fourier space, equation (2.12) is recast for all *k*=1,…,*d* as:
** ξ** to this equation and expanding the resulting triple products yields

*b*_{k}and summing the resulting equations for all

*k*=1,…,

*d*one finally obtains

**denotes the second-order identity tensor. Therefore, defining in Fourier space the dual periodic Green operator**

*I**ρ*featured in the set of

*d*auxiliary problems (2.12) satisfies

The solution (2.14) to the set of auxiliary problems is now employed to obtain the sought integral equation for the field *ρ*=*γ*^{−1}. Considering the original problem (2.11) with prescribed mean current *J*_{ℓk} and associated solution *j*_{ℓk} for a given index ℓ_{k}∈{1,…,*L*}, for all *k*=1,…,*d*, then the data terms in equation (2.12) are defined according to:

### Definition 2.3

For all *k*=1,…,*d* and ℓ_{k}∈{1,…,*L*} let

Substituting these data terms in equation (2.14) directly yields the following Lippmann–Schwinger equation for the unknown resistivity field *ρ*:

#### (ii) Duality principle

The previous developments that led to the primal integral equation (2.10) and its dual counterpart (2.15) are in agreement with a duality principle associated with the inverse problem considered here. Well known for forward solution methods (e.g. [25] and [26], §12.9), this principle allows one to apply any result pertaining to the integral equation (2.10) to its dual (2.15) owing to appropriate field and operator substitutions.

Relying on this duality principle, the remainder of this article focuses on the primal integral formulation and the results presented hereinafter remain valid also for the dual formulation owing to the equivalence (table 1). Details are provided in the given proofs when needed.

### (c) Series expansion and iterative algorithm

An integral equation such as (2.10), or its counterpart in Fourier space, is commonly encountered in scattering theory. It is well known that when the terms *δ**e*_{k}, for *k*=1,…,*d*, can be assumed to be small, in a sense discussed later on, then the featured integral operator can be inverted using a Neumann series [24,28]. Therefore, the unknown conductivity map is approximated by the following series expansion:
*n* is to be interpreted as the operator within parentheses applied *n* times. Next, we introduce short-hand notations with second-order tensors concatenating the available measurements:

### Definition 2.4

Let

On noting that one can interpret ^{n} denotes the inner product applied *n* times.

If it exists, computing the solution *γ* from the series expansion (2.16) is achieved using the following fixed-point iteration algorithm which constitutes the core of the proposed numerical method.

### Algorithm 2.5

*For all*

This iterative scheme features the product terms *γ*_{n}(** x**)

*δ*

*e*_{k}(

**) that are computed in real space for**

*x**k*=1,…,

*d*and a convoluted inner product with the Green tensor

**. As suggested in [18], this scheme can be recast by applying the direct and inverse Fourier transforms repeatedly in order to circumvent the costly computation of convolution terms. This yields a scheme that alternates between real space and Fourier space which contributes to reducing the overall computational complexity of the algorithm, as shown in the following section.**

*Γ*

### Remark 2.6

For completeness, it should be noted that, for the conductivity model considered here, the scalar field *γ* can also be inverted as follows: expanding equation (1.1) entails for all *k* = 1,…,*d*
*d* equations entails *γ*, one then obtains

## 3. Modified iterative algorithm

### (a) Reconstruction of normalized conductivity contrast

When focusing on the inverse problem it is crucial to work with a reconstruction algorithm that relies on the least amount of *a priori* information. Most existing numerical methods assume knowledge of the conductivity either at one reference point (integration-type approaches [29]) or over the entire domain boundary (variational formulation-based methods [12]). It is remarkable that the conductivity problem (1.1) is unchanged when multiplying the conductivity by a constant, which can also be seen in the Lippmann–Schwinger equation (2.10). Therefore, from the series expansion (2.16), it is preferable to work with the conductivity contrast normalized by the mean conductivity *n*≥0 are given by

### Algorithm 3.1

### (b) Convergence of successive approximations

The Neumann series (2.16) and thus algorithm 3.1 converge in a given Banach space

### Assumption 3.2

*The measurements satisfy* *for all* *k*=1,…,*d*.

We now consider the Hilbert space *g** being the complex conjugate of *g* and where the second equality is due to the Parseval theorem. Note that any real-valued function *g* satisfies

The Green tensor defined by (2.3) has unit norm

### Lemma 3.3

*If assumption* 3.2 *is satisfied, then the integral operator* *defined by equation* (2.17) *is bounded and*

### Remark 3.4

As discussed in §2b, the duality principle can be applied to lemma 3.3. Obtaining the key inequality (3.3) in the dual form relies on estimating the Frobenius norm of the dual Green tensor *d*=2 or 3

Lemma 3.3 is now used to establish that the proposed method of successive approximations converges. Similar proof of convergence for the inverse scattering series can be found in [21].

### Proposition 3.5

*If* *then algorithm* 3.1 *converges to a* *limit* *and the truncation error is given by the estimate*

### Proof.

By definition, algorithm 3.1 computes the normalized conductivity contrast associated with the series (2.16), so that one defines

### Remark 3.6

The sufficient convergence condition of proposition 3.5 can be expressed in terms of the measurements themselves. Indeed, owing to the definition 2.4 of the second-order tensor *b*_{k}} is an orthogonal basis of

### (c) Weak-contrast case

The weak-contrast case corresponds to the situations where the measured intensity maps are characterized by low-amplitude variations relative to the prescribed mean values. Therefore, as equation (2.16) is a nonlinear equation in measurements *δ**e*_{k} then, in such a case, an approximation *γ*_{w} of the sought conductivity can be computed by expanding (2.16) at first order in *δ**e*_{k}. This approximation is analogous to the Born approximation in scattering theory. Therefore, one defines
*γ*_{w} satisfies
** ξ**≠

**0**, one has

Finally, owing to the Cauchy–Schwarz inequality, the conductivity contrast *b*_{k}} is chosen in order to obtain the best weak-contrast approximation possible for a given dataset.

## 4. Analysis of the conductivity contrast reconstruction

### (a) Reconstruction stability

As customary in inverse problems, it is insightful to assess the robustness of the proposed reconstruction algorithm. Considering two sets of measurements that are concatenated in the tensors

By definition one has

### Theorem 4.1

*Let* *denote two measurement sets such that the quantity M defined by equation (*4.2*) satisfies M<1. Then the corresponding conductivity contrasts* *computed at the limit using algorithm* 3.1 *satisfy the stability estimate
*

According to the previous theorem, assuming that the featured data are constituted by intensity field measurements of solutions to the conductivity problem considered here makes the inversion stable in the *i*=1,2 according to definition 2.4 and using some measurements *k*=1,…,*d* and a common orthogonal basis {*b*_{k}} for simplicity, then owing to remark 3.6 one has

### Corollary 4.2

*Consider an orthogonal basis* {*b*_{k}} *and two datasets* *for k*= 1,…,*d and i*=1,2 *such that the associated second-order tensors* *satisfy M*<1 *in equation* (4.2). *Then the conductivity contrasts* *reconstructed at the limit satisfy the estimate*
*where the constant C*>0 *depends on M and* {∥*b*_{k}∥}.

### (b) Reconstruction error

In this section, the reconstruction errors associated with the proposed algorithms are analysed. In the following theorem, we first assess the properties of algorithm 3.1, which does not rely on any *a priori* information on the *true* normalized conductivity contrast *a priori* parameter *γ*_{N} and the true conductivity function *γ*.

### Theorem 4.3

*Let* *denote a measurement set such that* *. Then the normalized conductivity contrast computed using algorithm* 3.1 *satisfies
**Moreover, the error associated with the computation of the conductivity field defined by equation (*4.4*) with a priori data* *is given by the estimate
*

### Proof.

By construction, the true conductivity function *γ* satisfies the Lippmann–Schwinger equation (2.10) and the associated Neumann series (2.16) is convergent owing to the assumption that

To prove the second part of the theorem, we start by observing that, owing to equation (4.4) and the iteration method employed in algorithm 3.1, the computed conductivity field satisfies
*γ*, one arrives at

Theorem 4.3 highlights the satisfactory behaviour of the proposed algorithm 3.1. The computed normalized conductivity contrast converges uniformly in the *a priori* information in the form of the mean conductivity value, then the corresponding reconstruction error in (ii) is governed by the two contributions: a systematic bias proportional to the misfit |〈*γ*〉−*γ*_{0}| and a truncation error that tends to zero at convergence.

### (c) Conditions on the measurements

The sufficient convergence condition of proposition 3.5 can be directly checked from given full-field measurements to assess whether the reconstruction method converges for that dataset. Moreover, this condition can be used either to design experiments to be performed or to optimize the algorithm performance for a given dataset. These questions are discussed below.

Equation (3.5) motivates the choice of a number *L*=*d* of experiments corresponding to the imposed mean intensity fields **E**_{k} for *k*=1,…,*d* that are close to an orthogonal basis of _{k}=*k* and used to compute the reference orthogonal basis {*b*_{k}} by solving the following finite constrained min–max optimization problem:

This issue is illustrated in figure 1, which corresponds to the material configuration of figure 6*a,* where *a* along with fields of intensity vectors *e*_{1}(** x**),

*e*_{2}(

**) sampled from pixels of the solution images of figure 6**

*x**b*,

*c*respectively. As expected, each intensity field solution

*e*_{k}varies about the mean value

**E**

_{k}and the amplitudes of these variations are all the more important when

*b*_{1},

*b*_{2}} shown in figure 1

*b*is such that

Finally, if a number *L*>*d* of full-field measurements {*e*_{1},…,*e*_{L}} is available then the conductivity reconstruction is to be performed from the *best* subset of experiments. This means that the computation (4.5) of the orthogonal basis {*b*_{k}} is to be coupled with an outer optimization step in order to determine the experiment label ℓ_{k}∈{1,…,*L*} for *k*=1,…,*d* so as to minimize the radius of convergence for such a large dataset.

## 5. Numerical results

### (a) Preliminaries

In this section, we present a set of numerical results to assess the performance of the proposed algorithm. The measurements are constituted by a set of synthetic data. To avoid, as far as possible, connections between the forward solver computing the data and the inverse one that performs the conductivity parameter reconstruction from them, i.e. to guard against the so-called inverse crime, we choose to work with the finite-element method for the former and process images through an implementation of algorithm 3.1 in a distinct code.

In a first step, we consider the weak formulation of the conductivity problem (1.1) for a given objective periodic conductivity distribution ** x**↦

*γ*(

**). Following the discussion of §4c, we define a set of prescribed mean intensity vectors {**

*x***E**

_{k}} of

*e*_{k}(

**)=**

*x***∇**

*u*

_{k}(

**)+**

*x***E**

_{k}, where

*u*

_{k}belongs to the space

*H*

^{1}-functions over the unit cell

*k*=1,…,

*d*, one seeks

*d*problems are solved using the finite-element method, i.e. discretizing the domain

*d*=2 and

*u*

_{1},

*u*

_{2}are computed using P2 finite elements on this mesh.

The computed intensity maps *e*_{1}(** x**),

*e*_{2}(

**), possibly corrupted by some additive relative noise, constitute the dataset for the inverse problem. The key idea is that these maps are treated as**

*x**images*, so that all information about the finite-element computations are discarded and only numerical values on grids of dimension

*d*are stored, i.e. local values of the solutions at each pixel (two dimensions) or voxel (three dimensions).

The two-dimensional data considered here are constituted by a set of images that are sampled on a regular grid of *P*×*P* pixels with *P*=255. The corresponding discrete frequency values are given by

Following the discussion of §4c, an optimal orthogonal basis {*b*_{1},*b*_{2}} is computed for each of the material configurations considered. Optimization is performed by minimizing numerically the quantity **E**_{1},**E**_{2}} as the reference basis. Algorithm 3.1 is then implemented using a standard FFT package in order to compute, at iteration *N*, a reconstruction

### (b) Reconstruction quality

Provided that the condition *e*_{k}(** x**). Therefore, let the equilibrium residual

*r*

_{eq}(

*N*) for the reconstruction be defined as

Finally, as the objective conductivity map *ϵ*(*N*) at iterate *N* is computed as

### (c) Examples

In the following two-dimensional numerical examples, the spatial distribution of the objective contrast is generated by filtering spatially a periodic random Voronoi tessellation. The amplitude values of the conductivity contrast are varied about a chosen mean value so as to produce a conductivity contrast map

#### (i) Conductivity reconstruction

*Weak contrast*. The case of weak contrast is investigated in a first configuration, with ^{−3},4.9×10^{−3}], as shown in figure 2*a*. The two experiments considered are defined by prescribed mean intensity fields **E**_{1}=(1,0) and **E**_{2}=(0,1). Corresponding full-field maps of solution potentials *u*_{1}(** x**) and

*u*

_{2}(

**) are provided in figure 2**

*x**b*,

*c*, with

*scaled*arrows representing mean vectors

**E**

_{k}and superimposed streamlines corresponding to intensity fields

*e*_{k}(

**)=**

*x***∇**

*u*

_{k}(

**) +**

*x***E**

_{k}. The reconstruction using only the first iterate (3.6) of the modified algorithm is shown in figure 3 with reconstruction quality indicators provided in table 2. This shows that the first-order approximation of §3c yields a satisfying reconstruction for the weak-contrast configuration considered.

*Medium contrast*. In the second configuration considered, the spatial conductivity distribution is that of figure 2*a* but the relative contrast values are increased so that *a*. For the same experiments as in the previous example, the full-field data of figure 4*b*,*c* highlight larger fluctuations of the corresponding solution gradients **∇***u*_{k}(** x**) about the prescribed values

**E**

_{k}. After optimization, the convergence condition

*c*shows that the equilibrium residual decreases but then rapidly reaches a plateau. This phenomenon is related to an intrinsic accuracy limit associated with the image sampling considered, a characteristic also relevant to FFT-based methods pertaining to the forward problem [31]. It is a common understanding that increasing the image resolution improves the computation accuracy; see the discussion of §5d.

*Strong contrast*. A third material configuration is considered to address a case with larger contrast. The objective relative conductivity contrast considered is such that *a*), while the prescribed mean intensity values are no longer chosen to be orthonormal vectors with *b*,*c*. Yet, after optimization, the computed orthogonal reference basis {*b*_{k}} satisfies *a*, the reconstruction of figure 7*b* obtained after iteration 35 of the algorithm is satisfying. Figure 7*c* shows that the residual on equilibrium decreases with iterations but reaches a plateau, as seen in the previous example.

### (d) Noisy data and filtering

In this section, reconstructions from noisy full-field data are discussed. The configuration considered is that of the medium contrast case, i.e. figure 4 with **E**_{1}=(1,0) and **E**_{2}=(0,1). The synthetic full-field images are now polluted by a relative additive noise as *e*_{k}(** x**)(1+

*n*_{k}(

**)), for**

*x**k*=1,2, where

*n*_{k}(

**) denote normally distributed random spatial variables with zero mean and standard deviation**

*x**σ*.

The noise amplitude is varied as *σ*=0.05, 0.1 and 0.25 and optimization is performed to compute the optimal orthogonal basis {*b*_{k}} in each case. As the effect of measurement noise is expected to affect the high-frequency content, one introduces a truncated frequency domain as
*ξ*_{max} is a user-chosen truncation parameter. Then, algorithm 3.1 is employed with frequency filtering, i.e. the domain *ii*). For consistency, the equilibrium residual (5.2) and the relative error (5.3) are also computed using only those wavevectors that satisfy *P*×*P*, which contributes to reducing aliasing effects.

Quantitative results are provided in table 3 and corresponding conductivity field reconstructions are shown in figure 8. Remembering that the image sampling considered in equation (5.1) is such that ** ξ**∈[−127,127]

^{2}then the truncation value

*ξ*

_{max}is adjusted manually depending on the expected noise level. For the two cases

*σ*=0.05 and

*σ*=0.1, then the convergence criterion is satisfied after optimization and the reconstructions computed at iteration 50 are relatively accurate. For the noise distributions of higher amplitude,

*σ*=0.25, one obtains

For comparison with the original noise-free configuration of figure 4 and the associated reconstructions of figure 5, figure 9 quantifies the effect of varying the truncation parameter value *ξ*_{max}. The equilibrium residual *r*_{eq}(50) and the corresponding relative reconstruction error *ϵ*(50) at iteration 50 are computed for the various configurations considered. As expected, for any choice of *ξ*_{max} the indicators considered are larger when noisy data are employed than in the noise-free case. More importantly, these figures show that the indicators *r*_{eq}(*N*) and *ϵ*(*N*) have qualitatively the same behaviours. While this remains a simple observation at this point, this criterion can be valuable in practice since the quantity *r*_{eq}(*N*) can always be computed from the available dataset unlike the relative reconstruction error *ϵ*(*N*). Lastly, it should be noted that, even in the case of noise-free data, decreasing the truncation parameter *ξ*_{max} up to a certain threshold improves the reconstruction. Therefore, the proposed frequency filtering can be interpreted as an efficient regularization scheme.

## 6. Conclusion

A full-field image conversion method is proposed to solve the inverse conductivity problem from internal data. It relies on a Fourier-based algorithm for computing maps of material parameters from full intensity or current density field measurements. It is based on a Lippmann–Schwinger equation for the sought conductivity field, the kernel of which involves the fundamental Green tensor and the available full-field intensity data. Owing to a duality principle, it is shown that a similar treatment of internal current density measurements can be performed. The normalized conductivity contrast is then computed as a solution to the obtained integral equation using an iteration method and the FFT to reduce computational complexity. A sufficient convergence condition for the algorithm is obtained and it is shown how it can be checked and optimized for the measurement set at hand. For materials with small contrast, a first-order Born-like approximation is obtained as the first iterate of the reconstruction algorithm. The proposed method is then analysed through the derivation of stability and error estimates. Finally, a set of numerical results based on synthetic data highlights the performance of the overall approach. Reconstructions for material configurations with small and large contrasts are discussed, as well as reconstructions from noisy data using a regularization scheme based on frequency filtering.

Work prospects include extensions to the isotropic elasticity case and to anisotropic material configurations. Moreover, the reconstruction of real-life microstructures will be the subject of future work to blend relevant geometrical constraints with the proposed identification method.

## Data accessibility

This work does not have any experimental data.

## Authors' contributions

C.B. developed the method, performed the numerical computations and drafted the manuscript. H.M. contributed to the method development and to drafting the manuscript. C.B. and H.M. gave final approval for the publication.

## Competing interests

We have no competing interests.

## Funding

No funding was received for this study.

## Acknowledgements

C.B. is indebted to Pierre Suquet for many fruitful discussions and constructive comments during the course of this work. C.B. thanks Patrick Ballard, Sébastien Imperiale, Assad Oberai and Emmanuelle Sarrouy for useful discussions.

- Received July 16, 2015.
- Accepted February 5, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.