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Imaging With Nature: Compressive Imaging Using a Multiply Scattering Medium
Antoine Liutkus, David Martina, Sébastien Popoff, Gilles Chardon, Ori Katz, Geoffroy Lerosey, Sylvain Gigan, Laurent Daudet, Igor Carron
TL;DR
Compressive sensing seeks to recover structured signals from fewer measurements, but existing systems often depend on engineered randomization or sequential acquisition. The paper uses stable multiply scattering media as natural analog randomizers, calibrates their transmission matrix, and reconstructs objects from parallel measurements. The authors demonstrate the approach optically with a 300µm layer of Zinc Oxide white paint, reporting near-perfect and competitive reconstruction at sensor densities far below Shannon-Nyquist sampling.
Problem
Existing compressive-sensing implementations require carefully engineered randomization and often sequentially generate many patterns, limiting hardware simplicity and acquisition speed.
Method
The paper uses a stable multiply scattering medium as a natural analog randomizer, calibrates its transmission matrix, and reconstructs objects from parallel multiplexed measurements.
Results
Near-perfect reconstruction occurs at sensor densities much lower than classical Shannon-Nyquist sampling, with experiments using a 300µm Zinc Oxide white-paint layer.
Takeaways & Limitations
Multiply scattering media are promising candidates for efficient, compact compressive imagers that acquire a scalable number of measurements in parallel.
Abstract
from arXiv · showhide
The recent theory of compressive sensing leverages upon the structure of signals to acquire them with much fewer measurements than was previously thought necessary, and certainly well below the traditional Nyquist-Shannon sampling rate. However, most implementations developed to take advantage of this framework revolve around controlling the measurements with carefully engineered material or acquisition sequences. Instead, we use the natural randomness of wave propagation through multiply scattering media as an optimal and instantaneous compressive imaging mechanism. Waves reflected from an object are detected after propagation through a well-characterized complex medium. Each local measurement thus contains global information about the object, yielding a purely analog compressive sensing method. We experimentally demonstrate the effectiveness of the proposed approach for optical imaging by using a 300-micrometer thick layer of white paint as the compressive imaging device. Scattering media are thus promising candidates for designing efficient and compact compressive imagers.
Scattering Medium
Compressive sensing exploits signal structure to reduce measurements, but conventional implementations require engineered randomization or sequential pattern generation. This paper proposes using multiply scattering media as natural analog randomizers that multiplex object information in parallel.
- Motivation: Compressive sensing can reduce measurements by exploiting additional signal structure, including sparsity in a known representation.Random measurement matrices can support recovery with fewer samples when technical conditions such as incoherence are satisfied.
- Limitations of existing systems: Conventional optical implementations use engineered devices or sequential acquisition to create randomized measurements, increasing hardware complexity or acquisition time.Examples include digital micromirror devices, spatial light modulators, and rotating optical diffusers.
- Scattering Medium: Multiply scattering media naturally perform randomized multiplexing, so each sensor captures global information about the object without a designer-crafted randomizer.The approach replaces man-made emulated randomization with a stable complex material and measures its transmission matrix a posteriori.
- Scattering Medium: The proposed system acquires multiple compressive measurements in parallel rather than sequentially, using a 300µm layer of Zinc Oxide white paint in an optical implementation.If approximately 500 samples are needed, the scattering approach can acquire them simultaneously, whereas a single-pixel camera requires 500 individual measurements.
- Scattering Medium: Multiply scattering provides an underdetermined linear measurement model in which the object is observed through a known M × N matrix, with M < N enabling low sensor density.Recovery relies on object sparsity or near-sparsity and technical conditions on the measurement matrix.
M N ×
The study uses a calibrated multiply scattering medium as an analog randomizer for compressive imaging, replacing sequential engineered measurement patterns with parallel multiplexing. Experiments show sparse objects can be reconstructed from far fewer sensors than conventional sampling, with performance exhibiting a clear success–failure phase transition.
- Background: Compressive sensing can require far fewer measurements than Nyquist sampling when signals have exploitable structure.Sparse reconstruction algorithms such as OMP or Lasso recover objects under sparsity constraints.
- Method: A stable multiply scattering medium deterministically converts each input into a speckle pattern and acts as an analog multiplexer characterized by its transmission matrix.Each output measurement can be treated as a scalar product between the input and a corresponding transmission-matrix row.
- Method: The transmission matrix is estimated through least-squares calibration before imaging, after which Multichannel Orthogonal Matching Pursuit reconstructs sparse objects from M × P measurements.Optical calibration takes less than 1 minute in the reported setup, while applying the method at other wavelengths requires estimating the corresponding input–output mapping.
- Method: Experimentally measured transmission matrices resemble effective Gaussian sensing matrices, whose low coherence supports compressive-sensing recovery.Coherence measures maximal column colinearity, and lower coherence is better for compressive sensing.
- Results: Near-perfect reconstructions are obtained for canonical- and Fourier-sparse objects using sensor counts much smaller than the 1024-pixel object size.The experiments vary the number of measurements while testing both canonical-domain and Fourier-domain sparsity.
- Results: The recovery phase diagram shows a clear transition from failure to systematic success at sampling rates far below Shannon–Nyquist, with the transition close to noisy simulations.The grid summarizes more than 10^5 physical experiments, averaging approximately 50 measurements per point; the MMV protocol is more noise-robust than SMV.
- Conclusion: The approach shifts compressive-imaging complexity from engineered hardware and electronic control to calibration while enabling flat, few-detector, parallel acquisition.The system uses a simple natural scattering layer and can be implemented with few conventional lenses, with possible applications across optical, THz, RF, and ultrasound imaging.
- Conclusion: Experimental noise limits performance because it affects the measurement matrix, although standard calibration and sparse-reconstruction methods still produce competitive sampling rates.The reported method is presented as a truly analog compressive sampler with ease of implementation.
H A
The scattering medium is represented by an estimated transmission matrix that maps input optical fields to complex outputs. In matrix form, the output is expressed through this transmission matrix.
- The transmission matrix is estimated rather than directly known, while incorporating the scattering medium and the laser input.
- The complex output is written using the transmission matrix and the input field in matrix form.
- The input and output are represented as vectors, and the transmission matrix is a complex matrix relating them.
Estimation of the Transmission Matrix
The transmission matrix is calibrated using orthonormal inputs, with the Hadamard basis providing a simple inversion-based estimate. Additional calibration measurements enable least-squares estimation.
- The calibration procedure estimates the transmission matrix by probing the system with an orthonormal input basis.
- The Hadamard basis is advantageous because its entries and inverse have simple structure, yielding a straightforward transmission-matrix estimate.
- Additional calibration measurements can improve transmission-matrix estimation beyond the basic Hadamard procedure.
H H H YX XX
The method generalizes calibration through least-squares estimation and uses a combined Hadamard-random input matrix experimentally. A key limitation is the need to obtain linear output measurements.
- Least-squares estimation provides a more general transmission-matrix calibration framework that can account for noisy observations.
- The experimental calibration input combines a Hadamard matrix with a large random matrix whose entries are independently and uniformly distributed.
- The transmission matrix is estimated from the measured calibration outputs using the least-squares formula.
- The approach requires linear output measurements, whereas some scenarios provide only output magnitudes.
Virtually sparse intensity inputs
Because a phase-only spatial light modulator cannot directly create canonical sparse inputs, the experiments synthesize virtual sparse objects by differencing measurements from reference and modified phase patterns.
- The phase-only spatial light modulator prevents direct use of signals sparse in the canonical Dirac domain because its pixels share the same amplitude.
- Virtual sparse objects are generated by selecting a support and changing the phases at those entries relative to a random reference phase vector.
- The difference between the modified and reference complex outputs gives the system output for the corresponding sparse virtual input through optical linearity.
- The procedure generates sparse inputs with arbitrary sparsity, but the differencing process introduces additional doubled noise.
- Using the same support with different reference phases produces multiple outputs for one sparse object, implementing the Multiple Measurement Vector paradigm.
- The experiments repeat the construction to create many virtual objects spanning varying sparsity levels and corresponding outputs.
Algorithm for reconstruction using compressed sensing
Reconstruction treats measured outputs as random projections of a sparse input matrix after transmission-matrix calibration. The experiments use multichannel Orthogonal Matching Pursuit for estimation.
- The reconstruction assumes known sparsity and complex outputs from multiple illuminations of the same sparse input.
- Multiple outputs for a shared support are represented as measurement vectors, as illustrated in Fig. S2.
- After calibration, sparse-recovery algorithms can estimate the unknown input matrix from measured outputs related by the transmission matrix.
- The experiments use basic multichannel Orthogonal Matching Pursuit to reconstruct the sparse input matrix.
M X
Recovery succeeds by evaluating support correlation against ground truth, with Fourier-sparse signals handled by replacing the canonical measurement matrix with one appropriate to the alternative basis.
- Recovery is deemed successful when estimated support correlates above 0.9 with ground truth.This threshold indicates that at least 90% of the original support has been identified.
- For signals sparse in an alternative basis such as Fourier, the same recovery procedure uses a measurement matrix adapted to that basis.
HB H
The study maps compressed-sensing performance across sparsity and measurement counts, then compares experimental measurements with noisy synthesized Gaussian measurements.
- Approximately 25,000 inputs spanning sparsity levels from 1 to the maximum tested value were used to construct a complete phase transition.Each figure cell reports average performance over approximately 50 independent trials.
- A matched experiment used sparse inputs multiplied by a synthesized i.i.d. Gaussian matrix with noise averaging 17% of the clean output amplitude.The estimated matrix was used for compressed sensing in the same manner as for the experimental data.
Fourier-sparse inputs
The system was tested with directly measured inputs sparse in the 2D-Fourier domain, including constant-modulus plane waves, and its recovery performance was evaluated versus measurement count.
- A 2D plane wave provides a directly measured constant-modulus input with only one nonzero Fourier-transform element.
- Constructing constant-modulus 2D wavefronts with arbitrary Fourier sparsity is not straightforward, limiting direct tests across general sparsity levels.
- The Fourier-sparse recovery experiment plots success probability against measurement count for a plane wave and a superposition of many plane waves.Each plotted point averages 128 independent trials.
- 15 measurements suffice to properly recover a plane-wave input, demonstrating performance consistent with compressive-sensing theory.