Complex numbers and Laplace transform
Because of events over the last few years, Java applets no longer have a future on the web for the kind of application used here. Work is Sinusoid Relations. Fourier Series Electrostatic potential in a rectangular cavity. Laplace Transform . MTH, Differential Equations, Laplace Transforms and Fourier Analysis, . Relations and functions, Functions of single variable, their Domain, Range, Introduction to Java: History of Java, Java class Libraries, Introduction to java. A Java applet that performs graphical convolution of continuous-time signals on the screen A Java applet that displays Fourier series approximations and corresponding and produce the corresponding real and imaginary parts for comparison. Drag poles and zeros around the Laplace s-plane and observe changes in.
If the image contains pixels, it is represented by an matrix. The representation in the frequency domain of a digital image is obtained by applying the Discrete Fourier Transform DFT to the matrix representing the image. Spatial Representation of Images Gray scale images were used in this research. In this type of images, intensities of pixels are shades of gray ranging from black to white. This can be viewed clearly in Figure 4. One of the problems confronted is to find the appropriated range of gray values to graphically represent the output from the signal operators, especially if those outputs are negative numbers.
Jahne  gives an approach to solve this problem as follows: Each pixel in images of gray values occupies 8 bits, which causes negative pixels to take large positive values. Grayscale Image Essentially, gray values are regarded in this representation as a deviation from a mean value.
In order to have a better display on the screen, the gray values must be converted again to unsigned values by the inverse point operation": The classification was done according to the functionality of the operators. These operators are used to modify the gray values at specific pixels. They can be applied to correct the image illumination and contrast enhancement: Absolute Value, Clamp, Color Convert.
The basic operations that could be done over images include sum, subtraction, multiplication, and division. The function of these operators is to modify the position of the pixels: The operator cuts a portion of the image given the width and height of the section to be cropped. The operator transforms the image by adding or subtracting pixels. The operator makes a mirror of the image in a vertical or a horizontal direction. These operators are based on the convolution operation of image processing: Sharpening, Blurring, Embossing, and Edge Detection.
These operators are also based on the convolution operation and are mostly used for image enhancement, highlighting, or hiding features of the image: These operators have been implemented especially for those images whose pixels are complex numbers: Software Tools used for Implementation The implementation of all the components of the system was done using the programming language Java.
It was chosen because of the following features : It is object-oriented, which means that the programming of the state and behavior of a system is done by programming the state and behavior of the objects that compound the system.
It is platform-independent, or capable for running on different platforms such as Windows, Linux, or Macintosh. It is the most common programming language for web applications because it is the foundation of many developing frameworks.
Considering that it was assumed that the input data are complex signals, meaning that each sample is a complex number formed by a real number and an imaginary numberan open source library of operations that supports complex numbers was acquired.
This library is called Flanagan's Java Library flanagan. It includes basic arithmetic operations of complex numbers addition, subtraction, multiplication, and divisiontrigonometric operations, and special mathematical functions. Apache Tomcat was used as the servlet container which is used in the "official Reference Implementation for the Java Servlet and JavaServer Pages technologies, providing an environment for Java code to run in cooperation with a web server.
The JFree Chart is a free Java chart library that makes it easy for developers to display professional quality charts in their applications. This library consists of an API that supports a wide range of chart types, a flexible design that is easy to extend, and targets both server-side and client-side applications.
The OOP approach to software engineering is to start by identifying the objects involved in a problem and the messages that those objects should respond to. The program that comes out of it is a collection of objects, each with its own data and its own set of responsibilities. The CSP system has two basic structures of data input.
The one-dimensional 1-D signals of dimension and the two-dimensional 2-D signals of dimension. A 1-D signal is represented mathematically as a sequence of samples like this: Graphically, this sequence can be seen as a 1-D array, or as a vector data structure: A 1-D signal represented as vector A 2-D signal is represented mathematically as an matrix.
Graphically, this can be seen as a 2-D array data structure, as presented in Figure 6. A 2-D signal represented as matrix Making an extension to OOP, both the 1-D and the 2-D signals each represent a class with its own attributes and methods. Figure 7 illustrates the attributes that identify each class. The attribute complexSignal of Complex[ ] type is a vector that contains the complex samples read from the data file.
If the data are just real numbers, the imaginary part of each sample of the complex array is set to be zero. Figure 8 shows the class diagram of the Complex class. For each loaded signal there is a List of all the transformations resulting from the action of the operators over the signal. This list is represented by the attribute transformedList of the Signal class.
The attribute of RenderedImage type represents the image as a grid of pixels. With this representation, it is possible to obtain the values of the pixels as 2-D arrays, with int data type or as a matrix of complex numbers of type Complex[ ][ ]. As in the 1-D case, each image has its corresponding List of images that come from the action of the operators.
Operators are systems able to transform an input signal to produce an output signal. For the CIP, they were classified in unary operators and binary operators. Unary operators take only one signal on the input and produce one signal on the output. Figure 9 shows an example of a unary operator such as the Discrete Fourier Transform, and an example of a binary operator such as addition.
Example of Unary and Binary Operators Such a distinction of the unary and binary operators apply to both 1-D signals and 2-D signals. Figure 10 is the class diagram for both kinds of operators. The set of implemented operators is divided into two groups: Each group is classified as UnaryOperator or BinaryOperator, which in turn, are composed of all the different operators that fit into to each category.
Class Diagrams for Operators So, the relationship of the transformation of the Operator classes on the Signal classes can be seen in Figure Web Application Architecture A web application could be defined as a web system where user input navigation and data input affects the state of the logic of the system. The basic architecture of a web application includes browsers, a network, and a web server.
In  Conallen also states that "a web application uses a web site as the front end to a more typical application. Components of a Web Application The connection between the client and server only exists during a page request. Once the request is complete, the connection is broken. All the activity on the server occurs during the page request .
In this architecture, the client is the requester of services and the server is the provider of such services. Computer Architecture and Organization- John P. Patterson and John L.
Web Engineering introduces a structured methodology utilized in software engineering to Web development projects. The course addresses the concepts, methods, technologies, and techniques of developing Web Sites that collect, organize and expose information resources.
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Difference between Fourier Transform vs Laplace Transform
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- Convolution theorem
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