Discrete Wavelet Transforms of Haar's Wavelet

INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME 3, ISSUE 9, SEPTEMBER 2014 ISSN 2277-8616 Discrete Wavelet Transforms Of Haar’s Wavelet Bahram Dastourian, Elias Dastourian, Shahram Dastourian, Omid Mahnaie Abstract: Wavelet play an important role not only in the theoretic but also in many kinds of applications, and have been widely applied in signal processing, sampling, coding and communications, filter bank theory, system modeling, and so on. This paper focus on the Haar’s wavelet. We discuss on some command of Haar’s wavelet with its signal by MATLAB programming. The base of this study followed from multiresolution analysis. Keyword: approximation; detail; filter; Haar’s wavelet; MATLAB programming, multiresolution analysis. ———————————————————— 1 INTRODUCTION Definition 4. Let 퐻 be a Hilbert space i) A family 풙 ⊂ 푯 is an orthonormal system if Wavelets are a relatively recent development in applied 풋 풋∈푱 mathematics. Their name itself was coined approximately a decade ago (Morlet, Arens, Fourgeau, and Giard [11], Morlet 1 푖 = 푗, ∀푖, 푗 ∈ 퐽 ∶ 푥 , 푥 = [10], Grossmann, Morlet [3] and Mallat[9]); in recently years 푖 푗 0 푖 ≠ 푗. interest in them has grown at an explosive rate. There are several reasons for their present success. On the one hand, ii) A family 풙 ⊂ 푯 is total if for all 풙 ∈ 푯 the following the concept of wavelets can be viewed as a synthesis of ideas 풋 풋∈푱 which originated during the last twenty or thirty years in implication holds engineering (subband coding), physics (coherent states, renormalization group), and pure mathematics (study of ∀푗 ∈ 퐽 ∶ 푥, 푥푗 = 0 ⇒ 푥 = 0. Calderon-Zygmund operators). As a consequence of these interdisciplinary origins, wavelets appeal to scientists and iii) A total orthonormal system is called orthonormal basis. engineers of many different backgrounds. On the other hand, wavelets are a fairly simple mathematical tool with a great Definition 5. Let 퐻 and 퐻0 be Hilbert spaces and 푇: 퐻 → 퐻0 variety of possible applications. Already they have led to be a bounded linear operator. Then 푇 is an (orthogonal) exciting applications in signal analysis (sound, images) (some projection if 푇 = 푇∗ = 푇2. early references are Kronland-Martinet, Morlet and Grossmann [4], Mallat [6], [8]; more recent references are Definition 6. Let (푋, 훽, 휇) be a measure space. given later) and numerical analysis (fast algorithms for integral i) 푓: 푋 → 푅푛 is called measurable if 푓−1 퐵 ∈ 훽 for all Borel transforms in Beylkin, Coifman, and Rokhlin [1]); many other sets 퐵. The space of measurable functions 푋 → 퐶 is denoted applications are being studied. This wide applicability also by 푀(푋). contributes to the interest they generate. P.J. Wood in [15] developed the wavelet in the Hilbert C*-module case. Also J.A. ii) For 1 ≤ 푝 < ∞, the space 퐿푝 푋, 휇 = 퐿푝 (푋) is defined by Packer and M.A. Rieffel [12] and Z. Liu, X. Mu and G. Wu [5] 푝 푝 studies this concept in L2(Rd ). 퐿 푋 = {푓 ∈ 푀 푋 ∶ 푓 푥 푑휇 푥 < ∞ 푋 endowed with the 퐿푝 -norm 2 PRELIMINARIES 1 푝 푝 Detailed In this section we give some Definition and Theorem 푓 푝 = 푓 푥 푑휇 푥 that we need for the results. For more details about following 푋 concepts and proof of Theorems one can see [2, 7, 14]. In the sequel, we denoted the integer, real and complex numbers by For 풑 = ∞ we let 푍, 푅 and 퐶, respectively. ∞ 퐿 푋 = {푓 ∈ 푀 푋 ∶ 푓 ∞ < ∞} Where 푓 ∞ = inf 훼 ∶ 푓 푥 < 훼 푎. 푒. 퐿푝 푋 is complete with respect to the metric 푑(푓, 푔) = 푓 − 푔 ; this makes 퐿푝 푋 a Banach space. 퐿2 푋 is a Hilbert 푝 space, with scalar product _____________________________ 푓, 푔 = 푓 푥 푔 푥 푑휇 푥 . Bahram Dastourian, Department of Mathematics, 푋 Dehdasht Branch, Islamic Azad University, Dehdasht, Iran. E-mail: [email protected] For more details one can saw [13]. Shahram Dastourian, Elias Dastourian, Omid Definition 7. The Fourier transform of a function 푓 ∈ 퐿1(푅) is Mahnaie, Department of Mathematics, Dehdasht −2휋푖푤푥 1 Branch, Islamic Azad University, Dehdasht, Iran 푓 휔 = ∫ 푓 푥 푒 푑휇(푥), for 휔 ∈ 푅. If 푓 ∈ 퐿 (푅) then 푓 is 247 IJSTR©2014 www.ijstr.org INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME 3, ISSUE 9, SEPTEMBER 2014 ISSN 2277-8616 continuous with 푓 푥 = ∫ 푓 푤 푒2휋푖푤푥 푑휇(푤). The Fourier (inverse) scale or (with some freedom) frequency parameter. transform is the linear operator 퐹: 푓 → 푓 . Thus the inclusion property ―1‖ has a quite natural interpretation: Increasing resolution amounts to adding Definition 8. Let 휔, 푥, 푎 ∈ 푅, with 푎 ≠ 0. We define information. If we denote by 푃푗 the projection onto 푉푗 , we obtain the characterizations 2 푇푥 푓 푦 = 푓 푦 − 푥 , 푇푟푎푛푠푙푎푡푖표푛 표푝푒푟푎푡표푟 "2" ⇔ ∀ 푓 ∈ 퐿 푅 : 푓 − 푃푗 푓 → 0, 푎푠 푗 → ∞ −1/2 −1 2 퐷푎 푓 푦 = 푎 푓 푎 푦 , 퐷푖푙푎푡푖표푛 표푝푒푟푎푡표푟 "3" ⇔ ∀ 푓 ∈ 퐿 푅 : 푃푗 푓 → 0 , 푎푠 푗 → −∞ −2휋푖휔푦 푀휔 푓 푦 = 푒 푓 푦 , 푀표푑푢푙푎푡푖표푛 표푝푒푟푎푡표푟 푃푗 푓 can be interpreted as an approximation to 푓 with resolution 2푗 . Thus ―2‖ implies that this approximation converges to 푓, as All these operators are easily checked to be unitary on 퐿2(푅). resolution increases. Dilation and translation have a simple geometrical meaning. 푇푥 amounts to shifting a graph of a function by 푥, whereas 퐷 is a 푎 Definition 3. Let 푉푗 denote an MRA. A space 푉푗 is called dilation of the graph by a along the 푥-axis, with a dilation 푗 ∈푍 approximation space of scale 푗. Denote by 푊 the orthogonal by 푎 −1/2 along the 푦-axis: If 푠푢푝푝푓 = [0, 1], then 푗 complement of 푉푗 in 푉푗 +1, so that we have the orthogonal 푠푢푝푝(퐷푎 푓) = [0, 푎] and 푠푢푝푝(푇푥 푓) = [푥, 푥 + 1], where decomposition 푊 ⊕ 푉 = 푉 . 푊 is called the detail space of 푠푢푝푝 푔 = {푥 ∶ 푔 푥 ≠ 0}. 푗 푗 푗 +1 푗 scale 푗. Definition 9. If 푓, 푔 are complex-valued functions defined on 푅, their convolution 푓 ∗ 푔 is the function (푓 ∗ 푔) 푥 = 3 MAIN RESULTS ∫ 푓 푥 − 푦 푔 푦 푑휇(푦), provided that the integral exists. As Throughout this section we fix a multiresolution analysis 푉 with scaling function 휑, wavelet 휓 and scaling 푗 푗 ∈푍 Definition ???. Let 푚: 푅 → 퐶 be a bounded measurable coefficients 푎푘 푘∈푍. Given 푓 ∈ 푉0, we can expand it with function. Then the associated multiplication operator respect to two different orthonormal basis: 2 2 푆푚 : 퐿 푅 → 퐿 (푅), 푆푚 (푓)(푥) = 푚(푥)푓(푥) is a bounded linear operator. An operator 푄: 퐿2 푅 → 퐿2(푅) is called (linear) filter if ∞ 푓 = 푐0,푘 푇푘 휑 = 푑−1,푘 퐷2푇푘 휓 + 푐−1,푘 퐷2푇푘 휑 there exists 푚 ∈ 퐿 (푅) such that 푘∈푍 푘∈푍 푘∈푍 For all 푓 ∈ 퐿2(푅), 푄푓 ̂ 휔 = 푓 휔 푚 휔 . Here we use the notations Definition 1. A multiresolution analysis (MRA) is a sequence 푐푗 ,푘 = 푓, 퐷2−푗 푇푘 휑 , "푎푝푝푟표푥푖푚푎푡푖표푛 푐표푒푓푓푖푐푖푒푛푡" 2 of closed subspaces 푉 of 퐿 (푅) with the following list of 푗 푗 ∈푍 푑 = 푓, 퐷 −푗 푇 휓 , "푑푒푡푎푖푙 푐표푒푓푓푖푐푖푒푛푡" properties: 푗 ,푘 2 푘 1. ∀퐣 ∈ 퐙 ∶ 퐕퐣 ⊂ 퐕퐣+ퟏ, In the following we will use 푑푗 = 푑 and 푐푗 = 푐 . 2. 퐕 = 퐋ퟐ(퐑), −푗 ,푘 푘∈푍 −푗 ,푘 푘∈푍 퐣∈퐙 퐣 At the heart of the fast wavelet transform are explicit formulae 3. 퐣∈퐙 퐕퐣 = {ퟎ}, ퟐ for the correspondence 4. ∀퐣 ∈ 퐉, ∀퐟 ∈ 퐋 퐑 ∶ 퐟 ∈ 퐕 ⇔ 퐃 퐣 퐟 ∈ 퐕 , 퐣 ퟐ ퟎ 5. 퐟 ∈ 퐕ퟎ ⇒ ∀퐦 ∈ 퐙 ∶ 퐓퐦퐟 ∈ 퐕ퟎ, 푐0 ↔ (푐1, 푑1) 6. There exists 훗 ∈ 퐕ퟎ such that (퐓퐦훗)퐦∈퐙 is an orthonormal basis of 퐕ퟎ. For this purpose we require one more piece of notation. Definition 10. The up- and downsampling operators ↑ 2 and ↓ 2 In ―6‖ 휑 is called scaling function of 푉 . The properties ―1‖- 푗 푗 ∈푍 are given by ―6‖ are somewhat redundant. They are just listed for a better understanding of the construction. The following discussion ↓ 2푓 푛 = 푓 2푛 , "푑표푤푛" will successively strip down ―1‖-―6‖ to the essential. 0 푛 표푑푑, ↑ 2푓 푛 = "푢푝" Remarks 2. (a) Properties ―4‖ and ―6‖ imply that the scaling 푓 푘 푛 = 2푘. function 휑 uniquely determines the multiresolution analysis: We have We have (↑ 2)^ ∗= (↓ 2), as well as ↓ 2 표 ↑ 2 = 퐼, where 퐼 is the identity operator. In particular, ↑ 2 is an isometry. Theorem 11. The bijection 푐0 ↔ (푐1, 푑1) is given by 푉0 = 푠푝푎푛(푇푘 휑 ∶ 푘 ∈ 푍) 푗 +1 푗 푗+1 푗 by ―6‖ (which incidentally takes care of ―5‖) and 퐴푛푎푙푦푠푖푠: 푐 =↓ 2 푐 ∗ ℓ , 푑 =↓ 2 푐 ∗ 푕 And 푉푗 = 퐷2−푗 푉0 푗 푗 +1 ∗ 푗 +1 ∗ is prescribed by ―4‖. What is missing are criteria for ―1‖-―3‖ and 푆푦푛푡푕푒푠푖푠: 푐 = ↑ 2푐 ∗ 푕 + ↑ 2푑 ∗ ℓ ―6‖. However, note that in the following the focus of attention 1+푘 Here the filters 푕, ℓ are given by 푕 푘 = −1 푎 / 2 and shifts from the spaces 푉푗 to the scaling function 휑. 1+푘 ℓ 푘 = 푎−푘 / 2. (b) The parameter 푗 can be interpreted as resolution or 248 IJSTR©2014 www.ijstr.org INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME 3, ISSUE 9, SEPTEMBER 2014 ISSN 2277-8616 Remark 12. The ―Cascade Algorithm‖: Iteration of the analysis Lo_D is the decomposition low-pass filter. step yields 푐푗 , 푑푗 for 푗 ≥ 1. Hi_D is the decomposition high-pass filter.

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