PDE-BASED INTERPOLATION METHOD FOR OPTICALLY VISUALIZED SOUND FIELD. Kohei Yatabe and Yasuhiro Oikawa

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1 4 IEEE International Conference on Acoustic, Speech an Signal Processing (ICASSP) PDE-BASED INTERPOLATION METHOD FOR OPTICALLY VISUALIZED SOUND FIELD Kohei Yatabe an Yasuhiro Oikawa Department of Intermeia Art an Science, Wasea University, Tokyo, Japan ABSTRACT An effective way to unerstan the behavior of a soun fiel is to visualize it. An optical measurement metho is a suitable option for this as it enables contactless non-estructive measurement. After measuring a soun fiel, interpolation of the ata is necessary for a smooth visualization. However, conventional interpolation methos cannot provie a physically meaningful result especially when the conition of the measurement causes moiré effect. In this paper, a special interpolation metho for an optically visualize soun fiel base on the Kirchhoff-Helmholtz integral equation is propose. Inex Terms Soun fiel visualization, partial ifferential equation (PDE), laser Doppler vibrometer (LDV), Schlieren imaging.. INTRODUCTION Unerstaning the behavior of a soun fiel is one of the most important tasks for many acousticians. Microphones are usually use for such observation; however, the presence of measuring instruments insie the soun fiel prevents accurate measurement because a soun fiel epens on the spatial setting of the fiel. As an alternative, optical observation for auible soun fiel have been evelope recently incluing Schlieren imaging [, ] an Optophone [3, 4]. Our research group is eveloping the laser measurement metho using a laser Doppler vibrometer (LDV) [5, 6]. These methos enable a contactless non-estructive measurement of soun fiel by etecting fluctuations on optical quantity cause by the soun fiel such as iffraction an Doppler shift. Common characteristic among these optical methos is that the measure ata represent physical quantity of a soun fiel integrate along a beam of measuring light. Although some applications may require three-imensional information which leas to collaboration with compute tomography (CT) [5, 7, 8, 9, ], two-imensional visualization of the soun fiel is still a highly effective tool for its qualitative evaluation. This paper focuses on a visualize two-imensional soun fiel projecte by the optical methos. When a measure soun fiel is isplaye on a screen, postprocessing of the measure ata, typifie by interpolation, is necessary since it is ifficult to measure thousans or even millions of points for smooth appearance epicte on a several kilo- or mega-pixel isplay. Conventionally, polynomial interpolation, an extremely popular metho in image processing which fits a polynomial function to ata, is applie to optically visualize soun fiels for a smoother presentation. However, polynomial interpolation cannot always provie a goo approximation of an actual soun fiel especially when the conition causes a moiré effect. In this paper, a special interpolation metho for an optically visualize soun fiel base on the Kirchhoff-Helmholtz integral equation is propose. Experiments confirm its effectiveness an robustness against noise.. OPTICAL PROJECTION OF SOUND FIELD When a soun fiel is projecte by a light beam, physical quantity of the soun fiel, which in most cases is soun pressure variation, is integrate along the beam []. Let us consier a general moel of optical measurement of a soun fiel governe by the homogeneous Helmholtz equation as in Fig.. There are rigi parallel planes at z = an z = which reflect souns completely. (The actual meaning of these planes varies epening on which measurement metho is applie). Once a light beam is emitte from the plane at z = perpenicularly, the soun fiel is integrate through z =[,] as ( + k )p(r) z = () where p is soun pressure, k is the wave number, r R 3 is a position an = / x + / y + / z enotes the three-imensional Laplacian operator. Using the Leibniz integral rule, ( ) x + y +k p(r) z = α an this can be rearrange as v z (r) z ( D + k ) p(r) z = α [v z (r)] z= z, () z= (3) where v z (r) = /iωρ p(r)/ z is the z-way (which is also the normal irection to the planes) particle velocity, α = iωρ an D = / x + / y. As we assume complete reflection on the bounaries, the right-han sie of Eq. (3) becomes zero, an we obtain ( D + k ) p proj = (4) with p proj = p(r)z. This result inicates that a projecte soun fiel, originally governe by the three-imensional Helmholtz equation, is ominate by the two-imensional Helmholtz equation. Therefore, it is reasonable to apply the two-imensional Kirchhoff- Helmholtz equation to an optically projecte soun fiel. In the rest of this paper, p enotes the projecte soun pressure p proj an r R enotes a position in the interior of a two-imensional region Ω. 3. INTERPOLATION METHOD FOR OPTICALLY PROJECTED SOUND FIELD The Kirchhoff-Helmholtz integral equation [ p(r) = G(r, r b ) p(r b) n G(r, r ] b) p(r b ) r b (5) n Ω is the bounary integral form of the Helmholtz equation where r Ω, r b Ω, G is the funamental solution of the Helmholtz equation an / n enotes the outwar-irecte normal erivative at the bounary. This equation implies that the soun pressure at any points /4/$3. 4 IEEE 477

2 y z Light source + Detector (LDV) x Light source (Schlieren, Optophone) Light beam Rigi reflector (LDV) Detector (Schlieren, Optophone) Fig.. Geometry of a general moel of optical measurement of a soun fiel consiere in Section. in the interior of a region Ω can be calculate from the bounary conition since the funamental solution G an its normal erivative are known from positions r an r b as G(r, r b ) n G(r, r b )= j 4 H() (k r r b ) (6) = jk 4 H() (k r r b ) (r r b) n r r b in the two-imensional case where n is a unit normal vector at r b an H () enotes the Hankel function of the first kin of orer zero. Therefore, using the Kirchhoff-Helmholtz integral equation, an interpolation problem of an optically projecte soun fiel is reuce to the estimation problem of the bounary conition of an aitrarily selecte bounary surrouning the measure region. With appropriate iscretization, as in bounary element metho, Eq. (5) can be expresse as a simultaneous linear equation (7) p = Gp b (8) where G is a matrix of G an G/ n, p is a vector containing interior soun pressure p(r) an p b is the bounary conition to be estimate. This problem, however, cannot be solve simply because the conition number of G excees 5. In orer to solve this illpose problem, partial ifferential equation (PDE), the Helmholtz equation in this case, is utilize as an equality constraint on a least squares formulation. As a result, the problem to be solve becomes min p b Gp b p s.t. ( + k ) p = (9) where = / x + / y an enotes Eucliean norm. 3.. Discretization of Kirchhoff-Helmholtz integral equation One easy way to iscretize Eq. (5) is combining into a matrix equation Eq. (8) where p(r i)=g T i p b () p =[p(r ) p(r ) p(r N)] T, G =[g g g N ] T, g i =[G(r i,r b ) G(r i,r bm ) G (r i,r b ) G (r i,r bm )] T, p b =[p (r b ) p (r b ) p (r bm ) p(r b ) p(r b ) p(r bm )] T, () f enotes f/ n, N N\{} is the number of the measure points an M N\{} is the number of the aitrarily selecte points on a bounary for the iscretization. However, this formulation cannot hanle the Laplacian in Eq. (9) since it consiers the bounary conition with points only at the bounary; values of iscretization points only at the bounary cannot provie its graient an ivergence without knowing the unerlying formula. m + h m m m+ m Ω Fig.. Discrete points of the bounary conition p b± in Section 3.. / r c m m r b m+ m+ Ω Fig. 3. Arrangement of iscrete points aroun a corner in Eq. (4). In orer to approximate secon-orer erivatives from sample values, at least three points are require in the irection of the erivative. Thus, points near the bounary r b+ an r b, which are locate outsie an insie the region in normal-irection respectively, are incorporate into the bounary conition p b. Figure shows an example of the points at the bounary iscretize with the mipoint rule an its neighborhoos. In this paper, a square-shape bounary is chosen for a simpler notation of finite ifference approximation of its tangent an normal erivatives. 3.. Matrix formulation with finite ifference approximation Finite ifference approximation of the homogeneous Helmholtz equation ( + k ) p =can be written as [ h I ( h I+B+k I ) h I ] p b± =: Hp b± = () where h= r bm+ r bm = r bm r bm, I is the M-imensional ientity matrix, p b± is a vector of soun pressure at r b+, r b an r b p b±=[ p(r b+ ) p(r bm+ ) p(r b ) p(r bm ) p(r b ) p(r bm )] T, an B is the ban matrix approximating secon-orer erivatives as p () (r bm )= p(r b m+ ) p(r bm )+p(r bm ) (3) where = r bm+ r bm = r bm r bm an f (l) enotes l-th orer tangent erivatives. This ban matrix is moifie in orer to hanle corners of the box-shape bounary properly by aing corner points p(r c) which are only use for the constraint, not for the integration in Eq. (5), an using four-points ifference p () (r bm )= 6p(rcm ) 5p(r bm )+p(r bm+ ) p(r bm+ ) 5 (4) where r bm r cm = / as in Fig. 3. The error term of this four points formula is 4 p (4) (r bm )+O( 3 ) which is the same orer of the three points formula in Eq. (3) : p (4) (r bm )+O( 3 ). Similarly, the normal erivative of soun pressure at the bounary p (r bm ) in Eq. () can be approximate by finite ifference p (r bm )= p(r b m+ ) p(r bm ) h which escribes the relation between p b an p b± as p b = [ I h I h I ] p b± =: Dp b±. (5) 477

3 Therefore, incluing everything above, Eq. (9) finally becomes min GDp b± p s.t. Hp b± =. (6) p b± Although this kin of equality constraine quaratic programming problem can be solve irectly, a irect approach is not suitable for Eq. (6) because the equality constraint equation is not exact but approximate which leas to instability of the solution. Hence, the constraint is integrate as a penalty term with Tikhonov regularizer min GDp b± p + α Hp b± + β p b± (7) p b± where α an β are regularization parameters. The proceure of the propose metho is as follows:. Import measure ata p an calculate G as in Eq. () from Eq. (6) an Eq. (7).. Create H an D illustrate in Eq. () an Eq. (5). 3. Solve Eq. (7) to estimate the bounary conition p b±. 4. Calculate any points insie the region from the estimate bounary conition using Eq. (5). 4. EXPERIMENTS 4.. Numerical simulation A numerical simulation was conucte to confirm effectiveness of the propose metho. The simulation conition is liste in Table. A projecte soun fiel was sample by 6 6 sampling points, an a box-shape bounary was set aroun the sampling points as in Fig. 4. A point soun source was locate at the lower left, an the soun fiel was reconstructe by interpolating the sample points. Figure 5 shows an example of a reconstructe soun fiel compare with orinary polynomial interpolation methos: bilinear an thir-orer spline interpolation. Gaussian noise was ae to the soun fiel generate by a 5 Hz point soun source, an the noisy mixture was sample as test ata. Even though the sampling interval is shorter than the Nyquist interval, a moiré effect was observe on the results of polynomial interpolation methos, illustrate in Fig. 5() an 5(e). On the other han, it can be confirme that the propose metho clearly reconstructe the original fiel. For the quantitative assessment, signal-to-noise ratio (SNR) of interpolate soun fiels, x,y SNR result =log p x,y p ˆp (8) where p is the soun pressure of the original soun fiel an ˆp enotes the interpolate soun fiel, were calculate for several sample fiels with aitional noise whose SNR, x,y SNR sample =log p (9) x,y w where w enotes noise, set up aitrarily by ajusting the level of Gaussian noise. Figure 6 shows SNR of the interpolate soun fiels versus SNR of the sample fiels. SNR sample was calculate from 6 6 = 56 sampling points whereas SNR result was calculate from 3 3 = 96 interpolate points. The peak signal-tonoise ratio, PSNR = log { 96 p max/ x,y p ˆp }, was also illustrate in the same figure where p max enotes the maximum possible value of the soun pressure insie the sample region after normalization. For all of the ata, the regularization parameters in Eq. (7) were fixe to α =an β =.. Table. Simulation conition. Sampling points 6 6 (= 56) points Soun source position (-,-) Length of bounary elements 6.5 mm Small istance h. mm Sampling points interval. m Soun spee 34 m/s Spatial Nyquist frequency 7 Hz Soun source frequency, 5,, 4 Hz y [m] point soun source x [m] Fig. 4. Setting of the sampling points an the bounary for the numerical simulation in Section 4.. (a) Original soun fiel (b) Noisy soun fiel (c) Sample noisy fiel () Bilinear (e) 3r-orer spline (f) Propose metho Fig. 5. A visual example of the simulation for 5 Hz in Section 4.. Normalize soun pressure is epicte as white for an black for : (a) original soun fiel, (b) noisy soun fiel compose by aing (a) an Gaussian noise whose SNR was B, (c) sample noisy soun fiel, () interpolation results by bilinear interpolation, (e) thir-orer spline interpolation an (f) the propose metho. From the results, it can be confirme that the propose metho clearly outperforms bilinear an thir-orer spline interpolation for every frequencies. SNR result of the propose metho epicte in Fig. 5(f) is 5. B, which is an example showing that it is ifficult to fin a ifference between the original an the interpolate soun fiel visually when its SNR result is higher than aroun or 5 B. 4773

4 SNR result [B] Hz 5 Hz Hz 4 Hz Propose Bilinear Spline PSNR [B] LDV Speaker 4 m m m rigi wall with reflector Fig. 8. Setup of the experiment escribe in Section m SNR sample[b] Fig. 6. SNR result an PSNR of interpolate soun fiels by bilinear interpolation, thir-orer spline interpolation an the propose metho in Section 4.. (a) Sample soun fiel (b) 3r-orer spline (c) Propose metho Fig. 7. A visual example of the simulation for a soun fiel prouce by a 4 Hz point soun source locate as in Fig. 4. Table. Measurement conition. Place for measurement Reveeration Univ. Honjo campus Lou speaker YAMAHA MSP5 STUDIO Scanning LDV Polytec PSV-3 Measure points 5 5 (= 65) points Soun source frequency Hz For the soun fiel whose frequency is uner the spatial Nyquist frequency, 7 Hz in this case, it can be reconstructe accurately especially when the ata are less contaminate. On the contrary, the soun fiel exceeing the Nyquist frequency cannot be reconstructe from the ata sample by equal interval sampling points. However, the propose metho can still correctly sketch the image of the soun fiel exceeing the Nyquist frequency as shown in Fig Application to real ata The propose metho is applie to real ata recore by a LDV. The measurement conition an setup are illustrate in Table an Fig. 8. A Laser beam was emitte from the LDV to a light reflector stuck on the thick rigi cement wall of the huge ( m m 6.5 m) reveeration room. A stationary soun fiel riven by a Hz sinusoial wave was recore by 5 5 = 65 sampling points. The results are shown in Fig. 9. The measure ata were thinne out one-by-one an two-by-two in orer to confirm properness of the propose metho. As the ata reuce, thir-orer spline interpolation lost reproucibility of the measure soun fiel. In aition, spline interpolation was easily affecte by measurement noise. In contrast, the propose metho can reprouce highly similar interpo- (a) Measure soun fiel (b) 3r-orer spline (c) Propose metho Fig. 9. An actual soun fiel measure by the LDV in Section 4.. The top left figure illustrates the full 5 5 measure ata an its interpolation results are illustrate on the top row. The mile row represents the results for reuce 3 3 ata create by skipping the full ata one-by-one. The lowest row is the results for reuce 9 9 ata create by skipping the full ata two-by-two. late fiels from the reuce ata without influence of noise. This result shows that almost 9 % of the measuring points can be reuce to achieve a similar result in this situation. 5. CONCLUSIONS In this paper, a novel interpolation metho for an optically measure soun fiel was propose. The propose metho effectively utilizes the fact that the measure ata represent a soun fiel by formulating the least squares metho of the Kirchhoff-Helmholtz integral equation with a penalty term of the Helmholtz equation, while the conventional polynomial interpolation oes not consier the behavior of ata.the numerical simulation confirme that the propose metho clearly outperforme the conventional methos especially when the noise level of the ata was low. The effectiveness of the propose metho was also confirme by the real ata. Future work will inclue improvement on the formulation of the optimization problem to hanle the constraint of the Helmholtz equation more effectively. Furthermore, three-imensional reconstruction of a soun fiel from optically projecte two-imensional ata will be consiere. 4774

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