echantillonnage de champs acoustiquesplc/daudet2013.pdf · séminaire « méthodes mathématiques...

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

Laurent DAUDET

Institut LangevinUniversité Paris Diderot / IUF

[email protected]

Echantillonnage de champs acoustiques

mailto:[email protected]

mailto:[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

http://echange.inria.fr/

http://echange.inria.fr

http://echange.inria.fr

Acoustic fields and sampling : general framework

We want to measure an acoustic field over an area of interest

Measurements are punctual (spatial samples)

xx

x xx

x

xx

waves satisfy the Helmholtz equation

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

Physically-informed sampling-and-

interpolation problem

Acoustic fields and sampling : general framework

Outline of the presentation

• the 3D

• the 2D

• source localization in reverberant environments

• sensor calibration with convex optimization

Compressively sampling the plenacoustic function(with R. Mignot)

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 / m3 !

Ω

δx

n Samplingn with a cutoff frequency fc, time sampling is done at Fs > 2 fcn in space, the maximal sampling step δx is


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 hereleads 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

ωt

ωx

ωy


Numerical simulation in 2D


Sampled in 1D in space

−20−10

010

20

−20−10

0 10

20−20

−10 0

10

qx [m−1]

qy [m−1]

(d)

Sampled in 2D in space


ωt

ωx

ωy

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

provides sparsity

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


frequency

At low frequencies, rooms have a “modal” response

• sparsity in frequency

0100

200

300

400

500

600

05

10

15

20

| TF{ h(t) } |

Fre

quency (

Hz)

Schroeder frequency

• sparsity in timeEarly reflexions appear as

sparse spikesmixing time

At low frequencies, rooms have a “modal” response

0 100 200 300 400 500 6000

5

10

15

20| T

F{

h(t

) }

|

Frequency (Hz)


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


→ mixed norm ｜｜1,2

• sparsity in time

image-source model

Discrete reflections

T

Radius = Tc

energy

time

S0

R

(virtual) sourcesreceiver

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

Multi-resolution MP algorithm

experimental setup

Irregular sampling of a 2*2*2m cube

120 microphones


IJLRA, team MPIA

Microphone array experimentAnalysis with 119 microphones, interpolation of the 120th response.

• sparsity in time


full band, beginning of impulse reponses only

whole temporal support, low frequencies only


• How many microphones ?


12 1619 23 28 34 40 46 52 60 70 80 92 105

0

2

4

6

8

M (number of microphones for the analysis)

Sign

al−t

o−N

oise

Rat

io [d

B]

12 1619 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(with G. Chardon and A. Leblanc)

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

(fine sampling with laser vibrometer : 2 hrs)

Plane-wave approximation

Propagative modes

Evanescent modes

Dispersion relation

0 50 100 150 200 250 300 350

0.4

0.5

0.6

0.7

0.8

0.9

1

k (m−1)

ratio

but k0 is unknown ... Find best match (k0 s.t. measures are maximally projected on the set of

plane waves on the circle of radius k0)

“circle-MUSIC”

Recovering the modal shapes

Fourierinterp.

Structured sparsityinterp.

ReferenceRandom meas. pts

Fine grid Coarse grid

Structured sparsitydisambiguates aliasing


20453 Hz 15688 Hz 18951 Hz

19467 Hz 13597 Hz Sampling pattern

Recovering the dispersion relation

Regular grid

Random grid

Recovering the impulse responses

- measured IR• interpolated IR

time samples

Can be used for calibration of tactile interfaces


Fourier / Shannon

Proposed method

SNR


odd shape

Proposed methodwith evanescent waves

Fourier interp

Proposed methodwithout evanescent waves

SNR

Compressive Nearfield Acoustic Holography

Extension to NAH with sub-Nyquist measurements (collab. F. Ollivier and R. Gribonval teams)

Nearfield acoustical holography with a guitar

JASA paper + «reproducible research» web site NACHOS

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)

G. Chardon and LD, Computational Mechanics (2013, to appear) + online code

Source localization in reverberant environments

(with G. Chardon)


Very reverberant fields, few narrowband measurements

plate with arbitrary shape and boundary conditionsa few punctual sources

>>

= wavefield of free sources+ reverberant

field


well approximated by set of plane waves at

wavenumber k(w)

sparse in a dictionary of sources (Bessel fcs

of the 1st kind)


well approximated by set of plane waves at

wavenumber k(w)

sparse in a dictionary of sources (Bessel fcs

of the 1st kind)

unknowns

= wavefield of free sources+ reverberant

field

With discrete measurements at M positions xmdictionary 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 xm

unknown unknown, sparseAlgorithm 2 : “Greedy Source localization”

dictionary of sources located at xj

measured at xm


measured at xm

project on plane waves p find max

source


With discrete measurements at M positions xm

unknown unknown, sparseAlgorithm 2 : “Greedy Source localization”

dictionary of sources located at xj

measured at xm


measured at xm

project on plane wavesand identified

sources

p find max source


FEM simulated field sampling points

projection on plane waves residual

= +


experimental datamode at 30631 Hz

sub-nyquist sampling pattern

measurements

position of sources estimated by GBP


experimental datamode at 30631 Hz

sub-nyquist sampling pattern

position of sources estimated by GBP


It is possible to locate narrowband sources, at sub-nyquist precision, in unknown 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


Microphones only

o o o

oo

x x x

Microphones + virtual measurements

ooox

x x

real

detected

Optimizing sensor position(with G. Chardon and A. Cohen)

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 (with C. Bilen, G. Puy, R. Gribonval)

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)Y set of measurements

Calibration issues

Let’s assume M known up to a diagonal matrix D0

ex : unknown gain on each microphone

• Approach 1

• Approach 2 Δ = D-1

• non-convex• scaling issue

this is a convex problem !

Calibration issues

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

increasing de-calibration

Calibration issues

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

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

even at high decalibration (±10 dB)

How many training vectors ?

Here ~4-5 training vectors are neededSimilar «phase transitions» diagrams !

de-calibration

number of training vectors

(δ,ρ) fixed below the DT curve

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 problem solving

Conclusion

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 ...

echantillonnage de champs acoustiquesplc/daudet2013.pdf · séminaire « méthodes mathématiques...

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