Convolutional neural network for self-mixing interferometric displacement sensing
CConvolutional neural network for self-mixinginterferometric displacement sensing S TÉPHANE B ARLAND
AND F RANÇOIS G USTAVE Université Côte d’Azur - CNRS, Institut de Physique de Nice, 1361 route des Lucioles, F-06560, Valbonne,France DOTA, ONERA, Université Paris-Saclay, F-91123, Palaiseau, France * [email protected] Abstract:
Self mixing interferometry is a well established interferometric measurementtechnique. In spite of the robustness and simplicity of the concept, interpreting the self-mixingsignal is often complicated in practice, which is detrimental to measurement availability. Herewe discuss the use of a convolutional neural network to reconstruct the displacement of a targetfrom the self mixing signal in a semiconductor laser. The network, once trained on periodicdisplacement patterns, can reconstruct arbitrarily complex displacement in different alignmentconditions and setups. The approach validated here is amenable to generalization to modulatedschemes or even to totally different self mixing sensing tasks.
1. Introduction
Optical interferometric measurements are routinely used in science and engineering and manyschemes can be used to adapt the approach to the specific measurement to be performed. Oneparticularly interesting and well established method is the so-called self-mixing interferometry,which consists in realizing interference between the beam reflected by a target and a referencebeam inside the laser resonator emitting the reference beam (see eg [1–5] for reviews). Forits simplicity and versatility, many applications have been envisioned and perhaps the mostimmediate is that of displacement measurement. Two limit regimes are considered [6]: that ofvery small displacement (much smaller than the laser wavelength) or the opposite case where thedisplacement takes place over a very large number of wavelengths. In the first case, informationabout the target displacement can be retrieved from fitting the shape of the interferometric signal.In the latter case, most of the information is obtained by counting the fringes that are observed asa sawtooth signal whose symmetry depends on the direction of the motion. Despite its apparentsimplicity, this analysis is often complicated since the exact shape of the interferometric signaldepends on many factors including bias current, target reflectivity, alignment conditions [7], andmodal structure of the laser which may even lead to a double-peak structure in each fringe [8].Furthermore, on diffusive targets, speckle leads to an effective variation of the feebdack parametersand therefore a change of the signal shape in the course of the measurement. In practice, all theseeffects tremendously affect the availability of self-mixing measurement setups. This has led to anumber of hardware and software proposals to either improve the signal quality or the retrievalof the displacement from the interferometric signal [9–14].Computer neural networks are one of the many architectures which can be used for machinelearning tasks, whereby a computer is used to infer rules from a set of data and results insteadof providing results on the basis of an input and a priori known rules. The training of a neuralnetwork leads to the formulation of kind of statistical model [15], able to predict new resultson the basis of new data. These neural networks are already very widely used in everyday lifeand they are proving increasingly useful many areas of research and technology. In the specificcontext of interferometry, very few attempts exist to date. They have been used to identify andcount fringes in [16–18]. In [19] and [20] they have been used to pre-process self mixing tracesand in [21] they are used as a part of a self-mixing blood pressure measurement scheme.In the following, we discuss the use of a convolutional neural network for the direct recon- a r X i v : . [ phy s i c s . d a t a - a n ] J a n D driver OscilloscopeLD SpeakerRF amp Computer& sound cardCol.
Fig. 1. The self-mixing experimental setup consists of a laser diode (LD), short focallength collimator (Col.), laser diode current driver (LD driver) and voltage amplifier(RF amp). struction of a displacement signal across many different alignment conditions. We address aparticularly delicate regime which is the one of "few wavelengths" displacement. The neuralnetwork is first trained on a set of periodic data and in different alignment conditions. Itsperformance in reconstructing the displacement of a target from a self-mixing interferometricsignal is then validated on aperiodic times series whose continuous spectrum spans more thanthree octaves and under different alignment conditions, not used during training. We also analyzethe robustness of the reconstruction to the presence of very strong detection noise. Finally, weobserve that the neural network can, without any tuning, provide a sensible reconstruction of thedisplacement of a target obtained on a different experimental setup based on the same operatingprinciple. We then briefly discuss some details on the operation of such a network and somefurther possible uses of this approach in the context of self mixing. Thus, a reasonably simpleneural network such as the one used here can become one of the tools which contribute to therobustness and high availability of self-mixing interferometric setups.
2. Neural network design and training
The experimental arrangement, presented in Fig. 1, consists of a single transverse mode laseremitting at 𝜆 = 𝑛𝑚 (ML725B8F) whose threshold current is about 6 . 𝑚 𝐴 . In all theexperiments reported here the laser is driven at a constant current of 9 mA. The laser beam isfocused by a high numerical aperture (NA=0.7) lens on the central region of a basic computerspeaker located at about 20 cm from the laser. This speaker is put in motion via an electricalsignal produced by the sound card of a computer which can easily produce many kinds of patternsat a sampling rate of 44.1 kHz. The linearity of the speaker response has been assessed over arange of frequencies from 5 Hz to 100 Hz and over the range of voltage provided by the soundcard. Over this range, the speaker responds with constant 0 phase. Thus, in the range wherethe linearity of the displacement has been checked, one can use the voltage at the speaker as anindependent measurement of the position of the target with respect to some unknown origin.From that point on, we will therefore use this voltage as a proxy for the target position. Theself mixing signal is measured as a voltage at the laser electrodes, which is amplified by anAC-coupled amplifier with 10 amplification factor and several MHz bandwidth. We deliberatelydid not optimize the self-mixing signal quality, exactly because one of our aims is to check thatneural networks can help in making the measurement work even in sub optimal conditions. .2. Network setup The first thing to be noted is that in the "few wavelengths" range of displacement, the self-mixingsignal contains no information about the absolute position. Although one can be tempted toconsider that counting fringes (or some other equivalent technique) will lead to knowledgeof the exact position with respect to some unknown arbitrary origin, this approach is boundto diffuse in the long term: If for some reason a fringe is missed, the measurement systemhas no way to recover from this error because the physics of the system does not include thisinformation. Thus, (independently of how accurate the fringe counting is unless it is strictlyperfect ), a position measurement will unavoidably loose accuracy at a rate proportional to √ 𝑡 where 𝑡 is the measurement duration. Therefore, our aim here is to provide a measurement of thedisplacement within a prescribed time interval, a velocity .Once this is established, the setting of the architecture of the neural network is stronglyinfluenced by the specific question one wants to address. Here we assume that the self-mixingsignal is acquired at a much larger sampling rate than the Nyquist frequency of the displacementsignal one wants to measure. This is a very reasonable requirement in this context since mostapproaches address the question by counting fringes. Here we assume that the signal is sampled atleast 256 times faster than the Nyquist frequency of the displacement to be measured. Therefore,the reconstruction of the trajectory consists in inferring from 256 self-mixing signal points onesingle instantaneous velocity corresponding to the displacement of the target during the 256points acquisition. Then, in terms of machine learning the problem is reduced to a "regression"problem, where some algorithm must provide a single number on the basis of the availableinformation (a piece of time trace of length 256).The setting therefore consists in analyzing a sequence where temporal ordering matters andtherefore a recurrent neural network can be envisioned as a suitable architecture. However, thesenetworks are notoriously difficult to train and convolutional neural networks are known to be aneasier to train and valid alternative alternative. Therefore, we build a network based on a stack of1-dimensional convolutional layers with pooling layers between two convolutional layers. At theend of the stack, two fully connected layers convert the features identified by the convolutionallayers into a single number which is the inferred velocity of the target during the measurementsequence. More details are given in appendix, table 1. This global architecture was chosenfrom first principles of neural network design [15] and the model details where then determinedempirically. The network was implemented with the Keras library, which offers an excellenttradeoff in terms of complexity and versatility for our purpose [22]. Once the network architecture is chosen, the network must be trained with known data. Inpractice, that means providing the network a large number of pairs [ 𝑠 ( 𝑡 , ..., 𝑡 + 𝑑𝑡 ) , 𝑣 ] where 𝑠 ( 𝑡 ) is a self-mixing signal acquired during 256 sampling times 𝑑𝑡 and 𝑣 is the average velocity ofthe target during the duration of the interval 256 𝑑𝑡 . One must underline that neural networks areknown to be able to represent arbitrary functions provided a sufficient number of layers and cellsare present in the network [23]. Therefore, given enough computer time for training, a sufficientlylarge network will be able to perfectly reproduce the training data it has been shown. This meansthat a model trained this way achieves excellent accuracy . However, one of the key issues withself-mixing implementations is that the alignment conditions are sometimes different from onemeasurement to the next. Equivalently, speckle generated by the reflection of the beam on adiffusive target will lead to effective variations of the feedback strength parameter in the courseof the measurement. Therefore, the network trained here must be able to adapt to these changes.This is known as the capacity of the network to generalize the features which were learnt andidentify them in unseen data. To achieve this, we train the network on a deliberately limited set ofdata and observe the reconstruction of the network on a very different data set, both in terms of .00 8.002.50.02.5 S i g n a l ( a r b . u . ) D i s p l a c e m e n t ( / m s ) Fig. 2. Reconstruction of periodic displacements similar to those used to train thenetwork. Top row: interferometric signal corresponding to three different alignmentconfigurations. Bottom row: measured displacement signal (blue continuous line) andtrajectory predicted by the network. The orange crosses are predicted by the model andthe dashed line is a simple cubic interpolation. The shaded areas correspond to theinterferometric signals shown on the top row. the dynamics of the target (different displacement patterns) and in terms of the alignment of thebeam on the target. The training data consists exclusively of measurements of the interferometricsignal in response to periodic displacement of the target. We record self mixing signals in sixdifferent alignment conditions, in three of them a double peak is visible in each fringe. For eachof these alignment conditions, we record the self mixing signal for a set of 19 frequencies evenlyspaced between 10 and 100Hz. For each frequency we record 5 different amplitude signals.For each of these settings we record sinusoidal and triangular waveforms. The sampling rateof the oscilloscope is set to 250 kHz so that 𝑑𝑡 = 𝜇 s. In total the network is trained on about1 . × segments of 256 time steps, each of them of duration 256 ∗ 𝜇 s = .
024 ms. Theoperation regime of self-mixing sensing setups is often characterized in terms of the 𝐶 feebackparameter. Here, the alignment configurations we use are such that the system operates in theweak feedback regime 𝐶 <
𝐶 <<
3. Results
After training, one will assess the performance of the neural network (also "the model") bycomparing the displacement reconstructed from the interferometric signal and the voltage at thespeaker’s ends, used as a proxy of position. First, we check the model’s prediction accuracy inknown settings (periodic signals and known alignment conditions) and then in unseen settings.
We show on Fig. 2 how the network reconstructs examples of periodic traces after training. Thisexact sequence has not been used during training but these alignment conditions were used duringtraining and sequences with identical frequencies and amplitude were used during training.
50 100 150 2002.50.02.5 D i s p l a c e m e n t ( / m s )
94 96 98 100 102 104 1062.50.02.5 D i s p l a c e m e n t ( / m s )
94 96 98 100 102 104 106Time (ms)2.50.02.5 S i g n a l ( a r b . u ) Fig. 3. Reconstruction of unknown and complex displacements in alignment situationswhich are not in the training set. Top: the blue line is the displacement measuredfrom the speaker voltage, the orange line is the model’s prediction. Middle row: zoomaround the central area of the top row. The discontinuity close to 100 ms is where wenumerically connected the two time traces (see text). Bottom row: interferometricsignal corresponding to the middle row.
On the top row we show three examples of interferometric signal which correspond to three ofthe alignment conditions used during the training of the network and three different displacementfrequencies. On the bottom row, we show the displacement per time unit of the target, as it canbe measured from the voltage at the edges of the speaker (blue continuous line). Independentlyof that voltage measurement, we use the trained neural network to infer the displacement fromthe self-mixing signal. This is the orange dashed line, which is almost perfectly superimposed tothe actual displacement measured from the voltage at the speaker’s ends. This almost perfectreconstruction is not very surprising since, even if the network had not seen this exact piece oftime trace during training, it has seen periodic signals at these frequencies, these amplitudesand in these exact alignment conditions. That is however a confirmation that the training of thenetwork has worked to an excellent accuracy and under different alignment conditions . We check the capacity of the statistical model to adapt to unseen situations by preparing acompletely different displacement pattern. This pattern is obtained by applying a fifth orderbutterworth band-pass filter between 5 and 100 Hz to a delta-correlated gaussian randomnoise. This pattern is sent to the speaker in two different alignment conditions, none ofthem corresponding the the situations used during training. In one of the two situations, theinterferometric signal shows a double-peak structure. The two interferometric signals are thenconcatenated into a single time series and we use the model to reconstruct the displacement ofthe target corresponding to this concatenated time series. The results are shown on Fig. 3.As can be immediately appreciated, the reconstruction is excellent, the prediction matchingalmost perfectly the independently measured displacement. Of course the discontinuity close to100 ms, where the two measurements are artificially concatenated, cannot be predicted by thenetwork since it is absent in the measured interferometric signal. We just choose to emphasizethis region as it shows that the prediction is essentially insensitive to the alignment conditions,which change abruptly in the middle of the trace. As is evident from the lower panel of Fig. 3,reconstructing a trajectory from this interferometric signal would be very difficult due to the .0 2.5 0.0 2.5 5.0Displacement ( / ms )5.02.50.02.55.0 P r e d i c t i o n ( / m s ) ms )10 C o un t Fig. 4. The accuracy of prediction on unknown samples (at a given sampling rate)is conditioned by the training set. Left: displacement inferred by the model as afunction of the true displacement. The correlation is excellent but the model slightlyunderestimates the larger displacements (about 2.5 𝜆 /ms). Right: histograms of thedistribution of displacements in the training sets. The larger displacements (about2.5 𝜆 /ms) are very under-represented in the training set. presence of noise and very widely varying fringe shapes and repetition rates.From the above, one concludes that the model is able to generalize from its learning set toprovide an accurate reconstruction of the displacement in unseen alignment conditions and forvery complex time series, much more difficult to analyze than the simple periodic time tracesused during the training phase.To better appreciate the accuracy of the inference, one can plot the predicted displacementas a function of the actual displacement as shown on Fig. 4. A perfect reconstruction wouldbe the one shown by the orange line where the prediction is exactly equal to the truth. Wecan quantify the reconstruction quality by the Pearson’s correlation coefficient between thereconstruction and the ground truth which is here 0.90 and the absolute standard error which ishere 0 . 𝜆 / 𝑚𝑠 . Specifically, one can notice that the prediction is less good for the largest absolutevalues of displacement. This can be related to the statistical properties of the training set asshown on the right panel of Fig. 4. Here one can appreciate that absolute values of displacementlarger than 2.5 𝜆 / 𝑚𝑠 have been seen by the network during training only a few hundreds oftimes, while smaller displacements are much more frequent in our training set. Thus, the largedisplacements are very under-represented in the training set. This results in a lower precision ofthe reconstruction for larger displacements, which can also be appreciated on the top panel ofFig. 3 where the largest displacements are in general under estimated. One of the difficulties in reconstructing the displacement from the self mixing signal also comesfrom the fact that simple Fourier filtering is often not very efficient at separating the detectionnoise from the interferometric signal (although neural networks have been proposed to alleviatethis issue [20]). Here we check that the statistical model is very robust to the addition of noiseon top of the interferometric signal. To assess this robustness, we use the model to reconstructthe displacement corresponding to the complex interferometric signal described in 3.2 afteradding to this signal a gaussian white noise. As we show on Fig. 5, the model predictions areextremely robust. Since the added noise is 𝛿 − correlated, its standard deviation 𝜎 𝑛 is a measureof its power density. Here, we normalize the interferometric signal itself in absence of addednoise to its standard deviation 𝜎 𝑠 so that 𝜎 𝑠 =
1. We then vary the standard deviation of theadded noise 𝜎 𝑛 between 0 and 3 times 𝜎 𝑠 . On Fig. 5a), we observe that both the root meansquared and the absolute error remain very low up to 𝜎 𝑛 = n )0.30.40.50.60.70.80.9 M e a n e rr o r ( / m s ) Noise spectraldensity ( n )0.30.50.70.9 C o rr e l a t i o n c o e ff . D i s p l a c e m e n t ( / m s )
94 96 98 100 102 104 106Time (ms)505 S i g n a l ( a r b . u ) a) b)c)d) Fig. 5. Robustness against detection noise. a) Mean absolute (orange continuous) andRMS (blue dashed) error of the reconstruction as function of added noise in units ofthe interferometric signal’s standard deviation 𝜎 𝑛 . b) Pearson’s correlation coefficientbetween the prediction and the signal as function of added noise. c) Example ofprediction (orange line) and true displacement (blue line) for an added noise powerdensity 𝜎 𝑛 = . × 𝜎 𝑛 where 𝜎 𝑛 is the interferometric signal’s standard deviation. d)Interferometric signal in the same situation, zoom over the 94-106 ms region (the samesignal as Fig. 3, with added noise). excellent reconstruction of the displacement up to approximately 𝜎 𝑛 =
1. We show the predictionand the noisy interferometric signal for 𝜎 𝑛 = . The analysis above has shown that the statistical model is able to reconstruct the displacementfrom the self-mixing signal in a broad range of unknown conditions. However, all of the abovewas realized on a single experimental setup. Contrary to a physical model, which is constructed tocapture only the universal features of an experiment, an empirically constructed statistical modelsuch as the neural network we use may capture also non-universal and system-specific features.Thus, it is interesting to check what the model can predict on the basis of a different experiment,based on the same principle. To address this question, we prepare an "almost-twin" experiment,based on the same self mixing interferometry principle shown in Fig. 1 but featuring a differentlaser (HL6323MG, 𝜆 = 𝑛𝑚 , driven at 𝐼 = 𝑚 𝐴 for a threshold current 𝐼 𝑡ℎ = 𝑚 𝐴 ), adifferent speaker (with a different range of linear response), a different voltage amplifier for theacquisition of the laser diode voltage etc. Although this experiment is in principle the same, itdiffers in many of the details which should not be relevant to the physics, yet carry a significantrisk of distortion of the interferometric signal as compared to the one used in training set.To check the model’s ability to analyze this new self mixing experiment, we prepare a newdisplacement time series consisting of the sum of four different frequencies 𝑆 = (cid:205) 𝑖 = sin 2 𝜋 𝑓 𝑖 𝑡 .0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0Time (ms)10010 D i s p l a c e m e n t ( / m s ) D i s p l a c e m e n t ( / m s ) S i g n a l ( a r b . u ) Fig. 6. Reconstruction of the displacement on an unknown experiment. Top: com-parison between the displacement predicted by the model (orange line) and the actualdisplacement (blue line). There are no free parameters. Middle: zoom on one specificregion. Bottom: self mixing signal corresponding to middle panel. The green circlesshow where jumps between bistable states occur. with 𝑓 𝑖 = , , , 𝐻𝑧 . This time series is therefore in a very different (much higher)frequency band with respect to the training experiment. In order to provide the model withcomparable input data, the acquisition of the self-mixing signal is performed at a ten timesfaster rate than in the training experiment (2.5 MHz). The displacement per time unit of thespeaker is, as in the previous experiment, measured as a voltage at the edges of the speaker. Thecomparison between the displacement estimated from the speaker’s voltage and the displacementreconstructed from the interferometric signal is shown on Fig. 6.The agreement between the prediction and the measurement is strikingly good, especiallytaking into account that no free parameter exist: The model trained on experiment 1 canimmediately be used to infer displacements in units of 𝜆 / 𝑑𝑡 in experiment 2.It is important to underline once more the robustness of this process with respect to specificexperimental conditions. For instance, in this experiment, the interferometric signal showsclear signs of bistability between external cavity modes in forms of very fast jumps betweenstates (green circles on bottom panel of Fig. 6). These features are absent from the training set.Here one sees that the model essentially filters them out automatically. Besides this, it is alsoworth noting that, as compared to Fig. 3 for instance, the displacements and time scales are verydifferent. This shows that, provided an adequate sampling rate is chosen, the model can workat much higher frequency than the band it was trained in and for much larger displacementsper time unit. This feature is not unexpected since the network knows only about displacement"per 256 × 𝑑𝑡 ", without reference to the exact value of 𝑑𝑡 . Therefore, with adequate sampling,the measurement range can be tremendously extended with respect to the training range. Thisfeature is extremely useful since it allows training in an easily accessible range (displacementand frequency band) and prediction in very different range for more demanding applications.
4. Discussion
The results above clearly show that a convolutional neural network is a useful tool in thereconstruction of a target’s displacement, very robust to unknown displacement signal shapes,alignment conditions, electronic noise and even whole setups. We have also verified that anetwork trained in the 10-100 Hz frequency band can also meaningfully reconstruct displacementsncluding frequencies of hundreds of Hz, provided the measurement sampling rate is adapted. Thus,the neural network strength lies less in the absolute precision that it allows than in its robustnessagainst detailed experimental conditions and versatility across the "sub-wavelength/analog" and"beyond wavelength/digital" classification [6] for arbitrarily complex waveforms.One natural question which arises when preparing a neural network is that of model capacity [15].A network which does not possess enough cells or layers may be unable to take into account allthe complexity of the task. On the other hand, a network with a very large number of cells andlayers will sooner or later learn features of the experiment which should not be significant (forinstance, all the details about an amplifier used in the setup). This prevents the network fromgeneralizing, ie accurately predicting unknown data. This is in principle dealt with during thetraining phase [15] but it is only when the network processes fully new data that this issue canbe totally ruled out. Here this issue has been taken care of by predicting arbitrarily complextrajectories and also by using two different setups. In fact, the imperfect reconstruction in thecase of the unknown experiment is most probably due to the model learning some system-specificfeatures of the training experiment. This can be mitigated by a minor retraining of the final layerof the model on the new experiment (a procedure known as "fine tuning" in the deep learningcontext). We have noticed that a larger network featuring more than 10 coefficients instead ofthe 5 . × used here does not lead to better training and may even lead to worse predictions inthe unknown experiment.The performance of the network is of course strongly related to the training data set which isused. Here we deliberately use only a very limited set of displacements during training in orderto very clearly show the generalization phenomenon, the network being able to predict correctlydisplacement shapes it has never seen before. For real use beyond the proof of concept presentedhere, more refined training is possible: A training set featuring a more uniform distribution ofdisplacements will provide a more accurate reconstruction of the larger displacements for a givensampling rate . As an alternative, simply increasing the sampling rate at the prediction time mayalso be a sufficient solution to adapt the time series to the operating range of the model as wehave shown in 3.4. Care must be taken when training the network that the correct operatingrange of the neural network is set by a displacement per time unit, which includes limitationsin terms of displacement frequency and amplitude. Translating it in terms of counting fringes,that means that the network will saturate beyond a certain number of fringes during the 256 × 𝑑𝑡 measurement window. In terms of feedback range, here we have used only 𝐶 < 𝐶 where the interferometric signal is symmetric. At the predictionphase, the model is robust to 𝐶 slightly overcoming unity but when multistability becomes strongthe information of few wavelengths displacements is lost and the model has no chance to recoverit. Similarly, we have checked that when 𝐶 is so low that the signal is symmetric, the modelcannot predict accurately. A full characterization of performance degradation and the use ofmultichannel measurements to mitigate this issue is beyond the scope of this work.One particularly interesting avenue to circumvent the limitations of an experimental trainingset is to train the network on numerically generated data [24]. One drawback is that the networkwill not learn more than what is in the physical model used for the simulations, which may behard in complex settings such as multimode lasers [25, 26]. However, numerics may provide away to obtain a training set for which controlled laboratory experiments would be very hard torealize such as hard shocks or high frequency and high amplitude displacements. Experimentaldata may then be used to refine the training by using different sampling times as described in theprevious paragraph.Once trained, a convolutional neural network can be used in real time since no pre-processingof the data is required and prediction over thousands of interferometric measurements is very fast:As an example, 3 seconds of signal (749056 interferometric data points) are processed in 0.16seconds on a standard laptop. In addition, after the initial training, a neural network can relativelyasily be repurposed by retraining only its final layers even with a very limited set of data. Forinstance, it would be particularly interesting to assess the performance of the network trainedhere on self-mixing schemes which include bias current modulation towards some other sensingtask such as refractive index measurements. Alternatively, the input layer of the network can alsobe reworked at minor cost to take into account multichannel measurements and most interestingwould probably be to integrate this approach into multimodality imaging systems [27].
5. Conclusion
To conclude, we have presented a detailed analysis of how a reasonably simple convolutionalneural network can be used to reconstruct the displacement of a target on the basis of self mixinginterferometry. We believe that this approach can become one of the many tools which canbe used to tailor or enhance self-mixing coherent sensing setups. Far from being limited todisplacement measurement and single mode settings, we believe that computer neural networkscan become an extremely useful element of many sensing apparatus, especially self-mixingsetups. Finally, we stress that there are very few hard rules about the design of neural networksand this design is in itself often an area of research. The architecture we use here is essentiallya simple starting point and many refinements are possible. More specifically, one of the mostimmediate extensions of the work exposed above is to train a network which can provide anestimation of the accuracy of the reconstruction. This can be achieved by adding a calibratedregression stage on top of the convolutional base prepared here [28]. Other extensions mayinclude more complex network topologies, perhaps mixing convolutional and recurrent layers orincluding skip connections.
A. Network details
The key elements of the network are shown on table 1. We refer the reader to deep learningfundamentals for background information [15]. The total number of parameters (57 153) is muchsmaller than the number of time series segments in the data set even before augmentation. Networksof identical architecture with more cells per layer did not lead to significant improvements. Adropout layer is used here mostly as "safety net" since the data set is very large anyway whichmakes overfitting improbable. The training of the network takes about ten minutes on a simpleGPU (GeForce GTX 1060) and about four times more on CPU (Intel Xeon 3.8GHz).
Acknowledgments
The authors thank Dr. L. Columbo, Dr. M. Dabbicco and Dr. F. Pedaci for helpful discussions.
Disclosures
The authors declare no conflicts of interest.
References
1. G. Giuliani, M. Norgia, S. Donati, and T. Bosch, “Laser diode self-mixing technique for sensing applications,” J.optics A: Pure applied optics , S283 (2002).2. D. M. Kane and K. A. Shore, Unlocking dynamical diversity: optical feedback effects on semiconductor lasers (JohnWiley & Sons, 2005).3. S. Donati, “Developing self-mixing interferometry for instrumentation and measurements,” Laser & Photonics Rev. , 393–417 (2012).4. T. Taimre, M. Nikolić, K. Bertling, Y. L. Lim, T. Bosch, and A. D. Rakić, “Laser feedback interferometry: a tutorialon the self-mixing effect for coherent sensing,” Adv. Opt. Photonics , 570–631 (2015).5. J. Li, H. Niu, and Y. X. Niu, “Laser feedback interferometry and applications: a review,” Opt. Eng. , 050901(2017).6. S. Donati and M. Norgia, “Overview of self-mixing interferometer applications to mechanical engineering,” Opt.Eng. , 051506 (2018). etwork structure: sequentialLayer type Mainhyperparameters Trainableparameters Output shape1D convolutional kernel size: 7filters: 16 128 (250, 16)Max Pooling pool size: 2 0 (125, 16)1D convolutional kernel size:7filters: 32 3616 (119, 32)Max Pooling pool size: 2 0 (59, 32)1D convolutional kernel size: 7filters 64 14400 (53, 64)Max Pooling pool size: 2 0 (26, 64)Dropout 10% 0 (26, 64)1D convolutional kernel size: 7filters: 64 28736 (20, 64)Max Pooling pool size: 2 0 (10, 64)Fully connected units: 16 10256 16Fully connected units: 1 17 1 Table 1. Main parameters of the network used in this work. The network is a sequenceof 1-dimensional convolutional and dropout layers followed by two fully connectedlayers for the final regression. The total number of trainable parameters is 57 153.
7. R. C. Addy, A. W. Palmer, K. Thomas, and V. Grattan, “Effects of external reflector alignment in sensing applicationsof optical feedback in laser diodes,” J. lightwave technology , 2672–2676 (1996).8. L. Lv, H. Gui, J. Xie, T. Zhao, X. Chen, A. Wang, F. Li, D. He, J. Xu, and H. Ming, “Effect of external cavity lengthon self-mixing signals in a multilongitudinal-mode fabry–perot laser diode,” Appl. optics , 568–571 (2005).9. M. Norgia, S. Donati, and D. D’Alessandro, “Interferometric measurements of displacement on a diffusing target by aspeckle tracking technique,” IEEE J. quantum electronics , 800–806 (2001).10. U. Zabit, R. Atashkhooei, T. Bosch, S. Royo, F. Bony, and A. Rakic, “Adaptive self-mixing vibrometer based on aliquid lens,” Opt. letters , 1278–1280 (2010).11. O. D. Bernal, U. Zabit, and T. M. Bosch, “Robust method of stabilization of optical feedback regime by using adaptiveoptics for a self-mixing micro-interferometer laser displacement sensor,” IEEE J. Sel. Top. Quantum Electron. ,336–343 (2014).12. A. L. Arriaga, F. Bony, and T. Bosch, “Speckle-insensitive fringe detection method based on hilbert transform forself-mixing interferometry,” Appl. optics , 6954–6962 (2014).13. M. Usman, U. Zabit, O. D. Bernal, G. Raja, and T. Bosch, “Detection of multimodal fringes for self-mixing-basedvibration measurement,” IEEE Transactions on Instrumentation Meas. , 258–267 (2019).14. A. A. Siddiqui, U. Zabit, O. D. Bernal, G. Raja, and T. Bosch, “All Analog Processing of Speckle Affected Self-MixingInterferometric Signals,” IEEE Sensors J. , 5892–5899 (2017).15. I. Goodfellow, Y. Bengio, A. Courville, and Y. Bengio, Deep learning , vol. 1 (MIT press Cambridge, 2016).16. H. Li, C. Zhang, N. Song, and H. Li, “Deep learning-based interference fringes detection using convolutional neuralnetwork,” IEEE Photonics J. , 1–14 (2019).17. A. Reyes-Figueroa and M. Rivera, “Deep neural network for fringe pattern filtering and normalisation,” arXiv preprintarXiv:1906.06224 (2019).18. K. Kou, C. Wang, T. Lian, and J. Weng, “Fringe slope discrimination in laser self-mixing interferometry usingartificial neural network,” Opt. & Laser Technol. , 106499 (2020).19. L. Wei, J. Chicharo, Y. Yu, and J. Xi, “Pre-processing of signals observed from laser diode self-mixing intereferometriesusing neural networks,” in (IEEE, 2007), pp.–5.20. I. Ahmed, U. Zabit, and A. Salman, “Self-mixing interferometric signal enhancement using generative adversarialnetwork for laser metric sensing applications,” IEEE Access , 174641–174650 (2019).21. X.-l. Wang, L.-p. Lü, L. Hu, and W.-c. Huang, “Real-time human blood pressure measurement based on laserself-mixing interferometry with extreme learning machine,” Optoelectronics Lett. , 467–470 (2020).22. F. Chollet et al. , “keras,” (2015).23. K. Hornik, M. Stinchcombe, H. White et al. , “Multilayer feedforward networks are universal approximators.” Neuralnetworks , 359–366 (1989).24. R. Kliese, T. Taimre, A. A. A. Bakar, Y. L. Lim, K. Bertling, M. Nikolić, J. Perchoux, T. Bosch, and A. D. Rakić,“Solving self-mixing equations for arbitrary feedback levels: a concise algorithm,” Appl. optics , 3723–3736(2014).25. L. Columbo, M. Brambilla, M. Dabbicco, and G. Scamarcio, “Self-mixing in multi-transverse mode semiconductorlasers: model and potential application to multi-parametric sensing,” Opt. express , 6286–6305 (2012).26. L. Columbo and M. Brambilla, “Multimode regimes in quantum cascade lasers with optical feedback,” Opt. Express , 10105–10118 (2014).27. M. Brambilla, L. L. Columbo, M. Dabbicco, F. De Lucia, F. P. Mezzapesa, and G. Scamarcio, “Versatile multimodalityimaging system based on detectorless and scanless optical feedback interferometry—a retrospective overview for aprospective vision,” Sensors , 5930 (2020).28. V. Kuleshov, N. Fenner, and S. Ermon, “Accurate uncertainties for deep learning using calibrated regression,” in Proceedings of the 35th International Conference on Machine Learning, vol. 80 of