text:
Book ChapterDOI

# Finding recurrent patterns from continuous sign language sentences for automated extraction of signs

Sunita Nayak
01 Jan 2012-Journal of Machine Learning Research (Springer, Cham)-Vol. 13, Iss: 1, pp 2589-2615

AbstractWe present a probabilistic framework to automatically learn models of recurring signs from multiple sign language video sequences containing the vocabulary of interest. We extract the parts of the signs that are present in most occurrences of the sign in context and are robust to the variations produced by adjacent signs. Each sentence video is first transformed into a multidimensional time series representation, capturing the motion and shape aspects of the sign. Skin color blobs are extracted from frames of color video sequences, and a probabilistic relational distribution is formed for each frame using the contour and edge pixels from the skin blobs. Each sentence is represented as a trajectory in a low dimensional space called the space of relational distributions. Given these time series trajectories, we extract signemes from multiple sentences concurrently using iterated conditional modes (ICM). We show results by learning single signs from a collection of sentences with one common pervading sign, multiple signs from a collection of sentences with more than one common sign, and single signs from a mixed collection of sentences. The extracted signemes demonstrate that our approach is robust to some extent to the variations produced within a sign due to different contexts. We also show results whereby these learned sign models are used for spotting signs in test sequences.

Topics: Sign language (61%), Sign (mathematics) (56%)

### 1. Introduction

• Most of the existing work in sign language assumes that the training signs are already available and often signs used in the training set are the isolated signs with the boundaries chopped off, or manually selected frames from continuous sentences.
• The process is iterated till the parameter values converge to a stable solution.
• The authors also extract single signs from a mixed collection of sentences where there are more than one common sign in context.

### 2. Relational Distributions

• The authors use relational distributions to capture the global and relative configuration of the hands and the face in an image.
• The authors start from some level of segmentation of the object from the scene.
• It captures the global configuration of the low-level primitives.
• Figure 3(c) illustrates how motion is captured using relational distributions.
• Each bin then counts the pairs of edge pixels between which the horizontal and vertical distances each lie in some fixed range that depends on the location of the bin in the histogram.

### 3. Problem Formulation

• Sign language sentences are series of signs.
• Figure 4 illustrates the traces of the first vs. second dimension in the feature space, of three sentences S1, S2 and S3 with only one common sign, R, among them.
• Table 3 defines the notations that will be used in this paper.
• Also note that p(θ) is hard to compute or even sample from because it is computationally expensive to compute the denominator in Equation 2, as it involves the summation over all possible parameter combinations.
• In other words, the authors construct a probability density function of the possible starting points and widths in each sentence, given the estimated starting points and widths of the common pattern in all other sentences, that is, f (θi|θ(i)).

### 3.1 Distance Measure

• The distance function d in the above equations needs to be chosen carefully such that it is not biased towards the shorter subsequences.
• Here, the authors briefly describe how they compute the distance between two substrings using dynamic time warping.
• Let l1 and l2 represent the length of the two substrings and e(i, j) represent the Euclidean distance between the ith data point from the first substring and the jth data point from the second substring.

### 3.2 Parameter Estimation

• Gibbs sampling (Casella and George, 1992) is a Markov Chain Monte Carlo approach (Gilks et al., 1998) that allows us to sample the conditional probability density f (θi|θ(i)) for all the sequences sequentially and then iterate the whole process until convergence.
• ICM has much faster convergence, but it is also known to be heavily dependent on the initialization.
• The values for ai and wi are updated with those that maximize the conditional density f (θi|θ(i)).
• The vertical axis in the probabilities represents the starting locations and the horizontal axis represents the possible widths.
• Note that the probabilities are spread out in the first iteration for each sentence and it slowly converges to a fixed starting location for each of them.

### 3.3 Sampling Starting Points For ICM

• In order to address the local convergence nature of ICM, the authors adopt a uniform random samplingbased approach.
• The value for a0i is obtained by sampling a starting point based on uniform random distribution from the set of all possible starting points in the ith sequence, that is, from the set {1 · · ·(Li−w 0 i +1)}.
• ICM is run using each initial parameter vector generated and the most common solution is considered as the final solution.
• The authors run it the number of times equal to the average number of frames in each sentence from the given set of sentences for extracting the sign.
• Assign most frequently occurring value as the final value for each parameter, also known as comment.

### 4. Experiments And Results

• The authors present visual and quantitative results of their approach for extracting signemes from video sequences representing sentences from American Sign Language.
• The authors first describe the data set used then present the results of the automatic common pattern extraction.

### 4.1 Data Set

• The authors data set consists of 155 American Sign Language (ASL) video sequences organized into 12 groups based on the vocabulary (word that pervades the sentences of the group).
• The breakdown of these ‘pure’ groups and the number of sentences in each are as follows.
• The initial parameter vector for each ICM run was chosen independently using uniform random sampling.
• This data set was used to extract 12 common subsequences when the authors searched for the first most common sign, and 24 common subsequences when they searched for the second most common sign.
• All of the signs were performed by the same signer with plain clothing and background.

### 4.2 Common Pattern Extraction Results

• The authors present the results of their method for extracting common patterns from sign language sentences.
• The authors first present results for extracting the single most common sign and multiple common signs from the ‘pure’ sentence groups, followed by results for the most common patterns from the ‘mixed’ groups.

### 4.2.1 EXTRACTING THE MOST COMMON PATTERN

• The authors perform extraction of the most common patterns from the ‘pure’ sentence groups.
• The authors possess a priori knowledge of the most common word due to the organization of the sentence groups.
• As can be seen, the extracted patterns and the corresponding ground truth patterns are quite similar, except for a few frames at the beginning and end of the some of the patterns.
• Figure 10(b) shows the corresponding scatter plot for the end position of the patterns in the sentences.
• As can be seen most of the points in the scatter plots lie along the diagonal.

### 4.2.2 EXTRACTING MULTIPLE COMMON SIGNS

• In this section the authors present some visual results for the extraction of the two most common signs from the ‘pure’ groups of sentences.
• The authors focused on extracting only two signs because the shortest ASL sentence contained two signs.
• Figure 13 shows the results for the two most common signs extracted from the sentence ‘BAGGAGE THERE NOT MINE THERE’.
• The extracted subsequences correspond to the ASL words ‘BAGGAGE’ and ‘MINE’.
• The word ‘BAGGAGE’ appears in all the 14 sentences of the group, whereas the word ‘MINE’ (or ‘MY’) shows up in 11 sentences coinciding with what was expected.

### 4.2.3 EXTRACTING THE MOST COMMON PATTERNS FROM MIXED SENTENCES

• The authors perform extraction of the most common patterns from the collection of ‘mixed’ sentences as outlined in Section 4.1.
• Figure 15(a) shows the scatter plot of the ground truth start positions vs. the estimated start positions of the pattern extracted from each of the sentences.
• The frame width range for the sign ‘HAVE’ is between 4 and 6 frames with 4 being the minimum width and 6 being the maximum width.
• Combining these width ranges could be done using an average of the two or by selecting the minimum and maximum values between the two.

### 4.3 Sign Localization

• The same process that is used for training sign models is used for sign localization.
• The set of points representing the signeme were matched with the segments of the SoRD points from the test sentences to find the segment with the minimum matching score, which would represent the sign in the test sentence.
• The plot of the Start Offset vs. the End Offset is shown in Figure 16.
• The points for different signs are scattered in the four quadrants depending on the nature of the overlap between the ground truth sign and the retrieved signeme.
• The closer it is to the origin, the better the quality.

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Journal of Machine Learning Research 13 (2012) 2589-2615 Submitted 11/11; Revised 5/12; Published 9/12
Finding Recurrent Patterns from Continuous Sign Language
Sentences for Automated Extraction of Signs
Sunita Nayak SNAYAK@TAAZ.COM
Taaz Inc.
4250 Executive Square, Suite 420
La Jolla, CA 92037 USA
Kester Duncan KKDUNCAN@CSE.USF.EDU
Sudeep Sarkar SARKAR@CSE.USF.EDU
Department of Computer Science & Engineering
University of South Florida
Tampa, FL 33620, USA
Barbara Loeding BARBARA@USF.EDU
Department of Special Education
University of South Florida
Lakeland, FL 33803, USA
Editor: Isabelle Guyon
Abstract
We present a probabilistic framework to automatically learn models of recurring signs from mul-
tiple sign language video sequences containing the vocabulary of interest. We extract the parts of
the signs that are present in most occurrences of the sign in context and are robust to the variations
produced by adjacent signs. Each sentence video is ﬁrst transformed into a multidimensional time
series representation, capturing the motion and shape aspects of the si gn. Skin color blobs are ex-
tracted from frames of color video sequences, and a probabilistic relational distribution is formed
for each frame using the contour and edge pixels from the skin blobs. Each sentence is represented
as a trajectory in a low dimensional space called the space of relational distributions. Given these
time series trajectories, we extract signemes from multiple sentences concurrently using iterated
conditional modes (ICM). We show results by learning single signs from a collection of sentences
with one common pervading sign, multiple signs from a collection of sentences with more than
one common sign, and single signs from a mixed collection of sentences. The extracted signemes
demonstrate that our approach is robust to some extent to the variations produced within a sign due
to different contexts. We also show results whereby these learned sign models are used for spotting
signs in test sequences.
Keywords: pattern extraction, sign language r ecognition, signeme extraction, sign modeling,
iterated conditional modes
1. Introduction
Sign language research in the computer vision community has primarily focused on improving
recognition rates of signs either by improving the motion representation and similarity measures
(Yang et al., 2002; Al-Jarrah and Halawani, 2001; Athitsos et al., 2004; Cui and Weng, 2000; Wang
et al., 2007; Bauer and Hienz, 2000) or by adding linguistic clues during the recognition process
c
2012 Sunita Nayak, Kester Duncan, Sudeep Sarkar and Barbara Loeding.

NAYAK, DUNCAN, SARKAR AND LOEDING
(Bowden et al., 2004; Derpanis et al., 2004). Ong and Ranganath (2005) presented a review of
the automated sign language research and also highlighted one important issue in continuous sign
language recognition. While signing a sentence, there exists transitions of the hands between two
consecutive signs that do not belong to either sign. This is called movement epenthesis (Liddell and
Johnson, 1989). This needs to be dealt with ﬁrst before dealing with any other phonological issues
in sign language (Ong and Ranganath, 2005). Most of the existing work in sign language assumes
that the training signs are already available and often signs used in the training set are the isolated
signs with the boundaries chopped off, or manually selected frames from continuous sentences.
The ability to recognize isolated signs does not guarantee the recognition of signs in continuous
sentences. Unlike isolated signs, a sign in a continuous sentence is strongly affected by its context
in the sentence. Figure 1 shows two sentences ‘I BUY TI CKET WHERE?’ and ‘YOU CAN BUY
THIS FOR HER’ with a common sign ‘BUY’ between them. The frames representing the sign
‘BUY’ and the neighboring signs are marked. The unmarked frames between the signs indicate
the frames corresponding to movement epenthesis. It can be observed that the same sign ‘BUY’ is
preceded and succeeded by movement epenthesis that depends on the end and start of the preceding
and succeeding sign respectively. The movement epenthesis also affects how the sign is signed.
This effect makes the automated extraction, modeling and recognition of signs from continuous
sentences more difﬁcult when compared to just plain gestures, isolated signs, or ﬁnger spelling.
In this paper, we address the problem of automatically extracting the par t of a sign that is most
common in all occurrences of the sign, and hence expected to be robust with respect to the variation
of adjacent signs. These common parts can be used for spotting or recognition of signs in continuous
sign language sentences. They can also be used by sign language experts for teaching or studying
variations between instances of signs in continuous sign language sentences, or in automated sign
language tutoring systems. Furthermore, they can be used even in the process of translating sign
language videos directly to spoken words.
In a related work inspired by the success of the use of phonemes in speech recognition, the
authors sought to extract common parts in different instances of a sign and thus arrive at a phoneme-
analogue for signs (Bauer and Kraiss, 2002). But unlike speech, sign language does not have a
completely deﬁned set of phonemes. Hence, we consider extracting commonalities at the sentence
and sub-sentence level.
A different but a closely related problem is the extraction of common subsequences, also called
motifs, from very long multiple gene sequences in biology (Bailey and Elkan, 1995; Lawrence et al.,
1993; Pevzner and Sze, 2000; Rigoutsos and Floratos, 1998). Lawrence et al. (1993) used a Gibbs
sampling approach based on discrete matches or mismatches of subsequences that were strings of
symbols of gene sequences. Bailey and Elkan (1995) used expectation maximization to ﬁnd com-
mon subsequences in univariate biopolymer sequences. In biology, researchers deal with univariate
discrete sequences, and hence their algorithms are not always directly applicable to other multi-
variate continuous domains in time series like speech or sign language. Some researchers tried to
symbolize a continuous time series into discrete sequences and used existing algorithms from bioin-
formatics. For example, Chiu et al. (2003) symbolized the time series into a sequence of symbols
using local approximations and used random projections to extract common subsequences in noisy
data. Tanaka et al. (2005) extended their work by performing principal component analysis on the
multivariate time series data and projected them onto a single dimension and symbolized the data
into discrete sequences. However, it is not always possible to get all the important information in
2590

FINDING RECURRENT PATTERNS FROM CONTINUOUS SIGN LANGUAGE SENTENCES
(a) Continuous Sentence ‘I BUY TICKET WHERE?’
(b) Continuous Sentence ‘YOU CAN BUY THIS FOR HER’
Figure 1: Movement epenthesis in sign language sentences. Frames corresponding to the common
between marked frames represent movement epenthesis that is, the transition between
signs. Note that the sign itself is also affected by having different signs preceding or
following it.
the ﬁrst principal component alone. Further extending his work, Duchne et al. (2007) ﬁnd recurrent
patterns from multivariate discrete data using time series random projections.
Due to the inherent continuous nature of many time series data like gesture and speech, new
methods were developed that do not require approximating the data to a sequence of discrete sym-
bols. Denton (2005) used a continuous random-walk noise model to cluster similar substrings.
Nayak et al. (2005) and Minnen et al. (2007) use continuous multivariate sequences and dynamic
time warping to ﬁnd distances between the substrings. Oates (2002); Nayak et al. (2005) and Nayak
et al. (2009a) are among the few works in ﬁnding recurrent patterns that address non-uniform sam-
pling of time series. The recurrent pattern extraction approach proposed in this paper is based
2591

NAYAK, DUNCAN, SARKAR AND LOEDING
on multivariate continuous time series, uses dynamic time warping to ﬁnd distances between sub-
strings, and handles length variations of common patterns.
Following the success of Hidden Markov Models (HMMs) in speech recognition, they were
used by sign language researchers (Vogler and Metaxas, 1999; Starner and Pentland, 1997; Bowden
et al., 2004; Bauer and Hienz, 2000; Starner et al., 1998) for representing and recognizing signs.
However, HMMs require a large number of training data and unlike speech, data from native sign-
ers is not as easily available as speech data. Hence, non-HMM-based approaches have been used
(Farhadi et al., 2007; Nayak et al., 2009a; Yang et al., 2010; Buehler et al., 2009; Nayak et al.,
2009b; Oszust and Wysocki, 2010; Han et al., 2009). In this paper, we use a continuous trajectory
representation of signs in a multidimensional space and use dynamic time warping to match sub-
sequences. The relative conﬁguration of the two hands and face in each frame is represented by a
relational distribution (Vega and Sarkar, 2003; Nayak et al., 2005), which in itself is a probability
density function. The motion dynamics of the s igner is captured as changes in the relational distri-
butions. It also allows us to interpolate motion, if required, for data sets with lower frame capture
rates. It should also be noted that, unlike many of the previous works in sign language that perform
tracking of the hands using 3D magnetic trackers or color gloves (Fang et al., 2004; Vogler and
Metaxas, 2001; Wang et al., 2002; Ma et al., 2000; Cooper and Bowden, 2009), our representation
does not require tracking and relies on skin segmentation.
We present a Bayesian framework to extract the common subsequences or signemes from all
the given sentences simultaneously. Figure 2 depicts the overview of our approach. With this
framework, we can extract the ﬁrst most common sign, the second most common sign, the third
most common sign and so on. We represent each sentence as a trajectory in a multi-dimensional
space that implicitly captures the shape and motion in the video. Skin color blobs are extracted
from frames of color video, and a relational distribution is formed for each frame using the edge
pixels in the skin blobs. Each sentence is then represented as a trajectory in a low dimensional space
called the space of relational distributions, which is arrived at by performing principal component
analysis (PCA) on the relational distributions. There are other alternatives to PCA that are possible
and discussed in Nayak et al. (2009b). The other choices do not change the nature of the signeme
ﬁnding approach, they only affect the quality of the features. The starting locations (a
1
,...a
n
) and
widths (w
1
,...w
n
) of the candidate signemes in all the n sentences are together represented by a
parameter vector. The starting locations are initialized with random starting locations, based on
uniform random sampling from each sentence, and the initial width values are randomly selected
from a given range of values. The parameter vector is updated sequentially by sampling the starting
point and width of the possible signeme in each sentence from a joint conditional distribution that is
based on the locations and widths of the target possible signeme in all other sentences. The process
is iterated till the parameter values converge to a stable solution. Monte Carlo approaches like
Gibbs sampling (Robert and Casella, 2004; Gilks et al., 1998; Casella and George, 1992), which
is a special case of the Metropolis-Hastings algorithm (Chib and Greenberg, 1995) can be used for
global optimization while updating the parameter vector by performing importance sampling on the
conditional probability distribution. However, this has a high burn-in period.
In this paper, we adopt a greedy approach based on the us e of iterated conditional modes (ICM)
(Besag, 1986). ICM converges much faster than a Gibbs sampler, but is known to be largely de-
pendent on the initialization. We overcome this limitation by performing ICM a number of times
equal to the average length of the n sentences, with different initializations. The most frequently
occurring solution from all the ICM runs is considered as the ﬁnal solution.
2592

FINDING RECURRENT PATTERNS FROM CONTINUOUS SIGN LANGUAGE SENTENCES
Figure 2: Overview of our approach. Each of the n sentences is represented as a sequence in the
Space of Relational Distributions, and common patterns are extracted using iterated con-
ditional modes (ICM). The parameter set {a
1
,w
1
,...a
n
,w
n
} is initialized using uniform
random sampling and the conditional density corresponding to each sentence is updated
in a sequential manner.
The work in this paper builds on the work of Nayak et al. (2009a) and is different in multiple
respects. We propose a system that is generalized to extract more than one common sign from a
collection of sentences (ﬁrst most common sign, second most common sign and so on), whereas
2593

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Journal ArticleDOI
TL;DR: A simple explanation of how and why the Gibbs sampler works is given and analytically establish its properties in a simple case and insight is provided for more complicated cases.
Abstract: Computer-intensive algorithms, such as the Gibbs sampler, have become increasingly popular statistical tools, both in applied and theoretical work. The properties of such algorithms, however, may sometimes not be obvious. Here we give a simple explanation of how and why the Gibbs sampler works. We analytically establish its properties in a simple case and provide insight for more complicated cases. There are also a number of examples.

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text:
Problems for Chapter 3

Constitutive Models: Relations between Stress and Strain

3.6.  Viscoelasticity

3.6.1.      The uniaxial tensile stress-strain behavior of a viscoelastic material is idealized using the spring-damper systems illustrated in the figure, as discussed in Section 3.6.3.

3.6.1.1.            Derive the differential equations relating stress to strain for each system.

3.6.1.2.            Calculate expressions for the relaxation modulus for the Maxwell material and the 3 parameter model.

3.6.1.3.            Calculate expressions for the creep compliance of all three materials

3.6.1.4.            Calculate expressions for the complex modulus for all three materials.

3.6.1.5.            Calculate expressions for the complex compliance for all three materials.

3.6.2.      The shear modulus of a viscoelastic material can be approximated by a Prony series given by $G\left(t\right)={G}_{\infty }+{G}_{1}{e}^{-t/{t}_{1}}$.

3.6.2.1.            Find the creep shear compliance of the material

3.6.2.2.            Find the complex shear modulus of the material

3.6.2.3.            Find the complex shear compliance of the material

3.6.3.      A uniaxial tensile specimen is made from a viscoelastic material with time independent bulk modulus K, and has a shear modulus that can be approximated by the Prony series $G\left(t\right)={G}_{\infty }+{G}_{1}{e}^{-t/{t}_{1}}$.  The specimen is subjected to step increase in uniaxial stress, so that ${\sigma }_{11}={\sigma }_{0}$ $t>0$ with all other stress components zero.  Find an expression for the history of strain the specimen.  It is easiest to solve this problem by first calculating the creep shear compliance for the material.

3.6.4.      A floor is covered with a pad with thickness h of viscoelastic material, as shown in the figure.  The pad is perfectly bonded to the floor, so that ${\epsilon }_{11}={\epsilon }_{33}=0$.   The pad can be idealized as a viscoelastic solid with time independent bulk modulus K, and has a shear modulus that can be approximated by the Prony series $G\left(t\right)={G}_{\infty }+{G}_{1}{e}^{-t/{t}_{1}}$.  The surface of the pad is subjected to a history of displacement $u=u\left(t\right){e}_{2}$.

3.6.4.1.            Calculate the history of stress induced in the pad by   $u\left(t\right)=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}t<0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}u\left(t\right)={u}_{0}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}t>0$

3.6.4.2.            Calculate the history of stress induced in the pad by $u\left(t\right)=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}t<0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}u\left(t\right)={u}_{0}\mathrm{sin}\omega t\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}t>0\text{\hspace{0.17em}}\text{\hspace{0.17em}}$

3.6.4.3.            Assume that the pad is subjected to a displacement $u\left(t\right)={u}_{0}\mathrm{sin}\omega t\text{\hspace{0.17em}}$ for long enough for the cycles of stress and strain to settle to steady state. Calculate the total energy dissipated per unit area of the pad during a cycle of loading.

 27 0C 35 0C 35 0C 55 0C 65 0C Time (Sec) Modulus (GPa) Time (Sec) Modulus (GPa) Time (Sec) Modulus (GPa) Time (Sec) Modulus GPa Time (Sec) Modulus GPa 9 23.95 13 18.89 17 8.81 15 5.67 17 4.92 17 23.91 23 18.46 31 8.61 27 5.16 30 4.86 29 23.84 41 18.04 55 8.34 48 5.00 55 4.76 52 23.72 73 17.36 98 7.91 85 4.97 97 4.59 93 23.72 131 16.27 174 7.27 152 4.95 174 4.32 166 23.51 233 14.67 310 6.46 271 4.92 309 3.92 295 23.16 415 12.63 552 5.67 482 4.86 550 3.41 525 22.61 738 10.61 982 5.16 857 4.76 979 2.92 933 21.79 1312 9.29 1746 5.00 1524 4.59 1742 2.60 1660 20.73 2334 8.81 3106 4.97 2711 4.32 3097 2.50 2952 19.66 4151 8.61 5523 4.95 4822 3.92 5508 2.50 5250 18.89 7381 8.34 9822 4.92 8574 3.41 8000 2.50 9337 18.46 13126 7.91 17467 4.86 15248 2.92 17500 2.50 16600 18.04

3.6.5.      The table above lists measured relaxation (shear) modulus for a (fictitious) polymer at various temperatures.  The polymer has a glass transition temperature of 30 0C.

3.6.5.1.            Plot a graph of the modulus as a function of time for each temperature, using a log scale for both axes

3.6.5.2.            Hence, show that the data for various temperatures can be collapsed onto a single master-curve by scaling the times in each experiment by a temperature dependent factor $A\left(T,{T}_{1}\right)$, as described in Section 3.6.1.  Plot the master-curve corresponding to relaxation at 27 0C.

3.6.5.3.            Plot a graph of $\mathrm{log}\left(A\left(T,{T}_{1}\right)\right)$ as a function of temperature, and show that the data can be fit by a function of the form $\mathrm{log}\left[A\left(T;{T}_{1}\right)\right]=-{C}_{1}\left(T-{T}_{1}\right)/\left\{{C}_{2}+\left(T-{T}_{1}\right)\right\}$.  Determine the values of ${C}_{1},{C}_{2}$ that best fit the data

3.6.5.4.            Hence, determine the constants ${C}_{1}^{g},{C}_{2}^{g}$ for the material, as discussed in Section 3.6.1

3.6.5.5.            Hence, scale the times in the experimental data to plot the relaxation modulus at the glass transition temperature $G\left(t,{T}_{g}\right)$.

3.6.5.6.            Find a Prony series fit to $G\left(t,{T}_{g}\right)$.  Use four terms in the series, together with an appropriate value for ${G}_{\infty }$

3.6.6.      An instrument with mass m=10kg is mounted a set of rubber pads with combined cross sectional area A=5 $c{m}^{2}$ and height h=3cm as shown in the figure.  The pads are made from polyisobutylene, with properties listed in Section 3.6.6.  The base vibrates harmonically with amplitude ${Y}_{0}$ and (angular) frequency $\omega$, causing the instrument to vibrate (also harmonically) with amplitude ${X}_{0}$.

3.6.6.1.            Find an expression relating ${X}_{0}/{Y}_{0}$ to the harmonic modulus $G\left(\omega \right)$ of the material and m, A, h and $\omega$.  Assume that the pads are all subjected to a uniaxial state of stress.

3.6.6.2.            Find an expression for the harmonic modulus $G\left(\omega \right)$ in terms of the material properties ${G}_{\infty }$ and ${G}_{i},{t}_{i}$.

3.6.6.3.            Hence, plot a graph showing the variation of ${X}_{0}/{Y}_{0}$ as a function of frequency, for temperatures ${0}^{0}C,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{25}^{0}C,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{40}^{0}C$

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