Audible sound field visualization by using Schlieren technique

Transcription

1 Audible sound field visualization by using Schlieren technique Nachanant Chitanont, Kohei Yatabe and Yuhiro Oikawa Department of Intermedia Art and Science, Weda University, Tokyo, Japan Paper Number: O115 Abstract This paper introduces the Schlieren method coupled with spatio-temporal FIR filter banks for visualizing an audible sound field. A two-lens Schlieren system is used for observing the physical properties of sound. However, the pressure of the audible sound field is very small, making it difficult to visualize the audible sound field by using only the Schlieren technique. In our previous work, the spatiotemporal filters were proposed for enhancing visibility of the audible sound field. However, the information about the sound field can only be extracted if the frequency of the signal of interest, which is sound, w known. To extract the information of sound field without any prior knowledge about sound signal, the spatio-temporal filter banks are proposed. In order to evaluate the performance of the proposed method, a -dimensional Green s function is used to generate simulated Schlieren videos. The proposed method is applied on the simulated sound field videos to calculate the signal-to-noise ratio (SNR). The results indicate that the proposed method is effective for enhancing the visibility of the audible sound field. Index Terms Sound field visualization, digital filter, filter bank, spatio-temporal filtering, Schlieren imaging. 1. Introduction Sound visualization is an effective technique to understand the characteristics of sound. In the lt few decades, there have been many sound field meurement and visualization methods proposed by acousticians. For example, numerous microphones are generally used for acquiring the spatial information of sound [1]. Ler Doppler vibrometer (LDV) is used to meure sound field projections along a ler path [] without any microphone. However, visualization of sound field by using LDV technique needs to scan each point to obtain spatial information that can be applied only to a reproducible sound field. The Schlieren method is the technique of using an optical system and a high speed camera to visualize acoustic properties. It analyzes the density variation of air caused by the acoustic field. The advantage of the Schlieren method is that it can detect the change of sound properties in real-time from every point of the field. Normally, the Schlieren method is applied for visualizing ultronic sounds that have very large pressure [3]. The pressure of the audible sound field is very small and that makes it difficult to be visualized. Therefore, there are only few researchers that have proposed to use the Schlieren method for visualizing an audible sound field [4]. In our previous work, 1 khz and 15 khz sinusoidal sound fields were visualized by using spatiotemporal filters [5]. However, the information about the sound field can be extracted if the signal of interest, which is sound, w known. In this paper, spatio-temporal FIR filter banks are proposed for extracting sound field information. The proposed method is divided into 3 parts; analysis filter bank, sub-band processing and synthesis filter bank. The formulation of the method will be described in the next section. In order to evaluate the performance of the proposed method, the -dimensional Green s function is used to simulate the Schlieren sound field videos for calculating signal-to-noise-ratio (SNR). Simulated sound fields of 5 khz, 1 khz and 15 khz are used for testing its effectiveness. The SNRs show that the proposed method is effective for enhancing visibility of the sound fields. Moreover, the proposed method can extract the sound field information without any prior knowledge about sound signal.. Schlieren method.1. Relationship between sound and Schlieren system Visualizing sound from a Schlieren system is bed on refraction of light beams psing through a transparent media whose refractive index varies with distance. For ges, the linear relationship between the refractive index n and the g density ρ with the Gladstone-Dale constant for air G =.3 cm 3 /g is represented n = Gρ 1. (1) If the light beams travel in the z direction between two lens, the deflection angles in the x and ydirection ε x,ε y are represented ε x = GL ρ n x, ε y = GL n ρ y, () where n is the standard refractive index of air, L is the distance between lenses and ρ/ x, ρ/ y are 1th Western Pacific Acoustics Conference 15. Edited by Lim Kian Meng Copyright c 16 WESPAC 15 Organisers :: Published by Research Publishing, Singapore ISBN: doi: 1.385/ O115 5

2 Loudspeaker Fig. 1. Diagram of two-lens Schlieren system with f 1 = 1 cm, f = 1 cm and L = 1. the density gradients of air in the x and y direction. From the Newtonian equation of motion, the relationship between variation of the particle velocities v x / t, v x / t and the gradients of the sound pressure p/ x, p/ y are represented v x t = 1 p ρ x, v y t = 1 p ρ y. (3) If the sound pressure p is much smaller than the air pressure, the sound pressure is represented : p = K ρ ρ, (4) ρ where K is the bulk modulus and ρ is the standard density of air. From Eq. (), Eq. (3) and Eq. (4), the obtained deflection angles are represented ε x = ρ GL p n K x = v x t ρ GL Kn, (5) ε y = ρ GL p n K y = v y ρ GL. (6) t Kn From above equations, it indicates that the information of sound can be obtained from the deflection angles of the light beams... Schlieren recording system and apparatus The two-lens Schlieren system w used for observing properties of the audible sound field. It features two 1-cm diameter convex lenses, light source and knife edge. Figure 1 shows the diagram of the two lens Schlieren system that w used in our experiment. The original videos were recorded by NAC MEMRE- CAM HX-3 at 96, frames per second and pixel resolution. The audible sound field w emitted by a YAMAHA MSP-7 STUDIO loudspeaker. It psed through the test area of the Schlieren system. The sound field interaction with the air causes refraction of light. The knife edge w positioned at the focal point of the second lens. It blocked half of the refracted light ray and the remaining light w finally detected by the high speed camera. Figure shows the images of sound fields observed by the Schlieren system. It can be seen that it is very difficult to visualize the sound fields from the Schlieren videos. Fig.. Schlirern images of 1 khz, 15 khz and 4 khz sinusoidal sound fields detected by a high-speed camera. wave number k y wave number k x (a) (c) wave number k y wave number k x wave number k y wave number k y wave number k x wave number k x Fig. 3. Sound field spectrums. (a) -dimensional space and time dimension, (b) spatial spectrum of 1 khz sound field, (c) spatial spectrum of 15 khz sound field, (d) spatial spectrum of khz sound field. 3. The ideal sound field spectrum The ideal sound field in polar coordinate can be represented by the generalized harmonic polynomial for Helmholtz equation [6]. ψ k,m (r, ϕ) = kj m (kr)φ m (ϕ), (7) where r and ϕ are the radial and angular coordinate, k is the wave number and J m is the m order Bessel function. The angular function φ m (ϕ) is represented φ m (ϕ) = 1 exp {imϕ}, (8) π with m being an integer. The frequency spectrums obtained by the ideal sound fields from Eq. (7) are shown in Fig. 3. It shows that the radius of the spatial spectrum of sound field will be larger if the temporal frequency of sound is higher. Therefore, the shape of the spectrum of the -dimensional space and time dimension of sound is a cone. 4. Proposed method Spatio-temporal filter banks are proposed for extracting the sound field information and for noise re- (d) (b) 6

3 Raw image Analysis filter banks Removing DC component Spatio-tempotal FIR filter-banks Gaussian Bandps filters Analyzed image Synthesis filter banks Fig. 4. Procedure of visualization of an audible sound field. moval. It can extract the sound information while removing the noise outside the cone. The procedure of sound visualization of the audible sound field is shown in Fig. 4. Concepts of the proposed method are described in this section Removing DC component Large zero frequency components of the Schlieren images make the raw Schlieren video difficult to be visualized. Therefore, DC components of Schlieren videos are removed by evaluating the mean of each pixel in time direction of the Schlieren videos and subtract the mean from each pixel. 4.. Spatio-temporal FIR filter banks Spatio-temporal FIR filter banks are designed for extracting the audible sound field information. Figure 5 shows the tree structure of the designed filter banks. It is divided into 3 processes; analysis filter bank, sub-band processing and synthesis filter bank. It starts with two-channel filter bank. In analysis process, the highps H 1 (z) and lowps filter H (z) pair split the sound field signals into filtered channels. The sound field signal of each channel is decimated by two. The process is repeated n times until the number of desired channels, n, are acquired. After splitting the frequencies into each channel by the analysis filter, the frequencies above khz, inaudible sound, is removed. Next, in sub-band processing, the unwanted spatial frequencies of each channel are removed by Table 1. Parameters of the designed FIR filter banks order of filters 14 maximum stopband ripple of.5 analysis filters maximum stopband ripple of.1 synthesis filters Magnitude [db] Frequency [khz] Fig. 6. Magnitude response of the 8-channel analysis filter bank. using the Gaussian bandps filters. Finally, in the synthesis process, the filtered signals of each channel are interpolated and recombined by using the synthesis highps G 1 (z)and lowps G (z) filter pair Analysis-Synthesis Filter bank The process of perfect reconstruction is done by analysis-synthesis filter [7, 8]. It is achieved if the distortion transfer function satisfies: 1 {H (z)g (z)h 1 (z)g 1 (z))} = z l (9) and the function for aliing component term must be zero: 1 {H ( z)g (z)h 1 ( z)g 1 (z))} =. (1) Input signal Level 1 Level. Level n Level n. Gaussian filter.. Level Level 1 Output signal Analysis part Synthesis part Fig. 5. Tree structure of designed filter banks. 7

4 Solely FIR filter is selected for designing a prototype lowps filter H (z) of odd order N [7]. The relationship between analysis and synthesis filter is represented H 1 (z) =z N H ( z 1 ), G (z) =z N H (z 1 ), (11) G 1 (z) =z N H 1 (z 1 ), Simulated data Adding noise Calculating SNR Spatio-temporal FIR filter-bank filtered data where G (z) and G 1 (z) is synthesis lowps and highps filter, respectively. In this paper, the parameters of the designed filter banks are shown in Table 1. Figure 6 shows the magnitude response of the 8-channel designed analysis filter bank. It indicates that the filter splits frequencies of the sound field into each channel equally before going through the next process Gaussian Bandps Filter As we mention in the previous section, the spatial spectrum of sound field is a ring. The radius of the ring will be bigger if the temporal frequency of sound is higher. In this section, the Gaussian bandps filter is designed for removing the spatial frequency outside the ring. The radius of the Gaussian filter vary according to the upper band-limit of each temporal frequencyband. Let f n be a upper band-limit of filtered channel number n. The wave number k n is represented k n = πf n =πξ n, (1) c where c is speed of sound and ξ n is a spatial frequency. The frequency response of Gaussian bandps filter is given by { } D H n (u, v) =exp n (u, v) h n { } (13) exp 1 D n(u, v) ln, where h n is upper cut off frequency and l n is lower cut off frequency of the filter, respectively. D n (u, v) is a distance from the origin. The upper and lower cut off frequency can be represented h n = Nf n f s c,l n = N f s c, (14) where N is number of pixel of image and f s is sampling frequency of image. 5. Evaluation of the proposed method In order to evaluate the performance of the proposed method, the -dimensional Green s function is used to generate simulated Schlieren videos. The proposed method is applied on the simulated sound field videos to calculate the signal-to-noise ratio. Simulated Fig. 7. Procedure of calculation of SNR. (a) (b) (c) (d) (e) (f) Fig. 8. The images of simulated sound fields before and after adding noise. (a) 5 khz sinusoidal sound field, (b) 1 khz sinusoidal sound field, (c) 15 khz sinusoidal sound field, (d) 5 khz sinusoidal sound field with noise, (e) 1 khz sinusoidal sound field with noise, (f) 15 khz sinusoidal sound field with noise. sound fields of 5 khz, 1 khz and 15 khz are used for testing its effectiveness. The procedure of the evaluation of the proposed methods is shown in Fig. 7. The details of the evaluation process are described in this section Sound field simulation The simulated Schlieren videos is created by using -dimensional Green s function φ(x i,y)= i 4 H(1) (k x i y ), (15) where H (1) is a first kind of Hankel function, k is the wave number and x i y is the distance of each point in the field x i and point source y. The time direction is generated by multiplying the time factor e jωt. 5.. Adding noise to simulated sound field Gaussian noise is added to the simulated Schlieren videos to make it resemble the real Schlieren videos after removing the DC component. SNR -1 db is chosen to adjust the levels of the noise. The images 8

5 SNR after filtering khz 1 khz 15 khz channel of filter bank Fig. 9. SNR of simulated sound field after filtering (a) (b) (c) Fig khz sinusoidal sound field. (a) data without DC component (b) 1 khz sinusoidal sound field after being filtered by -channel spatio temporal filter banks (c) 1 khz sinusoidal sound field after being filtered by 64-channel spatio temporal filter banks of simulated 5 khz, 1 khz, 15 khz sinusoidal sound fields with noise and without noise are shown in Fig Signal-to-noise-Ratio To evaluate the performance of each proposed method, SNR of the Schlieren videos can be calculated Sdata SNR =1log 1 Sdata F data, (16) where S data is the Simulated Schlieren videos without noise and F data is the Simulated Schlieren videos with noise after being filtered by the proposed methods. 6. Result and Discussion The graph of SNR of the simulated sound field after filtering is shown in Fig. 9. It can be seen that the visibility of the sound fields is enhanced by using the proposed method. The slope of the graph is positive at lower channel and saturates to a constant level at higher channel. The real 1 khz sinusoidal sound fields obtained by our proposed methods with - channel and 64-channel filter banks are shown in Fig. 1. It can be seen that the audible sound field can be visualized more clearly if the number of channel incree. Therefore, the performance, SNR, of method is same for both simulated and real sound field. 7. Conclusions In this paper, spatio-temporal FIR filter banks are proposed for extracting sound field information and noise removal. In order to evaluate the performance of the proposed method, the -dimensional Green s function is used to simulate the Schlieren sound field videos for calculating SNR. Simulated sound fields of 5 khz, 1 khz and 15 khz are used for testing its effectiveness. The SNRs show that the visibility of the sound fields is enhanced by using the proposed method and the audible sound field can be visualized more clearly if the number of channel incree. Moreover, the proposed method can extract the sound field information without any prior knowledge about sound signal. In future work, we will focus on visualizing non-stationary sound field. References [1] Y. Yamaki and T. Itow, Meurement of spatial information in sound field by closely located four point microphone method, J. Acoust. Soc. Jpn.(E), vol.1, pp.11 11, [] Y. Oikawa, M. Goto, Y. Ikeda, T. Takizawa and Y. Yamaki, Sound field meurements bed on reconstruction from ler projections, Int. Conf. Acoust., Speech Signal Process. (ICASSP), pp , Mar. 5. [3] E.D. Mller, N. Degen and J. Dual, Schlieren visualization of ultronic standing waves in mmsized chambers for ultronic particle manipulation jet flow, J. Nanobiotechnology., vol.11, no.1, pp.1 5, Jun. 13. [4] M.J. Hargather, G.S. Settles and M.J. Madalis, imaging of loud sounds and shock waves in air near the limit of visibility, Shock Waves, vol., no.1, pp.9 17, Feb. 1. [5] N. Chitanont, K. Yaginuma, K. Yatabe and Y. Oikawa Visualization of sound field by means of Schlieren method with spatio-temporal filtering, Int. Conf. Acoust., Speech Signal Process. (ICASSP), pp , Mar. 15. [6] Q.Wang, O. Ronnebeger, and H. Burkhardt Rotational Invariance Bed on Fourier Analysis in Polar and Spherical Coordinates., IEEE Trans. Pattern Anal. Mach. Intell., vol.31, no.9, pp , 9. [7] S. K. Mitra, Digital Signal Processing. A Computer-Bed Approach, nd Ed., McGraw- Hill, New York, 1. [8] N. J. Fliege, Multirate Digital Signal Processing, John Wiley and Sons Ltd. Chichester,