How To Measure An Acoustic Field Over An Area Of Interest

Séminaire « Méthodes Mathématiques du Traitement d'Images » Laboratoire Jacques-Louis Lions, Paris, 27 juin 2013

Laurent DAUDET Institut Langevin

Université Paris Diderot / IUF [email protected]

Acknowledgements

• Joint work with Gilles Chardon, Rémi Mignot, Antoine Peillot, François Ollivier, Rémi Gribonval, Albert Cohen....

in the framework of the ECHANGE project (2009-2012), funded by the French Agence Nationale de la Recherche

Acoustic fields and sampling : general framework

We want to measure an acoustic field over an

area of interest

Measurements are

punctual (spatial samples) x x x x x x x x

waves satisfy the Helmholtz equation

boundary conditions and

sources (inside or outside the region of interest) are unknown

Physically-informed

Acoustic fields and sampling : general framework

Outline of the presentation

•

•

•

Compressively sampling the

plenacoustic function

Compressively sampling the plenacoustic function

• (abstract) notion of “plenacoustic function” (Ajdler / Vetterli, 2002)

p =

f

(S, R, t, room characteristics)

”How many microphones do we need to place in the room in order to completely reconstruct the sound field at any position in the room?”

• The acoustics of a room is characterized by the full set of impulse responses between sources and receivers.

• uniform sampling «a la Shannon» [Ajdler/Vetterli, IEEE TSP 2006]

• Sampling on a large 3D domain might require a very large number of measurements

• fc = 17 kHz → 106 _{measurements / m}3 _!

Ω

δx

n Sampling

n with a cutoff frequency f_c, time sampling is done at Fs > 2 f_c n in space, the maximal sampling step δ_x is

Compressively sampling the plenacoustic function

Use physics ! We are not sampling any random signal in [0, Fc] but solutions of the wave equation.

S fixed, for a given room f(x,t)

(Ajdler / Vetterli, 2002)

The useful information

lives here

leads to 1:2 savings in sampling along a line

Let us sample f along the x-axis

dispersion relation

but space is 3D

In the 4-D space (3-D + t), the useful information ideally

lives on a cone along the temporal frequency axis

We should “only” have to sample a 3-D surface in a 4-D space

ω

ω

ω

Numerical simulation in 2D

Compressively sampling the plenacoustic function

Sampled in 1D in space −20 −10 0 10 20 −20 −10 0 10 20 −20 −10 0 10 x [m −1_] y [m −1_] (d) Sampled in 2D in space

Compressively sampling the plenacoustic function

ω

ω

ω

If you sample in n-D a wave in n-D you get one dimension «for free»

Same gain as with integral theorems but with more freedom in the sampling scheme

(and therefore more control on the stability of numerical schemes)

... however, we cannot use this directly as we don’t sample in the 4D Fourier space, but this

The goal of this study is

to take advantage of this prior information on the signals

directly at the acquisition stage

in order to reduce the number of measurements, using the sampling paradigm of «compressed sensing».

In practice, 2 forms of (approximate) sparsity

time

Compressively sampling the plenacoustic function

frequency

At low frequencies, rooms have a “modal” response

• sparsity in frequency 0 100 200 300 400 500 600 0 5 10 15 20 | TF{ h(t) } | Frequency (Hz) Schroeder frequency • sparsity in time

Early reflexions appear as sparse spikes

mixing time

At low frequencies, rooms have a “modal” response 0 100 200 300 400 500 600 0 5 10 15 20 | TF{ h(t) } | Frequency (Hz) • sparsity in frequency

structured dictionary of plane waves, sparse in the number of modes

Compressively sampling the plenacoustic function

• sparsity in time image-source model Discrete reflections T Radius = Tc energy time S₀ R (virtual) sources receiver S₁ S₂ S₃ S₄ S9 S10 S7 S₆ S12 S₁₁ S₈ S5 For t < T : Diffuse reverberation

unstructured dictionary of monopole sources, sparse in the number of virtual sources

Toy example:

Simulation of 7 synthetic sources in a 2D free field.

microphones synthetic sources estimated sources

Toy example:

Simulation of 7 synthetic sources in a 2D free field.

microphones synthetic sources estimated sources

Toy example:

Simulation of 7 synthetic sources in a 2D free field. microphones synthetic sources estimated sources

Multi-resolution

MP algorithm

experimental

setup

Irregular sampling of

_{a 2*2*2m cube}

120 microphones

Compressively sampling the plenacoustic function

Microphone array experiment

Analysis with 119 microphones, interpolation of the 120th _response.

• sparsity in time

• sparsity in frequency

full band, beginning of impulse reponses only

whole temporal support, low frequencies only

•

Compressively sampling the plenacoustic function

12 16 19 23 28 34 40 46 52 60 70 80 92 105 0 2 4 6 8

M (number of microphones for the analysis)

Signal − to − Noise Ratio [dB] 12 16 19 23 28 34 40 46 52 60 70 80 92 105 76 82 88 94 100 Pearson Correlation [%] SNR [dB] Correlation [%]

• Yet to be done : fusion of the 2 approaches

time frequency

Here only a statistical behavior is necessary

(low-order ARMA model)

• Optimal spatial distribution of measurements ?

• ”How many microphones do we need to place in the room in order to completely reconstruct the sound field at any position in the room?”

➔ maybe not a crazy number !

Sparsity makes it realistic to experimentally sample the plenacoustic function in a whole 3D volume.

Sampling the vibrations of plates

D-shaped plate

known for chaotic

behavior

Extension to plate vibrations measurements

How about plates (Kirchoff-Love / Reissner-Mindlin) ?

How many point measurements are necessary

to recover the full vibration behavior ?

unknown dispersion relation

Plane-wave approximation

Propagative modes

Evanescent modes

Dispersion relation

but

k

is unknown ...

Find best match (k0 s.t.

measures are maximally projected on the set of plane waves on the circle

of radius k0)

Recovering the modal shapes

Random meas. pts Fine grid Coarse grid

Structured sparsity disambiguates aliasing

Recovering the modal shapes

20453 Hz 15688 Hz 18951 Hz

Recovering the dispersion relation

Regular

grid

Random

grid

Recovering the impulse responses

- measured IR

• interpolated IR

time samples

Recovering the impulse responses

Fourier /

Shannon

Proposed

method

SNR

Recovering the impulse responses

odd

shape

Proposed method

with evanescent waves

Fourier interp

Proposed method

without evanescent waves

Compressive Nearfield Acoustic Holography

Extension to NAH with sub-Nyquist measurements

Nearfield acoustical holography with a guitar

A new numerical method for plate eigenmodes

Generalization to plates of the Method of Particular Solutions

• approximation bounds for the plate equation • new formulation of the boundary conditions

➔ Finding plate eigenmodes can be done as a generalized eigenvalue problem

using a low-dimensional basis of solutions (plane waves or Fourier-Bessel fcs)

Source localization in

reverberant environments

Source localization in reverberant environments

Very reverberant fields, few narrowband measurements plate with arbitrary shape and boundary conditions

>>

=

reverberant

_field

₊

_{free sources}

wavefield of

Source localization in reverberant environments

well approximated by set of plane waves at

wavenumber k(w)

sparse in a dictionary of sources (Bessel fcs

Source localization in reverberant environments

well approximated by set of plane waves at

wavenumber k(w)

sparse in a dictionary of sources (Bessel fcs

of the 1st kind)

unknowns

With discrete measurements at M positions

x

dictionary of

sources located at xj

measured at xm

dictionary of plane waves, with wavenumber k,

measured at xm

Algorithm 1 : “Group Basis Pursuit”

unknown

unknown, sparse

With discrete measurements at M positions

x

unknown

unknown, sparse

Algorithm 2 : “Greedy Source localization”

dictionary of

sources located at xj

measured at xm

dictionary of plane waves, with wavenumber k, measured at xm project on plane waves

p

With discrete measurements at M positions

x

unknown

unknown, sparse

Algorithm 2 : “Greedy Source localization”

dictionary of

sources located at xj

measured at xm

dictionary of plane waves, with wavenumber k, measured at xm project on plane waves and identified sources

p

FEM simulated field sampling points

projection on plane waves residual

=

+

experimental data

mode at 30631 Hz

sub-nyquist

sampling pattern

measurements

experimental data

mode at 30631 Hz

sub-nyquist

sampling pattern

Source localization in reverberant environments

It is possible to locate narrowband

sources, at sub-nyquist precision, in

Source localization in reverberant environments

Does it still work in 3D ? (PhD research of T. Nowakowski)

Yes, but requires a very large number of measurements

➔ Use (partial) information about the room geometry

Ex : «virtual measurements» on wall near the antenna

100 « virtual measurements » where vy = 0 40 microphones (measure p) 1 source requires a hybrid model pressure - velocity

Source localization in reverberant environments

Microphones + virtual measurements

o o ox x x real detected

Optimizing sensor position

Sampling solutions of the Helmholtz equation

Back to basics : the Helmholtz equation

Vekua theory Fourier-Bessel fcs Plane waves

Fourier-Bessel plane waves

We wish to approximate u on a whole spatial domain,

with a budget of n point measurements:

• What order L to use ?

• L too small : bad approximation

• L too large : unstable approximations • Where to take the samples ?

Results by Cohen Davenport Leviatan (2011)

Example : functions defined on

sampling with uniform probability

sampling with

sampling with uniform probability

sampling with a fraction α of samples on the contour, the rest inside

approximation error (n = 400 samples)

Blind calibration for CS

Calibration issues

Our model assumes M perfectly known

What can you do if M only imperfectly known ? 1. Ignore the problem !

Calibration issues

Our model assumes M perfectly known

What can you do if M only imperfectly known ? 1. Ignore the problem !

2. Consider as additive noise 3. Supervised calibration

4. Unsupervised calibration

X unknown (but sparse)

Calibration issues

Let’s assume M known up to a diagonal matrix

D

ex : unknown gain on each microphone

•

•

Δ

= D

• non-convex

• scaling issue

Calibration issues

even for mild de-calibration (±1 dB), standard CS fails

Calibration issues

even for mild de-calibration (±1 dB), standard CS fails

with sufficiently many (unknown but sparse) training vectors, calibrated CS still succeeds

How many training vectors ?

Here ~4-5 training vectors are needed

Similar «phase transitions» diagrams !

de-calibration

number of training vectors

(

δ

,

ρ

) fixed

How many training vectors ?

Closer to the DT curve ~8-10 training

vectors are needed

Phase calibration

Now let’s assume known gains but unknown phases

ex : position of microphone not perfectly known

Similar to phase retrieval problems but here unknown phase shifts are common to all measurements

Calibration issues

But dimension of the problem gets squared

→ need fast suboptimal techniques

jointly solve for L=6 vectors Vector-by-vector

Compressively Sensing acoustic fields

• Compressed sensing is a way to put prior information about the signal at the acquisition stage (sparse regularization) and a way to design corresponding efficient measurements

Compressively Sensing acoustic fields

• Compressed sensing is a way to put prior information about the signal at the acquisition stage (sparse regularization) and a way to design corresponding efficient measurements

• Sensing sampling : crossroads between physics and signal processing. • Practical issues raise many interesting questions

calibration, algorithms for large data size, structured sparsity, dictionary design, optimal sampling distribution, ...

• The small print need specific design, sometimes heavy computational cost, need to know precisely the direct problem : very sensitive to model mismatches ...