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Procedia - Social and Behavioral Sciences 102 (2013) 454 - 463

6th International Forum on Engineering Education (IFEE 2012)

Linking Mathematics and Image Processing Through Common

Terminologies

Norhayati Bakri *, Ratnawati Ibrahim, Tuan Salwani Awang @ Salleh, Zalhan Mohd

Universiti Kuala Lumpur Malaysia France Institute,43650 Bandar Baru Bangi, Selangor, Malaysia

Abstract

The mathematics performance of students enrolling in the engineering technology subjects such as robotics, image processing, control systems and others have been degraded at an alarming rate during the recent years. One of the reasons for this scenario is their inability to relate the mathematical knowledge with the technical applications including image processing. The inconsistency of terminologies used in mathematics and technical subjects has been identified as one of the main sources that contribute to this problem. In this paper, the mapping of the terminologies used in mathematics and image processing was done. It is found that there are different terminologies used in both subjects carry the same meaning and also some same terminologies used in both subjects represent different meanings. Thus it is recommended that lecturers teaching both subjects to introduce the variety of terminologies in defining a newly taught concept for teaching and learning of mathematics and image processing subjects.

© 2013TheAuthors.PublishedbyElsevierLtd.

Selectionand/orpeer-reviewunder responsibility of ProfessorDr Mohd. ZaidiOmar, AssociateProfessorDrRuhizanMohammadYasin, DrRoszilahHamid, DrNorngainyMohd.Tawil, AssociateProfessorDrWan Kamal Mujani, Associate Professor Dr Effandi Zakaria.

Keywords: terminologies; mathematics; image processing.

1. Introduction and Background

Engineering technology is a field that requires the application of scientific and mathematics which are interrelated. The applications of these two main core subjects are crucial to help engineering technologist to transform the higher cognitive thinking ideas into reality through related suitable skills. The contents of some engineering technology subjects relied heavily on knowledge learnt in mathematics. However, inadequate mathematical skills present a widespread problem throughout engineering technology undergraduate programs.

* Corresponding author. Tel.: +6-03-89262022 E-mail address: norhayatibakri@mfLumkl.edu.my

1877-0428 © 2013 The Authors. Published by Elsevier Ltd.

Selection and/or peer-review under responsibility of Professor Dr Mohd. Zaidi Omar, Associate Professor Dr Ruhizan Mohammad Yasin, Dr Roszilah Hamid, Dr Norngainy Mohd. Tawil, Associate Professor Dr Wan Kamal Mujani, Associate Professor Dr Effandi Zakaria. doi: 10.1016/j.sbspro.2013.10.761

Well-documented examples of student difficulties are often lacking, and the exact nature of the difficulty is frequently uncertain. In most cases, many students also find that theoretical subjects such as mathematics, physics or programming are unattractive, difficult and useless. Most students have difficulties to relate the needs of these basic subjects to their engineering subjects. This usually de-motivates the students and decreases their study interest.

In fact, the importance of motivation and stimulation of study interest especially on mathematics and engineering subjects has often been recognized in the literature [1-4]. In Gail Rose[2], the authors believed that poor motivation in studying mathematics is one of the key reasons for the students' incomplete skills. The students do not see the real use for the mathematical theory that they have learned in mathematics classes. They also found that their students failed to acquire sufficient mathematics skills for them to further progress in mathematics-oriented engineering topics such as digital image processing. Even after their first year of study or sometimes after their second year they did not manage to grasp the required mathematical skills. Together with poor motivation and lack of interest in the topic, majority of the students are just taking mathematics subjects with minimum effort and therefore obtain only minimum grades or most of the time they failed the subjects.

Moreover, there is often little communication between engineering technology and mathematics department dedicated to or addressing mathematics skills related issues. It is rare to see the lecturers from both departments sit together and have discussion on these interrelate problems. This might due to the fact that engineering technology lecturers usually assume that all basic mathematical knowledge and skills should have been handled by mathematics lecturers. Meanwhile, mathematics lecturers only deliver mathematics lessons based on its syllabus without knowing what, where, how, why or when the topics could be useful for the students to apply in their engineering technology subjects. This represent a typical problem between mathematics and engineering technology lecturers that do not provide any benefits to the improvement in teaching and learning of both subjects. It can also be noticed that the engineering technology department usually assumes that certain concepts are taught in the mathematics courses, but their lecturer are often not familiar with the mathematics curriculum, or the methods utilized including the terminologies used and context of their specific applications.

As a result, students in the university involved in this research were found out to face problems to relate the mathematical knowledge with the engineering technology subjects including in Image Processing. Thus, the issue of a suitable teaching and learning strategy needs to be addressed carefully as it is critical to ensure a meaningful learning in both subjects. One of the possible approaches is the incorporation of knowledge delivery of both subjects, where one lecturer teaching both subjects. This approach may benefit one party but not the mathematics lecturers. Mathematics lecturers in this university argued that mathematics should be teaching in a proper sequence. Furthermore, this method will not teach students mathematics but only a set of rules and skills to a specific technological field [5].

In this study, a framework for a new approach to teach Mathematics and Image Processing were designed to satisfy both subjects. This framework was designed to help both lecturers in developing a new approach of the teaching and learning of these subjects in order to promote a meaningful learning experience. In its broadest definition, e-learning includes the delivery of instruction through electronic media such as the Internet, Intranets, Extranets, satellite broadcasts, audio/video tapes, interactive TV, and CD-ROMs [6]. E-learning offers a channel to facilitate information sharing and communication between educators and students. Educators can manage content materials, prepare assignments, prepare quizzes and tests and also engage in discussions with their students. Students can involve in virtual discussions through forums and chats.

Therefore, the focus of this research is twofold. Firstly, the mapping of terminologies in Mathematics and Image Processing subjects which was thoroughly done will be discussed in this paper. Secondly, the development of the framework for an e-learning for the teaching and learning of both subjects which has been carefully designed by both lecturers will be also elaborated in this paper. Lecturers from both subjects were collaboratively involved to develop a utility approach in order to ensure a meaningful learning experience for both subjects can be achieved.

2. Mapping the Terminologies and Definitions

The important task before any linkage to be established between two subjects is to study and identify the related topics, terminologies or definitions between both of them. The scope of this study was limited to mathematics and image processing subjects which are offered by UniKL MFI. There are three subjects involved, which are Engineering Technology Mathematics 1 (FKB10103), Engineering Technology Mathematics 2 (FKB20203) and Image Processing (FSB33503). These subjects are offered in semester 3, 4 and 5 respectively. FKB10103 and FKB20203 are the subjects owned by the Mathematics Department while FSB33503 belongs to the Industrial Automation Department. These three subjects are among the core subjects in Bachelor of Engineering Technology (Hons) in Industrial Automation and Robotics Technology program offered by UniKL MFI. Both FKB 10103 and FKB20203 serve as prerequisite subjects to provide adequate and necessary mathematical skills to students. FKB 10103 consists of five main topics; which are Linear Algebra, Complex Numbers and Polynomials, Vectors and Vector Applications while FKB20203 covers topics namely Pre-Calculus, Partial Derivatives, Integration of hyperbolic, inverse and inverse hyperbolic functions, Multiple Integral and Differential Equations. On the other hand, Image Processing has the purpose to introduce image processing techniques and technology with its topics covering digital image fundamental, image representation, transformation, enhancement, restoration, segmentation and compression.

As mentioned in earlier section, the focus given in this research was to identify and map the common and related terminologies and definitions in mathematics and image processing. Therefore, the communication between lecturers from different departments was established at the first place. Through detail discussions on main topics, sub-topics, terminologies and application examples in mathematics and image processing, it was found that similarities exist in both subjects. Some topics and sub-topics could have been related easily because they usually represent basic calculations in mathematics and at the same time, represent calculation applications in image processing. This includes the common and related terminologies used in these topics. An example is Matrix operations in mathematics and Image operations in image processing. In our case, the overall comparison and identification of related topics and sub-topics in ETM1, ETM2 and image processing is shown in Figure 1. It is interesting to discover that there are about 45% of ETM1 topics and 20% of ETM2 topics contribute directly to 90% of the basic calculations in image processing's topics. Since these topics are highly interrelated, the terminologies and definitions inside them have also been found to interrelate between each others. In general, this FSB33503 Image Processing can be said as highly dependent on mathematics because almost all of its topics are related to topics such as Linear Algebra in ETM1 and Pre-Calculus in ETM2.

Image Processing

> Linear Algebra

•Solving linear equation •Matrix operation -t,-,", scalar, inverse, multiplication •Matrix Algebra -eigenvectors, eigenvalues, diagonalization '

•M atrix Transformation - p enlargement/reduction, * translation, rotation I

^ 'Inverse Transformation ? ^■Complex Numbers and Polynomials

•Solving nth root of complex equations

•Polynomial: real and complex ^Vectors

•Vectors properties, sum of components

•Parallel and nonparallel lines •Scalar product and triple scalar products •Vector product and triple vector products ^■Vector applications

>lmage Representation Pixel relation, sampling, quantization

Distance, image operation averaging limage Transformation •Translation, rotation, scaling/resizing, cropping limage Enhancement •Filtering, smoothing, sharpening limage Restoration

•De-convolution, image filters, noise reduction limage Segmentation ^ 'Edge detection, thresholding limage Compression

l>Pre-Calculus

j 'Functions, limit, continuity, limit theorem 'Differentiation using first principle method |

>T)TFfe r e ntTaf! on s

■First order partial derivative •2nd order partial derivative > Integration

•Hyperbolic, inverse trigonometric and inverse hyperbolic functions ^Multiple Integral

•Double and triple integral ^Ordinary differential equation •1st order: Variable separable (SOVA), exact product, integrating factor •2nd order: Homogenous and non-homogenous, failure case

Fig. 1. The topics in ETM1, ETM2 and Image Processing subj ects that are identified as having significant relations in terms of terminologies

and definitions.

Grev&caleimage

Fig. 2. Digital image represented as matrix.

In order to study and identify the related topics, terminologies or definitions between mathematics and image processing subjects, understanding the fundamental knowledge of digital image processing is seen as very important because it can provide a platform of common understanding for lecturers from mathematics and

engineering technology department to work on it. For mathematics lecturer, it would be interesting to discover that some mathematical terminologies have been largely used in image processing. In image processing for example, an image has different type of categories: color, grey scale, and black and white images. Image in the latter category is also known as binary image. A grey scale image is what people usually consider as black and white image, but in reality, this type of image consists of many shades of grey. In terms of definition, an image is represented as an array, or a matrix, of square pixels (picture elements) arranged in columns and rows. The elements in these array or matrix also represent the pixel or intensity value of that image. Figure 2 shows a grey scale image with example of matrix image structure at a particular location shown by a circle. Here, a normal matrix structure in mathematics is known as matrix image in image processing and both of them use the same rows and columns terms.

On the other hand, in mathematical convention, the Cartesian coordinate system (Figure 3) has a special location called the origin. It corresponds to the center of the coordinate system. The Cartesian coordinate system has two straight lines; horizontal and vertical, that goes through the origin. Each line, known as an axis extends infinitely in both directions. The horizontal axis known as the x-axis has its positive x value directed to the right, while the vertical axis known as the y-axis has its positive y value directed up.

+ x ->

origin

Fig. 3. Cartesian coordinate system

A geometrical point is defined by two coordinates; namely x coordinate and y coordinate and written between an open and a close parentheses separated by a comma, (x,y). The point (x,y) in the Cartesian coordinate system is located x units horizontally followed by y units vertically from the origin. This kind of convention has been used by the students since their secondary school and the same convention is also applied in their mathematics subject in the university. When the students go to the image processing class, the concept of the origin and axes is still being used but in a slightly different way. In image processing, a digital image follows a somewhat unique coordinate convention [7]. The origin is located in the top left corner of the image. The y-axis goes from the origin to the right and the x-axis goes from the origin to the bottom. Each coordinate corresponds to the location of a quantized sample which is called picture element or pixel.

Origin

y-axis

Fig. 4. coordinate convention used in image processing

The xy plane is partitioned into grids, with a value at the center of each grid representing the picture element or pixel. Hence, the result of sampling and quantization in image processing is represented by a matrix of real numbers with a certain number of rows and columns.

Origin

Column

f(x,y) = pixel value

Fig. 5. xy plane is partitioned into grids in image processing

When the mathematics lecturers discuss a topic that is related to the axis and origin in two dimensional planes, they refer to the standard convention. This standard convention is normally fixed in the students' mind where the horizontal axis is always called x-axis with x positive pointed to the right and the vertical axis is always called y-axis with y positive pointed up. Meanwhile, the word origin refers to the center point. The same concept of the word axis is used in image processing but for the purpose of understanding what image is all

about, we allow some flexibility. The horizontal and vertical axes are still being used but the sense of the positive orientation is changed. The name of the axes is also changed to suit the need of the image itself. The word origin in mathematics corresponds to the center point where positive and negative directions from this point are allowed. In reality, where the image is concerned, only the positive direction is permitted.

After spending several years doing mathematics, the words axis and its sense of orientation and name, as well as the word origin become a fixed picture in their mind. Therefore when entering the image processing classes the students with a fixed idea of axes and origin do not understand the words axes and origin applied in image processing. When the xy plane is partitioned into grids, the result is actually a matrix of real number. While in mathematics the number in each row and column is called the element, in image processing this number is called the pixel value. Not being able to see that the word element and the word pixel value refer to the same thing, students are not able to relate the mathematics learned to image processing. The choice of words during the class can create a certain conflict and ambiguity to the students learning image processing. From a survey conducted in our university to explore the causes of the problems, one important factor observed was the inconsistency used of terminologies in mathematics and image processing subjects. Table 1 show some examples of the terminologies used in mathematics and image processing that could contribute to the stadents' confusion.

Table 1. Different terminologies and their meanings in mathematics and image processing

Term Mathematics Image Processing

origin • Center of the coordinate system • Initial location of the coordinate system represented

represented by (0,0) by (0,0)

axis • Both x and y axes have positive and • Both x and y axes have positive direction only

negative directions • x-axis is directed down from the origin

• x-axis can be directed to the left or right • y-axis is directed to the right of the origin

of the origin

• y-axis can be directed up or down from

the origin

coordinate • (x,y) is used to locate a point • (x,y) is used to locate the pixel element and also

represents the row and column in the digital image

representation

• The row and column are defined by 'i' • The row and column is represented by (x,y)

and 'j' respectively respectively

element • A number located at certain 'i' and 'j' • A value of pixel intensity at location (x,y)

matrix • Transformation matrices • Image transformation

• Enlargement/reduction • Resizing/scaling

• Translation • Translation

• Rotation • Rotation

• Not applicable (redefine rows and • Cropping

columns)

• P(x,y) is function P of two variables x • P(x,y) is the value of pixel intensity at location x row

and y and y column

3. Method of Linking Common Terminologies

Literatures have shown that many engineering or technical educators have proposed changes to the way that mathematics is taught to engineers or technicians. Some of the suggestions are mathematics should be taught by engineering lecturers rather than mathematics lecturers, integrating mathematics and engineering or technical subjects or even a more radical approach is by doing a revamp in engineering or technical curriculum [8]. At the same time, the mathematics community believes that by considering a broader notion of mathematics in developing mathematical thinking will be more advantageous to engineering or technical students [8]. Common needs between these two fields have to be investigated and achieved. Thus, this study aims to propose a new model of teaching mathematics for technical students by integrating the teaching of mathematics and its related

counterpart. A teaching method reform will be proposed from teaching mathematics and technical subjects separately to an integrating teaching approach. The main objective of this approach is to find one teaching technique that can cater both, mathematics and technical subject.

The collaboration of mathematics and technical subjects are likely to be an effective approach of teaching any mathematics subjects [9]. This integration method will give students a great opportunity to experience a more encouraging environment in learning mathematics concepts more deeply. The relevant technical activities and examples will help students to think critically about the related mathematical concepts. Furthermore, mathematics should be taught in conditions that lead to its real life applications.

In this study, there are two sub-topics of Linear Algebra involved, namely Matrices and Matrix Transformation. The applications of matrices in technical and engineering fields are very broad, but this study focuses on the application of Matrices and Matrix Transformation in Image Processing and emphasized on the terminologies and conventions used.

In our university, various teaching tools are provided to ease the lecturers' job to upload their teaching materials such as notes, assignments, and online quizzes. These tasks are done through an e-learning portal provided by the university and known as e-learn. All subjects have their own independent pages or modules in e-learn are accessible by students who have enrolled in any particular subjects as illustrated in Figure 6. For the technical subjects such as robotics, control systems, image processing and others that have calculations, mathematics will be the foundation for them to understand as the basics prior the learning of these subjects. The problems occurred when they did not perform well in technical subjects due to their failure to link these subjects effectively. So, one of the approach that could be undertaken is to enhance the current teaching practice of mathematics and image processing by building a "link" between these two subjects.

Fig. 6. The e-lern system with independent subject's web pages or modules for teaching and learning in UniKL MFI. The three subjects

studied are highlighted

By taking advantage of the usage of our institution's e-learn system, a new approach for the teaching and learning mathematics and image processing by linking common terminologies and conventions in these two subjects is proposed. These subjects will be linked through a newly developed page or module known as Maths-IP Room (Figure 7) which will contain common terminologies and conventions in these two subjects. In this module, students can refer to the terminologies and some related application examples for them to have better understandings in both subjects as well as to relate them. This should allow the students to create directly a "link"

between the theoretical mathematics and applications of image processing. For students who are studying ETM1 and ETM2 subjects, they could easily find the answer on where to apply the mathematical knowledge that they learn without waiting until they enroll in image processing subject later. This could give some early and useful idea for them and they should appreciate more mathematical knowledge and skills learnt. The students who are studying image processing meanwhile could review the mathematical knowledge and skills that they have learnt previously. On the other hand, the development and maintenance of this Maths-IP Room could also increase the level of communication and collaboration among the lecturers from mathematics and engineering technology departments. This could help them to understand better what are the similarities and differences in their subjects so that they could improve them for the benefit of the students.

Fig.7. New approach of e-learn using Maths-IP Room that contain common terminologies and conventions in mathematics and image

processing subjects

4. Conclusion and Future Works

In this research, an alternative concept of integrating teaching and learning in mathematics and image processing subjects which emphasized on common terminologies used in both subjects has been proposed. The students should be able to assimilate this new concept learnt as it provides a web-based module known as Maths-IP Room that link the terminologies and conventions used in image processing and mathematics. It will be more meaningful if the students understand the link between the concepts learned about common terminologies and their related applications in image processing. So, the lecturers themselves must also disseminate the meaning of this concept to the students and encourage them to use it. In this sense, the teaching of mathematics and image processing can be taught simultaneously. With the proposed concept, it is hoped that students can overcome the difficulties in learning image processing and mathematics through these common terminologies background. This could also help them to understand the relations between these two subjects and thus increase their study interest and motivation. Furthermore, lecturers will also be able to teach the contents of image processing to students in more stimulating and motivating environment.

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