REGULARIZATION METHOD APPLIED TO DECONVOLUTION PROBLEM IN REAL-TIME NEARFIELD ACOUSTIC HOLOGRAPHY Sébastien Paillasseur*1, Jean-Hugh Thomas 1, and Jean-Claude Pascal 1 1
Laboratoire d’Acoustique de l’Université du Maine, 72085 Le Mans Cedex 09, France sé
[email protected]
Abstract The purpose of this paper is to present a method to solve the deconvolution problem related to the forward projection of the pressure radiated by non-stationary sources in order to improve the Real-Time Nearfield Acoustic Holography (RTNAH) and to test its accuracy. The first step is to simulate measurements made on a microphone array in the nearfield of a source plane defined by non-stationary punctual sound sources. A spatial two dimensional Fourier transform is applied to each instantaneous pressure field yielding a time-dependent wavenumber spectrum. Taking the evanescente waves into account improves the spatial resolution of our solution but makes the deconvolution problem ”ill-posed”. It means that the inverse of the transfer function is neither unique nor stable. To solve this kind of problem, we studied regularization methods that consist in giving an other constraint to the solution we are seeking. In our case the standard Tikhonov regularization is used, which is based on the minimisation of the solution’s energy, combined with generalized crossed validation (GCV) to estimate the regularization parameter. In order to apply this method we first use a formulation that allows us to transform the time-wavenumber domain convolution into a matrix product. Once the backpropagated wavenumber spectrum is obtained, the inverse two-dimensional Fourier transform is applied to get a picture of the time-dependent pressure of the sources. It is then possible to localize and characterize the different sound sources.
Eds.: J. Eberhardsteiner, H.A. Mang, H. Waubke
S. Paillasseur, J.H Thomas, J.C Pascal
INTRODUCTION y
y x
y x
x
zs
zm
zf
Source plane
Measurement plane
Forward plane
z
Figure 1 – Geometry of the Acoustic Holography
The acoustic holography is a measuring process to localize sound sources from measurements made with an array of microphones located in the near-field of the acoustic source plane. This method called Near-field Acoustic Holography (NAH) was introduced in the heighties [1] and is used for stationary sources. In order to characterize non-stationary sources, some methods like Time Domain Holography [2] or Time Method [3] may be used. A new formulation has been introduced [4] to propagate signals on a forward plane z=zf (zf > zm) using a convolution product with an impulse response in the time-wavenumber domain. This formulation does not require any assumption about the stationary properties of the sources and can describe the time dependency of the propagated sound pressure field on the forward plane. The subject of this paper is to present a method to solve the deconvolution problem related to this formulation and thus to introduce the Real-Time Nearfield Acoustic Holography.
DIRECT PROBLEM The direct problem of the Real-Time Nearfield Acoustic Holography consists in propagating the sound pressure field measured on a plane z=zs to a forward plane z=zf. It was shown [4] that this problem can be solved by using the convolution product of the time-dependent wavenumber spectrum P(Kx,Ky,zm,t) with an impulse response h:
P ( K x , K y , z f , t ) = P ( K x , K y , zm , t ) ∗ h( K x , K y , z f − zm , t ) ,
(1)
where Kx and Ky are the wavenumber along the axis x and y. The time-dependent wavenumber spectrum P(Kx,Ky,z,t) is calculated by applying a two dimensional Fourier transform along the axis x and y on the sound pressure field p(x,y,z,t) :
ICSV13, July 2-6, 2006, Vienna, Austria
P( K x , K y , z, t ) = ∫
+∞
−∞
∫
+∞
−∞
p ( x, y , z , t )e
j ( kx x + k y y )
dxdy .
(2)
The expression of the impulse response h is obtained by considering the twodimensional Fourier transform of the acoustic propagation described by the wave equation (3) : ∆p( x, y, z , t ) −
1 ∂ 2 p ( x, y , z , t ) = 0, c2 ∂t 2
(3)
where c (m.s-1) is the velocity of sound in air. Solving this equation gives the expression of the impulse response h which depends on the propagation distance dz=zf - zm : dz 2 2 2 2 J1 c K x + K y t − 2 c dz 2 2 h( K x , K y , dz , t ) = δ t − − dz K x + K y c dz 2 t2 − 2 c
Γ t − dz c
(4)
where J1 is the first order Bessel function, δ(t), the Dirac delta function and Γ(t), the Heavyside function defined by : 0 for t < 0 Γ(t ) = 1 for t ≥ 0
(5)
Replacing the expression of h in the equation (1) leads to the time-dependent wavenumber spectrum on the forward plane P(Kx,Ky,zf,t). It is then possible to calculate the instantaneous spatial pressure in the forward plane p(x,y,zf,t) by applying the inverse two dimensional Fourier transform on P(Kx,Ky,zf,t).
INVERSE PROBLEM In the case of the Real-Time Neafield Acoustic Holography we are seeking the time-dependent pressure field on the source plane from measurements made by a microphone array in the nearfield of the source plane. In order to backpropagate the pressure field radiated by non-stationary sources it is necessary to solve the deconvolution problem of the equation (1) which can be written as : P ( K x , K y , zs , t ) = P ( K x , K y , zm , t ) ∗ h −1 ( K x , K y , zm − zs , t ) ,
(6)
S. Paillasseur, J.H Thomas, J.C Pascal
The effect of the backpropagation may be explained by separating the progressive and the evanescente waves of the radiated pressure field, however this separation is not done in the RT-NAH as all the components are convolued with the impulse response h.
Source plane
Measurement plane
Source plane
(a) Evanescente waves
Measurement plane
(b) Progressive waves
Figure 2 – Backpropagation of the evanescente and progressive waves
As shown on figure 2 , the backpropagation is equivalent to an amplification of the components corresponding to evanescente waves and a change of phase for the components corresponding to the progressive waves. This amplification, particulary in the presence of measurement noise may lead to erroneous values in the backpropagated pressure field. This problem is then said ”ill-posed”, processing the solution is needed in order to make it stable and unique.
RESOLUTION OF THE DECONVOLUTION PROBLEM The method we chose to solve this problem is the standard Tikhonov regularization [7] as it does not require any assumption on the processed signal. This method consists in adding a constraint on the solution which is the minimization of its energy. If we consider the linear system
H .x = y
(7)
where H, y are known and x is to be calculated. The standard Tikhonov regularized solution xλ of the equation (7) is given by :
{
2
xλ = min H .x − y 2 + λ 2 x
2 2
}
(8)
where ║x║2 is the L2 norm of x and λ is the regularization parameter which will influence the weigth of the regularization. This parameter is to be estimated by the use of methods like the Generalized Crossed Validation (GCV) [5] or the L-curve [6]. First of all it is necessary to discretize and rewrite the convolution product in equation
ICSV13, July 2-6, 2006, Vienna, Austria
(1) into a matrix product equivalent to the linear system in equation (7). If we consider the discretized convolution product
y ( n) = x ( n) ∗ h( n) .
(9)
0 x(1) y (1) h(1) 0 " y (2) h(2) % % # x(2) = . . # # % % 0 # y ( N ) h( N ) " h(2) h(1) x( N )
(10)
The equation (9) is equivalent to
It is then possible to apply the standard Tikhonov regularization for each pair (Kx,Ky). The Singular Value Decomposition (SVD) is a mathematical tool used to apply the regularization. The SVD of a matrix H is H = U .S .V H ,
(11)
where S is the diagonal matrix of the Singular values of H, U and V are the singular matrix associated and are orthonormal. VH is the conjugate transpose of the matrix V. The inverse of H is −1 1 1 H −1 = (U .S .V H ) = V .S −1.U H = V .diag ," , sN s1
H .U .
(12)
The solution of the equation (7) can be written as N
x=∑ i =1
uit . y .vi . si
(13)
It has been shown [8] that the regularization will act as a filter on the singular values of H N
xλ = ∑ f i . i =1
uit . y .vi , si
(14)
where xλ is the regularized solution and fi the coefficients of the regularization filter. In the case of the standard Tikhonov regularization the filter coefficients fi are fi =
si2 . si2 + λ 2
(15)
S. Paillasseur, J.H Thomas, J.C Pascal
To estimate the regularization parameter we used the General Crossed Validation, which consist in minimising the function G defined by: G=
H .xλ − y (1 − f )
2 2
2
,
(16)
where f is the regularization filter f =
H H
2
2 2
+ λ2 2
.
(17)
Once the regularization parameter λ is determined, the regularized solution is given by the equation (14) where the filters coefficient are calculated with the equation (15). The method presented in this section is used to solve the inverse problem related to the equation (1) yielding the backpropagated time-dependent wavenumber spectrum on the source plane.
SIMULATION The source plane is composed of one monopole at the position (0.2,0.2,0) that radiates a signal s (t ) = cos(2π ft )e −γ t .
(18)
The attenuation by an exponential function, characterized by the decreasing parameter γ, is used to ensure the non-stationary property of the source. The acquisition of the pressure field p(x,y,zm,t) is simulated by a 11x11 microphone array located on the measurement plane defined by z = 0.0322 m. A spatial two dimensional Fourier transform is applied to the pressure field simulated yielding the time-dependent wavenumber spectrum P(Kx,Ky,zm,t). For each pair (Kx,Ky) the backpropagation of the time-dependent signal is equivalent to the equation (9). It is then possible to apply the inverse method presented above to obtain the backpropagated time-dependent wavenumber spectrum P(Kx,Ky,zs,t). The regularized parameters λ estimated by the GCV method are shown in figure 3. We can see that the regularization will act as a filter that will attenuate the components with high values of Kr, which correspond to the evanescente waves. Kr = K x2 + K y2 .
(19)
ICSV13, July 2-6, 2006, Vienna, Austria
The regularization parameters are thus coherent with the theory introduced in the inverse problem section.
Figure 3 – Regularization parameters for each pair (Kx,Ky)
Once the regularization parameters are set, the standard Tikhonov regularization is applied. The backpropagated time dependent pressure field p(x,y,zs,t) is obtained by using the inverse two dimensional Fourier transform. Backpropagated pressure field (dz=0.0322 m) : Sample. 25 -> 0.0009 s 0.5
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Figure 4 – Backpropagated pressure field
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Figure 5 – Backpropagated time-dependent pressure (black) and reference (red)
As shown in figure 4 the localization of the source is effective, the backpropagated field pressure is coherent with the definition of the source plane. The figure 5 displays the comparison between the backpropagated time-dependent pressure at the position of the source and the signal radiated by this source.
S. Paillasseur, J.H Thomas, J.C Pascal
CONCLUSIONS The deconvolution problem related to the Real-Time Nearfield Acoustic Holography is an “ill-posed” problem and thus requires specialized processing. An effective method to solve this problem have been introduced in this paper using the standard Tikhonov regularization combined with the General Crossed Validation (GCV). The backpropagated time-dependent pressure field is coherent with the definition of the signal radiated by the source however we can notice a difference between the amplitudes which may be improved by oversampling the impulse response h as it was shown in [9].
REFERENCES [1] Williams E.G., Maynard J.D. and Skudrzyk E., “Sound source reconstructions using a microphone array”, J. Acoust. Soc. Am., 68, 340-344 (1980). [2] Hald J., Time domain acoustical holography. (Inter-noise, Newport Beach, 1995). [3] Deblauwe F., Leuridan J., Chauray J.L. and Béguet B., Acoustic holography in transient conditions. (ICSV, Copenagen, 1999). [4] Grulier V., Thomas J.-H., Pascal J.-C. and Le Roux J.-C., Time varying forward projection using wavenumber frmulation. (Inter-noise, Prague, 2004). [5] Iqbal M., “Deconvolution and regularization for numerical solutions of incorrectly posed problems”, Journal of computational and applied mathematics, 151, 463-476 (2001). [6] Hansen P.C. and O’Leary D.P., “The use of the L-curve in the regularization of discrete ill-posed problems”, SIAM J. Sci. Comput., 14, 1487-1503 (1993). [7] Williams E.G., “Regularization methods for near-field acoustical holography”, J. Acoust. Soc. Am., 110, 1976-1988 (2001). [8] Tanter M., Aubry J.F., Gerber J., Thomas J.L and Fink M., “Optimal focusing by spatiotemporal inverse filter”, J. Acoust. Soc. Am., 110, (2001). [9] Grulier V., Propagation directe et inverse dans l’espace temps-nombre d’onde: Application à une méthode d’holographie acoustique de champ proche pour les sources nonstationnaires. PhD thesis, Université du Maine, 2005.
