It is often hard to relate the sensor’s electrical output to the physical scenario when a multidimensional measurement is of interest. An artificial neural network may be a solution. Nevertheless, if the training data set is extracted from a real experimental setup, it can become unreachable in terms of time resources. The same issue arises when the physical measurement is expected to extend across a wide range of values. This paper presents a novel method for overcoming the long training time in a physical experiment set up by bootstrapping a relatively small data set for generating a synthetic data set which can be used for training an artificial neural network. Such a method can be applied to various measurement systems that yield sensor output which combines simultaneous occurrences or wide-range values of physical phenomena of interest. We discuss to which systems our method may be applied. We exemplify our results on three study cases: a seismic sensor array, a linear array of strain gauges, and an optical sensor array. We present the experimental process, its results, and the resulting accuracies.

1. Introduction

Reconstruction problems are of interest to the research community due to their importance in engineering applications. These have been studied with regard to heat source identification [15], shape reconstruction [6], wave source identification [79], inverse boundary conditions reconstruction [10, 11], and the problem of force identification on plates [1217]. A notable example is [14] by Chierichetti and Ruzzene which uses the confluence algorithm to numerically reconstruct the dynamic response field of the structure from a limited number of experimental measurements (for a survey on analytic methods, see [18]). Another reconstruction problem is considered in [19] where an optical sensor array is used to position an IR (reflected) light source.

This paper offers a new line of thought on such problems. We gather sensor inputs taken in simple scenarios and manipulate them in accordance with the physical governing equations to form synthetic data in order to reconstruct more involved scenarios.

A sensor measures a physical quantity or its derivatives (for example, the change of an electrical analog signal is converted into the corresponding physical unit which is of interest [20, 21]). We distinguish between two conceptual cases: (i)A sensor may be applied when a single quantity is to be detected. Examples include scalar quantities like velocity, temperature, distance, and light emission as the inputs to be estimated. Other examples are vector quantities like spatial orientation, detecting the location of a light source etc.(ii)A sensor may also be applied when simultaneous quantities are to be detected. Examples include scalar quantities like detecting all joint angles of a robotic arm [22], extracting the tensile stress field [6, 23], and measuring the motion of the sea bed [24]. Examples for simultaneous vector quantities to be detected include stress tensor field extraction and detecting the position of multiple light sources [25]

Using multiple sensors will not always detect simultaneous quantities. The reason for that is that in such cases, these quantities are aggregated into the sensor system—an array of sensors, in a complex manner. The same reasoning may apply also for the case where one is required to sense a wide range of a given physical quantity.

Artificial neural networks (ANN) are commonly used in computer vision [26] and regression problems. Though regression is a statistical process for estimating the relationships among variables, it can be used for computing arbitrary functions. Being available and due to the increasing computing power, the ANN usage has been accelerated in almost every domain of computer science [27, 28]. A neural network may be a good solution to analyze the sensor data described in (i) to estimate a physical quantity [29]. Of course, one needs to obtain a sufficiently large training set for convergence (c.f. in [30]). Obviously for simultaneous quantities to be detected, training the ANN would require to increase the training set exponentially in the power of which is unrealizable in many cases.

The training neural network for image recognition, 3D data, and video is typically an expensive task of collecting and annotating big data sets. The task of automatically generating such labeled data is an emerging field of research; see for example [3133]. A new different approach is called domain randomization [3437] where synthetic data is created by adding random textures to real images in order to force the ANN to learn the essential features. The underlying logic of this paper is somewhat similar to this approach.

In this paper, we introduce a way to analyze data from a sensor array by bootstrapping synthetic training data for an ANN. This way, the ANN “decomposes” the simultaneous quantity inputs and estimates its components individually. We also show how one can bootstrap synthetic training data for an ANN to analyze a wide range of input values.

Remark. The term bootstrap has various interpretations in literature. In computer science, the term may indicate a procedure for simulating discrete events [38] or be used to indicate a way to iteratively improve a classifier’s performance [39]. In statistics, bootstrap refers to a computational manipulation in order to establish certain statistical values [40]. Some other fields of research use the term in a totally different meaning. Here, we use the term bootstrap to indicate a procedure in which one manipulates real data from sensor arrays to form artificial sensor array data.

1.1. Contribution and Paper Organization

The main concern of this paper is to construct synthetic physical measurements as data sets for ANNs. This is essential for cases were harvesting experimental data is out of reach (for reasons of time consumption, experimental setup complexity, etc.). For this end, we shall use basic ANNs provided by MATLAB.

The synthetic training data sets are bootstrapped out of a certain experimental set. More precisely, data is first collected from experiments; we then manipulate the sets to create a new training data. We discuss when such a method can be applied and exemplify our method for three cases of sensor arrays (cantilever deflection sensor, seismic sensors on a metal plate, and an optical sensor array).

This paper is organized as follows: Section 2 provides a mathematical formulation for situations where our method may be applied. Section 3 presents three sensor array study cases. Results for these cases where a single stimulus is of interest are provided in Section 5. In Section 6, we apply our bootstrapping method, and we conclude the paper in Section 7.

2. A Mathematical Formalism

In what follows, we shall examine situations where one may desire to generate synthetic training data for a neural network. For clarity, we shall consider three examples for application and follow them throughout the paper:

2.1. Study Case I

A horizontal cantilever beam with an unknown, nonconstant cross-section is included. A set of sensors (e.g., strain gauges) located along its axis, each on a different segment, measures the beam’s curvature (Figure 1). The beam is thought to be loaded by a set of vertical forces in which their position and magnitude are of interest. This sensor array was simulated using a finite element method. We used the finite element method (FEM) as a ground truth to train the ANN. Specifically, we used it to calculate the stress field and the resulting error function for training the ANN.

2.2. Study Case II

A set of seismic sensors (as depicted in Figure 2) is positioned peripherally on an the aluminum plate with an unknown, nonconstant thickness. The plate is subjected to a single impulse with unknown magnitude in which its position is of interest.

2.3. Study Case III

A light sensor array built out of 6 light-dependent resistors (LDR) which are situated on an arc as depicted in Figure 3 is included. A set of six lights in which their identity and magnitude are of interest, individually illuminates the sensor. We shall be interested in detecting up to two simultaneous light sources.

The commutative diagram in Figure 4 formulates the various scenarios discussed in this paper. We consider a stimulus which is a map such that every sensor location is mapped to a physical quantity. So, is the space of that measured physical quantity (typically, ). For study case I, is the set of stain sensor positions and is the curvature value range (or deflection). In study case II, is the set of seismic sensor positions and is the plane’s peripheral movement space. In study case III, is the set of points where the LDRs are situated and is the full range luminescence, measured, for example, in Lum.

The sensor’s transfer function assigns an dimensional electrical output in to an dimensional stimulus (i.e., assigns a value to each sensor). The power map assigns for experiments. For study case I, computes the measured resistance in the ith segment as where is a constant (explicitly, where is the resistivity of the sensor and its cross-section) and xi is the strain in the ith segment. For study case II, is the amplitude and frequency of the plate in all sensor points due to an impulse of a unit magnitude (say, 37 gr dropped from 20 mm) and computes the corresponding measured capacitance in all the seismic sensors. Note that since in case study II, we refer only to a single impulse, we are actually interested only in a diagonal subset of (which can be thought of as several impulses assigned to the same position). For study case III, the light sensor is based on LDRs. is given by the LDR’s datasheet which provides the relation between resistance and luminous flux per unit area , measured in Lux.

For all linear systems, the net response of two or more simultaneous stimuli is simply their sum (i.e., is the addition operator). Note that for the analysis which will be introduced here, the commutativity of the diagram in Figure 4 is enough. In the most general case, is the map (not necessarily linear) from the stimuli to the net stimulus (which is of the same dimension as a single ), for example, for study case I that would be stating the superposition principle for small deflections (in other words, saying that the strain in the ith segment can be summed up). For case II, a naive assumption would be to state superposition again but this is, of course, not the case as wave interference plays here an important role given that the metal plate is not uniform.

For study case III when two light sources project light on an LDR, its resistance changes due the resulting luminous flux per unit area equation (1). Evidently, for the case of light sources, the resistance is with as their respective detected luminance:

In other words, given a prior knowledge of the resistance due to each light source independently, one may calculate the resistance in a case where two light sources project their light simultaneously.

The phenomena which are of interest are marked . For study case I, this corresponds to the amplitudes and positions of loads. The physical model is a mapping from the ’s to the stimulus and for the case of simultaneous phenomena . Note that there may be scenarios where may depend on the location of the sensors (e.g., nonisotropic plates, involving diodes etc.) In study case II, corresponds to dropping a unit magnitude weight. may be the transverse (shear) wave propagation principle while in study case I, this may refer to the Euler-Bernoulli beam deflection theory. For study case III, corresponds to the light intensity from the light sources. So, the map states the light propagation rules.

Finally, is a map that assigns a set of separate set of outputs to a single output . That is, constructs a single synthetic sensor output from a set of known sensor outputs (that correspond to a set of stimuli resulting from the physical phenomena), under the assumption that the stimuli are simultaneous.

We assume the function is a known decomposable multivariate function (e.g., is a simple sum or some other decomposable functions as as in [41], equation (1), which is exactly as in case III). A special case where a multivariate function is a linear combination of products of functions in single variables was examined by Neuman in [42].

In many cases where the relevant physics is too complicated to compute or out of reach for any reason, one may use the machine learning approach to estimate the phenomena of interest. In numerous engineering scenarios, the essential traits of the underlying measured object (e.g., deflected beam, vibrating plate, and sensor exact traits) cannot be attained, for example, (1) the inner geometry of the object contains nonnegligible holes and imperfections which cannot be inspected, (2) a portion of the object geometry is out of reach (e.g., pipe’s inner wall, pressure vessels, and tightly connected plates), (3) objects which suffer slow corrosion and therefore their geometry changes over time, and (4) objects which their governing equations are not fully known but could be measured (e.g., wooden and composite material beams).

Explicitly, one may use an artificial neural network if a single-stimulus data set of real experiments is at hand. For cases where real experiments of some simultaneous stimuli are available, an ANN will do. There are cases where one can extract synthetic data using to use as input of without conducting a full set of experiments. In other cases, one may use (i.e., real or synthetic data set) as an input to an ANN . This in turn is used to extract which then can be used as input for .

3. Sensor Array Examples

We shall now introduce the first step of our training scheme, that is, we consider a situation where a single quantity is to be detected. To analyze this input, we shall apply an ANN. This will serve as our benchmark in Section 6 where we shall use the ANN together with a preknowledge of our physical system to synthesize a virtual training set for out-of-reach scenarios.

3.1. Study Case I: Cantilever Beam Deflection Detection

In some cases, the frequency response of a loaded cantilever beam is examined to detect the load magnitude (as in [43] where femtograms are detected) and position [44]. Obviously, these cases use beams with known dimensions. Consider a case where the loads (magnitudes and positions) are to be detected on a beam which is already in use. So, the beam’s cross-section dimensions are not necessarily known for reasons mentioned above. Such a case can emerge if (1) internal cavities within the beam may exist, (2) a time-dependent wear is significant, and (3) cross-section measurements cannot be performed for any reason.

In this study, in order to find the loads’ magnitudes and positions along a cantilever beam, a set of strain gauges are used with the wiring depicted in Figure 5(a). If the beam’s cross-section is known, the strain field along the beam can provide the magnitude and position of a load. The expected stress field in this case would nullify from the free end to the load position. The force magnitude can be extracted using Euler-Bernoulli beam equation . Nevertheless, in the case where a second moment of area and the position from the neutral axis are unknown, this would not do. Therefore, one can conduct a set of experiments with a single load at a time; these would constitute a training set for an ANN that would be able to extract the load properties from the strain field. Figure 1 presents a cantilever beam with a nonuniform cross-section. The set of strain gauges is fixed along the beam axis on its bottom face. For simplicity, the loads that shall be considered are uniformly distributed as depicted to avoid torsion effect. Along the experiment, distributed forces of magnitude 5 − 15 kg were loaded one at a time on a 1 m long cantilever beam. At the second stage of this paper, we load the beam with two simultaneous forces of the same range of magnitudes.

3.2. Study Case II: Metal Sheet Impulse Detection

Solids support sound vibrations of different types. Seismic sensors are devices used to measure these vibrations by converting motion into a measurable electronic signal. Several attempts to develop an automatic means to detect and locate mechanical impacts are available in literature (see for example, [45] where inspections of metallic-based structures are proposed and in [46], a sensor system is devised to monitor impact damage of pressure vessels). The advantage of using an ANN for this application is realized for reasons mentioned above.

In this study, we attached six geophone seismic sensors to a aluminum plate. The sensor’s positions were selected arbitrarily and the wiring is depicted in Figure 5(b). A 75 mm long and 19 mm diameter brass rod was dropped on the plate from a 20 mm height. At each drop, the sensor data was collected using a microcontroller. For the sake of accuracy and repeatability, we used a ScorBot serial 5 DOF robot to drop the brass rod on the plate in a 5 mm resolution grid distribution. We repeated the experiment at the second stage of this paper where we dropped the same rod from a 40 mm height a few additional times in order to interpolate synthetic experiments (i.e., to extract data from nonexistent experiments performed from varying heights).

3.3. Study Case III: Light Sensor Array

In order to demonstrate a nontrivial , we devised a linear sensor array as a low-cost Lidar. The LEDs’ positions were not carefully aligned with the LDRs; the wiring is depicted in Figure 5(c). A similar optical light array concept is used in [19]; in their work, they used TSL1401 Linear Array of photodiodes which is commonly used for edge detection for printers. The researchers used two commercially available IR line image sensors and a array photo detector to measure the position of the finger and its gesture. For simplicity, here, we used LDRs.

4. ANN Architecture and Training

This section presents the ANNs and their training setups used for the study cases. For all cases, we used MATLAB’s neural network fitting as a generator with a two-layer feedforward network with Sigmoid hidden perceptrons and linear output perceptrons having 100 perceptrons in the hidden layer. The network was trained with Levenberg-Marquardt backpropagation algorithm. 70% of the experimental data was used for training, 15% for validation, and 15% for testing.

In study case I, the neural network is given a set of 50,000 experiments each having a vector of length 39 capturing the stress field under each loading. The output is a vector of length 2 representing the load’s position and magnitude. We used a finite element method to calculate the stress field and used mean square error of the output vector as the cost (error) function. We consider a uniform steel cantilever beam with a varying cross-section. The beam model is developed with the FEM Toolbox for MATLAB [47]; using elements, we set Poisson’s ratio to be 0.3 and .

In study case II, a set of 1,925 experiments each is a vector of size 6 capturing the geophone magnitudes. A vector of length 2 representing the impulse position on the plate constituted the output.

In study case III, the neural network is given a set of 50,000 real experiments each having a vector of length 6 capturing the voltage under each imminence. The output is a vector of length 6 representing the estimated magnitudes at the six possible positions of the six LEDs.

5. Results for the Single-Quantity Cases

In study case I, each FEM run took about 2 sec. Experimental data from 50,000 scenarios were collected (two days is a reasonable estimation for the time required for a real robotic experiment assuming ∼3 sec for each load). The training phase took about 10 minutes. Figure 6 plots the model error histogram across 50,000 unseen experiments. The average position error and the standard deviation extracted using were and , respectively. The average load error and the standard deviation extracted were and , respectively. In study case II, the data collection process took about 3.5 hours during which experimental data from 1,925 plate impulses in different positions were collected. Figure 7 plots the model error histogram across 131 unseen experiments. The average drop position error and standard deviation using were and , respectively. In study case III, the data collection process took about 2 hours during which data from 50,000 real experiments was collected. At each experiment, a single random LED was turned on with a random duty cycle. Figure 8(a) plots the magnitude error histogram across 20,000 unseen experiments. The histogram is given as a function of percentile error of the LED activation duty cycle (which corresponds to the light intensity one senses). In ∼85% of the cases, succeeded in assessing the true position of the light source. The duty cycle error average and standard deviation were and , respectively.

6. Bootstrapping Data Sets and ANN Training for the Multiple-Quantity Cases

In all cases, we used the same ANN architecture as described in Section 4.

The cantilever beam sensor in study case I is capable of locating multiple loads as well. For this end, the training task requires multiple simultaneously independent loadings. Nevertheless, recall that an ANN requires a large data set, for example, two loads would require a data set of to have the accuracy described in Section 5. This obviously is out of reach if this should be mechanically accumulated (i.e., actually loading the beam).

To overcome this, we bootstrap our single-load training data to that of multiple loads. In other words, we construct synthetic data from our already established one-load training data to train an additional ANN, which we marked in Figure 4, for multiple loads. Since we assume our loads do not exceed the elastic region, we may use superposition which corresponds to map of Figure 4. In other words, the resultant strain caused by several loads applied simultaneously is the summation of the strain caused by each load individually.

Figure 9 presents the results of synthetic training for two loads. The average position error and the standard deviation extracted using were and , respectively. The average load error and the standard deviation extracted using were and , respectively.

Study case II demonstrates our method for the case of finding the position of an unknown impulse magnitude (we do so since for study case II, simultaneous impulses will require a nonrealistic time proximity.). Recall that in Section 4, we dropped the brass rod from a constant height of 20 mm. Locating the impulse position for different impulse magnitudes requires data training for these height drops. Nevertheless, since we use a capacitor, an increase in the impulse magnitude will result in an increased output voltage of the sensor. Knowing the sensor’s output resulting from two different impulses located at the same position, one may use a simplified linear description (i.e., linear interpolation) to calculate the sensor’s output for an impulse in that range. To this end, we conduct 72 additional high-altitude drops (i.e., of large magnitude) to generate the synthetic data for ANN (see Figure 4). Explicitly, we define rectangular regions that cover the plate. For each region, we conducted an additional 3 drops at its center. These were then used to give 24 linear approximations, corresponding to map of Figure 4, for sensor data that corresponds to impulses of a wider magnitude range. Based on these experiments, we generated 194,000 synthetic data records that correspond to drops of a wide magnitude range. These were then used to train ANN . Figure 10(b) presents the results. The average position error and the standard deviation extracted using were and , respectively.

Remark. In order to demonstrate the efficiency of our method, we examined which was trained for only 20 mm height drops as in Section 5. The ANN was tested with experimental data from varying height drops. Figure 10(a) presents the results—an average position error and the standard deviation extracted were and , respectively.

The sensor in study case III is capable of locating multiple light sources as well. For this end, the training task requires multiple simultaneous luminescence. An experiment set for a single light source theoretically requires experiments which amount to only ∼7 minutes of data harvesting. But note that since we use here real experiments as training data, errors and inaccuracies are inevitable, so a limited data set did not suffice and we were forced to use 50,000 experiments to train the net. The experiment took ∼2 hours to complete.

Obviously, when two light sources are of interest, a much larger number of experiments are required. To overcome this, we used equation (2) to bootstrap our single light source training data to that of two lights. The bootstrapping procedure for generating 100,000 synthetic data “experiments” took 2 sec. This was used to train the ANN, which we marked in Figure 4, for multiple lights.

Figure 8(b) presents the results of the synthetic training of for the two light source cases across 20,000 unseen experiments. The average intensity error and the standard deviation were and , respectively (given in duty cycle percentage). The ANN provided a ∼70% success in accurately positioning two light sources with an additional ∼20% of the tests which showed an accurate estimation of one of the light source positions while the other was estimated neighboring the true position.

Remark. Note that the discussion above assumes that the number of loads/light sources is known. This is a rather easy task for an ANN. For example, to establish this for case I, one may use the simplest classifying ANN with a one-layer feedforward 50-perceptron architecture. We did so by training the ANN with the following data: (i) 25,000 acquired data entries from single-load experiments, with additional (ii) 25,000 synthetic data entries which simulate experiments with two loads. The ANN succeeded in categorizing 84% of the cases. Of course, once one knows how many loads participate in a given experiment, the ANNs described in the paper can be applied.

7. Conclusions

We have presented three sensor array scenarios. In Section 4, we used ANN to identify the nonlinear function between the sensor’s output voltages and the measured quantity (a static load, an impulse, and a light source detection). For the first and third sensor array cases, we demonstrated how one can bootstrap a synthetic data set of multiple stimuli from a data set of a single stimulus. For the second case, we showed how synthetic data for unknown impact magnitudes can be bootstrapped from a data set of an impact of a constant magnitude. Such a strategy is essential when a data set which should be extracted from a real experimental setup is not feasible.

Our results show that generating synthetic data while carefully considering the physical environment is efficient both in the sense of time resources and accuracy. For cases II and III for example, the time required for producing a training set for was about two orders of magnitude shorter.

In this paper, we used very simple and shallow ANNs and . Obviously, using more sophisticated architectures may improve the estimation accuracy if needed. A previous research work by the authors attempted to design a spatial shape sensor for flexible manifolds. Future work will apply the method presented here together with a deep-learning ANN for estimating spatial shapes in real time.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that there is no conflict of interest regarding the publication of this paper.