Implementation Of 2D DFT In MATLAB Image - Sample Solution

Question: The purpose of this project is to gain an understanding of the 2D discrete Fourier transform by applying it to the different images. 1. Create a vertical, horizontal triangle binary images. Check Fourier transforms properties Display the result 2. Take the forward Fourier transform and display the result. Use the fast Fourier transform operator. 3. Extract the center horizontal, vertical lines of the DFT. Determine the magnitude and phase components and plot the results. Answer: Introduction In this project, the 2D discrete Fourier transform has been studied and implemented, without using inbuilt functions. And their properties has been experimentally verified. This have been done in the software MATLAB, by the means of Image Processing. Problem Statement The purpose of this project is to gain an understanding of the 2D discrete Fourier transform by applying it to the different images. 1. Create a vertical, horizontal triangle binary images. Check Fourier transforms properties. Display the result. 2. Take the forward Fourier transform and display the result. Use the fast Fourier transform operator. 3. Extract the centre horizontal, vertical lines of the DFT. Determine the magnitude and phase components and plot the results. Print the DFT values of the extracted line. Programming in MATLAB All codes have been developed in MATLAB, which are attached separately with this document. The codes have been implemented to calculate the discrete Fourier transform of a two dimensional matrix. A two dimensional matrix if formed when a binary image is opened in the MATLAB. All the image processing techniques have been applied on the binary image, which is created in a photo-editor software, in the form of a vertical and a horizontal triangle. But to make sure that images are binary, the function im2bw has been used in MATLAB, at the start of every conversion. The properties are implemented by calling the custom made 2D DFT function from every program. The custom made DFT function has also been checked against the fft2 function of MATLAB, which is inbuilt function of MATLAB to implement 2 dimensional DFT. All the output generated has been popped out in form of an image using imshow function. The magnitude and phase of the DFT has also been plotted. Result and Conclusion 2D DFT has been successfully implemented and check against the MATLAB inbuilt function fft2. All the properties has been verified satisfactorily. All though the speed of our custom made function is slow, but it calculated the DFT exactly like the inbuilt function. References Oppenheim, Alan V.; Schafer, R. W.; and Buck, J. R. (1999). Discrete-time signal processing. Upper Saddle River, N.J.: Prentice Hall. Smith, Steven W. (1999). "Chapter 8: The Discrete Fourier Transform". The Scientist and Engineer's Guide to Digital Signal Processing (Second ed.). San Diego, Calif.: California Technical Publishing. Tim Morris (2004). Computer Vision and Image Processing. Palgrave Macmillan. Milan Sonka, Vaclav Hlavac and Roger Boyle (1999). Image Processing, Analysis, and Machine Vision. PWS Publishing.