NUMERICAL FORMULAE FOR TOA-BASED MICROPHONE AND ...

2014 14th International Workshop on Acoustic Signal Enhancement (IWAENC)

NUMERICAL FORMULAE FOR TOA-BASED MICROPHONE AND SOURCE LOCALIZATION Trung-Kien LE † and Nobutaka ONO †‡ †

National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo, 101-8430 Japan ‡ The Graduate University for Advanced Studies (SOKENDAI) {kien , onono}@nii.ac.jp ABSTRACT

since cost functions used in the iterative methods are nonlinear and non-convex, they can be easily trapped into local minima. However, in the localization problem, local minima are very far from true solutions. In the field of sound localization, therefore, determining closed-form solutions for microphone and source positions has gained a lot of attention.

This paper presents numerical formulae for the Time-ofArrival (TOA)-based microphone and source localization problem, which determines the positions of microphones and sources based on distances between each microphone and each source respectively. This is a purely geometrical problem in mathematics. Concretely, we show when the number of microphones or the number of sources is at least nine, the formulae of the microphone positions and source positions are given simply from the distance-matrix. A similar statement is given in the cases of at least eight microphones or sources if we know an extra information about the distance between any two microphones or two sources. The accuracy of these formulae are proven shortly by ten thousand independent experiments of which coordinates of points have an independent Uniform distribution.

Recently, by using a factorization of a difference of square-distance matrix generated by a distance-matrix (see in [12, 13]), Y. Kuang, et al. [10] proposed a novel method for estimating closed-forms of positions of microphones and sources on the basis of TOA-measurements. After utilizing the above factorization to reduce the number of unknown parameters from 3(N + M ) − 6 to 9, where N and M are the numbers of microphones and sources, respectively, the authors show that the remains of parameters are solutions of some multivariate quartic and cubic equations. They then use the Gröbner basis method (see in [14]) and the Macaulay2 software [15] to find out the closed-form of these unknownparameters. Their experiments prove that the closed-forms are very accurate in not only the noise-free case but also noisy cases. However, since the Gröbner basis method and the Macaulay2 software are complex and unfamiliar to nonmathematicians, their estimations are difficult to be used in a wide range of sciences.

Index Terms— Time-of-Arrival, Localization, Univariate quartic equation, numerical formula 1. INTRODUCTION Sound localization is a complex and cumbersome task that a wide range of acoustic signal processing problems, such as sound source separation, microphone array, robot audio, etc., face as a basic challenge. For at least 10 years, sound localization has received significant attention from scientists. In this paper, we would like to study this problem based on the Time-of-Arrival (TOA) measurements. Concretely, this paper focuses on estimating positions of microphones and sound sources in three dimensions when all the microphone-source distances are known. When the TOA-measurements are estimated, the microphone and source localization is a bipartite graph localization problem, a purely geometrical problem in mathematics. Several solutions to solve the sound localization based on TOA or Time-Difference-of-Arrival (TDOA) have been proposed. Some are iterative methods based on a least square criteria [1, 2, 3, 4, 5] or a maximum likelihood principle [6, 7, 8, 9], and some are non-iterative methods [10, 11]. Generally,

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In this study, numerical formulae for TOA-based microphone and source localization are presented. Similarly to Kuang’s estimation [10], the proposed method utilizes the factorization of the difference of square-distance matrix to reduce the number of unknown parameters from 3(N + M ) − 6 to 7. Then, we prove the remains of unknown parameters can be computed directly by solutions of a univariate quartic equation. More precisely, we show that this statement is true in the cases that the number of microphones or the number of sources is at least 9, and it is also true in the cases of at least 8 microphones or sources if an extra information about the distance between any two microphones or any two sources is known. These formulae can be improved for the cases 7 and 6. However, since the length of the paper is limited, we would like to focus on the cases 9 and 8. The cases 7 and 6 should be presented in another paper.

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= UT V, and there ex(3 × (N − 1))-matrix V such that D ists an invertible (3 × 3)-matrix L which X = L−T U and Y = LV. Since U, V are known, (x, y)-groups are known if we know L and y1 . Assuming that α = x1 − x2 2 , since yn − x1 2 = d1n , yn − x2 2 = d2n , the coordinate of yn is given based on α θn , hn sin θn , kn ) as follows y1 ≡ (0, h1 , k1 ) , yn ≡ (hn cos 1 (α2 +d21n −d22n ) and hn = d21n − kn2 . Thus, where kn = 2α if α is known, the position of y1 is known and the positions of yn , n 2 depend on the parameters θn . Moreover, let

2. TOA-BASED MICROPHONE AND SOURCE LOCALIZATION Let us consider two unknown groups of points in the space R3 , x-group containing M points {x1 , . . . , xM } and ygroup containing N points {y1 , . . . , yN }. The bipartite graph localization problem is stated simply as follows: Given distance-matrix D = (dmn )M ×N , where dmn = xm −yn 2 is the Euclidean distance between two points xm and yn , determining the positions of points in x-group and y-group. In the case of an acoustic signal processing problem, if x-group and y-group are groups of positions of microphones and sound sources, and distance-matrix D is given by measuring time of arrival (TOA) tmn from a microphone and a source (xm , yn ), i.e., dmn = vtmn , where v is the speed of measured acoustic signals, the localization of microphones and sources is exactly the bipartite graph localization problem. Thus, in the case of acoustic signal applications, the bipartite graph localization problem is named the TOA-based microphone and source localization, which determines the positions of microphones and sources from the time-of-arrival measurements. Without loss of generality, since the roles of x-group and y-group are symmetric, we assume that N M , and since the distance-matrix is invariant under any reflection, translation, and rotation, we also assume that x1 ≡ (0, 0, 0) , x2 ≡ (0, 0, α) , y1 ≡ (0, h1 , k1 ),

⎛

l1 L = ⎝l 4 l7

⎞ l3 l6 ⎠ , l9

if kn are known (α is known), l7 , l8 and l9 are known by the following presentation: The full-rank property of D (rank(V) = 3) confirms that there exists three in = dexes n1 , n2 , n3 ∈ {1, . . . , N } such that a matrix V (v|n1 , v|n2 , v|n3 ) is invertible, where v|n denotes the nth column of V. Since Y = LV, we have (l7 , l8 , l9 ) = −1 . Thus, if we know α and l1 , . . . , l6 (7 (kn1 , kn2 , kn3 )V variables), we know (x, y)-groups. From the formulae of hn and kn , α and l1 , . . . , l6 are determined by the following equations, for all n 2 2 2 (l1 , l2 , l3 )v|n−1 + (l4 , l5 , l6 )v|n−1 − 2h1 = 4h2n . (2) Let d1 = 2(d212 −d211 +d222 −d221 , . . . , d21N −d211 +d22N 2 d21 ), d2 = ([d221 − d211 ]2 − [d222 − d212 ]2 , . . . , [d221 − d211 ]2 [d22N − d21N ]2 ), and

(1)

where α, h1 0.

⎛

2 v12 2 ⎜ v22 ⎜ 2 ⎜ v32 ⎜ ⎜2v12 v22 ⎜ S=⎜ ⎜2v12 v32 ⎜2v22 v32 ⎜ ⎜ −4v12 ⎜ ⎝ −4v22 −4v32

Definition 1. Given distance-matrix D, a pair (x, y)-groups is called a solution of D-TOA-based microphone and source localization if xm − yn 2 = dmn for all m, n and it satisfies (1). Definition 2. D is called full-rank if it has a solution (x, y)groups such that the points in x-group do not lie on the same plane and a similarity states for y-group. This paper studies a problem that given a full-rank distance-matrix D, how to find its solutions. Since D is full-rank, there exists a solution of D which is denoted (x, y)-groups. The problem of finding (x, y)-groups has 3(N +M )−6 variables. However, we show that this problem is equivalent to another problem having 7 variables. The main idea comes from considering a difference of a square-distance matrix given in [12, 13] which is stated as follows: It can be verified that d2mn − d2m1 − d21n + d211 = −2xTm (yn − y1 ). Let X = (x2 , . . . , xM ), Y = −2(y2 − y1 , . . . , yN − y1 ) and with D(m, a difference of square-distance matrix D n) = 2 2 2 2 T dm+1,n+1 − dm+1,1 − d1,n+1 + d11 , we have D = X Y. The property full-rank confirms that the ranks of matri also has a rank ces X and Y are three, so the matrix D of three. Thus, by factorizing D using e.g. singular value decomposition, we have a (3 × (M − 1))-matrix U and a

l2 l5 l8

··· ··· ··· ··· ··· ··· ··· ··· ···

⎞ ⎞T ⎛ 2 2 v1N l1 + l42 2 2 2 ⎜ l2 + l5 ⎟ v2N ⎟ ⎟ ⎟ ⎜ 2 2 ⎟ ⎜ l3 + l62 ⎟ v3N ⎟ ⎟ ⎜ ⎜l 1 l 2 + l 4 l 5 ⎟ 2v1N v2N ⎟ ⎟ ⎟ ⎜ ⎜ ⎟ 2v1N v3N ⎟ ⎟ , L = ⎜l 1 l 3 + l 4 l 6 ⎟ ⎜l 2 l 3 + l 5 l 6 ⎟ 2v2N v3N ⎟ ⎜ ⎟ ⎟ ⎜ h1 l 4 ⎟ −4v1N ⎟ ⎜ ⎟ ⎟ ⎝ h1 l 5 ⎠ −4v2N ⎠ −4v3N h1 l 6

− −

(3)

where vij denotes the (i, j)-th element of V, equation (2) infers = d1 + 1 d2 . (4) LS α2 This is a pilot for our work in determining α and l1 , . . . , l6 . 3. DERIVATION OF NUMERICAL FORMULAE 3.1. Preparation for deriving the solution Let us introduce several important concepts. Given N polynomials f1 , . . . , fN in the n variables x1 , . . . , xn , and T, a vector of some monomials generated by x1 , . . . , xn . Definition 3. A matrix C is called a coefficients-matrix of (f1 , . . . , fN ) corresponding with T if f1 , . . . , fN = TCT for all x1 , . . . , xn .

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For example, given three polynomials f1 (x, y) = 2x2 y − 3x +y, f2 (x, y) = x3 +3x2 y−3y+1, f3 (x, y) = 4y 2 x−2x2 , and T = (x3 , x2 y, y 2 x, x2 , y, 1), a vector of some monomials. Then ⎞ ⎛

Note that cˆ7 c¯2 = cˆ6 c¯3 and cˆ3 c¯8 = cˆ8 c¯3 , etc.; thus,

2

0 C = ⎝1 0

2 3 0

0 0 4

−3 0 2

1 −3 0

c˘1 α8 + c˘2 α6 + c˘3 α4 + c˘4 α2 + c˘5 = 0, where c˘1 = cˆ1 c¯4 , c˘2 = cˆ1 c¯5 − cˆ4 c¯1 , c˘3 = cˆ1 c¯6 + cˆ2 c¯4 − cˆ4 c¯2 − cˆ5 c¯1 , c˘4 = cˆ1 c¯7 + cˆ2 c¯5 + cˆ3 c¯4 − cˆ6 c¯1 and c˘5 = cˆ5 c¯2 − cˆ4 c¯3 . As a sequence of the above expressions, α and l1 , . . . , l6 are determined by the following result

0 1⎠ 0

is a coefficients matrix of (f1 , f2 , f3 ) corresponding with T. T is called the order of monomials of (f1 , f2 , f3 ).

Proposition 1. Given D, U and V,

Definition 4. Given C = (cij )m×n is a coefficients matrix of m polynomials, and 1 k n. A matrix C(k) = (k) (cij )(m−1)×(n−1) is called a k-monomial-reducing of C if c1k = 0 and c if j < k ci+1,j − i+1,k (k) c1k c1,j cij = . ci+1,k if j k ci+1,j+1 − c1k c1,j+1

1. α2 is a solution of the following quartic equation: c˘1 x4 + c˘2 x3 + c˘3 x2 + c˘4 x + c˘5 = 0. 8

C∗ is called a (k1 , . . . , kr )-monomial-reducing of C, k1 < · · · < kr , if there exists C(kr ) such that C(kr ) is a kr monomial-reducing of C and C∗ is a (k1 , . . . , kr−1 )-monomialreducing of C(kr ) .

6

4

2

+¯ c6 α +¯ c7 α +¯ c8 2. l6 = − h11α2 c¯4 α +¯cc¯51 α , α4 +¯ c2 α2 +¯ c3

l4 a7 b p 3. = h11 + α12 7 − l6 7 ,

l5

4.

a8

2

l1 l22 l32

b8

=

a1 a2 a3

+

1 α2

b1 b2 b3

p8

− h1 l 6

p1 p2 p3

−

l42 l52 l62

,

such that l1 l2 = a4 + α12 b4 − p4 h1 l6 − l4 l5 and l1 l3 = a5 + 1 α2 b5 − p5 h1 l6 − l4 l6 , where pi , ai , bi denote the i-th elements of p, a and b, respectively.

3.2. Numerical formulae for N 9 ¯ = s|1 , . . . , s|8 , S0 = ¯s ; . . . ; ¯s , S1 = ¯s , where Let S 1 8 9 ¯ Let p = (S1 S−1 )T , ¯sn denotes n-th row of a matrix S. 0 ˜ 1 S−1 )T , b = (d ˜ 2 S−1 )T where d ˜ 1 and d ˜ 2 are veca = (d 0 0 tors of eight-first values of d1 and d2 respectively, equation (4) confirms 2 l1 + l42 , l22 + l52 , l32 + l62 , l1 l2 + l4 l5 , l1 l3 + l4 l6 , . . . T 1 . . . , l2 l3 + l5 l6 , h1 l4 , h1 l5 + ph1 l6 = a + 2 b. α

3.3. Numerical formulae for N = 8 Assuming that α is known. Let S0 = (s1 , . . . , s6 , h1 s7 ), −1 T T S1 = h1 (s8 , s9 ), and P = (S1 S−1 0 ) , b = (d1 S0 ) + −1 T 1 α2 (d2 S0 ) . Equation (4) infers that 2 l1 + l42 , l22 + l52 , l32 + l62 , l1 l2 + l4 l5 , . . . T l . . . , l1 l3 + l4 l6 , l2 l3 + l5 l6 , l4 , + P 5 = b. l6

Replacing the variables l4 , l5 by l6 , the parameters l12 , l22 , l32 , l1 l2 , l1 l3 and l2 l3 are presented by respectively with six polynomials corresponding with the following order of monomials l6 l6 1 1 1 1

T6,α = l62 , h1 l6 , , , 1, , , , h1 h 1 α 2 α2 h21 h21 α2 h21 α4 2 Since h21 α2 = − 14 α4 + 12 d211 + d221 α2 − 14 d211 − d221 , replacing l1 , l2 and l3 by l6 , h1 and α, we also have six polynomial equations in two variables x, α2 where x = h1 α2 l6 . Given the following order of thirteen monomials T

Replacing l4 by l5 , l6 , we present the parameters l12 , l22 , l32 , l1 l2 , l1 l3 and l2 l3 by six polynomials corresponding with the order of monomials T5,6 = l52 , l62 , l5 l6 , l5 , l6 , 1 . Again, replacing l1 , l2 and l3 by l5 and l6 , we have six polynomial equations presented by coefficient-matrix C corresponding with the order of ten monomials T = l53 , l63 , l52 l6 , l62 l5 , l52 , ¯, c ˆ be (1, 3, 4, 5, 7) and (1, 2, 3, 4, 5)l62 , l5 l6 , l5 , l6 , 1 . Let c monomial-reducing of C, we have l5 = − c¯13 c¯1 l63 + c¯2 l62 + c¯4 l6 + c¯5 and cˆ1 l62 + cˆ2 l5 l6 + cˆ3 l5 + cˆ4 l6 + cˆ5 = 0. Thus,

x3 , α4 x2 , α2 x2 , x2 , α6 x, α4 x, α2 x, x, α8 , α6 , α4 , α2 , 1

c˘1 l64 + c˘2 l63 + c˘3 l62 + c˘4 l6 + c˘5 = 0,

these polynomial equations are presented by the coefficients matrix C. Note that, generally these constraints lead us polynomials which have more than thirteen monomials. However, fortunately some monomials’ coefficients are zero, the poly¯, c ˆ nomials remains the above thirteen monomials T. Let c be (1, 2, 3, 4, 5) and (1, 2, 3, 4, 6)-monomial-reducing of C, respectively, we have

where c˘1 = cˆ2 c¯1 , c˘2 = cˆ2 c¯2 + cˆ3 c¯1 , c˘3 = cˆ1 + cˆ2 c¯3 + cˆ3 c¯2 , c2 c¯4 +ˆ c3 c¯3 and c˘5 = cˆ5 +ˆ c3 c¯4 . Therefore, l1 , . . . , l6 c˘4 = cˆ4 +ˆ are presented by α as follows Proposition 2. Given D, U, V and α,

c¯1 α4 + c¯2 α2 + c¯3 x + c¯4 α8 + c¯5 α6 + c¯6 α4 + c¯7 α2 + c¯8 = 0

cˆ1 α6 + cˆ2 α2 + cˆ3 x + cˆ4 α8 + cˆ5 α6 + cˆ6 α4 + cˆ7 α2 + cˆ8 = 0.

180

1. l6 is a solution of the following quartic equation: c˘1 x4 + c˘2 x3 + c˘3 x2 + c˘4 x + c˘5 = 0.


N=8

N=8 0.2

N=9

9 < N ≤ 200

0.2 0.1

10

Proportion

9 < N ≤ 200

0.1

0

−14

−12

−10

−8

−6

−4

−2

0 −4.5

N=8

−1

N=9

log (Average Error)

0.3

N=9 9 < N ≤ 200

−2 −3 −4 −5 −6

−4

−3.5

−3

−2.5

−2 −4

(a) : Noise free − log (Error)

−1.5

−1

(b) : Noisy case, std = 10 − log10(Error)

10

−0.5

−8

−7

−6

−5

−4

−3

−2

−1

(c) : Noisy cases − log (Noise level) 10

Fig. 1. Histograms of log10 (E) for (a): 10.000 independent Uniform noise free experiments, and (b): 1000 independent Uniform noisy cases with noise level std = 10−4 . (c): Simulation of the relationship between average errors and noise level of 1000 independent experiments. In all these experiments, the value of M is received uniformly on the set {4, 5, . . . , N }. 2. l5 = − c¯13 c¯1 l63 + c¯2 l62 + c¯4 l6 + c¯5 , 3. l4 = b7 − P71 l5 − P72 l6 2 4.

l1 l22 l32

=

b1 b2 b3

− l5

P11 P21 P31

− l6

P12 P22 P32

−

l42 l52 l62

and identity covariance. These results are given in Figure 1.b and 1.c. Note that, it is not possible to directly apply our formulae for noising-distance-matrix Dstd . We should find an approximating-distance-matrices D0 such that D0 Dstd 0 having ranks and its difference of square-distance matrix D of three, and then apply our formulae on D0 . From our experiments, we conclude that, firstly, the existences of S−1 and monomial-reducing of the coefficients 0 matrix C are believable; secondly, our results are accurate and stable. Since our formulae are numerical, our estimations are more easily understandable and usable than that given in [10]. Moreover, with some realistic applications when distance x1 − x2 2 is known, clearly our estimation works very well. Surprisedly, since in the case N = 8, we can control accuracy of α, the error of N = 8 is lower than that of N 9. Finally, scientists, who are familiar with Gröbner method, can combine this method with our formulae. Our formulae given in this paper will help the Gröbner method working faster and more stable with the TOA-based localization.

,

such that l1 l2 = b4 − P41 l5 − P42 l6 − l4 l5 and l1 l3 = b5 − P51 l5 − P52 l6 − l4 l6 , where Pij is the (i, j)-th element of P. 4. IMPLEMENTATION AND EVALUATION To evaluate the results, we consider a relative error of dis˘ − D2 /D2 , where D ˘ = d˘mn , tance, E = D M ×N ˘ dmn = xm − yn 2 , and xm , yn are the estimating points. In the cases N = 8, the relative error E is considered as a function of α, therefore α is determined by minimizing the relative error function. Fortunately, from the triangle inequality, parameter α is bounded by maxn |d1n − d2n | α minn (d1n + d2n ). Thus, optimal parameter α∗ can be found when we consider α as the running-parameter in function E. and Note that, we miss to check the existences of S−1 0 monomial-reducing of the coefficients matrix C of the results in section 3. The proving of these existences is inferred by the following experiments. We study for 10.000 independent experiments for each of three cases (i) N = 9, (ii) N = 8, (iii) 9 < N 200. For each experiment of each case, the positions in (x, y)-groups were simulated as independently and uniformly distributed points on unit cubic space. The errors between the original distance-matrix D which computed from the original positions and the approximating distance˘ which computed from the approximating positions matrix D are given in Figure 1.a. To study how our formulae work in a noisy environment, we measure errors for each above experiment when many different levels of independent Gaussian noise are added to the distance-matrix, i.e. Dstd ← D + std ∗ N (0, 1), where std denotes a different level, and N (0, 1) denotes a (M, N )-Gaussian matrix with zeros mean

5. CONCLUSION We have presented numerical formulae for the TOA-based microphone and source localization in the cases N 8. In the cases of N 9, the formulae determine their positions in a closed form without any complex polynomial solvers. In the case of N = 8, only one parameter, α is required for the closed-form solution, which can be estimated simply by grid search, or we can assume it is known in some applications. We confirmed the validity of our formulae by experiments on both of the noise free and noisy cases. The cases N = 7, N = 6 and N = 5 will be studied in near future. 6. ACKNOWLEDGEMENT This work was supported by a Grant-in-Aid for Scientific Research (B) (Japan Society for the Promotion of Science (JSPS) KAKENHI Grant Number 25280069).

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