Fractional Fourier Transform and Scaling Problem in Signals and Images

Abstract: Context: We identify a material or thing that can be seen and touched in the world as having structures at both coarser and finer levels of scale. Scaling problem presents in a branch of science concerned with the description, prediction understanding of natural phenomena and visual arts. A moon, for instance, may appear as having a roughly round shape is much larger than stars when seen from the earth. In the closer look, the moon is much smaller than the stars. The fact that objects in the world appear in different ways depending upon the scale of observation has important implications when analyzing measured data, such as images, with automatic methods [1]. The type of information we are seeking from a one-dimensional signal or two-dimensional image is only possible when we have the right amount of scale for the structure of an image or signal data. In many modern applications, the right scale need not be obvious at all, and we all need a complete mathematical analysis on this scaling problem. This thesis is shown how a mathematical theory is formulated when data or signal is describing at different scales. Objectives: The subtle patterns deforming in data that can foretell of a scaling problem? The main objectives of this thesis are to address the dynamic scaling pattern problem in computers and study the different methods, described in the latest issue of Science, are designed to identify the patterns in data. Method: The research methodology used in this thesis is the Fractional Fourier Transform. To recognize the pattern for a different level of scale to one or many components, we take the position and size of the object and perform the transform operation in any transform angle and deform the component by changing to another angle which influences the frequency, phase, and magnitude.  Results: We show that manipulation of Fractional Fourier transform can be used as a pattern recognition system. The introduced model has the flexibility to encode patterns to both time and frequency domain. We present a detailed structure of a dynamic pattern scaling problem. Furthermore, we show successful recognition results even though one or many components deformed to different levels using one-dimensional and two-dimensional patterns. Conclusions: The proposed algorithm FrFT has shown some advantages over traditional FFT due to its competitive performance in studying the pattern changes. This research work investigated that simulating the dynamic pattern scaling problem using FrFT. The Fractional Fourier transform does not do the scaling. Manipulating the Fractional Fourier transform can be helpful in perceiving the pattern changes. We cannot control the deformation but changing the parameters allow us to see what is happening in time and frequency domain.