Exploring Fractal Dimension Analysis as a Technique to Study the

Role of Intensity of Facial Expression and Viewing Angle

Braj Bhushan

and Prabhat Munshi

Department of Humanities & Social Sciences, Indian Institute of Technology Kanpur, India

Nuclear Engineering and Technology, Indian Institute of Technology Kanpur, India

Keywords: Facial Expressions, Geometric Feature Extraction, Fractal Dimension, Root Mean Square Pixel Difference,

Self-similarity.

Abstract: Fractal dimension analysis of the images of facial expressions has been reported earlier by Takehara and

colleagues We have performed a similar exercise for two Indian databases, the Indian dataset of basic

emotions and the Indian Affective Picture Database, to examine the relationship between the geometric

properties of the facial expressions vis-à-vis the intensity of expressions and the viewing angle. It is a first of

its kind in the Indian context. We analyzed the geometric pattern of three regions of the face, computed pixel

difference, and calculated fractal dimensions of the expressions for all the images of these two databases.

Thereafter, we analyzed the obtained outcomes of the geometric analyses and the reported unbiased hit rates

for these databases, respectively. Results suggest that recognition of facial expressions is independent of the

viewing angle. Further, happiness and anger are recognized best irrespective of their intensity followed by

more intense surprise and disgust. The Root Mean Square pixel difference shows identical pattern in the

expressions of happiness and disgust. Fractal dimensions indicate self-similarity among surprise, happiness,

and disgust.

1 INTRODUCTION

A substantial percentage of the information that we

intend to express is conveyed through facial

expressions. The analysis of feature point distance in

facial expressions is the key to quantification of

expression of emotions. However, most of the

behavioural studies have ignored the interaction

between facial morphology and facial expressions

(Hess et al., 2009).

Geometric methods have the potential to map

subtle changes on the face. They rely on the position

of facial features and the corresponding

displacement/ deformation. The higher the distance of

the feature vector of a given facial expression is from

the mean, the easier one finds recognizing it

(Valentine, 1991). Assuming face as geometric shape

implies extraction and analysis of facial components

such as eye brows, eyes, nose, and lips as geometric

features. The geometric feature-based methods use

geometric relationships between facial feature points

and extracts the features.

Fractal dimension is one of the robust but rarely

used techniques for the study of facial expression.

These applications also involve two-dimensional

images which are produced by medical and non-

medical machines. Fractal dimension has been

calculated using a pixel-intensity method in several

such studies (Athe et al., 2013). The same approach

has been adopted to compute fractal dimension from

photographs representing different facial expressions.

Takehara et al. (2002) have argued that recognition of

facial expressions of emotions is based in potential

regularity on the basis of fractal dimension.

Assuming the two-dimensional psychological space

homogenous, Takehara et al. (2002) have argued that

the expression prototype, and thus the geometric

relationship between these prototypes, should be

continuous. Fractal geometry is the best tool to

analyze uneven surface with irregularities of variable

sizes. They have analysed facial expressions in terms

of fractal dimension and found that recognition of

facial expression has fractal properties. However, the

two-dimensional psychological space is not

homogeneous. Some researchers have used fractal

analysis to explain fractal dimension changes in facial

expressions (Takehara et al., 2013). Except Takehara

et al. (2007) we did not find any study examining

Bhushan, B. and Munshi, P.

Exploring Fractal Dimension Analysis as a Technique to Study the Role of Intensity of Facial Expression and Viewing Angle.

DOI: 10.5220/0010124301030110

In Proceedings of the 16th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications (VISIGRAPP 2021) - Volume 2: HUCAPP, pages

103-110

ISBN: 978-989-758-488-6

103

facial expressions on the basis of fractals. The three

studies adopting fractal analysis of facial expressions

(Takehara et al., 2002, 2007, 2013) were conducted

on small sample and limited images. Considering the

merit of fractal mathematics, especially it being

independent of how the images are obtained, we

explored its application in the study of facial

expression vis-à-vis the intensity and viewing angles.

Researchers have mostly used high intensity static

stimuli (Lander & Butcher, 2015) and most of them

have not looked at varying intensity of expression. For

instance, the low, intermediate, and high intensity

expressions were created by Wingenbach et al. (2016)

by extracting consecutive frames. They found a linear

relationship between physical intensity of expression

and accuracy. Bould and Morris (2008) truncated the

unfolding of facial expression of emotions from neutral

to full intensity. Our day-to-day interaction involves

facial expression of emotions of low to intermediate

intensity (Motley & Camden, 1988) and intense

expressions are exceptions (Hess et al., 2009). Thus,

study of varying intensity of facial expression of

emotions can help understand issues related to emotion

recognition in normal as well as clinical population

such as Autism Spectrum Disorder (Harms et al.,

2010). Given this fact, it is important to combine the

thoroughness of geometric method and the

exquisiteness of behavioural method to examine what

makes us decipher facial expressions and recognize the

emotion as well as its intensity. Also, the angle of view

and its effect on the recognition of emotions needs to

be looked at. However, we did not come across any

such study and decided to work on it.

A careful screening of the datasets revealed that

three datasets (Bhushan, 2007; Bould & Morris,

2008; Wingenbach et al., 2016) contains variable

intensity of emotions based on extraction of

consecutive frames. However, all of them have only

front view of the expresser. Two other databases, the

Karolinska Directed Emotional Faces (Lundqvist et

al.,1998) and the Indian Affective Picture Database

(IAPD: Sharma & Bhushan, 2019), contains facial

expressions of the basic emotions taken from five

different angles. We looked at the three existing

Indian datasets reviewed by Sharma and Bhushan

(2019) and taking care of the cultural and ethnic

concerns selected two of the existing Indian datasets.

The choice of the database was based on three

criteria, the database should have facial expressions

of the basic emotions, it should have expressions of

variable intensity and/or facial expressions captured

from different viewing angles, and both of them

should have expressers and respondents from the

same culture. This was done to take care of issues

pertaining to cultural variations in recognition

patterns.

The objective of the study was to examine— (i)

the interplay of physical properties of facial

expression of emotions and their recognition, and (ii)

the relationship between physical intensity of

expression and response accuracy for front and

angular view of expressions. Accordingly, we

hypothesized that (i) the geometric properties of the

face would significantly affect the recognition of

facial expression of emotions, and (ii) the geometric

properties of the face for front view will differ from

the angular view of the facial expression and this will

affect recognition accuracy.

2 MATERIAL AND METHOD

2.1 Stimuli

As stated above, facial expressions of two datasets,

the Indian dataset of basic emotions (Bhushan, 2007)

and IAPD (Sharma & Bhushan, 2019), were used in

this study. The Indian dataset of basic emotions

comprise of 36 coloured images (6 emotions x 6

intensity) of a male expresser (see figure 1). They are

static frames of sequential changes derived from

videos. Depending upon their sequence of

occurrence, these static images are numbered from 1-

6 wherein the numbers also refer to the increasing

order of intensity of the expression (1= lowest

intensity – 6= highest intensity). All the images show

the front view of facial expression. This dataset has

been used to identify the areas of the face scanned

during emotion recognition (Bhushan, 2015).

IAPD contains 140 coloured pictures (7 emotions

X 5 angles X 4 expressers) of four expressers (two

males and two females) from five different angles—

-90 (full left profile), -45 (half left profile), 0

(straight), +45 (half right profile), and +90 (full right

profile) degrees (see figure 1).

The pictures of both the database were of same

size (562x762 pixels). Although, both the databases

have adopted Likert scale for rating the intensity they

are bit different. While the first dataset requires the

participants to recognize the emotion (happy, sad,

fear, anger, surprise, disgust, or neutral) and then rate

its intensity on a 5-point Likert scale (1= minimum,

5= maximum), IAPD has adopted 9-point scale

ranging from 1 (not at all intense) to 9 (completely

intense) for rating the intensity of the emotion.

HUCAPP 2021 - 5th International Conference on Human Computer Interaction Theory and Applications

104

Figure 1: Sample images of facial expressions of both, the

Indian dataset of basic emotions and the Indian Affective

Picture Database.

2.2 Procedure

Behavioural studies have relied on subjective

response of human participants by quantifying them

in terms of accuracy and/or response time. However,

we did not collect behavioural data; rather we

conducted three tier analyses of the images of both

the databases— analyzing geometric pattern in the

area attended on the face, computing root mean

square (RMS) pixel difference, and fractal

dimensions to examine the geometrical deviation

among images depicting facial expression of

emotions. Thereafter, we compared these outcomes

with the behavioural outcomes reported by the

respective authors of these two databases. RMS is

root-mean-square which is bread-and-butter term for

engineers. Scattered data points can be fit into a

straight line and root-mean-square (RMS) is the

standard techniques for making such a fit. Most

engineering application adopt this strategy to

represent seemingly large collection of individual

points by a straight line. The major advantage of such

a representation is that a straight line can be

quantified by only two parameters, the slope and the

intersect. This two-parameter system now represents

the set of several individual points and the RMS value

of zero (goodness-of-fit value of 1) represents a

perfect fit.

3 RESULT

3.1 Identifying Geometric Pattern

To see the interplay of physical properties of facial

expression of emotions and their recognition it was

important to extract embedded geometric pattern in

the photographs depicting facial expressions and see

the interplay of these physical properties and their

recognition accuracy. Based on eye-tracking Bhushan

(2007) has reported that the observers look at only

three regions while trying to understand human facial

expressions— the eyes (eyebrows, eyes, and eyelash),

nose (without the bridge) and lip regions (centre and

two diagonals). Thus, we investigated the geometric

pattern in all the 36 images of the Indian dataset of

basic emotions. The initial parameters were shortest

vertical and horizontal distance among the three facial

regions. Figure 2 illustrates the three regions. The

shortest distance between point ‘P’ and line ‘L’ is the

minimum vertical length and is denoted by ‘d’.

Distance ‘d’ can then be defined as the length of the

line segment that has P (x

, y

) as an endpoint and is

perpendicular to L (ax+by+c=0). It was calculated

using the formula—

𝑑=

𝑎𝑥0+𝑏𝑦0+𝑐

√

𝑎



+𝑏



(1)

The resultant horizontal distance is estimated by

= W

-W

(2)

where W

is the width of ith line and W

is the

width of jth line.

These equations (1 and 2) were worked out for all

six intensity levels of all the six basic emotions using

MATLAB

2014 software. Significant changes were

observed in all the three regions for surprise and

disgust. For surprise the eyebrows, lips centre and

diagonal changes play important role (higher value of

vertical distance) at higher intensity. For disgust the

eyebrows, eyelash, and nose play important role at

low intensity. Expansion (horizontal distance) of lips

centre is more expressive at lower intensity level,

whereas vertical distance of lips centre section is

more sensitive at higher intensity levels of disgust.

Fear had a distinct pattern. Each region of fear

expression becomes more prominent as the intensity

increases. For the remaining three emotions,

happiness, sadness, and anger, the horizontal-vertical

changes were not exhibited in all the three regions. In

happy expressions the expansion of lips centre

increases at higher intensity of happy expression. In

sadness the expansion of both nose and lips centre

become more prominent at higher intensity of

expression. The nose region plays important role at

low intensity of anger, whereas lips centre and

diagonal changes are more prominent (high values of

vertical distance) at high intensity of anger.

The overall comparison of the facial expressions

with respect to changes in the vertical and horizontal

distance points out that as far as horizontal distances

are concerned the findings suggest that the eyebrow,

eyelash, lips centre and diagonal sections of happy

emotion indicated distinguished pattern compared to

other emotions. Disgust facial expression show

Exploring Fractal Dimension Analysis as a Technique to Study the Role of Intensity of Facial Expression and Viewing Angle

105

variance in the eyelash, nose and lips centre section

compared to other emotions. Analysis of the vertical

distance indicated that the eyebrow section is the

most expressive part in surprise compared to the

remaining facial expressions of emotions. Eyelash,

lips centre and diagonal sections of happy emotion

indicate distinguished pattern compared to others

emotions. Disgust facial expression show anomaly in

nose and lips centre section compared to other

emotions.

Figure 2: The three regions of the face.

3.2 Root Mean Square and Average

Value

The root mean square (RMS) pixel difference was

estimated for facial expressions of all six intensities

of all the six emotions. RMS was compute using

following formula—

𝑅𝑜𝑜𝑡 𝑚𝑒𝑎𝑛 𝑠𝑞𝑢𝑎𝑟𝑒



𝑒

)







𝑓



)



−



𝑓



)



)







,

where the image size is M x M implying a total of

pixels and f

is the value of the gray-level for i

pixel. The six emotions are represented by the index j

which takes on integer values between 1 and 6. The

index k also takes on integer values between 1 and 6.

Whenever j=k we have RMS value as zero since the

image is compared by itself. The observer sees the

whole face and identifies the person concerned.

The findings suggest that RMS pixel difference

value of happy and disgust emotions are in increasing

order. Thus, these emotions show identical behaviour.

Similarly, fear and sad emotions show identical

pattern. The diagonal matrix shows all possible RMS

difference for all intensity levels. It is observed from

the matrix that fear facial shows maximum RMS

difference values with all combinations, whereas

happy and sad emotions show comparatively lower

values.

RMS pixel difference shows identical patterns in

the expressions of happiness-disgust and fear-

sadness. It suggests that changes in eyebrows (region

1) affect expression of surprise whereas changes in

nasal area (region 2) and lips (region 3) affect

expression of disgust. Happiness is affected by

changes in eye-lash (region 1) and lips (region 3).

We neither analyzed geometric pattern (of the

area attended on the face) nor computed root mean

square (RMS) pixel difference for IAPD images for

two reasons— (i) all expressions of any given

emotion in IAPD show only one static pose and

therefore doesn’t have variance, and (ii) except one

(straight), all other images of IAPD are from varying

degree which limitsgeometric comparison of select

regions.

3.3 Fractal Dimension

The mathematical formulation of fractal involves

analysis of distribution of these gray-levels and it is

independent of the source that makes the

image/photograph. This study is an attempt to

quantify emotions with the help of photographs taken

of the subject. These photographs have been taken

from different angles since observer in real life

situations may not be present exactly in front of the

subject. The effect of viewing angle thus becomes

important in this work. The geometric property

considered in the fractal theory is the change in pixel-

value (gray-level) from one pixel to another. This

notion of "self-similarity" exists in nature and we see

that facial photographs also exhibit this behaviour.

The facial muscles expand/contract in an interesting

way to produce facial expressions unique to the basic

emotions. This aspect has been captured by fractal

dimension computation. Fractal theory illustrates the

characteristics of images/photographs based on the

similarities of neighboring regions from the gray-

level perspective. The "measurement" aspects is

embedded in the fractal variable normalized-range-

scale (NSR) parameter which incorporates physical

distances in the formulations. These distances are in

pixel units since an image is composed of square

pixels. Each pixel in this study is 0.26 mm x0.26 mm

square shape and it is used in image processing steps

for computing fractal dimension.

3.3.1 The Indian Dataset of Basic Emotions

We analyzed fractal dimensions for all expressions

of the Indian dataset of basic emotions across all six

intensity levels. Table 1 summarizes the obtained

fractal dimensions whereas figure 3 illustrates the

HUCAPP 2021 - 5th International Conference on Human Computer Interaction Theory and Applications

106

change in fractal dimensions with increase in the

intensity of expression.

Table 1: Fractal dimensions for IDBE (all six intensity

levels) and IAPD (all five viewing angles) facial

expressions of all six basic emotions and respective

unbiased hit rates (Hu).

IDBE

Emotions

Intensity

Reported

1 2 3 4 5 6

Happy

2.58 2.59 2.59 2.59 2.59 2.61 0.64

Sad

2.61 2.64 2.64 2.65 2.62 2.63 .13

Fear

2.61 2.61 2.64 2.64 2.64 2.64 0.11

Anger

2.56 2.57 2.58 2.59 2.57 2.56 0.44

Surprise

2.58 2.58 2.58 2.59 2.59 2.60 0.25

Disgust

2.57 2.58 2.58 2.58 2.59 2.61 0.15

IAPD

Emotions

Viewing Angles

Reported

90- 45- 0 45+ 90+

Happy 2.29 2.23 2.42 2.34 2.37 0.88

Sad 2.33 2.16 2.39 2.29 2.37 0.8

Fear 2.3 2.18 2.33 2.35 2.44 0.86

Anger 2.38 2.17 2.42 2.41 2.31 0.92

Surprise 2.3 2.27 2.4 2.22 2.38 0.89

Disgust 2.26 2.2 2.35 2.34 2.41 0.9

Neutral 2.33 2.21 2.46 2.32 2.37 0.74

*Hu for IDBE has been reported separately for

all intensity levels across six emotions. Here average

Hu for the given emotions have been reported.

Figure 3: Fractal dimensions for the facial expressions of

the Indian dataset of basic emotions across six dimensions.

For clear understanding of the results, it is

presented in form of apolynomial fit. This apart can

further be used for detailed mathematical analysis of

emotions. The findings suggest very-very strong (say

3) self-similarity between surprise-happiness and

very strong (say 2) self-similarity between surprise-

disgust. Happiness and disgust also had strong (say

2.5) self-similarity. It indicates self-similarity which

is one aspect of complexity of a two-dimensional

function/image. Other mathematical functions can be

used also but fractal dimension has become very

popular in medical and industrial imaging, hence we

have tried it for understanding facial expressions also.

The surprise-happiness-disgust trio had self-

similarity across all six intensity levels, lowest to

highest. The self-similarity between surprise and

anger reached a non-distinguishable level as the

surprise-happiness-disgust trio between moderately

low (3) and moderately high (4) intensity levels. Fear

and sadness had a weak self-similarity. Sadness and

anger showed a distinct characteristic. They had

different base but shared same pattern (showing a

bump).

3.3.2 The Indian Affective Picture Database

We computed fractal dimensions for all the images of

this database. Table 1 summarizes the obtained fractal

dimensions.

The result suggests that the facial expressions of

IAPD images fall within a band ranging between 2.1

and -2.5 and this band is independent of the type of

emotion. The uncertainty band marginally changes (∆

.4) depending upon the angle of observation. The

manifestation of happy facial expression is isotropic,

i.e., they are angle independent. Further, there is no

sex difference as far as expression of happiness is

concerned. For neutral expressions the facial

expressions of the males do not change as much as the

females. Although IAPD has two male and two

female expressers, one of the male expressers does

not show much change in the expression, especially

neutral, surprise expressions.

4 COMBINING IMAGE

ANALYSIS OUTCOMES AND

REPORTED BEHAVIOURAL

OUTCOMES

Both the databases have reported the proportion of

correctly identified stimuli (hit rate), unbiased hit rate

(Hu), and rating of intensity. Having quantified the

geometric properties of the facial expressions we tried

mapping them together with the reported behavioural

response for the two respective databases to derive

some meaningful conclusion. The behavioural data

outcome of the Indian dataset of basic emotions

Exploring Fractal Dimension Analysis as a Technique to Study the Role of Intensity of Facial Expression and Viewing Angle

107

(Bhushan, 2007) clearly indicates high recognition

accuracy (hit rate) for happiness and disgust across

intensity levels. With increasing intensity recognition

accuracy of surprise and anger also increases. The

unbiased hit rate (Table 1) also suggests better

recognition of happy, anger, surprise, and disgust

expressions. RMS pixel difference show identical

pattern in the expressions of happiness and disgust.

On the other hand, fractal dimensions indicate self-

similarity among surprise, happiness, and disgust. It

also indicates self-similarity between surprise and

anger between moderately low and moderately high

intensity levels. It seems that human beings have a

natural inclination to search appropriate geometric

cues while deciphering facial expressions of

emotions.

The outcome of behavioural data for IAPD

(Sharma & Bhushan, 2019) suggests high recognition

rate (mean hit rate) for happiness (99.91). The other

emotions were also close (anger (96.21), disgust

(95.66), surprise (95.15), sadness (94.18), and fear

(92.91). They found that neutral expression (77.14)

was most difficult to recognize.

A comparison of both the databases on the basis

of hit rates indicates that happiness, disgust, anger,

and surprise are recognized better. When the Huof the

two databases were compared slight variation was

observed. For IAPD anger (0.92), disgust (0.90),

surprise (0.89), and happiness (0.88) had higher Hu.

The Hu for the Indian dataset of basic emotions have

been reported for all six intensity levels of the

respective emotions. Hence, we took the overall value

for each of the basic emotion. Here the Hu of

happiness (0.641) is far higher followed by anger

(0.443) and surprise (0.251). For remaining emotions,

the reported values are low.

The fractal dimension of IAPD suggests that

accuracy of recognition of emotion is not affected by

the viewing angle of the facial expression. The

findings confirm that geometric properties of the face

significantly affect the recognition of facial

expression of emotions. Further, these properties do

not affect recognition of facial expressions when

viewed from different angles.

5 DISCUSSION

This study examined the geometric changes in the

select regions of the face during expression of

emotions vis-à-vis increase in the intensity of

expression and the viewing angle. IAPD study has

reported happiness as ‘easily identifiable emotion’

and neutral to be difficult. Difficulty in distinguishing

fear-surprise and anger-disgust has also been reported

attributing it to similarity in‘configuration of the

facial muscles’. Further, recognition of emotion on

the face is not affected by the viewing angle. The

findings of IDBE study has also endorsed that the

accuracy of recognition depends on the intensity of

surprise, happiness, and disgust. Other studies have

also reported that happiness is very distinct compared

to other basic emotions and hence is distinctly

recognized. They show that happy and surprise

expressions are easily recognized (Calvo et al., 2014).

Recognition of happiness has also been found to be

faster compared to negative emotions (Leppanen &

Hietanen, 2004). Du and Martinez (2011) have

reported higher recognition of happiness and surprise,

poor recognition of anger and sadness and worst

recognition of fear and disgust. If we look at these

expressions in terms of larger face transformations,

surprise involves largest deformation followed by

disgust and fear. However, both disgust and fear are

very poorly recognized. (Calvo & Lundqvist, 2008)

found that fear is difficult to be recognized.

Another significant finding of the study was the

relationship between geometric changes and

recognition of facial expression. We found high

recognition accuracy as well as self-similarity

between surprise, happiness, and disgust. Neth and

Martinez (2009, 2010) have reported distance

changes between facial features in specific emotion

expression. Happiness involves change in the

curvature of mouth and surprise involves opening of

the eyes. These expressions share additional sclera.

The distance between eyebrows and eyes is large and

the face is thinner for surprise. The distance between

eyebrows, eyes, nose, and mouth is shorter for

disgust. Our findings are in consonance with these

results. The distance between eyebrows and mouth

and the width of the face are distinct to sad and anger

expressions. Larger distance has been reported

between eyebrows and mouth (Neth & Martinez,

2009) and a thinner face than usual (Neth & Martinez,

2010) for sadness. The distance between eyebrows

and mouth is much shorter and the face is wide for

anger. Further, the distinction between these two

expressions blurs with reduction in the computational

space between them.

Fearful expression had a distinct pattern but we

did not find self-similarity across intensity levels. The

evolutionary viewpoint has endorsed fair degree of

recognition of fear due to its survival significance but

later researchers have reported poor recognition of

fear in healthy population (Du & Martinez, 2011).

Susskind et al. (2008) have argued that the sensory

channels open in fear and closes in disgust.

HUCAPP 2021 - 5th International Conference on Human Computer Interaction Theory and Applications

108

These findings have several applications. The

understanding of facial expressions has application in

mental health setting where it can help identify

mental state, intensity of pain, deception of

symptoms, subjective experience of treatment/

interventions, automated counselling, and many more

areas. Such findings are also likely to affect human

computer interaction (HCI), interactive video, and

other related areas. Calder et al. (2001) have

classified emotion expression into three categories

and the take away for HCI research. Happiness and

surprise can be detected easily irrespective of the

distance between the expressor and the person

perceiving it. Anger and sadness are reasonably

detected from proximity. Fear and disgust constitute

the third group of emotions for which people are not

very good at recognizing. Although, we also found

the relationship between happiness and surprise, our

findings show little deviation from the findings of

Calder et al. (2001). These findings might be useful

for HCI researchers looking for systems that can at

least reasonably imitate human perceptual ability.

Some researchers suggest variability in the perception

of dynamic expressions in the clinical population

such as Pervasive Developmental Disorder (Uono,

Sato, & Toichi, 2010) and Asperger Syndrome

(Kätsyri et al., 2008). The stimulus used in the present

study has graded intensity level adding to the

dynamic nature of facial expression and thus might be

useful for study of the clinical population as well.

The advantage of the two databases analyzed in

this work is that they contain static stimuli extracted

from dynamic source that represents real life

condition. Thus, together they consist of facial

expression of emotions of all the six basic emotions

of six varying intensities and five different viewing

angles. However, there is an inherent limitation as

well. While IDBE consists of facial expressions of

only one male expresser, IAPD comprise of

expressions from five different viewing angles but not

of variable intensity. Although, the absence of larger

database limits the generalizability of specific

findings but it does establish that RMS and fractal

dimension can be very well applied in behavioural

science studies as well.

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