## 1 Introduction
### Imaging of Hyperechogenic Media  
The investigated objects, as fields perturbations, are classified into two categories: (i) penetrable objects, corresponding to contrast imaging; and (ii) nonpenetrable objects, corresponding to obstacle imaging. Let us consider an incident plane wave excitation of pulsation \(\omega\) and wave number \(k=\frac{\omega}{c_{0}}\):  
\[u(\boldsymbol{\alpha},t)=e^{ik(\omega)\cdot\boldsymbol{\alpha}-i\omega t} \tag{19}\]  
where \(c_{0}\) is the velocity of a wave in the known reference medium (the water in this work). A contrast-type perturbation for penetrable objects, independent of time, is written by separating the constant part \(p_{i}\) associated with the known reference medium from the contrast part \(\delta p_{i}\):  
\[p_{i}(\boldsymbol{\alpha})=p_{i}+\delta p_{i}(\boldsymbol{\alpha}) \tag{20}\]  
The integral representation involving the Green's kernels \(G\) of the problem associated with the Helmholtz operator \(L\):  
\[u=\int_{\mathbb{R}^{2}}GL_{p}u \tag{21}\]  
\[L_{p}(\boldsymbol{\alpha})=L(p_{i})-L(p_{i}+\delta p_{i}(\boldsymbol{\alpha})) \tag{22}\]\[L(p_{i})=\frac{\partial L}{\partial p_{i}}p_{i}+\frac{\partial^{2}L}{\partial p_{ i}^{2}}p_{i}^{2}+\ldots \tag{23}\]  
In addition, the diffraction transfer function \(T\) of a penetrable object associated with the perturbation \(\delta p_{i}\), such as soft tissues, is:  
\[T_{lin}(k,\ \mathbf{n}_{0},\mathbf{n})=\int_{\mathbb{R}^{2}}\int_{\mathbb{R}^{2}}e^{-i \mathbf{n}\mathbf{\cdot}\mathbf{\alpha}}G\big{(}k,\ \mathbf{\alpha}^{\prime},\mathbf{\alpha}\big{)}\delta p_{i} \big{(}\mathbf{\alpha}^{\prime}\big{)}\frac{\partial L}{\partial p_{i}}(e^{i\mathbf{n }\mathbf{\cdot}\mathbf{\alpha}^{\prime}})\ d\mathbf{\alpha}^{\prime}d\mathbf{\alpha} \tag{24}\]  
This formula corresponds to contrast imaging, also called Born imaging. Depending on the nature of the Green's kernels and the operator \(\frac{\partial L}{\partial p_{i}}\), the kernel can be reduced to a Radon kernel in diffraction, given in Section 2.1 in Equation (10). The far-field assumption is generally used. An obstacle-type perturbation is a finite volume in which the wave field is not defined. This volume is a form of hole bounded by a closed surface in the infinite reference medium. The diffraction behavior of an obstacle is different from that of a contrast. The wavefield \(u\) defined outside an obstacle denoted \(V\subset\ \mathbb{R}^{2}\) bounded by the surface \(S\) is expressed by the Kirchhoff identity written in integral form:  
\[u=\int_{\mathbb{R}^{2}}(-G\
abla u.\
abla f+u\
abla G.\
abla f) \tag{25}\]  
where \(f\) is the characteristic function of the volume \(V\). Depending on the boundary conditions at the surface of the object as an obstacle, the obstacle is called soft (Dirichlet condition); e.g., air gap in porous solid, or hard (Neumann condition); e.g., calcified structures in soft tissues. A distinction is thus made between the diffracted field in the case of a soft obstacle and in the case of a hard obstacle, and the two corresponding transfer function formulae in diffraction in the linear frame are written as:  
\[T_{soft,lin}(k,\ \mathbf{n}_{0},\mathbf{n})=\int_{\mathbb{R}^{2}}\int_{\mathbb{R}^{2}}e^{-i \mathbf{k}\mathbf{n}\mathbf{\cdot}\mathbf{\alpha}}G\big{(}k,\ \mathbf{\alpha}^{\prime},\mathbf{\alpha}\big{)}\
abla f \big{(}\mathbf{\alpha}^{\prime}\big{)}i\mathbf{k}\mathbf{n}_{0}e^{i\mathbf{k}\mathbf{n}_{0}\mathbf{ \cdot}\mathbf{\alpha}^{\prime}}d\mathbf{\alpha}^{\prime}d\mathbf{\alpha} \tag{26}\]  
\[T_{hard,lin}(k,\ \mathbf{n}_{0},\mathbf{n})=\int_{\mathbb{R}^{2}}\int_{\mathbb{R}^{2}}e^{-i \mathbf{k}\mathbf{n}\mathbf{\cdot}\mathbf{\alpha}}\
abla G\big{(}k,\ \mathbf{\alpha}^{\prime},\mathbf{\alpha}\big{)}\
abla f \big{(}\mathbf{\alpha}^{\prime}\big{)}e^{i\mathbf{k}\mathbf{n}_{0}\mathbf{\alpha}^{\prime}}d \mathbf{\alpha}^{\prime}d\mathbf{\alpha} \tag{27}\]  
The obstacle inversion problem is highly nonlinear, and it is very difficult to reconstruct the characteristic function \(f\).