When acoustic engineers make analysis of the sound propagation with numerical methods, they use values of the acoustic properties of the objects to describe the behavior of a given object when a sound wave hits its surface. In other words, these numerical methods often require the specification of boundary conditions that characterize the acoustic properties of the materials. For example, once the acoustic properties of the materials are known, numerical analysis such as boundary or finite element methods can be applied to predict and control the sound field by manipulation of the analyzed materials. In the present work, of particular interest is the development of a method to measure the acoustic property called ”normal acoustic impedance” of the interior surfaces of a room, (in what follows it will be referred sometimes as to simply ”impedance”). On the other hand, since the 3D model of the room is assumed to be known, we may simulate modifications of that room and use the estimated acoustic impedances to make predictions of the sound response that we would hear if we did the actual modifications in the real room. This is important for example in the early stages of the acoustic design of concert halls, seminar rooms, audio studios, etc. where an optimum acoustic design should be determined in advance before making costly expenses. The kind of problem addressed in this work deals with situations where samples of materials cannot be taken to an acoustic laboratory to measure their acoustic impedance with specialized devices. Therefore, if the impedance of those materials is desired, in-situ measurements must be performed. In the present work, two algorithms for the estimation of acoustic impedance are presented. Both algorithms are based on the solution of the Helmholtz Integral Equation (HIE) of the wave propagation in a homogeneous media. Hence, the underlying theory of these algorithms is the Boundary Element Method (BEM) and the Inverse Boundary Element Method (IBEM). Similar approaches based on these theoretical frameworks have been proposed for the identification of noise sources in a vibrating system. However, the present work represents a first attempt to estimate the acoustic impedances in interior spaces of arbitrary shapes (such as real rooms). The basic idea for a system of the inverse estimation of acoustic impedances is therefore as follows: a sound source is placed in a known position inside a room (e.g. an office, a conference room, a hall, etc.), then a harmonic tone is emitted. As the sound travels and reflects on the surfaces, a microphone is recording samples of sound while moving freely in space. After a number of samples are recorded the sound source stops. Now the problem is: given the 3D model of the interior space, the strength of the sound emitted by the source, and the set of recorded samples of sound, the objective is to estimate the acoustic impedance of the surfaces in that interior space. The approach proposed in this work to solve this problem consists of breaking the geometric model into N elements and using the measured sound samples. Then applying the IBEM theory, a large system of equations is constructed. The unknowns of this linear system are the boundary values of the HIE which happen to be the parameters that define the impedances at the surfaces. Therefore the solution of this linear problem leads to the sought acoustic impedance values. Nevertheless, the solution is not achieved straight forward since this kind of inverse problems are usually ill-posed, meaning that if one wants to find a solution to the linear system in the least-square sense, there exist many vectors in the solution space that minimize the residual norm of the least-squares. In other words, the system is not uniquely solvable and there maybe an infinite number of minimizing vectors near the desired solution. This is a direct consequence of the fact that the system of equations is rank deficient. Moreover, the ill-conditioning of the matrix makes the linear problem sensitive to the noise introduced to the data during the measurement process. Because of these reasons, extra information of the sought solution should be given in the form of constraints to the linear system (this process is usually known as regularization). A number of regularization methods have been proposed in the literature, being Tikhonov regularization the most widely used. Other regularization techniques are based on the singular value decomposition (SVD) of the linear system. But on the other hand, while the application of existent regularization methods improves the accuracy of the solutions, the estimation of the amount of regularization is usually a complex work resulting in extra computational cost. And as the dimensionality of the problem becomes large, the imitations imposed by the regularization step are more predominant. Hence, as an alternative to overcome these difficulties, the methods proposed in this dissertation attempt to solve the ill-conditioned linear system by exploiting a prior knowledge of the geometrical segmentation of the surfaces. This information is introduced as a physically-meaning constraint. The first proposed method consists of a non-linear Leastsquares optimization approach aims to find the sound pressure and the particle velocity (parameters that define the acoustic impedance) at each discrete element of the 3D mesh, consequently a good approximation (in terms of geometric resolution) to the distribution of the impedance values over the surfaces is obtained. A strong pitfall of this approach is that the solutions tend to have large variance as the dimension of the geometry grows. A second approach is an iterative optimization process that estimates directly the sought impedances under the assumption that the interior surfaces have homogenous impedance values. This assumption allows in addition a dramatic reduction of the dimensionality of the optimization problem, and therefore being able to keep an acceptable accuracy for large-scale problems. Another advantage of this method is its robustness to the perturbations in the measured data, due to the fact that the inversion of the ill-conditioned matrix is not required. A drawback of this second approach is its slow convergence. In the evaluation part, the performance of the methods proposed here is investigated by means of numerical simulations with basic geometrical shapes (such as a unitary cube) and with realistic 3D models (an office room). In addition to the simulations, validation experiments are realized by attempting to estimate the acoustic impedance of the interior walls of a reverberation chamber. Regarding the experimental setup system, the use of video cameras is introduced in this research work to perform 3D real-time tracking of the position of the microphone. This 3D tracking technique permits the acquisition of huge amounts of sound samples in the interior space. Results of the simulations and the experiments are presented and discussed in this dissertation.

発行年

2007-03-22

日本十進分類法

548

学位名

博士(情報理工学)

学位

doctoral

学位分野

Information Science and Technology (情報理工学)

学位授与機関

University of Tokyo (東京大学)

研究科・専攻

Department of Information and Communication Engineering, Graduate School of Information Science and Technology (情報理工学系研究科電子情報学専攻)

学位授与年月日

2007-03-22

学位授与番号

甲第22805号

学位記番号

博情第135号

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