Abstract
Compressive sensing (CS) is a technique to sample a sparse signal below the NyquistShannon limit, yet still enabling its reconstruction. As such, CS permits an extremely parsimonious way to store and transmit large and important classes of signals and images that would be far more data intensive should they be sampled following the prescription of the NyquistShannon theorem. CS has found applications as diverse as seismology and biomedical imaging. In this work, we use actual optical signals generated from temporal intensity chaos from externalcavity semiconductor lasers (ECSL) to construct the sensing matrix that is employed to compress a sparse signal. The chaotic time series produced having their relevant dynamics on the 100 ps timescale, our results open the way to ultrahighspeed compression of sparse signals.
Introduction
Compressive sensing (CS) defines a novel mathematical framework combining sampling and compression. This enables the sampling of bandlimited analog signals below the lowerbound rate set by the NyquistShannon theorem. Under such subsampling conditions, recent theoretical work^{1,2} has shown that exact reconstruction of subsampled data is possible under the assumption of sparsity, which refers to concise information content in a signal with respect to its occupied bandwidth. Applications of this new framework have already permeated various areas of physics and engineering such as reconstruction of threedimensional images with limited number of sensors^{3}, reconstruction of quantum states via compressive tomography^{4}, and prediction of dynamical network structures from limited data^{5}.
In CS, the measurement vector y of the sparse signal x is realized via the sensing matrix Φ such that y = Φx. This linear operation generates an incomplete set of information representative of the sparse signal that is then used to reconstruct the information via a minimization problem with respect to the norm (i.e. sum of absolute values of each vector component). This method works optimally if the sensing matrix satisfies the restricted isometry property (RIP), which means mathematically that all subsets (with a size dependent on the level of sparsity considered) of columns in the sensing matrix are nearly orthogonal^{6,7}.
Hence, generating sensing matrices with RIP is critical for achieving an optimal reconstruction of sparse signals with CS. It has been shown that randomness can lead to effective sensing mechanisms^{7}. For example, using Gaussian or Bernouilli random variables as elements of the sensing matrix will guarantee mathematically proven RIP^{1}. Other studies have also demonstrated that deterministic sensing matrices could be generated using advanced coding techniques^{8,9}.
Recently, Yu et al. have proposed the use of chaotic sequences generated by a logistic map to populate sensing matrices and have demonstrated mathematically that RIP was achieved with high probability for a level of performance comparable to those generated with Gaussian random variables^{10}. The interest in chaotic systems for CS relies on their broadband (tens of GHz) noiselike features and their potential hardware implementation with physical devices.
Externalcavity semiconductor lasers (ECSL) are known for generating largebandwith chaos used in many applications^{11,12}, such as secure communications^{13,14} and ultrafast generation of true random numbers^{15,16}. Nevertheless, the presence of the time delay in the dynamics induces temporally nonlocal correlations^{17} that could potentially hinder the level of performance of CS. Furthermore, the complexity of the optical chaos generated by an ECSL strongly depends on the choice of operating parameters (pumping current, feedback strength, and time delay)^{18} and also presents interesting scaling properties^{19,20} that could be useful for predicting parameter regimes, for which a chaotic sensing matrice has the RIP property empirically.
In this study, we demonstrate experimentally for the first time the possibility to harness optical chaos generated by an ECSL for the generation of a sensing matrix with an optimal level of performance for highspeed CS. We unveil the existence of a wide parameter range that allows for the reconstruction of onedimensional sparse signals (known as the basis pursuit problem) and image reconstruction. We also find that despite being statistically nonGaussian and strongly correlated for fast sampling rate, optical chaos guarantees similar performance in terms of reconstruction error and level of maximum sparsity.
Our experimental setup is described in Fig. 1(a). It consists of a singlelongitudinalmode DFB laser diode emitting at 1550 nm based on a InGaAsP quantum well for the gain medium. The threshold current for freerunning operation is J_{th} = 9.27 mA and the maximum cw power is 15 mW. The external cavity introduces a time delay ~4.3 ns and is comprised by a mirror (M) mounted on a translation state (TS), a polarizer (P) and a quarter wave plate (QWP), whose angle relative to the polarization direction of light is controlled by a piezoelectric actuator. The QWP angle is used to control the optical feedback strength η, which reaches up to ~20% of optical power, not taking into account coupling losses into the laser. A beam splitter (BS) is introduced to build a measurement arm that comprises an optical isolator (IO). A 12GHz amplified photodiode (PD) NewFocus 1544B, and a broadband oscilloscope Agilent DSO 80804B with 12 GHz bandwidth and 40 GSa/s sampling rate are employed.
As described in Fig. 1(b), the oscilloscope samples the intensity time series I (t) = E(t)^{2} with period Δt = 25 ps and quantizes it with 8bits precision to produce voltage V (t) ∝ I (t). The digitized chaotic time series is then subsampled by a factor d greater than the maximum decorrelation time of the chaotic intensity time series to ensure that two consecutive samples are maximally decorrelated. This guarantees approximate independent and identical distribution (iid) of the samples used to build the sensing matrix, but we will later show that this constraint can be relaxed in the construction of chaotic sensing matrices. We retain M × N samples with V_{i} = V (t_{i} = t_{0} + idΔt). Then, we form the chaotic intensity sensing matrix as
with σ_{V} the standard deviation of the chaotic optical intensity measured. The construction of this matrix follows similar principles to those of ref. 10.
We first analyze the statistical properties of three typical experimental chaotic time series of different pumping currents and feedback rates corresponding to dynamical regimes ranging from weakly to fully developed coherence collapse^{21}. We observe in Fig. 2(a) that the probability density functions (pdf) for each set of parameters have significantly different shapes, which may potentially impact the performance of CS. For example, at weak feedback rate η and pumping current J relatively close to threshold J_{th}, the distribution is sharpely peaked whereas at high pumping current the distribution is flattened with an approximate fourfold variance increase. For the various distributions, we have performed a onesample KolmogorovSmirnov test guaranteeing with a 5% significance level that optical chaos is not sampled from a Gaussian distribution.
We also analyze in Fig. 2(b) the spectral properties of the chaotic time series by computing the normalized autocorrelation function (ACF) defined mathematically by with 〈·〉 the time average, θ a time lag, and μ_{V} and σ_{V} the mean and standard deviation of the voltage time series V (t), respectively. We zoom in the vicinity of the zero timelag and observe oscillations reminiscent of the relaxationoscillation dynamics of the laser diode^{16}. For time lags greater than 25Δt, the correlation has pratically vanished retaining very little of the relaxation oscillation influence for the sets of parameters investigated. For 50Δt, the correlation has completely vanished thus making d = 50 a suitable default subsampling rate to create a sensing matrix Φ with RIP.
Next, to demonstrate CS with optical chaos, we consider the onedimensional basis pursuit (BP) problem. We generate a timesparse signal comprising only K < N randomly distributed nonzero spikes and with amplitude also randomly chosen to be equal to ±1 with probability 0.5. We construct the measurement vector , y = Φx to be used for the signal reconstruction. CS gives an accurate reconstruction of the subsampled data by solving the following linearprogramming optimization problem:
In our numerical analysis, we solve Eq. (2) with the Magic Software Toolbox^{22}. In the following, we use a similar benchmark to those employed in ref. 10 and observe the influence of the ECSL operational parameters.
Success in solving the BP problem with CS is given in a probabilistic sense with respect to the norm. Hence, we first characterize the recovery rate of timesparse signals for fixed size N. This is defined as the probability that the relative error of reconstruction with respect to the norm is below a given bound and is expressed mathematically as with ε = 0.01. We generate one hundred instances of timesparse signals of size N = 100 for each increasing value of sparsity level K ∈ [0, 40] and ten instances of the sensing matrix Φ leading to a measurement vector of size M = 50. We use this approach to obtain meaningful probability estimations.
Figure 3(a) shows the recovery rates using a chaotic sensing matrix obtained with optical chaos with pumping current J_{2} = 1.83J_{th}, feedback rate η_{2} = 12% of the light fed back, and subsampling factor d = 50. The curve has an Sshape with two plateaus corresponding to quasiperfect and failed reconstruction with probability one, respectively. This is compared to the recovery rate (in red) obtained using Gaussian random variables, which is known as providing asymptotically optimal reconstruction performance. We notice that the two curves are practically superimposed, hence showing that optical chaos can provide an optimal level of performance for the BP problem in CS and result in the sensing matrices having the RIP. As an illustration, we show in Fig. 3(b–c) examples of perfect and failed reconstruction of onedimensional sparse signals associated to the various level of sparsity at constant signal and observation sizes.
To gain additional insight on the performance of chaosbased sensing, we analyze the recovery rate as a function of both the number of measurements and sparsity level. In Fig. 4(a), we compute the recovery rate, with an identical threshold to that of Fig. 3(a), in the plane (γ = (N − M)/N, ε = K/N) and with sparse signal of size N = 100. For each couple of measurementsparsity values, we perform 1000 numerical experiments to estimate the probability of sucessful recovery. Similar to ref. 23, we observe a sharp and rapid transition from a recovery rate approximately equal to 1 to zero, which is also known as the DonohoTanner barrier. To compare the relative performance of optical chaosbased sensing and Gaussian sensing, we display in Fig. 4(b) the difference in recovery rates. We observe that the two sensing mechanisms lead to similar levels of performance for the BP problem. Specifically, the only differences, in the entire plane (γ, ε), occur in the phase transition, where small statistical fluctuations of magnitude ±0.04 are observed.
For practical concerns, investigating the impact of operational parameters such as pumping current and optical feedback strength is key to ensure the use of optical chaos robustly for CS applications. Figure 5(a) shows the recovery rate of the BP problem for seven different feedback strengths corresponding to a different dynamical complexity for the optical chaos. (This can be loosely quantified by the amplitude of the autocorrelation peak ACF (θ = τ) at the delay of the external cavity τ ~ 4.3 ns.) The range of possible recovery rates (shown by error bars) is maximally bound by ±0.020 and hence shows that a wide interval of feedback strengths provides performance similar to those of Gaussianbased sensing. The influence of pumping current (not shown) leads to similar results with errors bars bounded by ±0.038. We have also quantified the maximum level of fluctuation in recovery rate for a fixed set of operational parameters for the chaotic laser (identical to that of Fig. 3) and for one hundred realizations of the chaotic sensing matrices. We found that the fluctuations are bounded by ±0.027, thus demonstrating the robustness of the experimental generation of chaotic sensing matrices that provide a quasioptimal level of performance.
Another parameter to investigate is the subsampling factor d, which sets an upper bound on the maximum achievable rate for the exploitation of optical chaos in CS. For certain applications, such as random number generation, it is critical to remove temporal correlations to pass statistical tests^{15}. In CS theory, the sensing matrix should be constructed with iid random variables, hence suggesting that the correlation between optical chaos samples should be negligible as well. However, constructing the sensing matrix Φ columnwise allows one to relax this condition without impeding the level of performance^{24}.
Figure 5(b) shows that subsampling factors smaller than 50 still allow for optimal performance, despite a significant increase of correlation level. According to our analysis, even using unitary subsampling d = 1 corresponding to a correlation level greater than 0.6 does not result in a noticeable loss of performance on the BP problem. The range of recovery rate (given by error bars) is bounded by ±0.024 and the recovery curve is still well surimposed with the optimal curve. Hence, the difference between recovery rate for correlated and uncorrelated chaotic samples both lead to a sensing matrix Φ with empirical RIP. This observation corroborates simulations realized with correlated Gaussian iid variables to build sensing matrices^{24}.
An additional interest of our approach is that optical chaos can be readily exploited for CS without any additional postprocessing, while maintaining an optimal level of performance. This simplifies its potential use in information processing architectures and is in clear contrast with the more common use of chaotic lasers as energyefficient and ultrafast randomnumber generators^{11,25}. Indeed, this type of application usually requires significant postprocessing and sometimes even discarding some of the most significant bits resulting from the ADC process^{26} in order to pass standard randomness tests^{27,28}, although recent progress has allowed to keep more bits from the sampled waveform^{29}.
Finally, we use the chaotic sensing matrix in a CS problem consisting of reconstructing a 2D image based on a limited measurement rate. As defined by the CS framework, we need to introduce a sparsifying matrix realizing a change of basis [e.g. the wavelet transform]. To reconstruct an image, we modifed the linear program of Eq. (2) to
This linear program is searching for a sparse solution in the sparse basis leading to the obtained measurements. We consider the SheppLogan phantom image^{30} with N = 4096 pixels shown in Fig. 6(a) as the original data to be reconstructed using CS. We use a twolevel bidimensional discrete wavelet decomposition with Daubechies wavelets for our sparsifying transformation [see Fig. 6(b)]. We consider a measurement rate of 0.4 (M = 1638) and a chaotic sensing matrix with subsampling factor d = 1 and (J_{2}, η_{2}) as operating parameters. Finally, we solve Eq. (3) using the Magic Software Toolbox. The recovered image is shown in Fig. 6(c) and displays the main features of the original data despite imperfections [see Fig. 6(d)] due to the sparsity level (K = 1988) relative to the number of measurements M. The quality of the image is quantified using the peaksignaltonoise ratio , which denotes an accurate reconstruction. It is possible to achieve higher SNR using different linear programs, such as the TV Algorithm^{1}, tailored for certain image constructions where the image gradient is sparse (which is the case of the SheppLogan phantom image).
As a perspective, we envision that such a chaotic laser could replace the pseudorandom number generator based on an linearfeedback shift register (LFSR) which is traditionally used in the socalled random premodulation integrator (RMPI) scheme^{31}. In the original RMPI approach, a LFSR generates binary random numbers that are mixed with the analog signal of interest. The mixed signal is then integrated and sampled at a rate slower than the Nyquist rate. To make use of optical chaos in RMPI, one would need to preprocess the optical chaotic carrier to make it a piecewiseconstant signal prior to the analog mixing with the signal of interest. This operation could be realized electronically. The resulting chaotic piecewiseconstant signal could be then back converted into an analog signal to be mixed with the signal of interest. Being piecewiseconstant allows for a countable number of values to populate the sensing matrix and makes the compressivesensing reconstruction computationally tractable.
In this study, we have demonstrated experimentally that chaotic time series generated by a laser diode with optical feedback can be used for compressive sensing applications. Using the benchmark of the basis pursuit problem and reconstruction of the SheppLogan phantom image, we demonstrated that sensing matrices built with samples from the chaotic time series lead to very good and robust reconstruction of onedimensional sparse signals even in the presence of a high level of correlation between consecutive samples. This opens a new field of application for optical chaos in ultrafast information processing.
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How to cite this article: Rontani, D. et al. Compressive Sensing with Optical Chaos. Sci. Rep. 6, 35206; doi: 10.1038/srep35206 (2016).
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Acknowledgements
The authors acknowledge the financial support provided by the Conseil Régional de Lorraine. D.R. greatfully acknowledges the financial support of the Fondation Supélec and the Fonds Européen de Développement Régional (FEDER) through the projects PHOTON and APOLLO and the IAP P7/35 (BELSPO) with the Photonics@be project (2012–2017). The authors are grateful for fruitful discussions with Justin Romberg.
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D.R. conceived and designed the study and analyzed the data; A.L. designed the experimental setup; D.C. and C.Y.C. performed the experiments; D.R., A.L. and D.S.C. cowrote the paper.
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Rontani, D., Choi, D., Chang, CY. et al. Compressive Sensing with Optical Chaos. Sci Rep 6, 35206 (2016). https://doi.org/10.1038/srep35206
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