Our approach is summarized in Fig. 1(c) and its implementation in an optical experiment is depicted in Fig. 2. The coherent waves originating from the object and entering the imaging system propagate through a multiply scattering medium. Within the imager, propagation produces a seemingly random and wavelength-dependent interference pattern called speckle on the sensors plane. The speckle figure is the result of the random phase variations imposed on the waves by the propagation within the multiply scattering sample29. Scattering, although the realization of a random process, is deterministic: for a given input and as long as the medium is stable, the interference speckle figure is fully determined and remains constant. In essence, the complex medium acts as a highly efficient analog multiplexer for light, with an input-output response characterized by its transmission-matrix24,25. We highlight the fact that the multiple scattering material is not understood here as a nuisance occurring between the object and the sensors, but rather as a desirable component of the imaging system itself. After propagation, sensing takes place using a limited number M < N of sensors.

Figure 2 Experimental setup for compressive imaging using multiply scattering medium. Within the imaging device, waves coming from the object (i) go through a scattering material (ii) that efficiently multiplexes the information to all M sensors (iii). Provided the transmission matrix of the material has been estimated beforehand, reconstruction can be performed using only a limited number of sensors, potentially much lower than without the scattering material. In our optical scenario, the light coming from the object is displayed using a spatial light modulator. Full size image

Let x and y denote the N × 1 and M × 1 vectors gathering the value of the complex optical field at discrete positions before and after, respectively, the scattering material. It was confirmed experimentally24,25 that any particular output y m can be efficiently modeled as a linear function of the N complex values x n of the input optical field:

where the mixing factor corresponds to the overall contribution of the input field x n into the output field y m . All these factors can be gathered into a complex matrix [H] mn = h mn called the Transmission Matrix (TM), which characterizes the action of the scattering material on the propagating waves between input and output. The medium hence produces a very complex but deterministic mixing of the input to the output, that can be understood as spatial multiplexing. This linear model, in the ideal noiseless case, can be written more concisely as:

As can be seen, each of the M measurements of the output complex field may hence be considered as a scalar product between the input and the corresponding row of the TM. If multiply scattering materials have already been considered for the purpose of focusing, thus serving as perfect “opaque lenses”17,18, the main idea of the present study is to exploit them for compressive imaging. In other wavelength domains than optics, analogous configurations may be designed to achieve CS through multiple scattering. For instance, a collection of randomly packed metallic scatterers could be used as a multiply scattering media from the microwave domain up to the far infrared and the method proposed here could allow imaging at these frequencies with only a few sensors. A similar approach could be used to lower the number of sensors in 3D ultrasound imaging using CS through multiple scattering media.

In our optical experimental setup, we used a Spatial Light Modulator (an array of N = 1024 micromirrors, abbreviated as SLM) to calibrate the system and also to display various objects, using a monochromatic continuous wave laser as light source.

During a first calibration phase, which lasts a few minutes and needs to be performed only once, a series of controlled inputs x are emitted and the corresponding outputs y are measured. The TM can be estimated through a simple least-squares error procedure, which generalizes the method proposed in24,25, as detailed in the supplementary material below. In short, this calibration procedure benefits from an arbitrarily high number of measurements for calibration, which permits to better estimate the TM. It is important to note here that the need for calibration is the main disadvantage of this technique, compared to the more classical CS imagers based on pseudo-random projections, which have direct control on the TM. However, this calibration step here involves only standard least-squares estimation of the linear mapping between input and output of the scattering material24,25. In our experimental setup, the whole calibration is performed in less than 1 minute. While we here rely on optical holography to extract complex amplitude from intensity measurements, the TM measurement can be implemented in a simplified way for other types of waves (RF, acoustics, Terahertz), where direct access to the field amplitude is possible. It may not be so straightforward in practical situations when only the intensity of the output is available and where more sophisticated methods30 would be required.

After calibration, the scattering medium can be used to perform CS, using this estimated TM as a measurement matrix. Note that, in our experiment, the same SLM used for calibration is then used as a display to generate the sparse objects. This approach is not restrictive as any sparse optical field or other device capable of modulating light could equivalently be used at this stage. As demonstrated in our results section, using such an estimated TM instead of a perfectly controlled one does yield very good results all the same, while bringing important advantages such as ease of implementation and acquisition speed. Hence, even if the proposed methodology does require the introduction of a supplementary calibration step, this step comes at the cost of a few mandatory supplementary calibration measurements rather than at the cost of performance. This claim is further developed in our results and methods sections.

For a TM to be efficient in a CS setup, it has to correctly scramble the information from all of its inputs to each of its outputs. It is known that a matrix with i.i.d Gaussian entries is an excellent candidate for CS31 and the TM of optical multiple scattering materials were recently shown to be well approximated by such matrices22. The rationale for this fact is that the transmission of light through an opaque lens leads to a very large number of independent scattering events. Even if the total transmission matrix that links the whole input field to the transmitted field shows some non-trivial mesoscopic correlations32, recent studies proved that these correlations vanish when controlling/measuring only a random partition of input/output channels22. In our experimental setup, the number of sensors is very small compared to the total number of output speckle grains and we can hence safely disregard any mesoscopic correlation.

Several previous studies24,25 have shown on experimental grounds that TMs were close to i.i.d. Gaussians by considering their spectral behavior, i.e. the distribution of their eigenvalues. As a consistency check, we also verified that our experimentally-obtained TMs are close to Gaussian i.i.d., through a complementary study of their coherence, which is the maximal correlation between their columns with values between 0 and 1. Among all the features that were proposed to characterize a matrix as a good candidate for CS33,34,35, coherence plays a special role because it is easily computed and because a low coherence is sufficient for good recovery performance in CS applications36,37,38,39, even if it is not necessary40. In Fig. 3(a), we display one actual TM obtained in our experiments. In Fig. 3(b), we compare its coherence with the one of randomly generated i.i.d. Gaussian matrices. The similar behavior confirms the results and discussions given in22,24, but also suggests that TMs are good candidates in a CS setup, as will be demonstrated below.