A vector quantization-based compression scheme for wave field ...

A vector quantization-based compression scheme for wave field synthesis source signals Georgios N. Lilis1 1 Technical

University of Crete , Kounoupidiana, Chania, 73100, GREECE

Correspondence should be addressed to Georgios N. Lilis ([email protected]) ABSTRACT This paper introduces a lossy data compression scheme of the signals used for the synthesis of narrow-band virtual point wave fields. The fields are synthesized by a set of distributed point wave sources at fixed locations inside a wave medium. The data compression scheme is implemented using vector quantization. An application example referring the synthesis of a virtual point field, using three point sources, is considered. Its performance is assessed by defining and calculating a mean distortion measure, for different number of quantization levels. Extension to the synthesis of broad-band wave fields of arbitrary shape is straightforward.

1. INTRODUCTION 1.1. Wave field synthesis and signal compression The concept of synthesizing waves with specific spatial characteristics appears in many disciplines, ranging from creating virtual sound environments [3],[2] and optical holograms [9], to destroying cancer tumors by synthesizing focused ultrasound fields [11] . In all of these cases a wave field is induced by a number of distributed wave sources inside a wave medium. The goal of this process is the synthesized field to have certain temporal and spatial characteristics or to be as close as possible to a given desired wave field. Since the number of available wave sources is finite and limited not all wave fields can be synthesized perfectly. However given any wave source configuration, there exist source signals (synthesis signals), which when applied to the sources synthesizing the wave field (synthesis sources) the induced wave field is the closest possible (in the square error sense) to a given desired wave field. It is apparent that the information content of the synthesis signals is very rich and diverse depending on the configuration of the sources and the desired field characteristics. Therefore in order to enable any wave source configuration to synthesize any given wave field, an efficient method of encoding and transmitting the information content of the synthesis signals, must be established.

Such a communication scheme is displayed in Fig. 1. Based on a desired wave field and a given source topology, the transmitter sends a single codeword to each of the synthesis sources. Upon receiving this codeword every source generate its own wave field. The wave fields generated by all the sources add constructively in the acoustic medium in order to produce a field similar to the desired wave field.

Fig. 1: A general communication scheme for wave field synthesis applications 1.2. Prior Art Many researchers have applied signal processing techniques to encode acoustic environments based on sensed sound samples, transmit the encoded information to

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A VQ-based compression scheme for wave field synthesis signals

sound sources and regenerate these environments. Adjler et.al. [1] examined a spatio-temporal sampling and data compression technique using the field’s bi-dimensional Fourier spectrum. S.Servetto [10] studied the problem of wave field sampling and reconstruction from an information theoretic point of view, where the sampled wave field measurements are encoded and transmitted to a unique decoder which attempts to reconstruct the sensed wave field. Towards the same direction T.Kimura et al. [5] developed a signal processing technique in which moving acoustic sources were first sensed using microphones and then reproduced using a loudspeaker array. This method achieves high levels of compression when the number of primary sensed sound sources is considerably smaller than the number of secondary synthesizing sources. 1.3. Contribution Since any band and space limited wave field can be approximated by a summation of narrow band point wave fields, and assuming that the acoustic medium behaves linearly, the problem of reconstructing a composite band and space limited wave field can be reformulated as a problem of reconstructing its components (the individual narrow band point wave fields). In this sense, the knowledge of reconstructing narrow-band point wave fields is of paramount importance and becomes the focus of this paper. The problem of detection of the initial wave fields based on sensor measurements is a separate problem and is not discussed here. A lossy compression method of the data of the synthesis source signals required for the reconstruction of narrow band point wave fields is studied here. It appears that high levels of compression with relatively low distortion can be achieved. This holds a promise of designing compression schemes for encoding source signals required for the synthesis of composite narrow band wave fields, in general. 1.4. Wave field classification and data compression As the number of possible synthesized fields, using a specific source topology, is infinite, the amount of information required to be transmitted to the sources for every possible synthesis scenario is also infinite. However the class of all possible wave fields W which could be synthesized by a source topology can be partitioned into a finite number of subclasses of wave fields

Wp in the following manner: Every field belonging to the same subclass can be synthesized by the same synthesis signals at the expense of very small additional error. If the wave fields belonging to the same class do not differ substantially, the additional error introduced by this method is relatively small. The above process suggests a lossy compression scheme for the synthesis of all possible wave fields by a source configuration consisting of M sources. Figure 2 displays an example of such scheme. In this example the class of all possible wave fields W is partitioned into four subclasses Wd , d = 1, ..., 4. If a desired field P ∈ Wd is to be synthesized, the encoder transmits a codeword Cd to all the synthesis sources. Based on the received code word Cd synthesis source j, j = 1, ..., M, generates wave field Pjd . The wave fields from all the sources add inside the wave medium synthesizing a wave field Pd∗ which is close to the original field P and belongs to the same subclass Wd .

Fig. 2: General compression scheme for wave field synthesis applications The subclasses Wd , d = 1, ..., 4 define a partition PW = {W1 ,W2 ,W3 ,W4 } of the class of the synthesized fields W , as it is indicated in Fig. 2. Furthermore, for each subclass Wd there is a unique synthesized wave field Pd∗ associated with the subclass. Every wave field P from the subclass Wd is synthesized by inducing the wave field Pd∗ with some additional distortion. If we represent this difference (distortion) between wave fields P and Pd∗ with d(P, Pd∗ ) than the total distortion introduced by the above compression scheme equals to: 4

D=

∑ ∑

d(P, Pd∗ )

(1)

p=1 P∈Wd

Identifying compression schemes in the case of general wave fields is a very difficult task since the set of all possible wave fields can not be classified, easily. However

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the set of wave fields induced by an imaginary (virtual) point wave source called virtual point wave fields can be classified, as discussed in the next section. 1.5. Synthesis of virtual point fields It is common in many synthesis experiments the target wave field to be a wave field induced by a point source at a specific point inside the wave medium. In these cases, although there is not real source, the synthesized wave field gives the impression of the existence of a source at this point. Therefore the terms virtual source and virtual point field have been adopted to describe this synthesis process. Fig. 3 illustrates the previous synthesis scenario.

• M Distributed synthesis sources at points~r j ∈ S, j = 1, . . . , M. • A frequency band Ω. • An omni-directional virtual point wave field at~ro ∈ S with spectrum described at every point ~r ∈ S and at every frequency ω ∈ Ω by: − j ωc |~r−~ro |

e Pˆo (~ro ,~r, ω ) = 1 × 4π |~r −~ro |

(2)

The optimal in the l2 sense spectra Sˆl2 j (ω ), j = 1, ..., M of the distributed sources which synthesize a wave field with the least possible difference (in the square error sense) with the given virtual point field, are determined by the following vector of functions:

l2 {Sˆ1l2 (~ro , ω ), ..., SˆM (~ro , ω )} = [H(ω )−1 ]{l(~ro , ω )}

[H(ω )] pq =

∫

S

{l(~ro , ω )}q =

G p (~r, ω )G∗q (~r, ω )ds

∫

S

Pˆo (~ro ,~r, ω )G∗q (~r, ω )ds

p, q ∈ {1, ..., M} Fig. 3: Virtual point WFS performed by 16 synthesis sources on a circular array. Left: Energy distribution of the target field. Right: Energy distribution of the virtual point field focused at the center point (5,5) Knowing how to effectively synthesize virtual point fields at specific points in a wave medium, will enable us to synthesize more complex wave fields. The knowledge of synthesis of virtual point fields is associated with the calculation of the appropriate signals which must be transmitted to the synthesis sources. Towards this direction a virtual point field synthesis method has been developed [7]. This method determines the spectra of the signals of the synthesis sources which induce a wave field as close as possible in the l2-norm sense to a given virtual point wave field. More precisely:

where G j (~r, ω ) is the spectral distribution of the wave field induced by an omni-directional point wave source at point ~r j ∈ S. In three dimensions this distribution is the Greens function given by: ω

G j (~r, ω ) = 1 ×

∫ ∫

[ S

• A wave medium S characterized by wave speed c.

e− j c |~r−~r j | 4π |~r −~r j |

(4)

The star notation ∗ represents the complex conjugate. The solution given by (3) always exists and is unique based on the fact that the matrix [H(ω )] is positive definite [6],[7]. The synthesis error of the above process is defined as the second norm of the difference between the synthesized field Pˆs (~ro ,~r, ω ) and the virtual point field Pˆo (~ro ,~r, ω )[7]: E=

Given:

(3)

Ω

~ ]d~r |Pˆs (~r, ω ) − Pˆo (~ro ,~r, ω )|2 d ω

(5)

The above synthesis error can be expressed as a funcˆ ω )} = tion of the vector of the source spectra [6] ({S( {Sˆ1 (ω ), ..., SˆM (ω )}). If the source spectra obtain their

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optimal (in the l2 sense) values (Sˆl2 j (ω ), j = 1, ..., M given by (3)) then it can be shown [6] that the synthesis error for every frequency ω ∈ Ω is given by:

E({Sˆl2 (~ro , ω )}) = c(~ro , ω ) − {l(~ro , ω )}H {Sˆl2 (~ro , ω )} c(ω ) =

∫

S

|Pˆo (~ro ,~r, ω )|2 ds

(6)

where vector {l} and constant c, are defined as a function of~ro and ω in (3) and (6). H denotes transpose conjugate. 1.6.

Spectral dependency

There are two parameters which affect the optimal in l2 sense spectra of the source signals determined by (3). One of these parameters is the bandwidth Ω of the desired field. In the case of narrow-band fields the l2optimal spectra change insignificantly. This can be verified by the following simple example. Let us consider the synthesis of a two dimensional virtual point field at the center of a 10 × 10 wave medium indicated by the “x” at the center of the right plot of Fig. 4. The synthesis sources are located at three points indicated by three dots at the right plot of Fig. 4. In the left plot of Fig. 4 the complex values of the l2-optimal spectra (given by 3) are plotted for every 10 Hz, starting from 0 and ending at a frequency 5 kHz. It is apparent that for relatively narrow bandwidths (10 − 30Hz) the change in the complex values of the spectra is insignificant (≈ 10−2 ). Therefore for the synthesis of narrow band wave fields the source spectra can be assumed to be constant. The location of the virtual point ~ro with respect to the source topology is the second factor that affects the l2optimal spectra of (3). Therefore it appears useful to find an effective way to communicate these l2-optimal spectra in the case of the synthesis of a virtual field at any point in the wave medium. 2. 2.1.

Fig. 4: Virtual point field synthesis example performed by 3 point sources. Left: l2-optimal spectral values as a function of frequency. Right: Source Topology all the fields denoted by Pˆs (~ro ,~r, ω ) and synthesized using a source topology T and the corresponding l2-optimal source spectra Sˆl2 j (~ro , ω ) defined for every point ~ro ∈ S, by (3). Since each point inside the wave medium ~ro is associated with one synthesized virtual point wave field Pˆs (~ro ,~r, ω ) ∈ WT , the class WT can be partitioned into subclasses, based on the location of the virtual point. In this way, any partition PWT = {W1 , ...,WN } of the class WT defines a partition PS = {S1 , ..., SN } of the wave medium S into N subregions, as there is an 1-1 relation between a virtual point field and its location. Furthermore for each virtual point ~ro and each frequency ωo ∈ Ω a unique complex vector of l2-optimal spectral source values: {Sˆl2 (~ro , ωo )} = l2 (~r , ω )} is defined according to (3). {Sˆ1l2 (~ro , ωo ), ..., SˆM o o Furthermore if we assume that the band Ω is narrow the previous vector can be assumed to be constant with respect to the frequency (according to section 1.6), or: l2 (~r )}. {Sˆl2 (~ro )} = {Sˆ1l2 (~ro ), ..., SˆM o The information of the complex values of the previous vector must be encoded and transmitted to the distributed sources in order for the synthesis to be l2-optimal. 2.2. Distortion measure

COMPRESSION Virtual point wave field class

Based on the previous discussion a wave class WT can be defined as the class of all synthesized narrow band virtual point wave fields at every point inside the wave medium S, by a given source topology T. Essentially WT contains

If the spectral values of the synthesis signals {Sˆ1 (~ro ), ..., SˆM (~ro )} differ from the l2-optimal ones defined in (3), then additional distortion to synthesized virtual point field is added. According to the derivation developed in the appendix this additional distortion is equal to:

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ˆ ro )}, {Sˆl2 (~ro )}) = {δ Sˆl2 (~ro )}H [H]{δ Sˆl2 (~ro )} (7) d({S(~ where δ Sˆl2 is the divergence of the source spectra from their l2-optimal values: ˆ ro )} − {Sˆl2 (~ro )} {δ Sˆl2 (~ro )} = {S(~

(8)

Since matrix [H] is hermitian and positive definite [6] it admits a square root [K] defined by: [H] = [K]H [K]. Furthermore the distortion measure defined in (7) is similar to the Euclidean distance between two points in a vector space V defined by:

ˆ [H] = [K]H [K]} V = {{Vˆ } : {Vˆ } = [K]{S}

(9)

d({Vˆ (~ro )}, {Vˆ l2 (~ro )}) = ||{Vˆ (~ro )} − {Vˆ l2 (~ro )}||22 (10) with:

2.3.

∑

i=1~ro ∈Si

||{Vˆi } − {Vˆ l2 (~ro )}}||22

(12)

The fundamental question which arises here is the following: Given a total number N of subregions, how can one determine a partition PS∗ = {S1∗ , ..., SN∗ } and its alphabet {AVˆ∗ } = {{Vˆ1∗ }, ..., {VˆN∗ }} out of all possible partitions and alphabets, such that the total induced distortion determined by (12) is minimized. Equivalently: {PS∗ , AVˆ∗ } = min D(PS , AVˆ ) PS ,AVˆ

(13)

The answer to this question, is attempted by the vector quantization scheme presented next.

Or, in other words:

{Vˆ }l2 (~ro ) = [K]{Sˆl2 (~ro )},

N

D(PS , AVˆ ) = ∑

ˆ ro )} {Vˆ (~ro )} = [K]{S(~ (11)

Total distortion

Assuming a set of virtual points {~ro ∈ S} the unique l2optimal complex vectors Sˆl2 (~ro ) needed to be transmitted to the synthesis sources can be mapped to unique vectors Vˆ l2 (~ro ) via the mapping defined by (9). In order to design an efficient compression algorithm a partition of the wave medium into a finite number of subregions (PS = {S1 , ..., SN }) must be applied first. Then for every virtual point~ro inside subregion Si , i = 1, ..., N, a unique complex vector Vˆi (often called centroid) representing all the points of the subregion has to be transmitted to the synthesis sources. Generally it will be true that Vˆi 6= Vˆ l2 (~ro ). Therefore a distortion defined by (10) with Vˆi = Vˆ (~ro ) will be introduced. The total distortion induced by this general scheme will be a function of the partition PS = {S1 , ..., SN } and the centroid complex values contained in a set (called alphabet) AVˆ = {{Vˆ1 }, ..., {VˆN }}:

2.4.

Vector Quantization

A vector quantization algorithm initially proposed by Lloyd in 1957 [8] and later enhanced by Y.Linde A.Buzo and R.Gray [4] can be used in order to provide a good solution to the minimization problem described by (13). Initially a dense set of of virtual points {~ro ∈ S} must be determined. The existence and uniqueness of the solution {Sˆl2 } in (3) and the invertibility of the matrix [K] defined in (9), guarantees that the set {~ro ∈ S} can be ˆ ro )} and finally to a set of mapped to a set of vectors {S(~ vectors {Vˆ (~ro )} in the vector space V . Likewise there will be an one to one correspondence between the partitions PS , PSˆ and PVˆ and the alphabets ASˆ and AVˆ . This is illustrated in Fig. 5. Using vector quantization with a N quantization levels, the whole space V and also the set {Vˆ (~ro )} is partitioned into N subsets. Furthermore centroid complex values Vˆi i = 1, ..., N can be determined for every subset of the partition in such way that the total mean square error is minimized. Since the Euclidean distance in the space of vectors Vˆ is equivalent with the distortion measure defined in (7), the obtained complex values {Vˆ1 , ..., VˆN } will form the alphabet {AVˆ∗ } in (13). Furthermore, according to Fig. 5 the resulted (by the vector quantization) partition of the space V will define the partition PS∗ in the wave medium S. This becomes clear by the example of the next section. 2.5.

Example

In this example we examine the synthesis of virtual point fields at points ~ro inside a 10 × 10 2D wave medium S

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Fig. 5: One to one correspondence between the points of the medium S and the spaces of vectors Sˆ and Vˆ . at frequency f = ω2πo = 1 kHz. For simplicity the source topology of figure 4, is considered. The l2-optimal complex source spectral values {S1 (~ro ), S2 (~ro ), S3 (~ro )} are determined by (3) for ω = ωo and for virtual points ~ro ∈ S. The locations ~ro where chosen to be located on a 51 × 51 orthogonal grid defined by:

G = {(0.2(k − 1), 0.2(m − 1)), (k, m) ∈ {1, ..., 51}2 } (14) Ten different vector quantization scenarios, were considered, with a total of N = 2n , n = 1, 2, ..., 10 quantization levels each time. The Generalized Lloyd Algorithm (GLA) using a relative threshold stopping criterion with threshold value ε = 10−7 [8] was applied. Figure 6 displays the resulted partitions of the medium S for N=2,4,8, and 16 quantization levels, using different colors.

Fig. 6: Partitions of medium S, obtained by applying vector quantization with N=2,4,8,16 quantization levels. of Fig. 7 can be obtained. Since 2601 points were used for the synthesis of virtual point fields the compression ratio can be defined as the ratio of 2601 over the rate: CR =

2601 R

(15)

Remarks The plots of Fig. 6 indicate that areas close to the locations of the synthesis sources are not partitioned as the number of quantization levels increases. This is expected since the synthesis of virtual point fields near the location of a synthesis source is induced only by its signal which is far bigger than the synthesis signals of the other sources. In order to assess the performance of the proposed compression method the total induced distortion was calculated based on equations (7), (10) and (12). Since the we considered quantization levels to be powers of 2 the communication rate can be defined as R = log2 N in bits for each quantization scenario. By plotting the mean distortion as a function of the rate R, the rate distortion curve

Fig. 7: Mean distortion as a function of the communication rate.

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¯ Compression ratio (CR ) as Table 1: Mean distortion (D) functions of the rate (R) and the total quantization levels (N) R 1 2 3 4 5 6 7 8 9 10

N 2 4 8 16 32 64 128 256 512 1024

D¯ 0.119400 0.039900 0.020200 0.011300 0.006500 0.003700 0.002200 0.001500 0.000790 0.000438

CR 1300.5 650.25 325.12 162.56 81.28 40.64 20.32 10.16 5.08 2.54

Conceptually, this method provides a fundamental component for compressing the rich information content of the synthesis source signals in a general broad band wave field synthesis scenario. 4. ACKNOWLEDGMENTS This work is dedicated to my Ph.D. advisor Prof. Sergio Servetto, who left us unexpectedly in 24th of July 2007. 5. APPENDIX

Remarks: Given the rate distortion curve in Fig. 7 operating at rates R > 4 bits or with N > 16 quantization levels will result in total distortion smaller than 10−2 . Table 1 displays the mean distortion values D¯ and the compression ratios as functions of the communication rate R and the total quantization levels N. 3. CONCLUSIONS AND EXTENSIONS BROAD BAND ACOUSTIC SIGNALS

source signals of all of the respective narrow band components.

TO

In this section the distortion expression in (7) is derived. Given a specific source topology and source spectra deˆ = {S(~ ˆ ro , ω )} = scribed by the vector of functions {S} {Sˆ1 (~ro , ω ), ..., SˆM (~ro , ω )} the synthesis error can be described by the following quadratic form [6]:

ˆ = {S} ˆ H [H]H {S} ˆ + {l}H {S} ˆ + {S}{l} ˆ E({S}) + c (16) The matrix [H], vector {l} and constant c are specified by (3) as a function of the frequency ω . For simplicity the frequency ω and the point~ro are considered constant and are omitted in the derivation. The error expression (16) can be written as:

A lossy compression scheme of the source data required for the synthesis of virtual point wave fields inside a two dimensional wave medium which is based on vector quantization principles, was demonstrated. The performance of the proposed scheme was evaluated using an implementation example where synthesis of virtual point wave fields were performed by three point sources arbitrary distributed inside a two dimensional medium. It appears that this vector quantization based method achieves high levels of compression with relatively low distortion values. Finally, since broad-band wave fields can be approximated by superpositions of multiple narrow-band point wave fields and as the acoustic medium behaves linearly for relatively low acoustic signal power levels, the proposed method can be extended to include broad band acoustic fields of 20 KHz bandwidth. This can be achieved by compressing the source signals required for the synthesis of their multiple narrow-band components individually. In the reproduction stage, the wave field can be reconstructed by superimposing the compressed

H H ˆ = ([K]{S}−{m}) ˆ ˆ E({S}) ([K]{S}−{m})+c−{m} {m} (17)

with: [H] = [K]H [K] {l} = [K]H {m}

(18)

ˆ differ from the l2Lets assume that the spectra {S} l2 −1 ˆ l2 : ˆ optimal ones {S} = [H] {l} by amount δ {S} ˆ = {Sˆl2 } + {δ Sˆl2 } {S}

(19)

Since [H] = [K]H [K] it will be true that [H]−1 = [K]−1 [K H ]−1 . Furthermore since {l} = [K]H {m} the l2ˆ l2 = [H]−1 {l} can be expressed as: optimal spectra {S} ˆ l2 = [H]−1 {l} = [K]−1 [K H ]−1 [K]{m} = [K]−1 {m} {S} (20)

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Based on (20) (19) becomes: ˆ = [K]−1 {m} + {δ Sˆl2 } {S}

(21)

Substituting (21) into (17) taking into account that [H] = [K]H [K], leads to:

ˆ = {δ Sˆl2 }[H]{δ Sˆl2 } + c − {m}H {m} E({S})

(22)

Substituting (20) into (17), leads to E({Sˆl2 }) = c − {m}H {m}

(23)

The additional error or distortion can be defined as the ˆ − E({S} ˆ l2 ) which is always positive difference E({S}) and according to (22) and (24), is given by:

ˆ − E({Sˆl2 }) = {δ Sˆl2 }[H]{δ Sˆl2 } d = E({S})

(24)

[6] G.N. Lilis. Wave Field Synthesis Acoustics, Electromagnetics and LC Lattices. PhD thesis, Cornell University, 2008. [7] G.N. Lilis, D. Angelosante, and G.B. Giannakis. Sound field reproduction using the lasso. Audio, Speech, and Language Processing, IEEE Transactions on, 18(8):1902–1912, 2010. [8] S. Lloyd. Least squares quantization in pcm. Information Theory, IEEE Transactions on, 28(2):129– 137, 1982. [9] R. Piestun, B. Spektor, and J. Shamir. Wave fields in three dimensions: analysis and synthesis. JOSA A, 13(9):1837–1848, 1996. [10] S.D. Servetto and J.M. Rosenblatt. The multiterminal source coding problem for spatial waves. Proc. UCSD Wkshp. Inform. Theory App, 2006. [11] J.S. Tan, L.A. Frizzell, N. Sanghvi, S. Wu, R. Seip, and J.T. Kouzmanoff. Ultrasound phased arrays for prostate treatment. The Journal of the Acoustical Society of America, 109:3055, 2001.

6. REFERENCES [1] T. Ajdler, R.L. Konsbruck, O. Roy, L. Sbaiz, E. Telatar, and M. Vetterli. Spatio-temporal sampling and distributed compression of the sound field. In Proc. European Signal Processing Conference (EUSIPCO06), 2006. [2] A.J. Berkhout, D. de Vries, and P. Vogel. Acoustic control by wave field synthesis. The Journal of the Acoustical Society of America, 93:2764, 1993. [3] M. Camras. Approach to recreating a sound field. The Journal of the Acoustical Society of America, 43(6):1425–1431, 1968. [4] A. Gersho and R.M. Gray. Vector quantization and signal compression, volume 159. Springer, 1992. [5] T. Kimura, K. Kakehi, K. Takeda, and F. Itakura. Spatial coding based on the extraction of moving sound sources in wavefield synthesis. In Acoustics, Speech, and Signal Processing, 2005. Proceedings.(ICASSP’05). IEEE International Conference on, volume 3, pages iii–293. IEEE, 2005.

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