Microwave and millimeter-wave imaging applications are becoming increasingly numerous and cover a wide range of fields such as medical diagnostics [

1,

2,

3,

4], non-destructive testing [

5,

6,

7], and concealed weapon detection [

8,

9,

10]. However, all of these systems, in order to satisfy reasonable acquisition times, are constrained by the implementation of complex and redundant active systems, which represents a major economic constraint on the large-scale development of these applications. Faced with these limitations, a multiplicity of computational solutions have emerged, exploiting the availability of increasingly powerful and affordable digital processing units. These solutions are based on the development of predominantly passive components, capable of encoding and multiplexing radiated information in transmission and reception, thus reducing the amount of active channels required for imaging systems to function properly. The constraint is thus pushed back into the digital layer where the formulation and resolution of inverse problems represent new challenges that can make these solutions competitive. It has been demonstrated that such systems can be based on the use of electrically large cavities connected to conventional antenna arrays [

11,

12,

13], on the use of metasurfaces encoding information directly in the radiating aperture [

14,

15,

16], or even on hybrid solutions of leaky cavities that demonstrate interesting performance in many imaging modalities [

17,

18,

19]. Connections can also be made with earlier systems based on the use of frequency scanning antennas whose radiation patterns can encode a sum of information relative to the position of a target into a reduced number of signals [

20,

21,

22]. It is necessary to study all these systems in order to analyze them by means of a unified formalism taking into account the propagation of signals, their filtering within dispersive components, and their summation, as described in [

23] and

Figure 1.

Regardless of the passive computational system implemented, the objective is the estimate of the reflection function of the target

$f\left(\mathit{r}\right)$ considered invariant according to frequency in the operating bandwidth, from a single compressed signal represented here by

${\rho}_{\omega}$, where

$\omega $ is the pulsation. As a first approximation, it is possible to consider a scalar propagation model between two arbitrary positions

${r}_{a}$ and

${r}_{b}$ represented by free space Green’s functions

${G}_{\omega}({r}_{a},{r}_{b})=\mathrm{exp}(-jk|{r}_{a}-{r}_{b}\left|\right)/|{r}_{a}-{r}_{b}|$. The expression of the measured signal as a function of the signature of the target linearized assuming Born’s first approximation is as follows:

where

${\mathit{r}}_{\mathit{r}}$ corresponds to the coordinates of the radiating aperture and where

${r}_{t}$ is the location of the transmitting antenna. The most important element of this formula is the vector

${H}_{\omega}\left({\mathit{r}}_{\mathit{r}}\right)$, which corresponds to the response of the component encoding the received information. In the case of an electrically large cavity connected to an array of isotropic antennas,

${H}_{\omega}\left({\mathit{r}}_{\mathit{r}}\right)$ simply corresponds to the transfer functions of the cavity. In the case of a radiating metasurface, or in the case where it is not possible to neglect the impact of radiating elements connected to a cavity,

${H}_{\omega}\left({\mathit{r}}_{\mathit{r}}\right)$ stands for the near field radiated by this structure, which can be divided into a sum of secondary dipoles interacting with the target. In general terms, the principle of computational imaging applied to the microwave and millimeter-wave domains consists in using structured radiation patterns with a low degree of correlation in order to encode the information contained in the target space into electrical signals measured on a reduced number of ports in order to limit the costs and complexity associated with active systems. In contrast to synthetic aperture systems, which can also satisfy these constraints, computational systems also make it possible to quickly capture scenes to be imaged, making their usage compatible with real-time applications [

12]. The most comprehensive approach in this framework consists of formalizing the interaction between measured signals and space to be imaged by means of a matrix operator

M, giving rise to the expression of the following direct problem:

where

$\mathit{\rho}\in {\mathbb{C}}^{\phantom{\rule{0.166667em}{0ex}}{n}_{\omega}\times 1}$ and

$\mathit{f}\in {\mathbb{C}}^{\phantom{\rule{0.166667em}{0ex}}{n}_{r}\times 1}$ are, respectively, the vectorized measured signal including

${n}_{\omega}$ frequency samples and the reflection function of the target represented by

${n}_{r}$ spatial samples (for the sake of clarity, bold notation is used for all the vectors), and

$M\in {\mathbb{C}}^{\phantom{\rule{0.166667em}{0ex}}{n}_{\omega}\times {n}_{r}}$ is the sensing matrix accounting for the forward and backward wave propagations, as well as the response of the computational imaging component, as described in Equation (

1). Quite obviously,

${n}_{\omega}$ corresponds to the number of frequency samples and

${n}_{r}$ to the number of voxels of the discretized target space. This relation makes it clear that spatial information is encoded in a frequency signal and that the rank of the sensing matrix—directly related to the pseudo-orthogonality between the radiated patterns at each frequency—is the main limitation of the number of unknowns that can be reconstructed. This approach is undoubtedly the simplest and so far the most accurate way of reconstructing an estimation

$\widehat{\mathit{f}}$ by solving the inverse problem through, for example, a pseudo-inversion:

or through iterative reconstruction techniques, exploiting for example prior knowledge on the inherent sparsity of the interrogated scene [

24,

25,

26,

27]. Although this approach is particularly precise and simple to implement, it can suffer from prohibitive memory consumption and computing time, imposing great constraints on the processing units implemented in this framework [

23]. Alternative approaches proposed in previous work explored the possibility of breaking down the M measurement matrix into two operators of reduced dimensions, reconstructing in this context an estimate of the signals in the radiating aperture [

11,

12,

28]. The main advantage of this technique lies in the use of Fourier-based image reconstruction techniques on the estimated signals, exploiting the formidable computational efficiency of fast Fourier transforms [

13,

29]. Using the previous formalisms, the expression of the signals in the radiating aperture is defined as follows:

so that the measured signal can be written as follows:

The main limitation related to the current implementation of this technique is illustrated by writing the last equation in matrix form, spatially discretizing the radiating aperture into

${n}_{{r}_{r}}$ samples:

where

$H\in {\mathbb{C}}^{\phantom{\rule{0.166667em}{0ex}}{n}_{\omega}\times {n}_{{r}_{r}}}$ and

$S\in {\mathbb{C}}^{\phantom{\rule{0.166667em}{0ex}}{n}_{\omega}\times {n}_{{r}_{r}}}$ correspond, respectively, to the transfer functions of the computational imaging component and the signals in its radiating aperture, and where ⊙ stands for the Hadamard (or element-wise) product. Since the signal to be reconstructed

$\widehat{S}$ has more unknowns than the number of measured samples in

$\mathit{\rho}$, it seems impossible to calculate an accurate estimate of the latter unless the frequency dimension is sacrificed in a pseudo-inversion calculation, which would prevent the backpropagation computation for the estimate of

$\widehat{\mathit{f}}$. The approach suggested in the literature, initially inspired by time-reversal [

11], was to use a simple equalization of the transfer functions of the component using the pre-computed pseudo-inverse

${H}^{+}$ [

12,

13]:

where

$P=(\mathit{\rho},\dots ,\mathit{\rho})$ is the concatenation of

$\mathit{\rho}$ ${n}_{{r}_{r}}$ times to match the dimensions of

H. It is then possible to define an operator

$G\in {\mathbb{C}}^{\phantom{\rule{0.166667em}{0ex}}{n}_{\omega}.{n}_{{r}_{r}}\times {n}_{r}}$ taking into account the forward and backward wave propagation, linking the antenna signals with the reflectivity of the target as follows:

where

$\mathrm{vec}\left(S\right)\in {\mathbb{C}}^{\phantom{\rule{0.166667em}{0ex}}{n}_{\omega}.{n}_{{r}_{r}}\times 1}$ is vectorized to match the number of columns of

G. In this context, it is finally possible to obtain an estimate of

$\widehat{f}$ from the signals on the antennas reconstructed by equalization using the pseudo-inverse

${G}^{+}$:

Based on all these elements, a technique based on an identical decomposition of the sensing matrix M, compatible with the use of Fourier techniques for backpropagation but allowing more accurate reconstructions of the signals in the radiating aperture, is proposed and studied here. The theoretical principle, based on the exploitation of sparsity in the time-domain, is presented in the next section, which is followed by theoretical and experimental studies.