Instrumentation & Measurement Magazine 24-7 - 12

the normal velocity of a planar surface with the acoustic
pressure generated by the vibrating object. In particular, the
method is based on the k-space analysis (i.e., in the wavenumber
domain) with Fourier-based algorithms. Notice that
the exterior radiation generated by non-planar surfaces, and
in general by arbitrary shape geometries, is described by the
Kirchhoff-Helmoltz (KH) integral equation. Its direct discretization
leads to the well-known Boundary Element Method
(BEM), and therefore the inversion of BEM (IBEM) [16] offers
a numerical solution to the NAH problem using Tikhonov
regularization. Although IBEM provides accurate results,
even with complex geometries, it is highly demanding from
the computational standpoint, limiting its applicability. An
alternative to IBEM was proposed by Chardon et al. [17], exploiting
a sparse solution to the problem of NAH for convex
homogeneous plates. The main idea, based on Compressed
Sensing (CS), is to approximate the velocity field as a linear
combination of functions chosen in such a way to form
a proper basis, thus reducing the computational effort. This
technique is known as Nearfield ACoustic HOlography
with Sparse regularization (NACHOS), which is also able
to reduce the number of measurements that are needed, i.e.,
microphones, by taking advantage of the notion of CS. Unfortunately,
this approach can only be applied to planar plates,
where the approximation is valid.
An interesting approach to the NAH problem is the Equivalent
Source Method (ESM) [18], which mainly consists of
modeling the sound field generated by a vibrating structure
as if it were produced by set of point sources (equivalent
sources), placed within or beneath the structure itself. The
idea proposed by Zhang et al. was to perform NAH by inverting
the ESM, by first estimating the source weights, and then
retrieving the normal velocity on the surface. Although the
ESM offers a more efficient solution to the NAH problem than
IBEM, the computation of the optimal set of equivalent sources
in terms of numbers and location is still an open problem.
The Role of Learning Tools in NAH Analysis
In [19], Canclini et al. propose a modification to the standard
ESM, called Dictionary-based ESM (D-ESM), to reduce the
number of equivalent sources to a small and sparse subset. After
generating a compressed dictionary of possible equivalent
source weights, they perform NAH by seeking a sparse linear
combination of the entries of the dictionary. Although the velocity
reconstructions with D-ESM are improved, they are still
sensitive to noise and microphone positioning. Nevertheless,
this method shows how data-driven models can obtain an efficient
and parsimonious representation. Moreover, the recent
growth of Deep Learning techniques has proven the ability of
Deep Neural Networks to learn a set of powerful feature representations
directly from the data.
Olivieri et al. recently proposed a fully data-driven methodology
for the problem of NAH [20]. They focused on
Convolutional Neural Networks (CNN) and in particular on
Convolutional Autoencoders which are able to learn a latent
representation of the data by taking advantage of a pair of
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non-linear encoding and decoding functions. The main idea
is to let the network infer the inverse relation between the
acoustic pressure and the normal velocity fields, thus avoiding
explicit matrix inversions which characterizes the methods
mentioned earlier. In their preliminary study [20], the authors
train a CNN, inspired by the renowned UNet architecture,
with a synthetic dataset of rectangular plates generated with
Finite Element Method (FEM) simulations. Experiments have
proven the effectiveness of the devised methodology also
against noisy input data and errors in the microphone positioning
definition.
From these results, we are investigating the extension of
such technique to objects with more complex shapes, such as
the violin. Some reconstruction examples of the CNN-based
NAH applied to different violin top plates and different frequencies
are depicted in Fig. 1. In particular, it shows the
pressure data (IN) acquired on the hologram plane, the velocities
predicted by the neural network (OUT) and the
synthesized velocities used as ground truth (GT) to validate
the model. Moreover, the network can predict even vibrational
patterns at frequencies that were not part of the training
set. From these promising results, we foresee the application
to identification problems, thus offering a non-invasive technique
to prevent structural damages on objects with complex
shapes. Therefore, as a future work, we will integrate physicsinformed
CNN architectures to provide better reconstruction
accuracy in different scenarios where a limited number of microphones
can be positioned.
Feature-based Analysis on Vibrational
Data
The vibrational behavior of musical instruments in the frequency
domain has been studied for several decades. In
particular, the first definitions related to the Frequency Response
Function (FRF) and its acquisition techniques date
back to the first half of the 20th century. FRFs describe pointto-point
relations between applied force and consequent
vibration in a mechanical structure. In the case of the violin, the
bridge admittance gives a good description of the overall behavior
of the soundbox as 'seen' by the strings. The estimation
is performed by generating an impact on one side of the bridge
and collecting the response in terms of acceleration on the opposite
one. Acquired signals are then processed to improve the
Signal to Noise Ratio (SNR) by means of the so-called H1, H2
and Hv estimators, depending on the assumptions concerning
the presence of noise [7]. With the evolution of the sensors
adopted in the measurement of force applied and consequent
motion in mechanical structures, FRFs have become reliable
and objective estimators for vibration analysis. Indeed, FRFs
are still widely adopted and compared either to assess correlations
between different instruments or to describe changes in
the modes of vibration as a function of specific design parameters,
as done in [21]. However, even recent works still present
results in terms of visual comparisons, thus making their
interpretation too subjective and unable to report actual relationships
between one domain and the other.
IEEE Instrumentation & Measurement Magazine
October 2021
Instrumentation & Measurement Magazine 24-7

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