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Echantillonnage de champs acoustiquesplc/daudet2013.pdf · Séminaire « Méthodes Mathématiques...
Transcript of 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
Echantillonnage de champs acoustiques
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/
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
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 hereleads to 1:2 savings in sampling along a line
Let us sample f along the x-axis
dispersion relation
Compressively sampling the plenacoustic function
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
Compressively sampling the plenacoustic function
Numerical simulation in 2D
Compressively sampling the plenacoustic function
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
Compressively sampling the plenacoustic function
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».
Compressively sampling the plenacoustic function
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
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)
• sparsity in frequency
structured dictionary of plane waves, sparse in the number of modes
Compressively sampling the plenacoustic function
→ 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
Compressively sampling the plenacoustic function
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
IJLRA, team MPIA
Microphone array experimentAnalysis 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
• How many microphones ?
Compressively sampling the plenacoustic function
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.
Compressively sampling the plenacoustic function
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
Recovering the modal shapes
Recovering the modal shapes
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
Recovering the impulse responses
Fourier / Shannon
Proposed method
SNR
Recovering the impulse responses
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)
Source localization in reverberant environments
Very reverberant fields, few narrowband measurements
plate with arbitrary shape and boundary conditionsa few punctual sources
>>
= wavefield of free sources+ reverberant
field
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)
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
= 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
Source localization in reverberant environments
With discrete measurements at M positions xm
unknown unknown, sparseAlgorithm 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 find max
source
Source localization in reverberant environments
With discrete measurements at M positions xm
unknown unknown, sparseAlgorithm 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 wavesand identified
sources
p find max source
Source localization in reverberant environments
FEM simulated field sampling points
projection on plane waves residual
= +
Source localization in reverberant environments
experimental datamode at 30631 Hz
sub-nyquist sampling pattern
measurements
position of sources estimated by GBP
Source localization in reverberant environments
experimental datamode at 30631 Hz
sub-nyquist sampling pattern
position of sources estimated by GBP
Source localization in reverberant environments
It is possible to locate narrowband sources, at sub-nyquist precision, in unknown reverberant environments !
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 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
Sampling solutions of the Helmholtz equation
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)
Sampling solutions of the Helmholtz equation
Example : functions defined on
sampling with uniform probability
sampling with
Sampling solutions of the Helmholtz equation
sampling with uniform probability
sampling with a fraction α of samples on the contour, the rest inside
Sampling solutions of the Helmholtz equation
approximation error(n = 400 samples)
Sampling solutions of the Helmholtz equation
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 ...