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1 Image Reconstruction Using Fast Inverse Fourier Transform (IFFT) AHMAD FARIS BIN HAJI MOHD HASNAN Faculty of Electrical Engineering Universiti Teknologi MARA Malaysia 40450 Shah Alam, Selangor, Malaysia [email protected] Abstract ²This paper present the image reconstruction using Fast Fourier Transform (FFT) and Inverse Fast Fourier Transform (IFFT) technique. Two types of image are considered which are gray scales and binary images. The original images are being reference when reconstructing the original image and image with noise. The original images will be injected with Gaussian, random and salt & pepper noises using MATLAB toolbox. The mean square error will be calculated includes the peak signal to noise ratio (PSNR) for each type of images. Then, the PSNR for each type of images will be observed in terms of the effectiveness of IFFT as a reconstruction algorithm. The image reconstruction results have shown that the grayscale image has the slightly better quality compared to binary image when noiseless image being reconstructed using FFT. Keywords-componen ; FFT, IFFT, DFT, PSNR, Gaussian noise, random noise, salt & pepper noise I. I NTRODUCTION Digital image can be interpreted as a representation of a two-dimensional image using ones and zeros (binary) and it is depends on whether or not the image resolution is fixed. In photography and computing, a grayscale digital image is an image in which the value of each pixel is a single sample, that is, it carries only intensity information. Images of this sort, also known as black-and-white, are composed exclusively of shades of gray, varying from black at the weakest intensity to white at the strongest.[1] .The pixel values of this image runs from 0 to 255. A binary image is a digital image that has only two possible values for each pixel. [1] Typically the two colours used for a binary image are black and white though any two colours can be used [1]. The colour used for the object(s) in the image is the foreground colour while the rest of the image is the background colour [1]. Binary images are also called bi- level or two-level. This means that each pixel is stored as a single bit (0 or 1). For this project, it¶s describe about reconstruction of image which means, the attempts to retrieve the lost and obscured from imaging process itself. It is also can be interpreted as the problem of estimating a function from its Fourier-transform values. The Fourier Transform is an important image processing tool which is used to decompose an image into its sine and cosine components [2]. The output of the transformation represents the image in the Fourier or frequency domain, while the input image is the spatial domain equivalent [3]. Fourier analysis can be used to remove noise from a signal or image. Interpretation of the complex Fourier Transform is not always straightforward. Convolution and deconvolution are ³simple´ in Fourier transform space to restore or enhance images. Comparison between DFT and FFT Computation Speeds is made by Parul Goyal. From results it is evident that for a couple of samples of various lengths the time taken by normal DFT is higher than the time taken by FFT. Thus the speed of FFT is beneficial when large calculations are required [6]. In this paper the grayscale and binary images is perturbed with three types of noise: Gaussian , random, salt & pepper. The PNSR of the images is calculated from the mean square error (MSE) for binary (30x30) and grayscale (64x64) images respectively. The rest of this paper is organized as follows: Section II & III present a mathematical background of Fourier transform and the reconstruction algorithm, and mathematical background of error computation respectively. In section IV, the results are presented. Finally, the recommendation and conclusion being illustrated in section V. II. FAST FOURIER TRANSFORM FFTs were first discussed by Cooley and Tukey (1965), although Gauss had actually described the critical factorization step as early as 1805 (Bergland 1969, Strang 1993)[4].Fast Fourier Transform (FFT) is an efficient implementation of Discrete Fourier Transform (DFT) and is used, apart from other fields, in digital image processing. Fast Fourier Transform is applied to convert an image from the image (spatial) domain to the frequency domain. If the input signal is an im age then the number of frequencies in the frequency domain is equal to the number of pixels in the image or spatial domain. The inverse transform re-transforms the frequencies to the image in the spatial domain.
