APPLICATION OF CUSP SURFACE ANALYSIS USING WEB

APPLICATION TO VISUAL PERCEPTION

Yasufumi Kume

Department of Mechanical Engineerng, Kinki University, Japan

Keywords: Catastrophe theory, Server and client, Cusp surface analysis, Web application.

Abstract: Cusp surface analysis system is constructed to examine the effectiveness of the program of cusp surface

analysis by web server. According to operating on web server of cusp surface analysis, client can easily use

a cusp surface analysis program without preparing for an analysis program or system environment etc. This

paper examines the relationship of 'kinds of ambiguous figures', 'levels of detail figures ' and ' human visual

perception ' using 315 kinds of fisher's ambiguous figures. Fisher introduced ambiguous man/girl figure. An

array of 315 ambiguous man/girl figures has 21 steps of column from a man's face to girl's shape and has 15

row levels of detail. The experimental data are applied to cusp surface analysis. Summing up this paper, (1)

as the cusp surface analysis can be opened to the public on the web server, the clients input URL he or she

can perform the cusp surface analysis. (2) As the results of the experiments, the cusp catastrophe

phenomenon occurs as kinds of the ambiguous figures and the levels of detail figures in human visual

perception.

1 INTROUCTION

The catastrophe theory is applied to many fields

such as behavioral science, psychology, physics,

biology etc. French mathematician Rene Thom

classifies the elementary catastrophe of seven types

in catastrophe theory (Thom, 1975). The cusp model

with two control factors is frequently applied to

discontinuous phenomenon. Furthermore, Loren

Cobb proposed the cusp surface analysis. A cusp

surface is a statistical response surface model, based

on the cusp model of catastrophe theory

(Cobb,1998). Conventional cusp surface analysis

performed with a stand-alone computer. But, in this

paper, cusp surface analysis system is constructed to

examine the effectiveness of the program by

webserver (Kume, Morita,2006). Figure1 shows

transition of research for cusp model. Figure 2and

Figure 3shows 3D cusp catastrophe model and cusp

surface.

According to operating on web server of cusp

surface analysis, client can easily use a cusp surface

analysis program without preparing for an analysis

program or the system environment etc. This paper

examines the relationship of ‘kinds of ambiguous

figures’, ‘levels of detail figures’ and ‘human visual

perception’ using 315 kinds of ambiguous figures.

Fisher introduced ambiguous man/girl figure, and

Attneave made 8 figures by embedding the figure in

the sequence, and Tim Poston and Ian Stewart made

32 figures. 315 kinds of ambiguous figures by

interpolation of figure of Tim Poston and Ian

Stewart figure are made by Atsuo Murata etc

(Murata, Kume and Hashimoto, 1984). An array of

315 ambiguous man/girl figures has 21 steps of

column from a man’s face to girl’s shape, and has 15

row levels of detail. The experimental data are

applied to cusp surface analysis

2 CATASTROPHE THEORY AND

CUSP SURFACE ANALYSIS

In cusp surface analysis(Cobb,1998), it is assumed

that the variable expressed with X

, ・・・，X

exists in relation to parameter A(X),B(X),C(X). The

formula is changeable into the cusp catastrophe

model is used. Catastrophe theory is proposed by

Rene Thom in French mathematician as

mathematical method described development of

configuration in nature (Thom, 1975). Figure 3 and

Figure 4 show 3D cusp catastrophe model and cusp

surface. In cusp surface analysis, it is assumed that

251

Kume Y. (2009).

APPLICATION OF CUSP SURFACE ANALYSIS USING WEB APPLICATION TO VISUAL PERCEPTION.

In Proceedings of the International Conference on Knowledge Management and Information Sharing, pages 251-257

DOI: 10.5220/0002267602510257

 SciTePress

the independent variable expressed with X

, ・・・，X

exists in relation to parameter

A(X

),B(X),C(X). Then formula is changeable into the

form of the following formula.

() () ()

[]

(

)

[]

0 XCYDXCYXBXA −−−+=

()

XAXAXAAXA ++++= ……

22110

)

()

XBXBXBBXB ++++= ……

22110

()

XCXCXCCXC ++++= ……

22110

,where D is a constant, and X is independent variable

vector, three control factors are scalar values and

predicted value Y is decided, then ν is number of

independent variables. A (X

) is unsymmetrical

factor, B (X

) is bifurcation factor and C (X) is linear

factor. In parameter estimating, most likelihood

method is used. Statistical testing which is satisfied

criterion of catastrophe model is not single. Three

criterions are proposed by Cobb. This prediction

equation is a cubic polynomial in Y which means

that for each values of X there are either one or three

predicted values of Y.

Cobb’s stochastic catastrophe model 2D

3D representation of Cobb’s stochastic

catast ro phe mode l

Thom’s catastrophe qualitative model

Quantitative

3D represe ntation

Opening cusp surfa ce progra m o n web

Sever and client system

Thom.R

1975

Loren．C

1978

Kume.Y

2004

Kume.Y

2006

Figure 1: Transition of research for cusp mode.



Figure 2: Thom’s cusp model.

x St ate vari able

α Normal factor

β Splitting factor

Bifurcation set

Cusp Surface

Figure 3: 3D cusp models.

3 CUSP SURFACE ANALYSIS

SYSETM

The cusp surface analysis system as shown in Figure

4 is a system performing calculation in the cusp

surface analysis on a web server from cooperation of

Linux, Apache, Tomcat, PostgreSQL and Java.

Apache of web server software is installed in the

web server. In addition, Tomcat of the web server

software is carried to treat Java servlet and Java

Server Pages(JSP). Tomcat is done plug-in into

Apache. Tomcat independently plays a role of the

web server. But the reason of the plug-in is the

improvement of the processing performance and the

problem of security etc. If the client has web

browser and Java, client can use the cusp surface

analysis program on the web server. The giving and

receiving of the web server is performed in protocol

called the HTTP (Hyper Text Transfer Protocol).

The client request “the cusp surface analysis

program” as HTTP, and the web server response the

service of “the cusp surface analysis program” as

HTTP. Web browser on the client provides service

for the client as HTML(Hyper Text Markup

Language). Web browser plays a role to change

HTTP gotten from web server into HTML. The cusp

surface analysis system is available that the client

can save data to both database and local directory.

And saved data from both web server and local

directory are also able to open. Internet

Explorer(I.E) is started to start cusp surface analysis

program, and following URL is input.

https://163.51.55.140/cusp2008-DB/cuspmain.html

If the main screen of the cusp surface analysis

program is executed by input URL, the system

performs normally. If the main screen of the cusp

surface analysis program is executed, the system

KMIS 2009 - International Conference on Knowledge Management and Information Sharing

252

JRE

Web browser

C:\Cusp Test\data.txt

Read, Save

Client Container

Request

Response

Red Hat Enterprise

Linux 5

Fedora 8

JRE

Apache

Tomcat

(Servlet

Container)

webapps

Cusp

Surface

Analysis

program

Connector

Apache

Tomcat

(Servlet

Container)

Connector

PostgreSQL

Client(Windows) Web server(Linux) Database(Linux)

Figure 4: Structure of client, Web server and Database server.

Server Certificate

Server Public-key

Server Secret-key

Server Certificate

Server Public-key

make a client key

Client Public-key

Client Secret-key

Encrypt

Decrypt

Encrypt

Client Secret-key

Decrypt

Client Public-key

①

②

③

④

⑤

⑥

⑦

⑧

⑨

Figure 5: Structure of SSL.

performs normally.Cusp surface analysis system

uses SSL (Secure Sockets Layer) of a cryptographic

system as shown in Figure 5 that uses two keys to

encrypt data. The client and the server use the

public-key and secret-key to encrypt and decrypt the

data they send to each other and to secure the safety

of the data.

4 APPLICATION OF CUSP

SURFACE ANALYSIS SYSTEM

TO EXPERIMENT RESULT

This experiment shows the relationship among 21

kinds ambiguous figures which changes from man's

face to woman's shape, 15 levels detail figures, and

visual Figures size used in this experiment is

2.2×4.0cm. In the kind of the ambiguous figures, man's

face is A① and woman's shape is U①. Among 15

APPLICATION OF CUSP SURFACE ANALYSIS USING WEB APPLICATION TO VISUAL PERCEPTION

253

Experiment

1.5m

Man’s face？

Woman’s shape？

22cm

30cm

Visual perception degree（Y）

Y=0.0：Man’s face

Y=0.5: Yes and no

Y=1.0: Woman’s shape

Figure 6: Experimental methods.

1 6 11 16 21

K①

F① P①

U①

Woman’s shape

A①

Man’s face

•

let 21 kinds of ambiguous figures from A to U

support a number from A① sequentially.

Kinds of ambiguous

figures

The size of figure：4.0× 2.2cm

Figure 7: In the case of level ①of detail figures, 21 kinds of ambiguous figures.

Levels of detail

figures

A① A③ A⑦ A⑪ A⑮

Clearness

Indistinct

13 7 1115

•

let levels of detail figures from ① to ⑮

support a number from A ① sequentially.

The size of figure：4.0× 2.2cm

Figure 8: In the case of the kind A of ambiguous figures, ⑮ levels of detail figures.

levels of detail figure, level① is most detail figure

and most indistinct figure is level⑮. Two figures

past on the upper section of a black board so that the

subject can judge easily. Two figures are the highest

probability to be seen by man's face and woman's

shape.

In this experiment as shown in Figure 4, subjects

are shown at random 21 figures per one subject. And

the 315 data are taken from 15 subjects of 20's.

Visual perception change points are set as following:

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254

Case

1 6 4 1 106 8 10 1 211 16 9 1

2 2 4 0 107 17 8 1 212 11 3 0

3 3 4 0 108 9 14 1 213 19 9 1

41380.51097150.5214990

5 2 10 0. 5 110 14 10 0. 5 215 16 3 1

68120111781216550

714411122080217350

8196111314120.5218330

974111410140.5219210

10 1 13 0. 5 115 17 10 1 220 14 5 1

11214111610100.5221430

Table 1: The results of 21 kinds ambiguous figures, 15 levels detail figures, and visual perception (Case1~Case315)

Experimental data (21×15=315).

: Kinds of the ambiguous figures, X

: Levels of detail figures, Y：Visual perception.

96 18 12 1 201 15 7 0 306 8 7 0

97 9 13 1 202 4 9 0 307 10 9 1

98 1 2 0 203 6 9 0 30 8 9 1 1

99 5 8 0.5 204 21 9 0. 5 30 9 17 9 1

100 13 4 1 205 3 1 0 310 17 3 1

101 14 2 1 206 20 7 0.5 31 1 20 11 1

102 1 10 0.5 207 9 5 0 312 13 1 1

103 7 6 0 208 7 11 0 313 2 11 0

104 7 14 0.5 209 1 11 0 31 4 2 9 0

105 2 13 0.5 210 20 1 0.5 315 19 3 1

subject answers Man's face－ (0.0), Yes and no－

(0.5), and subject answers woman's shape－ (1.0).

The subjects are shown 1.5 meter far from figure.

Since back ground colors may influence subject's

judgment, 22×30cm black board is pasted on the

back of each figure. This getting data are used for a

cusp surface analysis.

In analysis, linear regression model’s parameter is

presumed. Linear model’s is estimated by the least

squqre method. The correlation coefficient is

obtained from data. In the result, correlation matrix

is obtained. And square multiple correlation

coefficient is obtained. Log liklihoooood of linear

regression model can be obtained from linear square

multiple correlation coefficien .This is the initial

values used by the maximum likelihood method

when it begins its iterative search for the best f itting

coefficient for the catastrophe model.

5 ANALYTICAL RESULT

This is the initial values used by the maximum

likelihood method when it begins its iterative

search for the best fitting coefficients for the

catastrophe model. The maximum of log likelihood

can be obtained by the Newton-Raphson method .

In the result of estimating parameter from data of

experiment results, it is shown in Table 2 as

follows.

Table 2: Standard partial regression coefficient.

Var

Const -0.11 7.113 0.097 6.591

1 0.673 -0.753 -0.014

2 -0.105 -3.006 -0.018

…

APPLICATION OF CUSP SURFACE ANALYSIS USING WEB APPLICATION TO VISUAL PERCEPTION

255

: Kinds of the ambiguous figures, X

: Levels of detail figures, Y：Visual perception

Figure 9: Visual perception by kinds of ambiguous figures in the case of the level ① of detail figures. And the probability

density function 3D figure according to kinds of ambiguous figures and visual perception in the case of the level ① of

detail figures.

Figure 10: Visual perception by kinds of ambiguous figures in the case of the level ⑧ of detail figures. And the probability

density function 3D figure according to kinds of ambiguous figures and visual perception in the case of the level ⑧ of

detail figures.

Figure 11: Visual perception by kind of the ambiguous figure in the case of the level  of detail figures. And the probability

density function 3D figure according to kinds of ambiguous figures and visual perception in the case of the level of detail.

’

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256

() ()

(

)

[]

(

)

[]

591.60 XCYXCYXBXA −−−+=

()

018.0014.0097.0

006.3753.0113.7

105.0673.011.0

XXXC

XXXB

XXXA

−−=

−+−=

In figure 11 visual perception dramatically changes

A→B→C→D from man's face to woman's shape.

Thus, when the level of detail figures is ⑮, visual

perception is suddenly changed at near Q⑮ (X

=17)

of ambiguous figures. The cusp catastrophe

phenomenon occurs as kinds of ambiguous figures

and levels of detail figures in human visual

perception.

The figure of equal probability is Fisher's figure

of K①. But the experiment test result shows that the

ambiguous figure to have equal probability exist

between J① and K① in this case.

Figure 12: Probability density function of ambiguous

figure J in level① of detail figures. Figure 10 and 11 show

the probability density function that the variety of

ambiguous figures at the level ① of detail figures.

Figure 13: Probability density function of ambiguous

figure K in level ① of detail figures.

6 CONCLUSIONS

(1) As the cusp surface analysis can be opened to the

public on the web server, the clients input URL he or

she can perform the cusp surface analysis.

(2) It is clarified that the data in local PC and

database server are loaded and saved by cusp surface

analysis program. However, when the client

transfers data of 73 rows or more from database

server to local PC, the cusp surface analysis system

cannot be executed normally. Also, when the data

saved into the database server are 365 rows or more,

‘S.Read’ button cannot be executed.

(3) The data transmission of the cusp surface

analysis program is encrypted for security by SSL

and communicates with web server.

(4) As the results of the experiments, the cusp

catastrophe phenomenon occurs as kinds of the

ambiguous figures and the levels of detail figures in

human visual perception.

REFERENCES

Cobb, L., 1998. An Introduction to Cusp Surface Analysis.

http://www.aetheling.com/models/cusp/Intro.htm

Kume, Y., Morita,Y., 2006.Construction of Web

Application for Cusp Surface Analysis, J. school Sci.

Eng. Kinki Univ.42 , pp.45-51(in Japanese).

Murata, A., Kume, Y. and Hashimoto,F., 1984. Cusp

Catastrophe Phenomenon in Visual Organic Functions,

Japan Industrial Management Association ,Vol. 35,

No.3, pp.150-155(in Japan).

Thom, R., 1975. Structural stability and morphogenesis,

New York:Benjamin-Addison Wesley.

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