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Total Variation and Compressive Sensing (TV-CS) techniques represent a very attractive approach to inverse scattering problems. In fact, if the unknown is piecewise constant and so has a sparse gradient, TV-CS approaches allow us to achieve optimal reconstructions, reducing considerably the number of measurements and enforcing the sparsity on the gradient of the sought unknowns. In this paper, we introduce two different techniques based on TV-CS that exploit in a different manner the concept of gradient in order to improve the solution of the inverse scattering problems obtained by TV-CS approach. Numerical examples are addressed to show the effectiveness of the method.

The capability of solving in a fast and accurate fashion inverse scattering problems has an enormous interest in fields as different as biomedical imaging, nondestructive evaluation, and subsurface sensing. In all these applications, including the cases concerned with the use of radar or radar-like sensor for subsurface imaging and through the wall imaging, it makes sense to look for methods which allow to reduce as much as possible the number of measurements/sensors while still achieving accurate reconstructions. In this respect, the Compressive Sensing theory (CS) [

In fact, as long as the sought function is known to be sparse or compressible in a given basis, namely it is represented in an exact or anyway accurate fashion through a limited number of nonzero coefficients, the number of measurements actually needed for an accurate reconstruction can be much less than the overall number of unknowns and, moreover, it is possible to obtain nearly optimal reconstructions, as well as a kind of “superresolution” [

As well known, the inverse scattering problem, which is a possible framework for quantitative GPR and through the wall imaging, amounts to recover the geometry and the electromagnetic properties of unknown scattering objects, starting from the knowledge of the incident fields and the measurement of the corresponding scattered fields. Unfortunately, the problem is both ill-posed and nonlinear [

Very many different approaches exist to tackle such a problem, ranging from qualitative methods [

The Compressive Sensing theory is well developed for the case of linear problem and, as a result, it is usually used jointly with simplified models, such as the Born or Rytov approximations [

In the following, we consider the joint exploitation of CS and TV approach and try to generalize this latter to improve the reconstruction of objects with discontinuities having different orientation and shape.

The paper is organized as follows. In Section

For the sake of simplicity, let us assume that the investigated domain

As can be seen in (

For a fixed contrast function, the scattered field is linearly related to the incident fields. Hence, a linear superposition of the

In particular, one is able, for different “pivot points” located inside the scatterer, to realize virtual scattering experiments wherein the internal fields are focused around the pivot points (see [

Let us consider a reference system

In (

The approach described in (

The two-cylinder example. (a) Real and (b) imaginary part of the contrast reference profile. (c) Normalized logarithmic LSM indicator with the selected pivot points superimposed as dots. The retrieved profile by means of the approach (

In particular, we introduce a new objective function which allows to identify additional discontinuities located at + or −45° with respect to the coordinate axes. In such a way, one will have more accurate reconstruction of discontinuities having a generic or even circular shape.

In practice, we consider an additional term defined as the discretized version of the directional derivative

As a second contribution, we asked ourselves if we can have a still better procedure for profiles where the discontinuities are actually parallel to the

A simple yet original solution to such a problem is to exploit sparsity in terms of the second order mixed partial derivative. In fact,

In order to show the validity and to investigate the performances of the two proposed techniques, which aim at improving the TV-CS approach, some numerical examples with simulated data are addressed, each one dealing with a different type of scatterer.

In each example, we have first linearized the scattering equation, following the procedure described in Section

The double L scattering system. (a) Real part of the contrast reference profile. (b) Normalized logarithmic LSM indicator with the selected pivot points superimposed as dots. The retrieved profile by means of the approach (

The inhomogeneous square example. (a) Real part of the contrast reference profile. (b) Normalized logarithmic LSM indicator with the selected pivot points superimposed as dots. The retrieved profile by means of the approach (

The ring square example. (a) Real part of the contrast reference profile. (b) Normalized logarithmic LSM indicator with the selected pivot points superimposed as dots. The retrieved profile by means of the approach (

Moreover, in performing the numerical analysis, the presence of the convex function

In the following examples, the region of interest is a square of side

The receivers and transmitters are spaced on a circumference of radius

For all these numerical examples, we have considered, as indicator of accuracy, the reconstruction error defined as

In order to show performances of the first proposed approach, (equation (

The second example deals with two lossless L-shape targets with different dielectric permittivity (

In the third example, we consider an inhomogeneous square scatterer with

In the last example, a square ring scatterer with

In this paper, we have introduced two new CS-TV approaches which, together with a recently introduced linear scattering model for quantitative profile inversion, allow us to achieve nearly optimal reconstructions of arbitrarily shaped and piecewise nonweak targets. In this respect, it has been shown that it is possible to improve performances of TV-CS approaches by introducing new cost functions based on directional derivatives to pursue accurate reconstructions of nonsquared objects as well as on second order derivatives to further enhance sparsity of the unknown in the case of scatterers constituted by a superposition of squares and rectangles. Joint exploitation of these concepts and their extensions to the 3D case is currently under investigation.

All that authors state that the research herein described is not influenced by secondary interests (such as financial gain) and that they have no conflict of interests concerning all terms used in this paper.

The authors would like to thank Professor T. Isernia and Dr. L. Crocco for the fruitful discussions from which the present paper has taken great profit.