MATLAB小波分析與圖像壓縮的文獻翻譯

摘要小波分析提供了一種強大的和非常靈活的工具集，來處理在科學(xué)和工程方面基本的問題，比如音頻消噪、信號壓縮、目標(biāo)探測和指紋壓縮、圖像去噪、圖像增強、圖像識別、診斷心臟病和語音識別等等。在這里,我們要集中精力解決應(yīng)用程序領(lǐng)域的小波圖像壓縮,以觀察如何實現(xiàn)小波變換的應(yīng)用到一個圖像壓縮的過程中,如何利用數(shù)學(xué)方面的小波影響壓縮的過程和結(jié)果。小波圖像壓縮的小波進行各種已知與不同的數(shù)學(xué)性質(zhì)。我們研究的見解如何實現(xiàn)小波在數(shù)學(xué)的方式來適應(yīng)圖像壓縮的工程模型。1介紹小波函數(shù),允許數(shù)據(jù)分析的信號或圖像,根據(jù)鱗片或決議。處理的信號,通過小波算法在factworks一樣的人類的眼睛并;或數(shù)碼相機處理視覺尺度的決議,以及中間的細(xì)節(jié)。但同樣的原則也抓住了手機信號,甚至數(shù)字化彩色圖像用于醫(yī)藥。小波是真正的使用在這些領(lǐng)域,例如在逼近數(shù)據(jù)與劇烈波動如波濤洶涌的信號,或者圖片,大量的邊緣。小波也許一章函數(shù)論,我們證明了算法,結(jié)果是關(guān)鍵的處理數(shù)字,或更準(zhǔn)確的信息的數(shù)字化、信號時間序列,電影,顏色,圖片等等。因此,應(yīng)用小波變換的想法包括大的部分對信號和圖像、數(shù)據(jù)壓縮、指紋編碼,和其他很多領(lǐng)域的科學(xué)和工程。本論文重點對彩色圖像的處理與使用定制的設(shè)計小波算法,和數(shù)學(xué)閾值過濾器。盡管有最近的一系列文章,對小波算子理論,需要一個教程這解釋了一些應(yīng)用往往從起步到運營商理論家。小波分析作為一門學(xué)科是高度跨學(xué)科和它吸引了至關(guān)重要的途徑從外部世界的想法。我們的目的是列出各種希爾伯特空間幾何之間的聯(lián)系和圖像處理。因此,我們希望能幫助學(xué)生和研究人員從一個地區(qū)了解正在進行中的其他。溝通的困難之一是一個巨大的地區(qū)差異在術(shù)語、行話、數(shù)學(xué)術(shù)語。有親自動手實驗,我們的報紙是為了幫助創(chuàng)造一個更好的了解,雙方之間的聯(lián)系,數(shù)學(xué)和圖像。這是一個微妙的平衡決定包含什么。在選擇我們所想要的學(xué)生在算子理論,強調(diào)解釋,不容易找在《文學(xué)。我們的論文結(jié)果擴展以前是已知的,我們希望收益率新的見解的縮放和表示的彩色圖像;特別是,我們已經(jīng)為更好的算法。最后對一組計算機生成的圖像以說明我們的想法和我們的算法,并使用生成的圖像壓縮1.1綜述小波圖像處理使計算機存儲圖像在許多規(guī)模的決議,因此圖像分解成不同層次和類型的細(xì)節(jié)和近似具有不同價值的決議。因此,使其能夠放大,以獲取更多細(xì)節(jié),樹木的葉子,甚至一只猴子在樹頂。小波圖像壓縮允許使用更少的存儲空間和更詳細(xì)的圖像。作為數(shù)學(xué)上3.3所示,一個圖像可以分解成近似、水平、垂直和對角的細(xì)節(jié)。N分解層級的完成。在那之后,量化工作是在分解的圖像位置不同的量化也許做不同的組件,從而最大限度地發(fā)揮需要的數(shù)量細(xì)節(jié)和忽視“not-so-wanted的細(xì)節(jié)。這是通過一些系數(shù)閾值的像素值圖像是“扔掉”或設(shè)置為零或者一些“平滑”效應(yīng)的工作是在圖像矩陣。這個過程是用在JPEG2000標(biāo)準(zhǔn)。1.2目標(biāo)在許多論文和書籍,主題在小波圖像處理中所討論的主要是在一個極端,即在工程方面的方面或小波進行了討論,在條件操作符沒有被明確的提到它是如何被用于其在實際工程中的應(yīng)用。在本文中,作者補充說到[Sko01],[Use01]和[Vet01]更深入的數(shù)學(xué)性質(zhì),如屬性從算子理論、泛函分析等,對小波起主要作用的結(jié)果在小波圖像壓縮編碼。我們研究的目的在建立(如果尚未建立或改善之間的連接的數(shù)學(xué)方面的小波圖像處理等方面的應(yīng)用。同時,我們的論文探討圖像是用計算機程序?qū)崿F(xiàn),以及如何進行小波分解的數(shù)字化圖像的計算機程序而言,在數(shù)學(xué)方面,希望數(shù)學(xué)和工程學(xué)之間的溝通將會得到改善,從而將帶來更大的收益,數(shù)學(xué)家和工程師。2彩色圖像小波壓縮整個過程的小波圖像壓縮是執(zhí)行如下:一個輸入圖像被計算機,前鋒小波變換進行了數(shù)字圖像,閾值可以對數(shù)字圖像,熵編碼工作是在圖像在必要的地方,因此壓縮圖像是在電腦上完成。然后使用壓縮圖像,重建圖像小波變換完成,然后逆小波變換進行圖像,因此圖像重建。在某些情況下,分塊算法[Sha93],并且用已知有更好的壓縮和分塊算法實現(xiàn)了,但它不是。2.1提出小波變換.各種小波變換用于這一步。即Daubechies小波、Coiflets,雙正交小波、和Symlets。這些不同的轉(zhuǎn)換是實現(xiàn)觀察各種數(shù)學(xué)性質(zhì),如對稱,號碼消失的時刻和正交壓縮圖像的不同結(jié)果。優(yōu)點是它可以保存短支持本地。使用正交小波的Daubechies,所以做Coiflets。Symlets擁有的特性接近對稱。雙正交小波的是垂直的,但不是必須正交提供了更多的選項來各種過濾器如對稱過濾器從而允許他們擁有對稱的財產(chǎn)。MATLAB有一個叫做wavedec2子例程執(zhí)行分解的圖像為你到給定的期望水平(N)與特定需要小波(wname)。因為有三個組件來處理,應(yīng)用小波變換componentwise。“wavedec”是一個二維小波分析功能。[C,S]=wavedec2(X,N,的wname”)返回小波分解矩陣的X在級別N,利用小波中指定字符串'wname”將是分解矢量C和correspondingbookkeeping矩陣[MatlabUG]。這里的圖象為X的矩陣。2.2閾值因為這個項目的目的是比較每個圖像壓縮的性能使用不同的小波、固定閾值被使用。MATLAB里被稱作wthrmngr的子程序，已經(jīng)叫它可計算全局閾值或依賴閾值水平根據(jù)選項和方法?？捎玫倪x項是全球性的閾值和閾值水平依賴,和全球閾值是程序中使用。然而,一個固定閾值是用給定的條件相同的每個小波變換來比較不同條件下的性能。在這里,固定閾值10,20人被使用。對于無損壓縮0作為閾值,原因很明顯。2.3重建圖像小波變換和逆小波變換在這一步,意義的地圖是采取與振幅小波系數(shù)的非零價值,小波轉(zhuǎn)換圖像重建。逆小波變換.小波參數(shù)轉(zhuǎn)換為一個圖像幾乎相同,原始圖像。有多少相同,那么它們將取決于是否有損壓縮或無損。2.4Coiflets設(shè)計Coiflets設(shè)計,以維護一個接近的匹配值之間的趨勢和原始信號的值。所有的coiflets,CoifI,我=6、12、18,24歲,30定義在一個類似的方式,但他們有一些Daubechies小波不同的屬性。Coif6轉(zhuǎn)換產(chǎn)生一個更接近的比分subsignals之間趨勢和原始信號的價值觀匹配的DaubJ轉(zhuǎn)換可以產(chǎn)生。這意味著Coifman小波系統(tǒng)類似于Daubechies小波系統(tǒng)(等級2),都擁有最大數(shù)量的消失的時刻,但是消失的時刻是平均分布在這個尺度函數(shù)和小波函數(shù)。相比之下的案例Daubechies小波、沒有公式的Coiflets任意屬,沒有正式的證明它們的存在為任意屬在這個時候。有數(shù)值解的方法,用牛頓的工作很好,直到出現(xiàn)舍入錯誤給問題,約20(roundoff屬錯誤也是一個問題在計算數(shù)值超過這個Daubechies縮放向量相同的范圍與光譜分解,即使公式并給出一個arevalid存在定理為每一個種類(Res98]。如果我們使用小波Daubechies以同樣的方式,一個人不能獲得同樣的近似結(jié)果,除了低階。2.5小波支撐小函數(shù)定義在一個有限的時間間隔和擁有一個平均的值為零。基本的想法的小波變換來表示任意函數(shù)f(x)的疊加一組這樣的小波或基函數(shù)。這些基本功能是獲得原型叫做母親英格爾小波小波ψ(x),通過dilations或縮放和翻譯。小波基非常擅長有效地代表函數(shù)平滑除了少量的間斷。2.6雙正交小波雙正交小波分析的基礎(chǔ),在定義弱定義依據(jù)正交小波基地。盡管正交小波濾波器的self-duality只有,雙正交小波濾波器的二元性。自正交小波濾波器的使能源保護證實在[Wal99],雙正交小波不是能源保護。當(dāng)前壓縮系統(tǒng)使用雙正交小波代替正交小波,盡管事實上它不是能源保護。雙正交小波的事實不是能源保護不是一個大問題,因為有線性相位雙正交濾波器系數(shù)是“接近”被正交[Use01]。的主要優(yōu)點是雙正交小波變換,它允許使用更高級別的過濾器,這類包含對稱過濾器。雙正交小波變換是有利的,因為它可以使用線性相位濾波器提供對稱輸出當(dāng)面對對稱輸入。這種轉(zhuǎn)變被稱為對稱小波變換和它解決了問題的擴張和邊界斷層系數(shù)。這里的圖像進行小波分解的次數(shù)的圖像能夠除以2ie。(地板(log2(min(大小的圖像))))倍。平均的上一級的圖像分解成四個subimages在每個級別的小波圖像分解。進一步對圖像進行小波分解應(yīng)用在圖2將會導(dǎo)致圖像圖3和圖4。請注意,上的圖片左上角最角落得到模糊“平均”時,還要注意水平、垂直、對角線圖像的組件。一個更好的例子,其中的水平、垂直和對角組件更明確地顯示在圖6和圖圖像7.注意,水平、垂直和對角組件在矩形撣子圖中。3圖像的數(shù)學(xué)表示法在這一節(jié)中,我們將探討了數(shù)字圖像背后表示和數(shù)學(xué)。MATLAB是一個互動的系統(tǒng),它的基本數(shù)據(jù)元素是一個數(shù)組,它不需要尺寸。這使得制定解決許多技術(shù)的計算問題,特別是那些涉及矩陣表示,在很短的需要花費一些時間來編寫一個程序在一個標(biāo)量交互式語言如C或Fotran。MATLAB的名字代表矩陣實驗室。在大學(xué)環(huán)境中,MATLAB是標(biāo)準(zhǔn)的計算工具和高級課程介紹數(shù)學(xué)、工程和科學(xué)。在工業(yè)上,MATLAB是計算選擇的工具對研究、開發(fā)和分析。MATLAB輔以一個家庭的特定于應(yīng)用程序的解決方案稱為工具箱;在這里,用小波分析工具(Gon04]。3.1數(shù)字圖像表示法一個圖像被定義為一個二維的函數(shù)ie。一個矩陣,f(x,y),x和y會是空間坐標(biāo),振幅的f在任何一對坐標(biāo)(x,y)稱為強度或灰度圖像的點。彩色圖像是由結(jié)合個人二維的圖像。例如,在RGB顏色系統(tǒng)、彩色圖像由三個即紅色,綠色和藍(lán)色的個別組件圖片。因此許多技術(shù)開發(fā)的黑白圖像可以擴展到彩色圖像進行處理的三個組件分別圖像。當(dāng)x,y和振幅值f的都是有限的,離散的物理量,映像稱為數(shù)字圖像。數(shù)字圖像處理領(lǐng)域是指處理數(shù)字圖像通過數(shù)字計算機。一個數(shù)字圖像是由一個有限數(shù)目的元素,每個元素都有一個特定的位置和價值。這些元素被稱為圖像元素,圖像元素,針對和像素。因為像素是最廣泛使用的術(shù)語,這些元素將被標(biāo)記為像素從現(xiàn)在開始。一個圖像也許持續(xù)對y-coordinatesx和,并在振幅。數(shù)字化坐標(biāo)以及振幅會影響這種形象的轉(zhuǎn)換到數(shù)字形式。在這里,數(shù)字化的坐標(biāo)值是稱為采樣;數(shù)字化振幅值被稱為量化。一個數(shù)字圖像是由一個有限數(shù)目的元素,每個元素都有一個特定的位置和價值。數(shù)字圖像處理領(lǐng)域是指處理數(shù)字圖像通過數(shù)字計算機。3.2MATLAB里讀取圖片圖像是讀進MATLAB環(huán)境下使用函數(shù)叫做imread。語法如下:imread(文件名)這里,文件名是一個字符串,該字符串包含完整的圖像文件的名字包括任何適用的擴展。例如,命令行>>f=imread(lena.jpg);讀取JPEG圖像莉娜分為圖像數(shù)組或圖像矩陣f。因為有三種顏色組成的形象,即紅色,綠色和藍(lán)色分量,圖像分為三個不同顏色矩陣fr，fg和fb。3.3圖像的小波分解。顏色轉(zhuǎn)換，圖像壓縮的過程中,應(yīng)用壓縮圖像的RGB組件會導(dǎo)致不良的顏色變化。因此,圖像轉(zhuǎn)化成它的強度,色調(diào)和色彩飽和度組件。顏色轉(zhuǎn)換用于JPEG-2000標(biāo)準(zhǔn)[Sko01]已經(jīng)被采用。4結(jié)果和討論4.1該計劃的實施這個計劃執(zhí)行,使用MATLAB與不同的子程序,使得小波變換、圖像壓縮和閾值計算信號的小波變換的工具包。4.2討論有損壓縮，有多種影響因素的圖像壓縮。正如上面提到的第二部分,nonorthogonality小波可能會引起壓縮是有損的。當(dāng)閾值應(yīng)用于壓縮,一些“微不足道的系數(shù)被扔掉從而導(dǎo)致有損壓縮。同時,層級的數(shù)量將小波變換應(yīng)用會影響壓縮質(zhì)量。盡管lossiness所引起的非正交小波是不可以避免某些小波時被使用,試圖最小化了lossiness數(shù)量的層部分下降到單個像素級當(dāng)小波變換應(yīng)用(floor(log2(min(大小的圖像))))。除了應(yīng)用各種閾值lossiness觀察。一個有損壓縮方法傾向于生產(chǎn)中的錯誤解壓映像。有損壓縮方法時使用這些錯誤是如此的小,以致他們難以察覺。如果那些無法察覺的錯誤是可以接受的技術(shù)是有利的損耗比無損的,就可以達到更高的壓縮比。為了支持所宣稱的比較結(jié)果的圖像和理論知識,我們獲得了文本特征,比較了數(shù)值。他們的壓縮率,均方根誤差,rms,兩個規(guī)范的相對差異,D,和峰值信噪比,PNSR。各種小波變換具有兩種不同thresholdings被用來壓縮和8位lena.png彩色圖像。有一件事可以馬上指出通過查看圖像,圖像壓縮與較小的閾值為10看起來更接近原始圖像。現(xiàn)在,考慮在每個小波變換的表演獲得相同的閾值,bior2.2(雙正交小波),sym5(Symlet)和Coif3(Coiflet)似乎已經(jīng)產(chǎn)生了更少的完美的圖像壓縮的小波相比,所有其他的在小波db2似乎產(chǎn)生了最完美的圖像壓縮;同意上面討論什么是在db2Daubechies小波被更好的信號壓縮比db1(哈爾)。考慮到錯誤和壓縮比的感知sym5形象將是最好的選擇小波,在那些被用于圖像壓縮。因此,在這種情況下,sym5非常接近對稱小波做了一份更好的工作,在圖像壓縮。同樣,讓額外的屬性如前所述Coiflets下段由在圖像壓縮Coif3執(zhí)行得更好。有biorothogonal屬性似乎也導(dǎo)致更好的圖像壓縮。另一方面正交小波Daubechies似乎并不比coiflets表現(xiàn)更好,symlets雙正交小波。而且,有再支持比例為小波的順序,似乎導(dǎo)致企業(yè)績效的不斷惡化的圖像壓縮。與閾值10,當(dāng)一個Daubechies小波,壓縮比使用db134.2627而db238.4340了。一個CoifletCoif1導(dǎo)致壓縮率為37.0173,而發(fā)型3導(dǎo)致壓縮率為26.8321。雙正交小波bior1.1和bior2.2給了34.2627和30.2723的壓縮率分別。Symletssym2和sym5導(dǎo)致壓縮率分別為38.4340倍和34.3523倍?，F(xiàn)在,用更高的閾值,因為更多的日期被丟失,壓縮比的增加而增大。然而,圖像的質(zhì)量的同時減少了。4.3結(jié)論小波壓縮確實展現(xiàn)出非凡的性能，尤其是較小的閾值，它并不是在原始圖像之間的可微的,那么圖像壓縮為某些情況下。然而,還可以進行更多的改進。作為里提到[Sko01]有更多的改進的空間通過添加更多的階段到壓縮如量化、熵編碼等。同時,我們沒有涉及到所有的小波,就在那里,它不能決定是哪一個表現(xiàn)最佳的圖像壓縮。數(shù)學(xué)方面的小波扮演一個重要的不同結(jié)果的工程應(yīng)用。我希望研究的數(shù)學(xué)性質(zhì)及其應(yīng)用小波在不同地區(qū)的工程研究。

AbstractAbstract:Waveletsprovideapowerfulandremarkablyflexiblesetoftoolsforhandlingfundamentalproblemsinscienceandengineering,suchasaudiode-noising,signalcompression,objectdetectionandfingerprintcompression,imagede-noising,imageenhancement,imagerecognition,diagnostichearttroubleandspeechrecognition,tonameafew.Here,wearegoingtoconcentrateonwaveletapplicationinthefieldofImageCompressionsoastoobservehowwaveletisimplementedtobeappliedtoanimageintheprocessofcompression,andalsohowmathematicalaspectsofwaveletaffectthecompressionprocessandtheresultsofit.Waveletimagecompressionisperformedwithvariousknownwaveletswithdifferentmathematicalproperties.Westudytheinsightsofhowwaveletsinmathematicsareimplementedinawaytofittheengineeringmodelofimagecompression.1.IntroductionWaveletsarefunctionswhichallowdataanalysisofsignalsorimages,accordingtoscalesorresolutions.Theprocessingofsignalsbywaveletalgorithmsinfactworksmuchthesamewaythehumaneyedoes;orthewayadigitalcameraprocessesvisualscalesofresolutions,andintermediatedetails.Butthesameprinciplealsocapturescellphonesignals,andevendigitizedcolorimagesusedinmedicine.Waveletsareofrealuseintheseareas,forexampleinapproximatingdatawithsharpdiscontinuitiessuchaschoppysignals,orpictureswithlotsofedges.Whilewaveletsisperhapsachapterinfunctiontheory,weshowthatthealgorithmsthatresultarekeytotheprocessingofnumbers,ormorepreciselyofdigitizedinformation,signals,timeseries,movies,colorimages,etc.Thus,applicationsofthewaveletideaincludebigpartsofsignalandimagepro-cessing,datacompression,fingerprintencoding,andmanyotherfieldsofscienceandengineering.Thisthesisfocusesontheprocessingofcolorimageswiththeuseofcustomdesignedwaveletalgorithms,andmathematicalthresholdfilters.Althoughtherehavebeenanumberofrecentpapersontheoperatortheoryofwavelets,thereisaneedforatutorialwhichexplainssomeappliedtendsfromscratchtooperatortheorists.Waveletsasasubjectishighlyinterdisciplinaryanditdrawsincrucialwaysonideasfromtheoutsideworld.WeaimtooutlinevariousconnectionsbetweenHilbertspacegeometryandimageprocessing.Thus,wehopetohelpstudentsandresearchersfromoneareaunderstandwhatisgoingonintheother.Onedifficultywithcommunicatingacrossareasisavastdifferenceinlingo,jargon,andmathematicalterminology.Withhands-onexperiments,ourpaperismeanttohelpcreateabetterunderstandingoflinksbetweenthetwosides,mathandimages.Itisadelicatebalancedecidingwhattoinclude.Inchoosing,wehadinmindstudentsinoperatortheory,stressingexplanationsthatarenoteasytofindinthejournalliterature.Ourpaperresultsextendwhatwaspreviouslyknown,andwehopeyieldsnewinsightintoscalingandofrepresentationofcolorimages;especially,wehaveaimedforbetteralgorithms.Thepaperconcludeswithasetofcomputergeneratedimageswhichservetoillustrateourideasandouralgorithms,andalsowiththeresultingcompressedimages.1.1.Overview.WaveletImageProcessingenablescomputerstostoreanimageinmanyscalesofresolutions,thusdecomposinganimageintovariouslevelsandtypesofdetailsandapproximationwithdifferentvaluedresolutions.Hence,makingitpossibletozoomintoobtainmoredetailofthetrees,leavesandevenamonkeyontopofthetree.Waveletsallowonetocompresstheimageusinglessstoragespacewithmoredetailsoftheimage.Theadvantageofdecomposingimagestoapproximateanddetailpartsasin3.3isthatitenablestoisolateandmanipulatethedatawithspecificproperties.Withthis,itispossibletodeterminewhethertopreservemorespecificdetails.Forinstance,keepingmoreverticaldetailinsteadofkeepingallthehorizontal,diagonalandverticaldetailsofanimagethathasmoreverticalaspects.Thiswouldallowtheimagetoloseacertainamountofhorizontalanddiagonaldetails,butwouldnotaffecttheimageinhumanperception.Asmathematicallyillustratedin3.3,animagecanbedecomposedintoapproximate,horizontal,verticalanddiagonaldetails.Nlevelsofdecompositionisdone.Afterthat,quantizationisdoneonthedecomposedimagewheredifferentquantizationmaybedoneondifferentcomponentsthusmaximizingtheamountofneededdetailsandignoring‘not-so-wanted’details.Thisisdonebythresholdingwheresomecoefficientvaluesforpixelsinimagesare‘thrownout’orsettozeroorsome‘smoothing’effectisdoneontheimagematrix.ThisprocessisusedinJPEG2000.1.2.Motivation.Inmanypapersandbooks,thetopicsinwaveletsandimageprocessingarediscussedinmostlyinoneextreme,namelyintermsofengineeringaspectsofitorwaveletsarediscussedintermsoperatorswithoutbeingspecificallymentionedhowitisbeingusedinitsapplicationinengineering.Inthispaper,theauthoraddsonto[Sko01],[Use01]and[Vet01]moreinsightsaboutmathematicalpropertiessuchaspropertiesfromOperatorTheory,FunctionalAnalysis,etc.ofwaveletsplayingamajorroleinresultsinwaveletimagecompression.Ourpaperaimsinestablishingifnotalreadyestablishedorimprovetheconnectionbetweenthemathematicalaspectsofwaveletsanditsapplicationinimageprocessing.Also,ourpaperdiscussonhowtheimagesareimplementedwithcomputerprogram,andhowwaveletdecompositionisdoneonthedigitalimagesintermsofcomputerprogram,andintermsofmathematics,inthehopethatthecommunicationbetweenmathematicsandengineeringwillimprove,thuswillbringgreaterbenefitstomathematiciansandengineers.2WaveletColorImageCompressionThewholeprocessofwaveletimagecompressionisperformedasfollows:Aninputimageistakenbythecomputer,forwardwavelettransformisperformedonthedigitalimage,thresholdingisdoneonthedigitalimage,entropycodingisdoneontheimagewherenecessary,thusthecompressionofimageisdoneonthecomputer.Thenwiththecompressedimage,reconstructionofwavelettransformedimageisdone,theninversewavelettransformisperformedontheimage,thusimageisreconstructed.Insomecases,zero-treealgorithm[Sha93]isusedanditisknowntohavebettercompressionwithzero-treealgorithmbutitwasnotimplementedhere.2.1ForwardWaveletTransform.Variouswavelettransformsareusedinthisstep.Namely,Daubechieswavelets,Coiflets,biorthogonalwavelets,andSymlets.Thesevarioustransformsareimplementedtoobservehowvariousmathematicalpropertiessuchassymmetry,numberofvanishingmomentsandorthogonalitydiffertheresultofcompressedimage.Advantagesshortsupportisthatitpreserveslocality.TheDaubechieswaveletsusedareorthogonal,sodoCoiflets.Symletshavethepropertyofbeingclosetosymmetric.Thebiorthogonalwaveletsarenotorthogonalbutnothavingtobeorthogonalgivesmoreoptionstoavarietyoffilterssuchassymmetricfiltersthusallowingthemtopossessthesymmetricproperty.MATLABhasasubroutinecalledwavedec2whichperformsthedecompositionoftheimageforyouuptothegivendesiredlevel(N)withthegivendesiredwavelet(wname).Sincetherearethreecomponentstodealwith,thewavelettransformwasappliedcomponentwise.“wavedec”isatwo-dimensionalwaveletanalysisfunction.[C,S]=wavedec2(X,N,‘wname’)returnsthewaveletdecompositionofthematrixXatlevelN,usingthewaveletnamedinstring‘wname’.Out-putsarethedecompositionvectorCandthecorrespondingbookkeepingmatrixS[MatlabUG].HeretheimageistakenasthematrixX.2.2Thresholding.Sincethewholepurposeofthisprojectwastocomparetheperformanceofeachimagecompressionusingdifferentwavelets,fixedthresholdswereused.MATLABhasthissubroutinecalledwthrmngrwhichcomputestheglobalthresholdorleveldependentthresholdsdependingontheoptionandmethod.Theoptionsavailableareglobalthresholdandleveldependentthreshold,andtheglobalthresholdisusedintheprogram.However,afixedthresholdvalueswereusedsoastohavethesamegivenconditionforeverywavelettransformtocomparetheperformancesofdifferentconditions.Here,fixedthresholds10and20wereused.Forthelosslesscompression0isusedasthethresholdforanobviousreason.2.3ReconstructionofWaveletTransformedImageandandInverseWaveletTransformationAtthisstep,thesignificancemapistakenandwiththeamplitudesofthenon-zerovaluedwaveletcoefficients,thewavelettransformedimageisreconstructed.Thewaveletparametersareconvertedbackintoanimagealmostidenticaltotheoriginalimage.Howmuchidenticaltheyarewillbedependentuponwhetherthecompressionwaslossyorlossless.2.4Coiflets.Coifletsaredesignedsoastomaintainaclosematchbetweenthetrendvaluesandtheoriginalsignalvalues.Allofthecoiflets,CoifI,I=6,12,18,24,30aredefinedinasimilarwayasDaubechieswaveletsbuttheyhavesomedifferentproperties.Coif6transformproducesamuchclosermatchbetweentrendsubsignalsandtheoriginalsignalvaluesthanthematchthatanyoftheDaubJtransformscanproduce.Thismeansthatthe.CoifmanwaveletsystemsaresimilartoDaubechieswaveletsystems(inrank2)inthattheyhaveamaximalnumberofvanishingmoments,butthevanishingofmomentsareequallydistributedbetweenthescalingfunctionandthewaveletfunction.IncontrasttothecaseforDaubechieswavelets,thereisnoformulaforCoifletsofarbitrarygenus,andthereisnoformalproofoftheirexistenceforarbitrarygenusatthistime.TherearenumericalsolutionsusingNewton’smethodwhichworkwelluntilround-offerrorgivesproblems,uptoaboutgenus20(roundofferrorisalsoaproblemincalculatingtheDaubechiesscalingvectornumericallybeyondthissamerangewithspectralfactorization,eventhoughtheformulasarevalidandgiveanexistencetheoremforeverygenus[Res98].IfweusedDaubechieswaveletsinthesameway,onecannotgetthesameapproximationresults,excepttoloworder.2.5WaveletsCompactlysupportedwaveletsarefunctionsdefinedoverafiniteintervalandhavinganaveragevalueofzero.Thebasicideaofthewavelettransformistorepresentanyarbitraryfunctionf(x)asasuperpositionofasetofsuchwaveletsorbasisfunctions.Thesebasisfunctionsareobtainedfromaingleprototypewaveletcalledthemotherwaveletψ(x),bydilationsorscalingandtranslations.Waveletbasesareverygoodatefficientlyrepresentingfunctionsthataresmoothexceptforasmallsetofdiscontinuities.2.6BiorthogonalThebiorthogonalwaveletshavebasesthataredefinedinawaythathasweakerdefinitionofthebasesoforthogonalwaveletbases.Thoughtheorthogonalwavelet’sfilterhasself-dualityonly,thebiorthogonalwavelet’sfilterhasduality.Sincetheorthogonalityofthefiltermakesthewaveletenergypreservingasprovenin[Wal99],thebiorthogonalwaveletsarenotenergypreserving.Currentcompressionsystemsusebiorthogonalwaveletinsteadoforthogonalwaveletsdespitethefactthatitisnotenergypreserving.Thefactthatbiorthogonalwaveletsarenotenergypreservingisnotabigproblemsincetherearelinearphasebiorthogonalfiltercoefficientswhichare“close”tobeingorthogonal[Use01].Themainadvantageofthebiorthogonalwavelettransformisthatitpermitstheuseofamuchbroaderclassoffilters,andthisclassincludesthesymmetricfilters.Thebiorthogonalwavelettransformisadvantageousbecauseitcanuselinearphasefilterswhichgivessymmetricoutputswhenpresentedwithsymmetricinput.Thistransformiscalledthesymmetricwavelettransformanditsolvestheproblemsofcoefficientexpansionandborderdiscontinuities.See[Use01].3DigitalImageRepresentationandMathematicsbehindItInthissectionwewillexplorethedigitalimagerepresentationandMathematicsbehindit.MATLABisaninteractivesystemwhosebasicdataelementisanarraythatdoesnotrequiredimensioning.Thisenablesformulatingsolutionstomanytechnicalcomputingproblems,especiallythoseinvolvingmatrixrepresentations,inafractionofthetimeitwouldtaketowriteaprograminascalarnon-interactivelanguagesuchasCorFotran.ThenameMATLABstandsformatrixlaboratory.Inuniversityenvironments,MATLABisthestandardcomputationaltoolforintroductoryandadvancedcoursesinmathematics,engineeringandscience.Inindustry,MATLABisthecomputationaltoolofchoiceforresearch,development,andanalysis.MATLABiscomplementedbyafamilyofapplication-specificsolutionscalledtoolboxes;here,WaveletToolboxisused[Gon04].3.1DigitalImageRepresentation.Animageisdefinedasatwo-dimensionalfunctionie.amatrix,f(x,y),wherexandyarespatialcoordinates,andtheamplitudeoffatanypairofcoordinates(x,y)iscalledtheintensityorgrayleveloftheimageatthepoint.Colorimagesareformedbycombiningtheindividualtwo-dimensionalimages.Forexample,intheRGBcolorsystem,acolorimagesconsistsofthreenamely,red,greenandblueindividualcomponentimages.Thusmanyofthetechniquesdevelopedformonochromeimagescanbeextendedtocolorimagesbyprocessingthethreecomponentimagesindividually.Whenx,yandtheamplitudevaluesoffareallfinite,discretequantities,theimageiscalledadigitalimage.Thefieldofdigitalimageprocessingreferstoprocessingdigitalimagesbymeansofadigitalcomputer.Adigitalimageiscomposedofafinitenumberofelements,eachofwhichhasaparticularlocationandvalue.Theseelementsarereferredtoaspictureelements,imageelements,pelsandpixels.Sincepixelisthemostwidelyusedterm,theelementswillbedenotedaspixelsfromnowon.Animagemaybecontinuouswithrespecttothex-andy-coordinates,andalsoinamplitude.Digitizingthecoordinatesaswellastheamplitudewilltakeintoeffecttheconversionofsuchanimagetodigitalform.Here,thedigitizationofthecoordinatevaluesarecalledsampling;digitizingtheamplitudevaluesiscalledquantization.Adigitalimageiscomposedofafinitenumberofelements,eachofwhichhasaparticularlocationandvalue.Thefieldofdigitalimageprocessingreferstoprocessingdigitalimagesbymeansofadigitalcomputer.3.2.ReadingImages.InMATLAB,imagesarereadintotheMATLABenvironmentusingfunctioncalledimread.Thesyntaxisasfollows:imread(filename)Here,filenameisastringcontainingthecompletenameoftheimagefileincludinganyapplicableextension.Forexample,thecommandline>>f=imread(lena.jpg);readstheJPEGimagelenaintoimagearrayorimagematrixf.Sincetherearethreecolorcomponentsintheimage,namelyred,greenandbluecomponents,theimageisbrokendownintothethreedistinctcolormatricesfRfGandfB。3.3.WaveletDecompositionofanImage.ColorConversion.Intheprocessofimagecompression,applyingthecompressiontotheRGBcomponentsoftheimagewouldresultinundesirablecolorchanges.Thus,theimageistransformedintoitsintensity,hueandcolorsaturationcomponents.ThecolortransformationusedinJPEG2000standard[Sko01]hasbeenadopted.Forthelossycompression,equations(3.2)and(3.3)wereusedintheprogram.4.ResultsandDiscussion4.1.ImplementationoftheProgram.TheprogramwasimplementedusingMATLABwithvarioussubroutinesthatenablesthewavelettransformation,imagecompressionandthresholdcomputationfromtheWaveletToolkit.4.2.Discussion.LossyCompression.Therearevariousfactorsthatinfluencetheimagecompression.Asmentionedaboveinsection2,nonorthogonalityofthewaveletmaycausethecompressiontobelossy.Whenthresholdisappliedtothecompression,someofthe’insignificant’coefficientsarethrownoutthustheresultinginlossycompression.Also,thenumberoflevelsthewavelettransformisappliedwouldinfluencethecompressionquality.Althoughthelossinesscausedbythenonorthogonalwaveletwasnotavoidablewhencertainwaveletswereused,anattempttominimizethelossinesswasmadeforthenumberoflevelspartbygoingdownallthewaytothesinglepixellevelwhenthewavelettransformwasapplied(floor(log2(min(sizeofImage)))).Inadditionvariousthresholdvaluesareappliedtoobservethelossiness.Alossycompressionmethodtendtoproduceinaccuraciesinthedecompressedimage.Lossycompressionmethodisusedwhentheseinaccuraciesaresosmallthattheyareimperceptible.Ifthoseimperceptibleinaccuraciesareacceptablethelossytechniqueisadvantageouscomparedtothelosslessonesforhighercompressionratioscanbeattained.Inordertosupporttheclaimsmadebycomparisonoftheresultingimagesandthetheoreticalknowledgethatweobtainedfromthetexts,somenumericalcomparisonsaremade.Theyarethecompressionratio,therootmeansquareerror,rms,therelativetwonormdifference,D,andthepeaksignaltonoiseratio,PNSR.Theformulasusedareasfollows:Variouswavelettransformswithtwodifferentthresholdingswereusedtocompresstheand8-bitcolorimagelena.png.Theresultsareasfollows:Onethingthatcouldbenotedrightawaybylookingattheimagesisthattheimagescompressedwithsmallerthresholdvaluethatis10lookclosertotheoriginallena.pngcomparedtotheimagescompressedwiththresholdvalue20overall.Now,lookingattheperformancesofeachwavelettransformsgiventhesamethresholdvalue,bior2.2(Biorthogonalwavelet),sym5(Symlet)andCoif3(Coiflet)seemtohaveproducedthelessflawlesscompressedimagescomparetoalltheotherwavelets.WithintheDaubechieswaveletsdb2appearstohaveproducedtheleastflawlesscompressedimage;thatagreeswithwhatwasdiscussedaboveinDaubechieswaveletsthatdb2isbeingbetterinsignalcompressionthandb1(Haar).Consideringtheerrorsandcompressionratiosaswellastheperceptionoftheimagesym5wouldbethebestchoiceofwavelets,amongtheonesthatwasusedfortheimagecompr