INTERSPEECH 2016
September 8–12, 2016, San Francisco, USA

Acoustic-to-Articulatory Inversion Mapping based on Latent Trajectory
Gaussian Mixture Model
Patrick Lumban Tobing1 , Tomoki Toda2 , Hirokazu Kameoka3 , Satoshi Nakamura1
1

Graduate School of Information Science, Nara Institute of Science and Technology, Japan
2
Information Technology Center, Nagoya University, Japan
3
NTT Communication Science Laboratories, NTT Corporation, Japan
patrick.lumbantobing.pf3@is.naist.jp, tomoki@icts.nagoya-u.ac.jp,
kameoka.hirokazu@lab.ntt.co.jp, s-nakamura@is.naist.jp

Abstract

ity density of articulatory parameters. Hidden Markov model(HMM) based inversion mapping has been proposed in [14],incorporating dynamic constraints and also linguistic contexts.
In this paper, we focus on the study of Gaussian mixture
model (GMM) based inversion mapping [15], capable of representing multiple mixtures of probability density and incorporating dynamic constraints without textual input. Albeit, in termsof dynamic constraints, this method bears an inconsistency issue. In its training phase, the model parameters are optimized
with respect to the likelihood of joint static and dynamic features of acoustic-articulatory data. While in the mapping phase,given a sequence of acoustic parameters, a sequence of articulatory parameters is estimated with respect to the conditional likelihood of their static features, where the inter-frame correlationis considered by imposing a constraint between static and dynamic features. Because of the inconsistency of optimized feature space between the training and mapping phases, the trained
GMM is not optimum for the mapping phase.
To overcome this inconsistency problem, a trajectory training method has been proposed, known as the trajectory HMM[16]. To train the trajectory model, optimization is performed
with respect to the likelihood of the static features of the training data. This likelihood is obtained by imposing a constraint
between static and dynamic features, making it possible to incorporate inter-frame correlation in the training phase. It has
been reported that this method can be further extended to incorporate additional constraints, e.g. global variance [17] andmodulation spectrum [18]. However, in this standard trajectory training method, it is somewhat difficult to optimize several
model parameters, e.g. 1) it is not straightforward to optimize
the covariance matrices due to the difficulties of obtaining theiranalytical ML estimates and 2) it is basically difficult to include
all possible mixture component sequences or even to decide thebest mixture component sequence to maximize the likelihood.
In this paper, as an alternative approach for solving the inconsistency issue, inspired by the latent trajectory HMM [19],we propose a novel inversion mapping method based on latent
trajectory GMM (LT-GMM). In the proposed method, a time
sequence of joint static and dynamic features is treated as a latent variable, while a time sequence of static features is treatedas an observed variable. A likelihood function of the observed
variable is obtained through marginalizing out the latent variable by imposing a soft constraint between static and dynamic
features. Based on such likelihood function, parameters are easily optimized by using the EM algorithm. Experimental results
indicate that the proposed LT-GMM gives higher accuracy forthe inversion mapping compared with the conventional GMM.

A maximum likelihood parameter trajectory estimation basedon a Gaussian mixture model (GMM) has been successfully
implemented for acoustic-to-articulatory inversion mapping. In
the conventional method, GMM parameters are optimized by
maximizing a likelihood function for joint static and dynamic
features of acoustic-articulatory data, and then, the articulatory
parameter trajectories are estimated for given the acoustic databy maximizing a likelihood function for only the static features,
imposing a constraint between static and dynamic features toconsider the inter-frame correlation. Due to the inconsistencyof the training and mapping criterion, the trained GMM is not
optimum for the mapping process. This inconsistency problem
is addressed within a trajectory training framework, but it becomes more difficult to optimize some parameters, e.g., covariance matrices and mixture component sequences. In this paper,
we propose an inversion mapping method based on a latent trajectory GMM (LT-GMM) as yet another way to overcome theinconsistency issue. The proposed method makes it possible
to use a well-formulated algorithm, such as EM algorithm, tooptimize the LT-GMM parameters, which is not feasible in the
traditional trajectory training. Experimental results demonstratethat the proposed method yields higher accuracy in the inversion
mapping compared to the conventional GMM-based method.
Index Terms: Gaussian mixture model, inversion mapping, latent trajectory model, EM algorithm, inter-frame correlation

1. Introduction
Speech can be characterized not only by the acoustic spectrum of the vocal tract but also by the more slowly varying parameters, such as articulatory movements [1]. Indeed,utilization of underlying articulatory movements from speechsounds has been studied in many applications, e.g. speech
analysis/synthesis [2, 3], speech coding [4], speech recognition[5, 6], speech pathology [7, 8], speech visualization [9, 10], etc.
Hence, the need of a robust system to convert acoustic data into
articulatory data, so called inversion mapping, grows rapidly.
Lately, the advancement in recording devices that enablea simultaneous recording procedure of acoustic-articulatory
data has been inducing many works on statistical methods ofacoustic-to-articulatory inversion mapping. Codebook based
inversion mapping has been proposed in [11, 12]. Moreover, in[12], it has been reported that the accuracy of inversion mapping
is significantly improved by introducing dynamic constraints.
Neural network based inversion mapping has been proposed in[13], stating the importance of multiple mixtures of probabil-

Copyright © 2016 ISCA

953

http://dx.doi.org/10.21437/Interspeech.2016-1196

2. Conventional GMM-based
acoustic-to-articulatory inversion mapping

Then, the estimated articulatory parameter sequence ŷ is determined as follows:

Let us assume a time sequence of Dx -dimensional static> >acoustic feature vectors as x = [x>and that
1 , · · · .xT ]of Dy -dimensional static articulatory feature vectors as y => >[y >At frame t, a 2Dx -dimensional acoustic
1 , · · · , yT ] .
> >feature vector is denoted as X t = [x>t , ∆xt ] , a 2Dy dimensional articulatory feature vector is denoted as Y t => >[y >t , ∆y t ] , where ∆xt and ∆y t denote the dynamic feature vector of acoustic parameters and that of articulatory parameters, respectively, and their joint feature vector is denoted> >as Z t = [X >t , Y t ] . Then, time sequences of acoustic feature vectors, articulatory feature vectors, and their joint fea> >ture vectors are denoted as X = [X >1 , · · · , XT ] , Y =
>
> >
>
> >
[Y 1 , · · · , Y T ] , and Z = [Z 1 , · · · , Z T ] , respectively.

ŷ = arg max P (Y |X, m̂, λ(Z) ), s.t. Y = W y y,where the ML estimate can be analytically determined [21].
Note that, in this mapping phase, the inter-frame correlation
is taken into consideration, thanks to the usage of the constraint
between static and dynamic features, i. e. Y = W y y. On
the other hand, this constraint is neglected while optimizing the
GMM parameters in the training phase.

3. Proposed LT-GMM-based
acoustic-to-articulatory inversion mapping
Let the observed variable be a time sequence of joint static fea>>
> >ture vectors z = [z > , · · · , z >T ] , where z t = [xt , y t ] , andthe latent variable be a time sequence of joint static and dynamic
feature vectors Z. The observed and latent variables are related
through a soft constraint as follows:

2.1. Training phase
The joint probability density of the acoustic and articulatory
feature vectors is modeled by a GMM as follows:
P (Z|λ(Z) ) =

T X
M
Y

(Z)
(Z)
αm N (Z t ; µm
, Σm),

Z ' W z = [W x , W y ][x> , y > ]> .

(1)

t=1 m=1

P (Z|z, Σ) = N (Z; W z, Σ)
1∝ exp − (Z − W z)> Σ−1 (Z − W z) , (7)2where its covariance matrix (Σ = diag [Σx , Σy ]) represents
the variations of margin errors, compensating the soft constraint. By completing the square in the exponent of the above
pdf, the conditional probability density of the observed variable
z given the latent variable Z is written as

where the mean vectors of the acoustic parameters and that of
the articulatory parameters for the mth mixture component are(Y ), respectively. The covariance maand µm
denoted as µ(X)mtrices of the acoustic parameters and that of the articulatory pa(XX)rameters for the mth mixture component are denoted as Σm(Y Y )and Σm , respectively. The cross covariance matrices of theacoustic and the articulatory parameters for the mth mixture
(XY )
(Y X)
component are denoted as Σmand Σm . These model
parameters are optimized with EM algorithm [20].

(9)where λ(z) is the set of LT-GMM parameters, consisting of thecovariance matrix Σ and model parameters λ(Z) . Note that thecovariance matrix Σ depends on only dimension of Z t , i. e. independent of both time frames and mixture components. The
model parameters can be optimized with variational EM algorithm [19].
In this paper, an approximation of the above joint pdf is employed by using the sub-optimum mixture component sequencem̂. The approximated joint pdf is then written as follows:Z
P (z|m̂, λ(z) ) = P (z|Z, Σ)P (Z|m̂, λ(Z) )dZ#!
  " (z|Z) # " (zz)
Z
(zZ)
z
µm̂
Σm̂
Σm̂
,
= N
;
dZ, (11)
(Z)
(Zz)
(ZZ)
Z
µm̂
Σm̂
Σm̂

(3)

where W y is a transformation matrix to expand a sequence ofstatic feature vectors into its sequence of joint static and dynamic feature vectors. The ML estimate can be determined with
EM algorithm [15].
In this paper, an approximation of the above conditional
probability density is employed by using a single mixture component sequence m = {m1 , · · · , mT }. First, the sub-optimum
mixture component sequence m̂ is determined as follows:
P (mt |X t , λ(Z) ).

Λ = W > Σ−1 W .

The joint probability density of the acoustic and articulatory
feature vector sequences is modeled by an LT-GMM as follows:Z
P (z|λ(z) ) = P (z|Z, Σ)P (Z|λ(Z) )dZ,
(10)

y

m

(8)

3.1. Training phase

In the conventional GMM-based mapping, given an acoustic parameter sequence X, the estimated articulatory parameter sequence ŷ is determined as follows:TY

P (z|Z, Σ) = N (z; Λ−1 W > Σ−1 Z, Λ−1 ),where

2.2. Mapping phasem̂ = arg max(6)

Employing the above soft constraint, the conditional probability density of the latent variable Z given the observed variable z is written as

where the Gaussian distribution with mean µ and covariance Σ
is denoted as N (·; µ, Σ). The mixture component index is m.
The total number of mixture components is M . The weight ofthe mth mixture component is αm . The set of GMM parameters is λ(Z) , consisting of weights, mean vectors and covariancematrices of all mixture components. The mean vector µ(Z)
m and
(Z)
covariance matrix Σm of the mth mixture component are written as#" (X) 
(XY )
(XX)
Σm
Σm
µm
(Z)
(Z)
(2)
µm =
(Y X)
(Y Y ) ,(Y ) , Σm =
µm
Σm
Σm

ŷ = arg max P (Y |X, λ(Z) ), s.t. Y = W y y,

(5)

y

(4)

t=1

954

Conventional GMM
Mapping
Training
t = 1...T t = 1...T

where the mean vectors are written as
(z|Z)
(Z)
µm̂ = Λ−1 W > Σ−1 µm̂ ,
>

h

(Z)

(Z)

>

(Z)

µm̂ = µm̂1 , · · · , µm̂T

(12)
i>

,

(13)

Σm̂ = Λ−1 W > (Σ−1 + Σ−1 Σm̂
(zz)

(zZ)

Σm̂

(ZZ)

= Λ−1 W > Σ−1 Σm̂(ZZ)Σ−1 )W Λ−1 , (14)

,

h

Xt

(15)

(Zz)
(ZZ)
Σm̂ = Σm̂ Σ−1 W Λ−1 ,(ZZ)(Z)Σm̂ = Σm̂ = diag

i

.

=

, λ̂

Z


(Z) 
P (Z|z, λ(z) ) log P (z|Z, Σ)P (Z|λ̂ ) dZ. (18)
and

x

y

(Z)>

(Z)−1

r t + µ̂m̂t Σ̂m̂t(Z)µ̂m̂t .

1,0,

if trueotherwise.

(25)

3.2. Mapping phase
In the proposed LT-GMM-based mapping, given an acoustic
feature sequence x, the estimated articulatory feature sequenceŷ is determined as follows:

T

(Z)−1

(
δ(m̂t = m) =


(Z)
(Z)−1
(Z)−1 (Z)1X
log Σ̂m̂t + tr Σ̂m̂t Rt − r >µ̂m̂tt Σ̂m̂t
2 t=1
(Z)>

Yt

Figure 1: Simplified graph representations of the training andthe mapping phases for the conventional GMM and the proposed latent trajectory GMM (LT-GMM).

)− µ̂m̂t Σ̂m̂t

Xt
yConsidering both the terms that depend on only the model parameters λ(Z) and the use of the sub-optimum mixture component sequence m̂, the auxiliary function can be written as
Z
(Z)
(Z)
Q(λ(Z) , λ̂ ) ∝ P (Z|z, m̂, λ(z) ) log P (Z|m̂, λ̂ )dZ
=−

Xt

(17)

(Z)

Q(λ

Yt

(16)

(Z)
(Z)
Σm̂1 , · · · , Σm̂T

Using the likelihood in Eq. (10), we maximize an auxiliary
function defined as
(Z)

mt

mt

mt

and the covariance matrices are written as

Proposed LT-GMM
Training & Mappingt = 1...Tŷ = arg max P (y|x, λ(z) ),

(19)

(26)

y

In the expectation step (E-step), the expected values of> >the latent variable, i.e. r = [r >and R =
1 , · · · , rT ]diag [R1 , · · · , RT ], are estimated as follows:

where the conditional pdf is written as
ZZ
P (y|x, λ(z) ) = P (y|Y , Σ) P (X, Y |x, λ(z) )dXdY. (27)r = E[Z|z, m̂, λ(z) ]
(Z)

(Zz)

(zz)−1= µm̂ + Σm̂ Σm̂(z|Z) z − µm̂,This ML estimate can be determined with variational EM algorithm.
In this paper, an approximation of the above conditional
pdf is employed by first determining the sub-optimum mixture
component sequence m̂. Then, the estimated articulatory feature sequence ŷ is determined as follows:(20)

R = E[ZZ > |z, m̂, λ(z) ]
(ZZ)

= Σm̂

(Zz)

(zz)−1− diag Σm̂ Σm̂
2D

(zZ)

Σm̂


+ rr > ,(21)

where diag2D yields a block diagonal matrix with each block
having the size of 2D = 2(Dx + Dy ).
In the maximization step (M-step), by maximizing the auxiliary function with respect to the updated mean vector µ̂(Z)m forthe mth mixture component, its ML estimate is given by
µ̂(Z)
m =

T
1 X
δ(m̂t = m)r t ,γm t=1ŷ = arg max P (y|x, m̂, λ(z) ),where the approximated conditional pdf is written as
ZZ
P (y|x, m̂, λ(z) ) = P (x|Y , Σ) P (Y |X, m̂, λ(Z) )(22)

P (X|x, m̂, λ(z) )dXdY(y|x)= N (y; µm̂

and by maximizing the auxiliary function with respect to the up(Z)dated covariance matrix Σ̂m for the mth mixture component,

(y|x)

µm̂

γm =

δ(m̂t = m),

),

(29)

(y|x)

= Σm̂

(Y |x)

,

(30)

−1
W>
y Σy µm̂

(Y |x)(Y )(Y X) (XX)−1µm̂
= µm̂ + Σm̂ Σm̂

(23)

(X)

(E[X|x, m̂, λ(z) ] − µm̂ ), (31)where the total number of frames γm belonging to the mth mixture component is given by
T
X

(y|x)

, Σm̂

where

its ML estimate is given by
T
(Z)
1 X
(Z)>
Σ̂m =
δ(m̂t = m)Rt − µ̂(Z),m µ̂m
γm t=1

(28)

y

(y|x)

Σm̂

−1
−1
= (W >.
y Σy W y )(32)Thus, it is apparent that the ML estimate of the articulatory feature sequence ŷ is the mean vector of the above conditional pdf,
(y|x)
i.e. µm̂ , which can be analytically determined.

(24)

t=1

955

GMM
LT-GMM

Average RMSE [mm]

1.9
1.85

Average Correlation Coefficient

1.95
(16)
(32)

1.8

(32)

1.75
1.7

(64)

(64)

(64)

(64)
(128)

1.65
1.6
1.55

50

100 150 200 250 300 350
Number of training utterances

400

0.78
GMM
LT-GMM

0.76

(64)

(128)

100 150 200 250 300 350
Number of training utterances

400

0.74

(64)

0.72

(32)

0.7
0.68
0.66

(64)

(64)

(32)
(16)

50

Figure 2: Average root-mean-square error of estimated articulatory data. Numbers in parentheses are optimum number of
mixture components.

Figure 3: Average correlation coefficient of estimated articulatory data. Numbers in parentheses are optimum number of
mixture components.

Note that, in the LT-GMM, the inter-frame correlation iswell considered in both training and mapping phases by incorporating the soft constraint in Eq. (6). Figure 1 shows the difference of consistency of the training and mapping phases between
the conventional GMM and the proposed LT-GMM.

GMM, we exploited the structure of banded matrix, which is
implied at the huge linear equations in Eq. (20) and Eq. (21).
Specifically, we used the multifrontal massively parallel sparse
direct solver (MUMPS) [24, 25]. The computational time fortraining the LT-GMM is about 15 times slower compared to theconventional GMM.

4. Experimental evaluation

4.2. Experimental results

4.1. Experimental conditions

Figures 2 and 3 show the average values of the RMSE and thatof the correlation coefficient over all 14 dimensions of estimatedarticulatory parameters through all 20 testing utterances. The
proposed LT-GMM method gives lower values than the conventional GMM in terms of average RMSE. The LT-GMM method
gives overall higher values for also the average correlation coefficient compared to the conventional GMM. These results indicate that the proposed LT-GMM method improves the accuracyof the acoustic-to-articulatory inversion mapping. Such good
performance can be achieved because of the consideration of
the inter-frame correlation while optimizing the LT-GMM parameters, whereas in the conventional GMM, this consideration
is taken into account only in the mapping phase.

We used a set of simultaneously recorded speech and articulatory data provided in MOCHA [22], spoken by a single British
male speaker. The total number of utterances was 460. Speech
data was sampled at 16 kHz. EMA data was used as the articulatory data. The frame shift was set to 5 ms.
As the acoustic parameters, we used the 1st through 24th
mel-cepstral coefficients converted from the spectral envelope,which was extracted at each frame by using STRAIGHT analysis method [23]. As the articulatory parameters, we used the
14-dimensional EMA data, which were converted to z-scores.
The EMA data represented the movements of seven articulators, i. e. upper lip, lower lip, lower incisor, tongue tip, tongue
body, tongue dorsum, and velum, defined within x- and y- coordinates on the midsagittal plane.
The constant positive-definite matrix Σ was set to the diagonal matrix of global variances. A trained model from the conventional GMM was used as an initial model for the LT-GMM
training. In the training phase of LT-GMM, the sub-optimum
mixture component sequence m̂ was initialized using the initialmodel and held fixed. In the mapping phase of LT-GMM, thesub-optimum mixture component sequence m̂ was determinedusing also the initial model.
We conducted an objective evaluation by calculating theroot-mean-square errors (RMSEs) and the correlation coefficients between the estimated articulatory parameters and themeasured ones. The number of training utterances was varied
to 50, 100, 150, 150, 200, 250, 300, 350 and 400. The number of mixture components was varied to 16, 32, 64 and 128.
Optimum number of mixture components for each number oftraining utterances was determined by using the conventional
GMM, as given in both Fig. 2 and Fig. 3. The number of testing utterances was 20.
In order to efficiently perform the training process of LT-

5. Conclusions
This paper presents an acoustic-to-articulatory inversion mapping system based on latent trajectory Gaussian mixture model(LT-GMM). In the proposed LT-GMM method, the consistency
between training and mapping phases is preserved by imposing
a soft constraint between static and dynamic features, where
the inter-frame correlation is taken into account. In the training
phase, the parameters can be conveniently optimized by using
EM algorithm. The experimental results demonstrate that the
proposed LT-GMM method improves the accuracy of the inversion mapping compared to the conventional GMM. For the
future work, we plan to optimize also the mixture component
sequence and incorporate acoustic segment feature consistingof multiple frames of input features.

6. Acknowledgements
This research was supported in part by JSPS KAKENHI Grant
Number 26280060.

956

7. References[18] S. Takamichi, T. Toda, A. W. Black, and S. Nakamura, “Modulation spectrum-constrained trajectory training algorithm for GMMbased voice conversion,” In Proc. ICCASP, pp. 4859–4863, Brisbane, Australia, Apr., 2015.

[1] S. Parthasarathy, J. Schroeter, C. Coker, and M. M. Sondhi, “Articulatory analysis and synthesis of speech,” In Fourth IEEE Region 10 International Conference, pp. 760–764, Bombay, India,
Nov. 1989.

[19] H. Kameoka, “Modeling speech parameter sequences with latent
trajectory hidden Markov model,” In Proc. The 25th IEEE International Workshop on Machine Learning for Signal Processing(MLSP2015), pp. 1–6, Boston, USA, Sept. 2015.

[2] Z. -H. Ling, K. Richmond, J. Yamagishi, and R. -H. Wang, “Integrating articulatory features into HMM-based parametric speechsynthesis,” IEEE Transactions on Audio, Speech, and Language
Processing. Vol. 17, No. 6, pp. 1171–1185, 2009,[20] D. A. Reynolds and R. C. Rose, “Robust text-independent speaker
identification using Gaussian mixture speaker models,” IEEE
Transactions on Speech and Audio Processing, Vol. 3, No. 1,
pp. 72–83, 1995.

[3] P. L. Tobing, T. Toda, G. Neubig, S. Sakti, and S. Nakamura,“Articulatory controllable speech modification based on Gaussian mixture models with direct waveform modification using
spectrum differential,” In Proc. INTERSPEECH, pp. 3350–3354,
Dresden, Germany, Sep. 2015.

[21] T. Toda, A. W. Black, and K. Tokuda, “Voice conversion based on
maximum likelihood estimation of spectral parameter trajectory,”
IEEE Transactions on Audio, Speech and Language Processing,Vol. 15, No. 8, pp. 2222–2235, 2007.

[4] J. Schroeter and M. M. Sondhi, “Speech coding based on physiological models of speech production,” In Advances in Speech Signal Processing, ed. by S. Furui and M. M. Sondhi, Marcel Dekker,
New York, pp. 231–267, 1992.

[22] A. Wrench, “The MOCHA-TIMIT articulatory database,”
http://www.cstr.ed.ac.uk/artic/mocha.html, Queen Margaret University College, 1999.

[5] A. Wrench and K. Richmond, “Continuous speech recognition using articulatory data,” In Proc. ICSLP, Vol. 4, pp. 145–148, Beijing, China, Oct. 2000.

[23] H. Kawahara, I. Masuda-Katsuse, and A. de Cheveigne, “Restructuring speech representation using a pitch-adaptive time frequency
smoothing and an instantaneous-frequency-based F0 extraction:
possible role of a repetitive structure in sounds,” Speech Communication, Vol. 7, No. 3–4, pp. 187–207, 1999.

[6] J. Frankel, K. Richmond, S. King, and P. Taylor, “An automatic
speech recognition system using neural networks and linear dynamic models to recover and model articulatory traces,” In Proc.
ICSLP, Vol. 4, pp. 254–257, Beijing, China, Oct. 2000.

[24] P. R. Amestoy, I. S. Duff, J. Koster, and J. -Y. L’Excellent, “A
fully asynchronous multifrontal solver using distributed dynamicscheduling,” SIAM Journal on Matrix Analysis and Application,
Vol. 23, No. 1, pp. 15–41, 2001.

[7] H. Ackermann and W. Ziegler, “Articulatory deficits in parkinsonian dysarthria: an acoustic analysis,” Journal of Neurology, Neurosurgery & Psychiatry, Vol. 54, No. 12, pp. 1093–1098, 1991.

[25] P. R. Amestoy, A. Guermoche, J. -Y. L’Excellent, and S. Pralet,“Hybrid scheduling for the parallel solution of linear systems,”
Parallel Computing, Vol. 32, No. 2, pp. 136–156, 2006.

[8] H. Ackermann, I. Hertrich, I. Daum, G. Scharf, and S. Spieker,“Kinematic analysis of articulatory movements in central motor
disorders,” Movement Disorders, Vol. 12, No. 6, pp. 1019–1027,
1997.
[9] M. Cohen, D. W. Massaro, “Modeling coarticulation in synthetic
visual speech,” In Models and Techniques in Computer Animation, Springer, 1993, pp. 1019–1027.
[10] I. Steiner, K. Richmond, and S. Ouni, “Using multimodal speech
production data to evaluate articulatory animation for audiovisual
speech synthesis,” In Proc. The 3rd Symposium on Facial Analysis and Animation (FAA2012), No. 2, p. 1, Vienna, Austria, Sept.
2012.
[11] J. Hogden, A. Lofqvist, V. Gracco, I. Zlokarnik, P. Rubin, and
E. Saltzman, “Accurate recovery of articulator positions fromacoustics: new conclusions based on human data,” Journal of the
Acoustical Society of America, Vol. 100, No. 3, pp. 1819–1834,
1996.
[12] S. Suzuki, S. Okadome, T. Honda, “Determination of articulatory
positions from speech acoustics by applying dynamic articulatory
constraints,” In Proc. ICSLP, pp. 2251–2254, Sydney, Australia,
1998.
[13] K. Richmond, K. King, and P. Taylor, “Modelling the uncertainty
in recovering articulation from acoustics,” Computer Speech andLanguage, Vol. 17, No. 2, pp. 153–172, 2003.
[14] S. Hiroya, and M. Honda, “Estimation of articulatory movementsfrom speech acoustics using an HMM-based speech production
model,” IEEE Transactions on Speech and Audio Processing,Vol. 12, No. 2, pp. 175–185, 2004.
[15] T. Toda, A. W. Black, and K. Tokuda, “Statistical mapping between articulatory movements and acoustic spectrum using a
gaussian mixture model,” Speech Communication, Vol. 50, No. 3,
pp. 215–227, 2008.
[16] H. Zen, K. Tokuda, and T. Kitamura, “Reformulating the HMM
as a trajectory model by imposing explicit relationship between
static and dynamic feature vector sequences,” Computer Speechand Language, Vol. 21, No. 1, pp. 760–764, 2007.
[17] T. Toda, and S. Young, “Trajectory training considering global
variance for HMM-based speech synthesis,” In Proc. ICCASP,
pp. 4025–4028, Taipei, Taiwan, Aug, 2009.

957