Data-driven reconstruction of rough surfaces from acoustic scattering

Michael-David Johnson 1

Department of Mechanical Engineering, The University of She ffi eld, She ffi eld, United Kingdom

Anton Krynkin Department of Mechanical Engineering, The University of She ffi eld, She ffi eld, United Kingdom

Jacques Cuenca Siemens Digital Industries Software, Interleuvenlaan 68, B-3001 Leuven, Belgium

Mansour Alkmim Siemens Digital Industries Software, Interleuvenlaan 68, B-3001 Leuven, Belgium

Laurent De Ryck Siemens Digital Industries Software, Interleuvenlaan 68, B-3001 Leuven, Belgium

Yue Li Siemens Digital Industries Software, Interleuvenlaan 68, B-3001 Leuven, Belgium

Giulio Dolcetti Department of Civil, Environmental and Mechanical Engineering, University of Trento, Trento, Italy

Abstract

This work investigates the use of data-driven approaches for reconstructing rough surfaces from scattered sound. The proposed methods stands as alternatives to matrix inversion, which requires a linearisation of the dependence on the surface parameters. Here, a large dataset was formed from scattered acoustic field, estimated through the Kirchho ff Approximation. Limiting this work to the reconstruction of a static surface, K-Nearest Neighbors, Random Forests and a stochastic approach are compared to recover a parameterisation of surfaces using the scattered acoustical pressure as input. The models are then validated against a laboratory experiment alongside methods highlighted in Dolcetti et. al., JSV, 2021. The models are tested at a frequency that best fits the lab uncertainties, then tested on a broad frequency range. This scheme provides relatively accurate results in comparison to the approaches tested. Estimation errors as well as robustness in the presence of noise are discussed.

1. INTRODUCTION

Machine Learning has been a highly active section of research in recent years, proving more than capable in many fields, including acoustics. Bianco et al. [1] highlighted some key areas in acoustics where Machine Learning has been used. Namely, source localisation, bioacoustics and reverberation. However, as the number of Machine Learning solutions increase, questions are rising about the reproducibility of results [2–4]. Typically, when evaluating the performance of a model or method in inverse scattering, there are some base cases which are tested. For example, Fan et al. [5] successfully applied deep learning methodologies utilising the Helmholtz equation in the recovery of the shape and placement of multiple scatterers in two di ff erent settings, including a seismic imaging setting where the source and receivers

1 mjohnson9@she ffi eld.ac.uk

were above the scatterers and receivers were in a linear array. The scatterers were placed randomly and formed from a number of shapes such as triangles, squares or ellipses. These results were then verified against positions and shapes that were known. As well as in Johnson et al. [6] where the surface was created in laboratory, and those measurements were used to predict the shape of the surface relatively accurately for di ff erent noise levels [6] (Figure 1). However, although these results are accurate, it is only known to be accurate because of the setup used for validation. The aim of this work is to investigate a further method of knowing if the model’s prediction is correct. This is done by leveraging the potential benefits that a broadband frequency source can bring, more typical regression models (Linear regression, K-nearest neighbours, and random forests) are trained on every frequency in a broadband frequency range to estimate consistency in prediction, in order to give some confidence in prediction. Following this, the Metropolis Markov Chain Monte Carlo (MCMC) algorithm is shown on a single frequency case, yielding more information than merely a single point prediction. The layout of this paper is as follows: Section 2 holds information about the experiment as well as the properties of the surface tested, section 3 presents the Kirchho ff Approximation, which is used to generate data for the models, as well as a discussion on the data processing. Section 4 showcases results and discussions for the models at a broadband frequency range. Section 5 introduces the Metropolis algorithm, as well as results and discussions. Section 6 contains the conclusions.

Figure 1: Surface reconstruction using a random forest approach, trained with 15% noise included, as highlighted in [6].

2. EXPERIMENT SETUP

For these calculations, the source and receiver locations were chosen in accordance with the existing experimental data [7] which was used for validation in this paper. The source location was at ( x 1 , y 1 ) = ( − 0 . 20 , 0 . 22) m. The angle ϕ 0 of the source main axis to the Ox axis was 60 degrees. The receivers were located at a height of approximately y 2 = 0 . 28 m in the y -axis and 34 receivers were distributed evenly with x 2 taking values from -0.13 m to 0.53 m in the x -axis, leading to an average distance between the receivers of 0.02 m. The real-world data used was collected with 34 1 / 4 ′′ microphones (G.R.A.S. 40PH) and a loudspeaker (Visaton G 25 FFL), arranged with the same geometry discussed. A sinusoidal surface

(with amplitude ζ 1 = 0 . 0015 m and wavelength ζ 2 = 0 . 05 m) was machined from an aluminium block. A signal was produced from the source and recorded simultaneously at all microphones, with a sampling rate of 102.4 kHz. The amplitude at each microphone was calculated by a Fourier transform applied to 0.02 s segments, and averaged over 2000 segments using Hann windowing. This yields information beginning at 10,000Hz and ending at 51,150Hz. The data was calibrated by comparing measurements of the acoustic field reflected by a flat surface with the corresponding Kirchho ff approximation, following the procedure outlined in Dolcetti et al. [7]. The residual di ff erence between the measured pressure field after calibration and the one predicted by the Kirchho ff approximation is shown in Figure 3.

3. DATA GENERATION THROUGH THE KIRCHHOFF APPROXIMATION

Defining ψ s ( R ) as the acoustic pressure field at a point R produced by a source with co-ordinates ( x 1 , y 1 ), scattered by a sinusoidal rigid surface with profile

ζ ( x ) = ζ 1 cos " 2 π

ζ 2 ( x + ζ 3 ) # , (1)

Figure 2: The geometry of the problem where the rough surface is defined by a function ζ ( x ) from equation Equation 1. Surface is not to scale.

The Kirchho ff approximation is assumed to be valid if the following condition is satisfied [10]:

sin( ϕ ) > 1 ( kh ) 1 / 3 , (2)

where h is the local radius of curvature of the surface, k is the acoustic wavenumber and ϕ is the angle of incidence of the acoustic wave. The validity of this approximation for the conditions investigated in this work was demonstrated in Krynkin et al. [8,9]. The Kirchho ff approximation is suitable for the calculation of large amounts of data, which is required for machine learning problems, while being fast to compute. With this approximation, the scattered 2D acoustic pressure ψ s is calculated as [9]:

Z ∞

A ( x , 0) √ R 1 R 2 e i k ( R 1 + R 2 ) ( q y − q x γ )d x , (3)

ψ s ( R ) = 1 2 k π i

−∞

where, as shown in Figure 2, the values R 1 and R 2 are the Euclidean distance from the source at ( x 1 , y 1 ) and receiver at ( x 2 , y 2 ) to a given point ( x , ζ ( x )) on the surface, respectively:

( x 1 − x ) 2 + ( y 1 − ζ ( x )) 2 , (4)

R 1 = p

( x 2 − x ) 2 + ( y 2 − ζ ( x )) 2 . (5)

R 2 = p

R = ( x 2 , y 2 ), γ = d ζ ( x ) / d x ; q x and q y are the x and y components of q = − k ∇ S ( R 1 + R 2 ) with the gradient ∇ S = ( ∂/∂ x , ∂/∂ y ). The directivity term A ( r ), the far-field radiation from a ba ffl ed piston is given by [11]:

A ( r ) = 2 J 1 ( ka sin( ϕ ( r ) − ( − ϕ 0 + π/ 2)))

ka sin( ϕ ( r ) − ( − ϕ 0 + π/ 2)) , (6)

where a is the aperture, J 1 is the Bessel function of the first kind, ϕ 0 is the angle of inclination of the source main axis to the Ox -axis, and ϕ ( r ) is the angle between the vector produced from the location of the source and the point r with the Oy -axis. The phase is then removed from the application of the Kirchho ff approximation to simulate the scattered field by applying the modulus:

p ( R ) = | ψ s ( R ) | . (7)

Taking into account the receiver locations in an array of M receivers, phase-removed acoustic pressure used in the random forest algorithm is given by the following matrix,

p = { p l  R ( j )  | j = 1 .. M , l = 1 .. N } , (8)

where the rows of the matrix are formed from p l (an ensemble containing the absolute array pressure for a given ζ l ), and R ( j ) form the columns (receiver locations defined with respect to the origin of the Oxy plane). Further to this, noise can be added to the signal through row-wise operations on 8:

˜ p l = p l + ϵ l , (9)

where ϵ l j ∈ ϵ l , ϵ l j ∼N (0 , σ ) is drawn from a normal distribution independently for each receiver.

4. RECOVERING MODEL CONFIDENCE USING A BROADBAND FREQUENCY

A way of attempting to recover a measure of confidence in machine learning predictions that are not inherently stochastic is to use a broadband acoustic signal. Therefore, for each frequency one can generate a dataset using that frequency in the Kirchho ff approximation, then train a model on each frequency. Table 1 presents the frequencies used, as well as the sampling used for data generation. For brevity, the datasets did not have any noise present in the training, and ζ 3 was fixed to be 0. For each of the 824 frequencies tested, the mean-squared error of the Kirchho ff approximation against the data observed in the laboratory are shown in Figure 3. A comparison was made between random forests, k-nearest neighbours, and linear regression, implemented through scikit-learn [12]. The metric to measure model performance was chosen to be the coe ffi cient of determination ( R 2 ) and is presented in figure 4. In order to evaluate the potential confidence factor of this method, a scatter plot as well as histograms of the recovered parameters for linear regression, random forests, and k-nearest neighbours are presented in Figures 5 and 6. The random forests were trained using 300 trees in the forest, and the k-nearest neighbours were trained with scikit-learn’s default hyperparameters. From figure 4, it is clear that linear regression is not suited to multiple parameter recovery as expected. For random forest and k-nearest neighbours, the coe ffi cient of determination decreases as

Minimum Maximum Num. of samples Increment

Frequency 10,000Hz 51,150Hz 824 50Hz

Amplitude -0.01m 0.01m 90 0.0002m

Wavelength 0.035m 0.15m 90 0.0012m

Table 1: Bounds for each parameter in the data generation stage as well as the number of samples generated within those bounds, for each frequency tested.

the frequency increases. This could be due to the removal of phase information, with stronger impact at higher frequencies. The k-nearest neighbours slightly outperforms the random forests, where the largest deviation is approximately at 35,000Hz. Figure 5 presents the predictions for the model as scatter points compared to the true parameter values from the experiment. For linear regression (Figure 5a), the estimated amplitude parameter and wavelength parameter are much larger than the actual values, the predictions are consistently bad for all frequencies. For the random forest parameter (Figure 5b), the amplitude seems to mostly be underestimated, and the wavelength parameter seems to deviate from the actual value following the mean-squared error of the predictions presented earlier in Figure 3. K-nearest neighbours (Figure 5c) appears to be the only model presented in this work that consistently predicts the correct parameters, with only a few outliers up until near 30,000Hz. This is reflected in Figure 6, where only the k-nearest neighbour’s histogram has an extremely large mode at the true parameter value. For the data used in this paper, the k-nearest neighbours could be used with a broadband acoustic signal to get some measure of confidence in prediction.

Figure 3: Mean-squared error of the Kirchho ff approximation against the data collected in real-world experiments.

Figure 4: Coe ffi cient of determination scores for random forests, k-nearest neighbours and linear regression (higher is better).

5. RECOVERING MODEL CONFIDENCE THROUGH SINGLE FREQUENCY EXCITATION USING A METROPOLIS SCHEME

Instead of training models on each frequency in order to get a measure of confidence, one can also leverage existing stochastic methods on a single-frequency case. Namely, the Metropolis algorithm. As ϵ is drawn from a probability distribution, equation 9 can be interpreted as:

P ( θ | ˜ p l ) ∼ P ( ˜ p l | θ ) P ( θ ) (10)

The left-hand side is referred to as the posterior distribution and allows for distributions to be taken over parameters, P ( ˜ p l | θ ) is called the likelihood function, and P ( θ ) is called the prior distribution and allows for prior belief on the distribution of parameters. Now that the posterior distribution is found, the main concern is how to sample from this posterior distribution. The Metropolis algorithm will be used for this, the Metropolis-Hastings algorithm is described in Algorithm 1.

Algorithm 1 Metropolis-Hastings Algorithm [13]

Initialise θ 0 for n = 0 , . . . , N − 1 do

Sample u ∼ U (0 , 1) Sample θ ′ ∼ Q ( θ ′ | θ n ) if u ≤ α ( θ n , θ ′ ) = min 1 , P ( θ ′ | ˜ p l ) Q ( θ n | θ ′ )

P ( θ n | ˜ p l ) Q ( θ ′ | θ n )
then θ n + 1 = θ ′ ▷ Accept sample. else

θ n + 1 = θ n ▷ Reject sample. end if end for

Note that to get the Metropolis algorithm from Algorithm 1 requires a symmetric proposal distribution Q ( θ n | θ ′ ) = Q ( θ ′ | θ n ). The choice of the proposal distribution was an selected from the Adaptive Metropolis (AM) algorithm [14] with a targeted acceptance rate of 0.2. The prior distribution was assumed to be independent for each parameter. The amplitude component of the

prior was defined to be uniform with a lower bound at 0 and an upper bound at y s − 3 λ the near-end of the far-field for the acoustic source height. The wavelength was also uniform, with a lower bound of 0.08 and an upper bound of 0.4. The o ff set was also uniform, with a lower bound of zero and an upper bound of the wavelength. If the proposal distribution proposed a sample outside the Kirchho ff criteria, the prior was set to 0. As ϵ is drawn from a Normal distribution, the likelihood function was defined to be a multivariate normal:

P ( ˜ p l | θ ) = exp h − 0 . 5( ψ s ( R ) − ˜ p l ) T Σ − 1 ( ψ s ( R ) − ˜ p l ) i

(2 π ) 34 | Σ | (11)

p

Where Σ - the covariance matrix, is a 34x34 identity matrix with ϵ on the diagonal. For the model, ϵ was taken to be 10% of the mean value from the real-data sample at 14,000Hz. As the Metropolis algorithm has been applied to one frequency, the o ff set parameter is also inferred. Figures 7 and 8 plot the resulting traces and corner plot of the obtained distribution over the three parameters for 14,000Hz source excitation, with 62,000 iterations to the algorithm. Figure 7 highlights the accepted samples from the Metropolis algorithm, after a burn-in period of 5,000 samples, which is represented in grey with a vertical line indicating the cut-o ff point. The burn-in period is used to avoid any skew in distributions, as the accepted samples tend to the target distribution. The traces appear to be consistent, and do not jump between two discontinuous heights. Figure 8 present the resulting histograms of the density of each parameter on the diagonal, and the joint density between two parameters. The resulting densities are very clear single modal, almost Gaussian, distributions. The wavelength has a mode exactly at the true wavelength parameter, and the amplitude’s mode is overestimated by a millimetre. Interestingly, although the phase is 0, the o ff set’s mode is at the wavelength parameter, which relates to a phase of 2 π . So although the o ff set is unexpected, as the surface is periodic the distribution is expected. The resulting distributions seem to provide more easily interpretable information than the broadband frequency investigation highlighted earlier.

6. CONCLUSIONS

To conclude, utilising a broadband acoustic source could be used in order to find some metric of confidence in the model’s predictions. In this case, the peak mode given from the K-NN constant prediction at the true parameter values could be leveraged in order to have a more informed prediction for real-time predictions. However, this leveraging does not produce “clean" Gaussian-like behaviour, such as what was observed using the Metropolis scheme. If true stochastic information is needed, then the Metropolis scheme is highly recommended over the broadband signal. However, due to the time taken for the Metropolis scheme to run (approximately 1 hour), this would not be able to yield real-time predictions. It is also important to note that there have been some limitations in creating the datasets for this paper, the most important of which are: not including noise inside the data, which limits generalisation, and a limited number of samples, which allows for a more dense domain knowledge for the model. It is also important to note that this method, which fits a dataset on every frequency, will be a ff ected by the curse of dimensionality [15] where the data required for higher dimensional surface recovery will require much more data to be able to give any reasonable predictions.

7. ACKNOWLEDGEMENTS

The authors are grateful to Dr. Timo Lähivaara at the University of Eastern Finland for valuable guidance on the statistical sampling procedure.

REFERENCES

[1] Bianco, M. J., Gerstoft, P., Traer, J., Ozanich, E., Roch, M. A., Gannot, S., and Deledalle, C. A. (2019), “Machine learning in acoustics: Theory and applications,” The Journal of the Acoustical Society of America , 146(5), 3590-3628. [2] Beam, A.L., Manrai, A.K. and Ghassemi, M., 2020. Challenges to the reproducibility of machine learning models in health care. Jama, 323(4), pp.305-306. [3] McDermott, M., Wang, S., Marinsek, N., Ranganath, R., Ghassemi, M. and Foschini, L., 2019. Reproducibility in machine learning for health. arXiv preprint arXiv:1907.01463. [4] McDermott, M.B., Wang, S., Marinsek, N., Ranganath, R., Foschini, L. and Ghassemi, M., 2021. Reproducibility in machine learning for health research: Still a ways to go. Science Translational Medicine, 13(586), p.eabb1655. [5] Y. Fan and L. Ying, “Solving inverse wave scattering with deep learning,” arXiv preprint arXiv:1911.13202 , 2019. [6] M.-D. Johnson, A. Krynkin, G. Dolcetti, M. Alkmim, J. Cuenca, and L. D. Ryck, “Application of machine learning to recover surface parameters from phaseless scat- tered acoustic data,” Acoustical Society of America, vol. (Under review). [7] G. Dolcetti, M. Alkmim, J. Cuenca, L. De Ryck and A. Krynkin, “Robust reconstruction of scattering surfaces using a linear microphone array,” Journal of Sound and Vibration , vol. 494, p. 115902, 2021. [8] A. Krynkin, K. V. Horoshenkov, and T. V. Renterghem, “An airborne acoustic method to reconstruct a dynamically rough flow surface,” The Journal of the Acoustical Society of America , vol. 140, pp. 2064-2074, Sept. 2016. [9] A. Krynkin, G. Dolcetti, and S. Hunting, “Acoustic imaging in application to reconstruction of rough rigid surface with airborne ultrasound waves,” Review of Scientific Intruments , vol. 88, p. 024901, Feb. 2017. [10] I. F. F. G. BASS, “Wave scattering from statistically rough surfaces,” pp. 220-227, 1979. [11] P. M. Morse and K. U. Ingard, Theoretical acoustics . Princeton university press, pp. 381, 1986. [12] F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V. Dubourg, J. Vanderplas, A. Passos, D. Cournapeau, M. Brucher, M. Perrot, and E. Duchesnay, “Scikit-learn: Machine learning in Python,” Journal of Machine Learning Research , vol. 12, pp. 2825-2830, 2011. [13] Hastings, W.K., 197,. Monte Carlo sampling methods using Markov chains and their applications. [14] Haario, H., Saksman, E. and Tamminen, J., 2001. An adaptive Metropolis algorithm. Bernoulli, pp.223-242. [15] Kuo, F.Y. and Sloan, I.H., 2005. Lifting the curse of dimensionality. Notices of the AMS, 52(11), pp.1320-1328.

(a)

(b)

(c)

Figure 5: Scatter plots for the parameter recovery at all frequencies for (a) linear regression, (b) random forests, and (c) k-nearest neighbors. The blue horizontal line indicates the true parameter values from the experiment.

(a)

(b)

(c)

Figure 6: Histograms for the parameter recovery at all frequencies for (a) linear regression, (b) random forests, and (c) k-nearest neighbors.

Figure 7: Traces of the resultant Metropolis-Hastings scheme for the amplitude, wavelength, and o ff set. Including the cuto ff for the burn-in period.

Figure 8: Corner plot of the Metropolis-Hastings scheme for the amplitude, wavelength, and o ff set.

PPPPP PPOPP FIPF =<] 6/8 eee SS Sk aS