how to (compressively) sample an acoustic ﬁelda few references • mignot r., daudet l. and...

ESI workshop - Time-frequency methods for the applied sciences, 4 dec 2012

Laurent DAUDET

Institut LangevinUniversité Paris Diderot

[email protected]

How to (compressively) sample an acoustic field ?

mailto:[email protected]

mailto:[email protected]

Acknowledgements

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

in the framework of the ECHANGE project, funded by the French Agence Nationale de la Recherche

http://echange.inria.fr/

http://echange.inria.fr

http://echange.inria.fr

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 [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

Sampling with a cutoff frequency fc, time sampling is done at Fs > 2 fc 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

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

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


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


‖=‖

q,r

2-step algorithm: find frequencies (joint sparsity across measurements) then modes

• sparsity in time

image-source model

Discrete reflections

T

Radius = Tc

energy

time

S0R

(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


Source interpolation : the « basis image method »

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


• Dependency on distance to the centre


• 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)

• ”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.

• Applications : virtual acoustics, source localization, dereverberation ...• Many open questions : calibration, algorithms for large data size, structured

sparsity, dictionary design, optimal sampling distribution, ...


A few references

• Mignot R., Daudet L. and Ollivier F., Compressed sensing for acoustic response reconstruction: interpolation of the early part, Proceedings of the IEEE Workshop on Applications of Signal Processing to Audio and Acoustics, WASPAA (2011)

• Mignot R., Chardon G. and Daudet L., Compressively sampling the plenacoustic function, Proceedings of the SPIE XIVth Conference on Wavelets and Sparsity, San Diego, vol.8138-08, invited paper (2011).

• Journal papers submitted (please email me).

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 ?

With Gilles Chardon and Albert Cohen

Sampling solutions of the Helmholtz equation

Example : functions defined on

sampling with uniform probability

sampling with


Results by Cohen Davenport Leviatan (2011)

sampling with uniform probability

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


approximation error(n = 400 samples)


model order

«Auditory sketches» : A preliminary perceptual experiment

with A. Drémeau, collaboration D. Pressnitzer team at ENS (C. Suied, S. Shamma)

A preliminary perceptual experiment

We want to test if humans still perceive a sound «identity» after a sparse approximation

• how many atoms must we keep ? • what selection rule ?

• This experiment is based on «emotions»

• 4 emotions : anger, disgust, happiness, sadness• Equalized for duration (~) and rms level• Unprocessed sounds: No noise / SNR = -6 dB• Features are atoms in an auditory time-frequency

representation (no other psychoacoustic model).• 3 levels of approximation : 10, 100,1000 features/s• 2 selection rules :

• Approx. best N-atoms approximation • Largest N local maxima

• 10 participants



less sparsevery sparse

original

Best N-atoms approximation

(maximum energy)

local maxima

Local maximaSparse approximation

SNR is not a good criteria to select atoms

Time-frequency spreading is also

important

Clean original sounds


Local maximaSparse approximation

... But peak-picking is not a good denoising method at high rates

noisy -6dB sounds


work in progress...

THE END

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how to (compressively) sample an acoustic ﬁelda few references • mignot r., daudet l. and...

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