1. Automatic Position Calibration of Multiple Microphones
Vikas Chandrakant Raykar | Ramani Duraiswami
Perceptual Interfaces and Reality Lab. | University of Maryland, CollegePark

2. Motivation
Multiple microphones are widely used for applications like source localization, tracking and beamforming.
Most applications need to know the precise locations of the microphones.
Small uncertainity in the sensor location could make substantial contribution to the overall localization error.
In ad-hoc deployed arrays it is tedious and often inaccurate to manually measure using a tape or a laser device.
In this paper we describe a method to automatically determine the three dimensional positions of multiple microphones.

3. Automatically fix a coordinate systemZ

4. If we know the positions of 3 speakers….Y
Distances are not exact
Need atleast 3 speakers
in 2D. Can use more
speakers
Find the intersection in
the least square sense ? X

5. If positions of speakers unknown…
Consider M Microphones and S speakers.
What can we measure?
Distance between each speaker and all microphones.
Or Time Of Flight (TOF)
MxS TOF matrix
Assume TOF corrupted by Gaussian noise.
Can derive the ML estimate.
Calibration signal

6. Nonlinear Least Squares..
More formally can
derive the ML estimate
using a Gaussian
Noise model
speed of sound
Find the coordinates of both the microphones as speakers which minimizes

7. Maximum Likelihood (ML) Estimate..
If noise is Gaussian
and independent
ML is same as
Least squares
we can define a noise model
and derive the ML estimate i.e. maximize the likelihood ratio
observation
model
parameters to be
estimated
Gaussian noise

8. Reference Coordinate System
Positive Y axis
Similarly in 3D
1.Fix origin (0,0,0)
2.Fix X axis
(x1,0,0)
3.Fix Y axis
(x2,y2,0)
4.Fix positive Z axis
x1,x2,y2>0
Origin
X axis
Which to choose? Later…

9. Nonlinear least squares..
Levenberg Marquadrat
method
Function of a large number of parameters [ 3(M+S)-6 ]
Unless we have a good initial guess may not converge
to the minima.
Approximate initial guess required.
If we have M microphones and S speakers
[ 3M+3S–6 ] parameters to estimate. [ MS ] TOF observations
[ MS ] >= [ 3M+3S – 6 ] If M=S=K then K>=5
Why do we consider M=S ? Later..

10. Closed form Solution..
Say if we are given all pairwise distances between N points can we get the coordinates. 1 2 3 4 X

11. Classical Metric Multi Dimensional Scaling
dot product matrix
Symmetric positive definite
rank 3
Say given B can you get X ?....Singular Value Decomposition
Same as
Principal component Analysis
One hitch.. we can measure
only the pairwise distance matrix

12. How to get dot product from the pairwise distance matrix…Cosine Lawk j

13. MDS... If given pairwise distances between cities we can build a map.
Instead of pairwise distances we can use pairwise “dissimilarities”.
When the distances are Euclidean MDS is equivalent to PCA.
Eg. Face recognition, wine tasting
Can get the significant cognitive dimensions.
Steyvers, M., & Busey, T. (2000). Predicting Similarity Ratings to Faces using Physical Descriptions. In M. Wenger, & J. Townsend (Eds.),
Computational, geometric, and process perspectives on facial cognition: Contexts and challenges. Lawrence Erlbaum Associates

14. Can we use MDS.. ? 1. We do not have the complete pairwise distancesX
1. We do not have
the complete
pairwise distances
UNKNOWN
UNKNOWN

15. Forming microphone speaker pairs…
Now we know the locations of speakers and microphones close to them.
Problem is essentially same as with position of speakers known.
Can get a closed form solution using least squares technique.
Can refine all the values further by a further ML estimation.

16. Microphone and speaker
The complete algorithm…
Approximation
Approx
Distance matrix
Between
Microphone
Speaker pairs
TOF matrix
MDS
Approx. microphone
and speaker
locations
Nonlinear
minimization
Nonlinear
minimization
Exact. microphone
and speaker
locations
Microphone and speaker
locations
Approx.
Microphone
locations

17. Sample result in 2D…

18. Algorithm Performance…
The performance of our algorithm depends on
Noise variance in the estimated distances.
Number of microphones and speakers.
Microphone and speaker geometry
One way to study the dependence is to do a lot of monte carlo simulations.
Else can derive the covariance matrix and bias of the estimator.
The ML estimate is implicitly defined as the minimum of a certain error function.
Cannot get an exact analytical expression for the mean and variance.
Can use implicit function theorem and Taylors series expansion to get approximate expressions for bias and variance.

19. Where to place loudspeakers..

22. Monte Carlo Simulations…

23. Calibration Signal…

24. Time Delay Estimation…
Compute the cross-correlation between the signals received at the two microphones.
The location of the peak in the cross correlation gives an estimate of the delay.
Task complicated due to two reasons
1.Background noise.
2.Channel multi-path due to room reverberations.
Use Generalized Cross Correlation(GCC).
W(w) is the weighting function.
PHAT(Phase Transform) Weighting

25. Experimental Setup…

26. Results

27. Related Previous work…
J. M. Sachar, H. F. Silverman, and W. R. Patterson III. Position calibration of large-aperture microphone arrays. ICASSP 2002
Y. Rockah and P. M. Schultheiss. Array shape calibration using sources in unknown locations Part II: Near-field sources and estimator implementation. IEEE Trans. Acoust.,Speech, Signal Processing, ASSP-35(6): , June 1987.
R. Moses, D. Krishnamurthy, and R. Patterson. A self-localization method for wireless sensor networks. Eurasip Journal on Applied Signal Processing Special Issue on SensorNetworks, 2003(4): , March 2003.
J. Weiss and B. Friedlander. Array shape calibration using sources in unknown locations a maximum likelihood approach. IEEE Trans. Acoust., Speech, Signal Processing , 37(12): , December 1989.

28. Our Contributions… Locations of the speakers need not be known.
Only constraint is that there showld be a microphone close to a loud speaker.
In a practical setup attach a microphone to a louspeaker.
Derived the theoretical variance of the estimator.
Where to place the loudspeakers?

29. Acknowledgements… Dr. Dmitry Zotkin for building the microphone array.
Dr. Elena Grassi and Zhiyun Li for the data capture boards,

30. Thank You ! | Questions ?