毕业论文基于小波变换的数字图像处理

1、 基于小波变换的数字图像处理摘 要：本文先介绍了小波分析的基本理论，为图像处理模型的构建奠定了基础，在此基础上提出了小波分析在图像压缩，图像去噪，图像融合，图像增强等图像处理方面的应用,最后在MATLAB环境下进行仿真，验证了小波变化在图像处理方面的优势。关键词：小波分析；图像压缩；图像去噪；图像融合；图像增强引 言数字图像处理是利用计算机对科学研究和生产中出现的数字化可视化图像信息进行处理，作为信息技术的一个重要领域受到了高度广泛的重视。数字化图像处理的今天，人们为图像建立数学模型并对图像特征给出各种描述，设计算子，优化处理等。迄今为止，研究数字图像处理应用中数学问题的理论越来越多，包括概率

2、统计、调和分析、线性系统和偏微分方程等。小波分析，作为一种新的数学分析工具，是泛函分析、傅立叶分析、样条分析、调和分析以及数值分析理论的完美结合，所以小波分析具有良好性质和实际应用背景，被广泛应用于计算机视觉、图像处理以及目标检测等领域，并在理论和方法上取得了重大进展，小波分析在图像处理及其相关领域所发挥的作用也越来越大。在传统的傅立叶分析中，信号完全是在频域展开的，不包含任何时频的信息，其丢弃的时域信息可能对某些应用同样非常重要，所以人们对傅立叶分析进行了推广，提出了很多能表征时域和频域信息的信号分析方法，如短时傅立叶变换，Gabor变换，时频分析，小波变换等。但短时傅立叶分析只能在一个分辨

3、率上进行，所以对很多应用来说不够精确，存在很大的缺陷。而小波分析则克服了短时傅立叶变换在单分辨率上的缺陷，在时域和频域都有表征信号局部信息的能力，时间窗和频率窗都可以根据信号的具体形态动态调整。 本文介绍了小波变换的基本理论，并介绍了一些常用的小波函数，然后研究了小波分析在图像处理中的应用，包括图像压缩，图像去噪，图像融合，图像增强等,本文重点在图像去噪，最后用Matlab进行了仿真1。1小波分析理论小波分析的思想最早出现在1910年Haar提出了小波规范正交基。1981年，Stromberg对Haar系进行了改造，为小波分析奠定了基础。1986年Meyer和Lemarie提出了多尺度分析的思

4、想。后来信号分析专家Mallat提出了多分辨分析的概念，给出了构造正交小波基的一般方法，并以多分辨分析为基础提出了著名的快速小波算法Mallat算法。Mallat算法的提出标志着小波理论获得突破性进展，从此，小波分析从理论研究走向了应用研究。通过小波分析，可以将各种交织在一起的由不同频率组成的混合信号分解成不同频率的块信号，能够有效地解决诸如数值分析、信号分析、图像处理、量子理论、地震勘探、语音识别、计算机视觉、CT成像、机械故障诊断等问题。1.1 小波及小波变换小波的核心作用是用小波及其伸缩和平移来表示函数和信号，不但具有局部化时频分析能力，而且时间分辨率和频率分辨率均可以调整。定义：设 ,

5、其傅立叶变换为，当满足允许条件（完全重构条件或恒等分辨条件） < 时，我们称为一个基本小波或母小波。将母函数经伸缩和平移后得 称其为一个小波序列。其中a为伸缩因子，b为平移因子。对于任意的函数的连续小波变换为 其重构公式（逆变换）为 把连续小波变换中的尺度参数a和平移参数b进行离散化：，其中,为了方便起见，总是假设a0>0，则得到离散小波函数 相应的离散小波变换 其重构公式为由于基小波生成的小波在小波变换中对被分析的信号起着观测窗的作用，所以还应该满足一般函数的约束条件 故是一个连续函数。这意味着，为了满足完全重构条件式, 在原点必须等于0，即 为了使信号重构的实现在数值上是稳定的

6、，除完全重构条件外，还要求小波 的傅立叶变化满足下面的稳定性条件： 式中0AB。 1.2常用小波基介绍3(1)Haar小波Haar于1990年提出一种正交函数系,定义如下： 这是一种最简单的正交小波，即 (2)Daubechies（dbN）小波系该小波是Daubechies从两尺度方程系数出发设计出来的离散正交小波。一般简写为dbN，N是小波的阶数。小波和尺度函数中的支撑区为2N-1。的消失矩为N。除N1外(Haar小波)，dbN不具对称性(即非线性相位)，没有显式表达式(除N1外)。但的传递函数的模的平方有显式表达式。假设，其中，为二项式的系数，则有 其中 (3)SymletsA（symN）

7、小波系Symlets函数系是由Daubechies提出的近似对称的小波函数，它是对db函数的一种改进。Symlets函数系通常表示为symN（N=2，3，8）的形式。1.3 双尺度关系与分解关系与的两尺度关系：与的分解关系：其中 1.4 小波进行分解与重构两尺度函数的两尺度关系是 由两尺度关系，得序列再由得则有小波函数的两尺度关系是进一步得分解关系由分解算法得由重构算法得2 图像处理的模型分析2.1 图像的数学模型4物体反射或投射的物质能量在空间上的分布在数学上可以表述为一能量场E(x,y,z,t)，其中x，y，z表示在几何空间中点的坐标，为辐射波长，t为时刻。适当选取坐标系使取图平面垂直于z

8、轴，设截距为Z0，图像可看作只是记录在平面z=z0上的能量分布2，实际中这种物质能量的记录值往往用亮度值表示。则式中x，y为像平面中点的坐标。图像的数学模型是一个二元函数f(x,y)，它反映了图像上点坐标f(x,y)与该点上的能量值之间的对应关系。对于一幅灰度照片，它上面点的明暗程度需用不同的数值代表。图像在某点处的函数值称为图像在该点的灰度或亮度。由于f的值是能量的记录，故其是非负有界的实数，即一幅实际图像的尺寸是有限的，一般定义(x,y)在某一矩形域中，即有2.2 图像处理的小波模型通常情况下，图像处理可以被抽象为一个输入输出系统，即以各种形式的算符Q来对图像F进行处理，算符Q的形式决定于

9、图像处理的目的以及图像F的数学模型其中，T代表图像处理算子，如去噪、锐化、分割，压缩或图像修复。输入数据Q0可以是一幅图也可以是图像序列，输出数据Q是所有希望得到的图像性质。图像处理的两个基本问题是作为输入的Q0和算子T的建模，它们是相对独立但又紧密相连的：算子T的性能很大程度上取决于输入数据的模型。为了能够有效地处理图像，首先需要知道如何从数学的角度理解和表示图像。图像模型和它的表示方法在很大程度决定了图像的处理模型。把图像看作像素点集合的基础上，以像素点集合为基础的数学模型可表达三种不同的模型，即随机场模型，小波模型和正则空间模型，它们分别从概率统计、小波分析和偏微分程这三方面来研究图像处

10、理。小波应用图像处理的本质是一个函数逼近问题，小波可以构成Hilbert空间的规范正交基，给出了图像的多尺度表示： 对图像的稀疏逼近为图像的高效压缩等图像处理提供了可能。2.3 小波分析在图像处理中的应用小波分析在图像处理中的应用主要表现在以下几个方面：图像去噪噪声的产生是一个随机过程，噪声分量灰度值是一个随即变量，其统计特性由概率密度函数表征，白噪声、高斯噪声、泊松噪声是三种形式常见的重要噪声；设长度为N的信号fn被噪声en所污染，所测得的含噪数据为：去噪的目标是从含噪数据X得到信号x的一个逼近信号x,使得在某种误差准则估计下x是x的最佳逼近。小波去噪的基本策略是将含噪信号进行多尺度小波变换

11、，从时域变换到小波域，然后在各尺度下尽可能提取信号的小波系数，而去除属于噪声的变换系数，然后由小波逆变换重构信号。阀值去噪的主要理论依据是，小波变换具有很强的数据去相关性，能够使信号的能量在小波域集中在少量的大的小波系数中，而噪声却分布在整个小波域，经小波分解后，信号的小波系数的幅值要大于噪声的小波系数的幅值，于是可以采用阀值的办法把信号的小波系数保留，而使大部分噪声的小波系数减少为零，阀值去噪主要思想是对小波分解后的除了最低频以外的各层系数模对大于和小于某阈值系数分别处理，然后对处理完的小波系数再反变换重构出一幅经去噪后的图像。在小波阈值收缩去噪法中最为重要的就是如何选择阈值和阈值函数，大多

12、数阀值选择过程是针对一组小波系数，即根据本组小波系数的统计特性，计算出一个阈值。阈值函数主要可以分为如下三种： 硬阈值函数它假定幅值大于门限T的小波系数都是由信号贡献的，而小于门限T的小波系数来至于噪声。实验表明这种取门限的方式有时并不太令人满意。软阈值函数硬阈值方法可以很好保留图像边缘等局部特征，但图像会出现振铃、伪吉布斯效应等视觉失真，阈值方法处理结果相对平滑得多，但是软阈值方法会造成边缘模糊等失真现象。半软阀值函数该方法通过选择合适的阈值T1和T2，可以在软阀值方法和硬阈值方法之间达到很好的折中。阀值去噪算法阀值去噪具体的算法步骤如下：选择合适的小波函数，对于长度为N(设N=2J)的含噪

13、信号(常采用周期延拓方法)进行小波变换，利用小波交换的快速算法获得低分辨率L(O0<J)下的尺度系数VL,K K=1,2,2L及各分辨率下的小波系数Wj,k，j=L,L+1,J-1,k=1,2,2L)其中尺度系数和小波系数共N个。为保持信号的整体形状不变，保留所有的低频系数VL,K K=1,2,2L，对小波系数进行非线性阈值处理。对每个小波系数，采用软阈值和硬阈值方法进行处理。进行逆小波变换。由所有低频尺度系数，以及经由阈值处理后的小波系数作逆小波变换进行重构，得到恢复的原始信号的估计值。图像压缩 小波分析用于信号与图像压缩是小波分析应用的一个重要方面。它的特点是压缩比高，压缩速度快，压

14、缩后能保持信号与图像的特征不变，且在传递中可抗干扰。由于小波分析固有的时频特性，我们可以在时频两个方向对系数进行处理，这样就可以对我们感兴趣的部分提供不同的压缩精度。图形增强图像增强问题主要通过时域和频域处理两种方法来解决。时域方法通过直接在图像点上作用算子或掩码来解决，频域方法通过修改傅立叶变换系数来解决。这两种方法的优劣很明显，时域方法方便快速但会丢失很多点之间的相关信息，频域方法可以很详细地分离出点之间的相关,但计算量大得多。小波分析是以上两种方法的权衡结果，傅立叶分析在所有点的分辨率都是原始图像的尺度，但我们可能不需要这么大的分辨率，而单纯的时域分析又显得太粗糙，小波分析的多尺度分析特

15、性为用户提供了更灵活的处理方法，可以选择任意的分解层数，用近可能少的计算量得到我们满意的结果。图像融合 图像融合是将同一对象的两个或更多的图像合成在一幅图像中，以便它比原来的任何一幅图像更容易为人们所理解。这一技术可应用于多频谱图像理解以及医学图像处理等领域。在这些场合，同一物体部件的图像往往是采用不同的成像机理得到的。3MATLAB仿真下面先利用小波分析对含噪的二维图像wbard进行去噪处理，对比去噪效果，然后对去噪后的图像进行锐化处理。再对原二维图像wbarb进行压缩处理，之后再对wbarb和woman图像进行融合处理。Matlab程序如下：load wbarb%下面进行噪声的产生init

16、=3718025452;rand('seed',init);Xnoise=X+50*(rand(size(X);%显示原始图像及它的含噪声的图像colormap(map);figure(1);subplot(2,2,1);image(wcodemat(X,192);title('原始图像X')subplot(2,2,2);image(wcodemat(Xnoise,192);title('含噪声的图像Xnoise');%用sym5小波对图像信号进行二层的小波分解c,s=wavedec2(X,2,'sym5');%下面进行图像的去噪

17、处理%使用ddencmp函数来计算去噪的默认阈值%使用wdencmp函数来实现图像的压缩thr,sorh,keepapp=ddencmp('den','wv',Xnoise);Xdenoise,cxc,lxc,perf0,perfl2=wdencmp('gbl',c,s,'sym5',2,thr,sorh,keepapp);%显示去噪后的图像subplot(223);image(Xdenoise);title('去噪后的图像');%对去噪后的图像做锐化处理blur2=Xdenoise;%对图像做2层的二维小波分解c

18、,l=wavedec2(Xdenoise,2,'db3');csize=size(c);%对低频系数进行放大处理，并抑制高频系数for i=1:csize(2); if(abs(c(i)<300) c(i)=c(i)*2; else c(i)=c(i)/2; endend%通过处理后的小波系数重构图像blur2=waverec2(c,l,'db3');subplot(224);image(wcodemat(blur2,192);colormap(gray(256);title('锐化去噪后的图像');%图像压缩处理figure(2);sub

19、plot(221);image(X);colormap(map)title('原始图像');disp('压缩前图像X的大小：');whos('X')%对图像用bior3.7小波进行2层小波分解c,s=wavedec2(X,2,'bior3.7');%提取小波分解结构中第一层低频系数和高频系数ca1=appcoef2(c,s,'bior3.7',1);ch1=detcoef2('h',c,s,1);cv1=detcoef2('v',c,s,1);cd1=detcoef2('d&

20、#39;,c,s,1);%分别对各频率成分进行重构a1=wrcoef2('a',c,s,'bior3.7',1);h1=wrcoef2('h',c,s,'bior3.7',1);v1=wrcoef2('v',c,s,'bior3.7',1);d1=wrcoef2('d',c,s,'bior3.7',1);c1=a1,h1;v1,d1;%显示分解后各频率成分的信息subplot(222);image(c1);title('分解后低频和高频信息');%保留

21、小波分解第一层低频信息，进行图像的压缩%第一层的低频信息即为ca1，显示第一层的低频信息%首先对第一层信息进行量化编码ca1=appcoef2(c,s,'bior3.7',1);ca1=wcodemat(ca1,440,'mat',0);%改变图像的高度ca1=0.5*ca1;subplot(223);image(ca1);colormap(map);title('第一次压缩');disp('第一次压缩图像的大小为：');whos('ca1')%保留小波分解第二层低频信息，进行图像的压缩，此时压缩比更大%第二层的低

22、频信息即为ca2，显示第二层的低频信息ca2=appcoef2(c,s,'bior3.7',2);%首先对第二层信息进行量化编码ca2=wcodemat(ca2,440,'mat',0);%改变图像的高度ca2=0.25*ca2;subplot(224);image(ca2);colormap(map);axis squaretitle('第二次压缩');disp('第二次压缩图像的大小为：');whos('ca2')%图像融合figure(3);X2=X;map2=map;subplot(221);image(X

23、2);colormap(map2);title('wbarb');load woman;X1=X;map1=map;%画出原始图像subplot(222);image(X1);colormap(map1);title('woman');%用小波函数sym4对X1进行2层小波分解c1,s1=wavedec2(X1,2,'sym4');%对分解系数进行处理以突出轮廓部分，弱化细节部分sizec1=size(c1);for i=1:sizec1(2) c1(i)=1.2*c1(i);end%用小波函数sym4对X2进行2层小波分解c2,s2=waved

24、ec2(X2,2,'sym4');%下面进行小波变换域的图像融合c=c1+c2;%减小图像亮度c=0.5*c;%对融合的系数进行重构xx=waverec2(c,s1,'sym4');%画出融合后的图像subplot(223);image(xx);title('wbarb和woman的融合图像');去噪和锐化仿真结果：图像压缩仿真结果：Matlab输出：压缩前图像X的大小： Name Size Bytes Class X 256x256 524288 double arrayGrand total is 65536 elements using 5

25、24288 bytes第一次压缩图像的大小为： Name Size Bytes Class ca1 135x135 145800 double arrayGrand total is 18225 elements using 145800 bytes第二次压缩图像的大小为： Name Size Bytes Class ca2 75x75 45000 double arrayGrand total is 5625 elements using 45000 bytes图像融合仿真结果：3结束语本文详细讨论了小波在图像处理领域的应用，并用Matlab进行了仿真，Matlab语言对数字图像进行处理时具

26、有编程简单、处理速度快的特点，结果显示用小波变换理论进行图像处理可以取得较好的效果。在本文中只是实现了图像处理的基本功能,对于处理效果还有待改善,在后续进一步研究中可以选用更合适的小波基和算法进行处理,以使处理效果更佳,参考文献1 李朝晖、张弘编.数字图像处理及应用M.北京:机械工业出版社，2004（06）.2 张亶、陈刚编.基于偏微分方程的图像处理M.北京:高等教育出版社，2004(01).3 崔锦泰(著)，程正兴(译)小波分析导论M西安：西安交通大学出版社，19954 龚声蓉、刘纯平、王强编.数字图像的处理与分析M北京：清华大学出版社2006您好，为你提供优秀的毕业论文参考资料，请您删除以

27、下内容，O(_)O谢谢！A large group of tea merchants on camels and horses from Northwest China's Shaanxi province pass through a stop on the ancient Silk Road, Gansu's Zhangye city during their journey to Kazakhstan, May 5, 2015. The caravan, consisting of more than 100 camels, three horse-drawn carri

28、ages and four support vehicles, started the trip from Jingyang county in Shaanxi on Sept 19, 2014. It will pass through Gansu province and Xinjiang Uygur autonomous region, and finally arrive in Almaty, formerly known as Alma-Ata, the largest city in Kazakhstan, and Dungan in Zhambyl province. The t

29、rip will cover about 15,000 kilometers and take the caravan more than one year to complete. The caravan is expected to return to Jingyang in March 2016. Then they will come back, carrying specialty products from Kazakhstan A small art troupe founded six decades ago has grown into a household name in

30、 the Inner Mongolia autonomous region. In the 1950s, Ulan Muqir Art Troupe was created by nine young musicians, who toured remote villages on horses and performed traditional Mongolian music and dances for nomadic families. The 54-year-old was born in Tongliao, in eastern Inner Mongolia and joined t

31、he troupe in 1975.He says there are 74 branch troupes across Inner Mongolia and actors give around 100 shows every year to local nomadic people. "I can still recall the days when I toured with the troupe in the early '80s. We sat on the back of pickup trucks for hours. The sky was blue, and

32、 we couldn't help but sing the folk songs," Nasun says. The vastness of Inner Mongolia and the lack of entertainment options for people living there, made their lives lonely. "The nomadic people were very excited about our visits," Nasun recalls. "We didn't have a formal

33、stage. The audience just sat on the grass. Usually, the performances became a big party with local people joining in." For him, the rewarding part about touring isn't just about sharing art with nomadic families but also about gaining inspiration for the music and dance. Ulan Muqir literall

34、y translates as "red burgeon", and today's performers of the troupe still tour the region's villages and entertain nomadic families, but their fame has spread around the world. On May 16 and 17, nearly 100 singers and dancers from the troupe performed at Beijing's Poly Theater.

35、 Their show, titled Ulan Muqir on the Grassland, depicted the history and development of the art troupe. "Being from the region allowed me to embrace the culture of Inner Mongolia and being a member of the troupe showed me where I belonged," Nasun, the art troupe's president, who is al

36、so a renowned tenor, tells China Daily. During a tour in 1985, he went to a village and met an elderly local man, who told him a story about his friendship with a solider from Shenyang, capital of Northeast China's Liaoning province, decades ago. The solider gave the old man a handmade saddle wh

37、en they bid farewell. The story inspired Nasun to write Carved Saddle, a song that later became one of his most popular numbers. Now, every year, Nasun recruits young singers and dancers for the troupe. The troupe has also designed a new repertoire, which is mostly based on the daily lives of Mongol

38、ian people, especially the lives of nomadic families, and has combined contemporary musical elements with folk songs of the region. Haimu, a 25-year-old khoomei (a local variant of overtone singing) singer, joined the troupe three years ago. Along with a six-member band, he performs fast songs and s

39、oft ones that he writesall while playing the horse-head fiddle."Although I learned the piano since childhood and grew up listening to various kinds of music, to me, the folk music of Inner Mongolia is the root," he says. "Performing in remote villages is pleasant. I feel at home on th

40、e boundless grasslands, and the warm people there make me feel fulfilled." The first round of spring auction season in Beijing ended last week, but it failed to create much spring in the art market. Although two pieces of Chinese painting fetched more than 100 million yuan, the decline in tradi

41、ng volume and sale rate showed a downturn this year. In the “Grand View: Chinese Painting Highlight” session at China Guardian 2015 spring auctions, Pan Tianshous representative work Eagle, Rock and Flora hit a record auction price of 279 million yuan, while Li Kerans masterwork Jinggang Mountain fe

42、tched 126.5 million yuan, an unexpected high in recent years. However, the trading volume fells to 1.87 billion yuan from 2.22 billion yuan in the same period the year before. The Huangchen 2015 Spring Auctions, which recorded 42.5 million yuan in total sales, experienced the same. The section numbe

43、r went down to 5 from 12 compared to last year. According to expert Shao Jianwu, the art market did not attract much excitement this year due to the booming stock market and the persistent problems of forgery and fake deals. The two pieces of Chinese painting notched up high price this spring due to

44、 their own value not due to a revival in the market. "The successful transaction of two works with a hammer price of more than 100 million yuan once again confirms that valuable and rare works of high quality always earn the long-lasting high price. Chinese painting and calligraphy is still the

45、 pillar of the market. However, the overall sales rate has dropped slightly, which reflects the demand for the ordinary auction goods has weakened,” said the president of China Guardians Hu Yanyan. "Besides, the entry of new buyers also makes certain changes to the direction of purchasing. In a

46、 word, the overall market trend is very unpredictable. Although China Guardian 2015 Spring Auctions has achieved remarkable results, it does not mean that the market has walked out of the predicament. We still have to wait for the big economic boom to invigorate the art market.” Even so, the hot pur

47、suit of some special items appeared this spring. A special session of China Guardian Spring Auctions titled “Fine GiltBronze Buddhist Images” achieved a great deal with total sales volume of 41.89 million yuan and 92 percent sale rate. The 5th Shamar Rinpoche Statue from 16-17th century of Tibet was

48、 sold for 5.06 million yuan. The Beijing Chengxuan Auctions featured almost 2,800 items of coins and stamps in three sessions with good sale rate. The Huachen Auctions also set a special session of photographs under the condition of large-scale decline of auction sessions. There was a palpable dull

49、thud of disappointment that accompanied the return of the imperial entourage of Zhen Huan to her homeland. It followed a couple years of hushed excitement as Chinese fans were fed tidbits about their proud concubine who was supposed to conquer the high ground of the North American market. Zhen Huan

50、is, of course, the title character of The Legend of Zhen Huan, a 2011 television series that swept China off its feet and later took other Asian countries by storm. Two years ago, it was reported that HBO, a premium cable service headquartered in the United States, was going to air it in North Ameri

51、ca after some modification. Now, a condensed version that provides English subtitles but no dubbing has finally been made available on Netflix for online streaming. This version, highly anticipated as a milestone in China's cultural foray overseas, has been widely panned by its home audience. Re

52、titled Empresses in the Palace, the American version has been shortened from its original 76 episodes at 45 minutes each, to six 90-minute episodes. The quick pacing threw off many native viewers, who are accustomed to a more leisurely daytime-soap-style narrative rhythm. (Chinese TV stations would

53、run two or three episodes every day.) I did not finish the full-length version and found the truncated one not difficult to follow. What's lost, I believe, are the interesting setups and pauses that illuminate the Chinese art of storytelling. Much of the plot is still there. It is the flavor tha

54、t was sacrificed. The American edition uses the framework of the Empress Dowager in her senior years reminiscing at the beginning and the end of each episode, hinting at what's to come and recapping the key points. This device, not used in the original, is culturally understandable but artistica

55、lly mediocre. What puzzles me is the two new songs for the opening and end credits. They were written in English, but sung by Chinese with an uncomfortable accent. They were obviously designed to appeal to an English-speaking base, but do not jibe with the Chinese dialogue. Speaking of the dialogue,

56、 the English translation, picked apart by some Chinese, is too literal for my taste. I can imagine a typical American hit by a flurry of royal ranks, addresses and greetings, even multiple names and titles for the same person. The first half hour must be a swamp to wade through, very much like my ex

57、perience of getting through a Tolstoy tome with its endless inflections of names transliterated into lengthy Chinese. I see the choice of verbatim translation as an effort for conveying exotica. It is fairly competent, with no error that I could detect, but fails to rise above words or capture the e

58、ssence of the language. A cultural product usually crosses over to a foreign territory first by an emphasis on the commonalities. But whether inside or outside China, the temptation to sell it for the differences is just too great. Sure, the sumptuous sets and costumes are a big attraction, but the narrative technique has b