## Abstract

A high-speed optical measurement for the vibrating drumhead is presented and verified by experiment. A projected sinusoidal fringe pattern on the measured drumhead is dynamically deformed with the membrane vibration and grabbed by a high-speed camera. The shape deformation of the drumhead at each sampling instant can be recovered from this sequence of obtained fringe patterns. The membrane vibration of Chinese drum has been measured with a high speed sampling rate of 1,000 frames/sec. and a standard deviation of 0.075 mm. The restored vibration of the drumhead is also presented in an animation.

©2005 Optical Society of America

## 1. Introduction

Optical three-dimensional (3D) non-contact profilometry has been widely used for 3D sensing, machine vision, robot simulation, industrial monitoring, dressmaking and biomedicine. Among those extensively studied methods, Fourier Transform Profilometry (FTP) [1–3], introduced by Takeda at first, is particularly unique and more noticeable for its high-speed measurement with its prominence characters of requiring only one image of the deformed fringe pattern to reconstruct the 3D shape of the measured dynamic object [4–6].

The drum plays an important role in many musical styles. When you hit a drum sharply with a drumstick, the drumhead vibrates rapidly. As the drum vibrates, it disturbs the air around it in such a way that sound waves are produced. The sound is resonated, or increased by the hollow drum shell. When the sound waves reach your ears, you hear the sound made by the drum. Understanding the characteristics of the vibration of the membrane of the drum is important in studying the acoustic characteristic and improving the manufacture technique of the drum. The drumhead has been studied to a certain extent using various methods. M. Bertsch [7] employed Laser Interferometry, a Polytec Scanning Vibrometry, to measure the vibration patterns of the Viennese timpani by quickly scanning 110 points on the membrane. His method is the non-contact measurement technique and can achieve high accuracy, about nanometers, in vibration amplitudes measurement. Dan Russell [8] and ISVR [9] completed the mathematic calculations and modal analysis of an idea membrane using Bessell functions. The investigators have paid most of their attentions to the resonant sound, the pitch, the effect of air loading and the influence of tension and diameter of membrane. They have found some significant differences between various skins of drumhead and the most important factors to produce a clear, focused pitch and to achieve a good resonant sound.

In this paper, we propose an optical 3D shape measurement of the vibrating drumhead based on FTP. A sinusoidal structured pattern, which is projected onto the surface of a measured drum, is dynamically deformed with the vibration of the membrane. The sequence of the deformed fringe images is grabbed by a high-speed CCD camera, and then a series of wrapped phase maps will be produced after processed by Fourier transform, filtering and inverse Fourier transform. Unwrapping these phase maps in 3D phase space along three space-time axes x,y,t, the shape deformation of the measured drumhead can be recovered completely and the whole vibration pattern can also be restored clearly.

Compared with the vibration pattern measurement by Laser vibrometry [7], the method proposed in this paper has lower accuracy, but has its own obvious advantage that all the data of the whole membrane at one sampling instant can be obtained simultaneously, furthermore its information recording time can last along with the whole vibration process to demonstrate the whole vibration from uncreated to fade away. Our method only needs a simple experimental setup and a well-known data batch processing flow.

## 2. Fundamental concepts

#### 2.1 FTP used in shape measurement for dynamic objects

FTP for the dynamic shape measurement is usually implemented as the following. A Ronchi grating or sinusoidal grating is projected onto an object. Then, a sequence of dynamic deformed fringe images, which contain the information of the object’s height distributions, can be grabbed by a CCD camera and saved in a computer rapidly for post-processing. Next, the data are processed with three steps. Firstly, by using Fourier transform, we obtain their spectra, which are isolated in the Fourier plane when sampling theorem is satisfied. Secondly, by adopting a suitable bandpass filtering, for example, a 2D Hanning window, in spatial frequency domain, all frequency components are eliminated except fundamental component. And by calculating inverse Fourier transform of fundamental component, a sequence of phase-maps can be obtained. Thirdly, by applying the phase unwrapping algorithm in 3D phase space, the continuous and natural phase distributions as well as the shapes of the measurement object in different times can be constructed under a perfect phase-to-height conversion. In this paper the experimental results of a drum show that this method can efficiently deal with the vibration measurement.

The optical geometry of the measurement system for membrane vibration measurement of a drum is similar to the traditional FTP, as shown in Fig. 1, in which the optical axis *E*_{p}
-${{E}_{p}}^{\prime}$ of a projector lens crosses the optical axis *E*_{c}
-${{E}_{c}}^{\prime}$ of a camera lens at point *O* on a reference plane. The reference plane is fictitious and serves as a base plane for measuring the object height *h*(*x*,*y*) of the measured object. *d* is the distance between the projector system and the camera system, *l*_{0}
is the distance between the camera system and the reference plane.

By projecting a grating fringe onto the reference plane, the grating image (with period *p*_{0}
) on the reference plane observed through a CCD can be represented by

where *r*_{0}
(*x*,*y*) is a non-uniform distribution of the reflectivity on the reference plane, *A*_{n}
is the weighting factors of Fourier series, *f*_{0}
(*f*_{0}
=1/*p*_{0}
) is the fundamental frequency of the observed grating image, and *ϕ*_{0}
(*x*, *y*) is the original phase on the reference plane (i.e. *h*(*x*,*y*)=0). The coordinate axes are chosen as shown in Fig. 1.

When the measured object is in static, the image intensity, which obtained by the CCD camera, is independent of the time and expressed usually as *g*(*x*,*y*). When a dynamic 3D object, whose height distributions are changing with time, is placed into the optical field, the intensity of these fringe patterns is obviously a function of time and can be marked as *g*(*x*,*y*,*z*(*t*)), and the phase distribution which implicates the height variation of the measured dynamic object is also a function of time and can be noted as *ϕ*(*x*,*y*,*t*). The intensity distributions of these fringe patterns in difference time can be expressed as

where *r*(*x*,*y*,*t*) and *ϕ*(*x*,*y*,*t*) represent a non-uniform distribution of the reflectivity on the object surface and the phase modulation resulted from the object height variation at different times, respectively, *m* is the number of all fringe images grabbed by the CCD camera. The interval between two neighboring frame is dependent on the sampling rate of the camera.

Fourier transform, filtering only the first order term (*n*=1) of Fourier spectra, and inverse Fourier transform are carried out to deal with each fringe pattern grabbed by CCD at different time. Complex signals at different time can be calculated.

The same operations are applied to the fringe pattern on the reference plane to obtain the complex signal.

Noting the geometry relationship between the two similar triangles, Δ*E*_{p}*HE*_{c}
and Δ*CHD*, in Fig. 1, we can write

The phase shift produced by the object height distribution is

Substituting Eq. (5) into Eq. (6) and solving it for *Z*(*x*,*y*,*t*), the formula of height distribution can be obtained.

## 2.2 3D phase calculation and unwrapping in 3D phase space

From the complex signal in Eq. (3) and Eq. (4), the phase distribution *ϕ*(*x*,*y*,*t*) and *ϕ*
_{0}(*x*,*y*) are usually calculated by employing the function of arc tangent, so the result is consequentially wrapped in the range of (-*π*, *π*), which must be recovered to a natural and continuous distribution in order to restore the correct shapes of the measured object. In dynamic measurement, this wrapped phase is also a function of time, so we call it 3D wrapped phase. 3D phase unwrapping is conducted not only along the *x* and *y* directions, but also along the *t* direction necessarily. Some points of discontinuity, which are resulted from noise, shadow and under-sampling, fail to be unwrapped along the *x* or *y* direction in its own frame, can be unwrapped successfully along the *t* direction. So compared with 2D unwrapping [10–14], 3D phase unwrapping is easier and more accurate.

By calculating the product of every two frame complex signals *g*ĝ*(*x*,*y*,*t*-1) with *g*ĝ (*x*, *y*, *t*) in two neighboring sampling instants, we can obtain their 3D phase difference distribution like this

In evidence, when the sampling rate in *t* axis is high enough, the time interval between two grabbed frame fringe patterns will be small, as a result, the phase difference between two neighboring pixels with the same (*x*,*y*) coordinates will be less than π in *t* direction, i.e. the phase difference *ϕ*(*x*,*y*,*t*)-*ϕ*(*x*,*y*,*t*-1) is always smaller than *ϕ*(*x*,*y*,*t*)-*ϕ*(*x*,*y*,0). It provides a new approach for 3D phase unwrapping.

Corresponding to 3D phase calculation, 3D phase unwrapping will be done based on the phase difference between two frames in neighboring sampling times. Only one phase map, which is reliable everywhere and easy to be unwrapped in 2D spatial space (*x*,*y*), is selected to be unwrapped by 2D phase unwrapping procedure and to calculate its natural phase Φ_{uw1}(*x*,*y*,1). The unwrapped phase Φ_{uwk} at any time (*t*=*k*) can be obtained by calculating the sum of the *k*-1 phase differences from *t*=1 to *t*=*k* along the *t* direction. It can be described as

Where Δ*ϕ*_{i}
(*x*,*y*) represents the phase difference between two frames *t*=*i* and *t*=*i*-1.

In this method, only one frame of the wrapped phase needs to be unwrapped in 2D phase space, and the precise phase values in the whole 3D phase space can be obtained by calculating the sum of the phase differences along *t* direction. So the whole unwrapping procedure is very simple and quick.

## 3. Experiment and results

A schematic diagram of the experimental setup in Fig. 2 is used to verify this method. The measured object is a Chinese double-side drum with a diameter of 250 mm and a height of 145 mm, shown in Fig. 3. An image of a sinusoidal grating, 2 lines/mm, is projected and focused onto the drum membrane, which is located at a distance of 780 mm (*l*_{0}
) from the exit pupil of the projection system. The deformed grating image is observed by a high frame rate CCD camera (SpeedCam Visario, made in Switzerland, the sampling rate speed is up to 10,000 frames/sec.) with a zoom lens (Sigma Zoom, Φ82 mm, 1:28 DG, 24~70 mm, made in Japan). The distance between the projection system and the high-speed camera is 960 mm (*d*). A sheet of dark and featureless cloth is placed farther behind the drum to minimize the glare and decrease the intensity of the reflected light, which will distract the attention from the drum, so that it is easy to segregate the drum from background.

We hit the batter side of the drumhead only one time with a wood drumstick and measure the whole vibration of the resonant side. During the whole vibration process, we have recorded near 1,000 fringe images of vibrating membrane in 1536×1024 picture elements (its corresponding view field is near 426.7×284.5 mm) with 1,000 frames/sec. in sampling rate speed. We selected 300 frames of them, which integrally demonstrate the whole vibration from uncreated to fade away, and scaled down each frame to 900×900 pixels by abandoning the unwanted parts. One of the resulted images of this primary analysis is shown in Fig. 4. Considering the size of the multimedia file, the animation only contains the fore part (frame number: 1~140, their lasting time is 140 ms) of the data set.

Figure 5 shows the positions of the center point of the drumhead in vibration. We can see that the motion of the drumhead is damped, which maybe change the vibration modes, and we can work out the principal vibration frequency of this drumhead about 163.34 Hz.

The height distributions of the vibrating drumhead at the corresponding sampling instants are exactly restored. The vibration range is from -2.20 to 2.28 mm. The standard deviation of the restored height is 0.075 mm, which is ~2% of the equivalent wavelength. Fig. 6 gives the profiles of the center rows of six sampling instants in one period; the number labeled above each line is the corresponding sampling instant respectively.

Figure 7 shows the six mesh charts of the corresponding instants in Fig. 6 and their sampling instants are given respectively as the subheads. The whole reconstructed vibration with the duration of 300 ms is given as an animation.

## 4. Conclusion and discussion

In one period of the principal drumhead vibration, there are two modes, (1,0) and (1,1), which indicates that the point we hit is not the exact center of the drumhead. Furthermore, there are nonlinear effects (such as the uniform surface tension across the entire drumhead), which exert their own preference for certain modes by transferring energy from one vibration mode to another.

The results of the theoretical analysis and experiment on a Chinese drum indicate that FTP with good 3D phase unwrapping algorithm can efficiently deal with the vibration measurement and restored the vibration of the drumhead. With the development of high-resolution CCD cameras and high frame rate frame grabbers, this method should be one promising in the high-speed vibration phenomena research and be more helpful in studying the acoustic characteristics and the manufacture techniques of the percussion instrument.

This method has the advantage of high-speed full-field data acquisition and analysis, so its application can be expanded into other fields, such as 3D dynamic shape measurement of moving, rotating, inflating or deflating objects.

## Acknowledgments

This project was supported by the National Natural Science Foundation of China and the Chinese Academy of Engineering Physics (No. 10376018).

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