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Wavelet Analysis Applied in Image Denoising Using MATLAB 1 Brikena Xhaja, (Phd Student) 1Ligor Nikolla, (Prof

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Wavelet Analysis Applied in Image Denoising Using MATLAB 1 Brikena Xhaja, (Phd Student) 1Ligor Nikolla, (Prof Journal of Multidisciplinary Engineering Science and Technology (JMEST) ISSN: 3159-0040 Vol. 2 Issue 6, June - 2015 Wavelet Analysis Applied In Image Denoising Using MATLAB 1 Brikena Xhaja, (PhD Student) 1Ligor Nikolla, (Prof. As.) Department of Mathematics Faculty of Mathematics’ and Physics’ Department of Mathematics Engineering, Polytechnic University of Tirana, Faculty of Mathematics’ and Physics’ Albania Engineering, Polytechnic University of Tirana, [email protected] Albania [email protected] Abstract—Wavelet analysis has drawn a great could superpose sines and cosines to represent other attention from mathematical sciences in various functions. However, in wavelet analysis, the scale that disciplines. The subject of wavelet is to create a we use to look at data plays a special role. Wavelet common link between mathematicians, physicists, algorithms process data at different scales or and electrical engineers. Wavelets are resolutions. If we look at a signal with a large mathematical functions that cut up data into "window," we would notice gross features. Similarly, if different frequency components, and then study we look at a signal with a small "window," we would each component with a resolution matched to its notice small features. More clearly, we can say that scale. They have advantages over traditional the result in wavelet analysis is to see both the forest Fourier methods in analyzing physical situations and the trees. This makes wavelets interesting and where the signal contains discontinuities and useful. For many decades, scientists have wanted sharp spikes. Wavelets were developed more appropriate functions than the sines and cosines independently in the fields of mathematics, which comprise the bases of Fourier analysis, to quantum physics, electrical engineering, and approximate choppy signals. By their definition, these seismic geology. Interchanges between these functions are non-local (and stretch out to infinity). fields have led to many new wavelet applications They therefore do a very poor job in approximating such as image compression, turbulence, human sharp spikes. But with wavelet analysis, we can use vision, radar, and earthquake prediction. This approximating functions that are contained neatly in paper introduces wavelets to the interested finite domains. Wavelets are well-suited for technical person outside of the digital signal approximating data with sharp discontinuities. Other processing field. It is described the history of applied fields that are making use of wavelets include wavelets beginning with Fourier, compare wavelet astronomy, acoustics, nuclear engineering, sub-band transforms with Fourier transforms, state coding, signal and image processing, properties and other special aspects of wavelets, neurophysiology, music, magnetic resonance imaging, and finish with some interesting applications such speech discrimination, optics, fractals, turbulence, as image compression, musical tones, and de- earthquake-prediction, radar, human vision, and pure noising noisy data. mathematics applications such as solving partial differential equations. Keywords—wavelets; signal processing; orthogonal basis functions; wavelet applications; II. FOURIER ANALYSIS denoising; A. Fourier transforms I. INTRODUCTION Fourier's representation of functions as a The wavelet transform is an expansion that superposition of sines and cosines has become decomposes a given signal in a basis of orthogonal ubiquitous for both the analytic and numerical solution functions. In this sense, we can set a complete of differential equations and for the analysis and analogy with the Fourier Transform. While the Fourier treatment of communication signals. Fourier and Transform uses periodic, smooth and unlimited basis wavelet analysis have some very strong links. functions (i.e., sines and cosines), the wavelet transform uses non-periodic, non-smooth and finite support basis functions, allowing a much more The Fourier transform's utility lies in its ability to meaningful representation through multi-resolution analyze a signal in the time domain for its frequency analysis. content. The transform works by first translating a Wavelets are functions that satisfy certain function in the time domain into a function in the mathematical requirements and are used in frequency domain. The signal can then be analyzed representing data or other functions. Approximation for its frequency content because the Fourier using superposition of functions has existed since the coefficients of the transformed function represent the early 1800's, when Joseph Fourier discovered that he contribution of each sine and cosine function at each www.jmest.org JMESTN42350803 1389 Journal of Multidisciplinary Engineering Science and Technology (JMEST) ISSN: 3159-0040 Vol. 2 Issue 6, June - 2015 frequency. An inverse Fourier transform does just what you'd expect; transform data from the frequency 1 t u domain into the time domain. Wx(u, s) x, x(t) dt (4) u,s The Fourier transform decomposes a signal in s s complex exponential functions at different frequencies. The equations used in the decomposition By looking at the equation above we can conclude and reconstruction part will be given below: that the Wavelet Transform can be seen as a convolution between the signal to be analyzed and the X()() x t ejt dt (1) 1 t reverse function, s (t) derived from s s 1 the Mother Wavelet. x(t) X ()e jt d (2) 2 III. COMPARISON OF WAVELET TRANSFORM WITH Where, t stands for time, 2f for frequency, x FOURIER TRANSFORM denotes the signal in the time domain and X denotes The fast Fourier transform (FFT) and the discrete the signal in the frequency domain (also known as the wavelet transform (DWT) are both linear operations spectrum of the original signal). that generate a data structure that contains Projecting the signal on complex exponentials leads to log 2 n good frequency analysis, but no time localization. The segments of various lengths, usually filling and poor time localization is the main disadvantage of the transforming it into a different data vector of length 2 n Fourier transform, making it not suitable for all kind of . applications. B. Wavelet Transform The mathematical properties of the matrices involved in the transforms are similar as well. The inverse Based in the limitations of the Fourier Transform (poor transform matrix for both the FFT and the DWT is the time localization) Grossman and Morlet gave in 1984 transpose of the original. As a result, both transforms the formulation of the Continuous Wavelet Transform. can be viewed as a rotation in function space to a Unlike, the Fourier transform that decomposes the different domain. For the FFT, this new domain signal into a basis of complex exponentials, the contains basis functions that are sines and cosines. Wavelet Transform decomposes the signal over a set For the wavelet transform, this new domain contains of dilated and translated wavelets [1]. more complicated basis functions called wavelets, mother wavelets, or analyzing wavelets. This difference confers to the wavelet transform the Both transforms have another similarity. The basic advantage of performing a multiresolution analysis, functions are localized in frequency, making meaning that it processes different frequencies in a mathematical tools such as power spectra (how much different way. By using this technique, the time power is contained in a frequency interval) and resolution is increased when we analyze a high scalegrams useful at picking out frequencies and frequency portion of the signal, and the frequency calculating power distributions. localization is increased when analyzing a low- frequency part of the same signal[2][3]. This type of analysis is suitable for signals that have both low- The most interesting dissimilarity between these two frequency components with long time duration and kinds of transforms is that individual wavelet functions high-frequency components with short time duration, are localized in space. Fourier sine and cosine which is the case of most signals. If we consider a functions are not. This localization feature, along with 2 wavelets' localization of frequency, makes many function (signal) x L (R) and for analysis we use functions and operators using wavelets "sparse" when the mother wavelet (t) , with its scaled and transformed into the wavelet domain. This translated versions of sparseness, in turn, results in a number of useful applications such as data compression, detecting features in images, and removing noise from time series. 1 tu us, ()t , (3) s s IV. APPLICATION IN THE IMAGE PROCESSING Now, we can write the wavelet transform of x(t) at In diverse fields from planetary science to molecular spectroscopy, scientists are faced with the problem of time u and scale s as: recovering a true signal from incomplete, indirect or noisy data. Wavelets can help to solve this problem, through a technique called wavelet shrinkage and www.jmest.org JMESTN42350803 1390 Journal of Multidisciplinary Engineering Science and Technology (JMEST) ISSN: 3159-0040 Vol. 2 Issue 6, June - 2015 thresholding methods that David Donoho has worked >> [m,n,k]=size(A) on for several years [5]. m = 2776 n = 1971 The technique works in the following way. When you k = 3 decompose a data set using wavelets, you use filters subplot(131); imagesc(A(:,:,1)); title('R'); that act as averaging filters and others that produce subplot(132); imagesc(A(:,:,2)); title('G'); details [6]. Some of
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