Wavelets For Kids B
Unit 1Science Fiction 单词词汇讲解-高中英语人教版(2019)选择性必修第四册
declaration n. 声明；宣告 Universal Declaration of Human Rights 《世界人权宣言》
whereas conj. 然而；但是；尽管 while 强调对比 He is a worker while I am a doctor. 他是工人，我是医生。 whereas 强调相反 Some of the studies show positive results, whereas others do not. 有一些结果令人满意，然而其它的则不然。
on the basis of/that... 根据；在...的基础上；基于
Market researchers often segment the population on the basis of age and social class.
They believe if AI can handle tasks like calcultating and driving for us, we will have more time for creating, thinkin, and “being human”.
He always treated her with dignity. dignity n. 庄重；庄严；尊严 The real winnier in sports are those who know how to persevere and to behave with dignity-- whether they win or lose a game. 在体育运动上，真正的赢家是那些知道如何坚持和表现得有尊严的人--无论他们 在比赛中是赢还是输。
2021年高考英语的阅读理解专项训练及答案
2021年高考英语的阅读理解专项训练及答案一、高考英语阅读理解专项训练1．阅读理解Smart Kids Festival EventsSmart Kids is a collection of one hundred events scheduled in October. This year, it is experimenting with Pay What You Decide (PWYD). That is, you can decide to pay what you want to or can afford, after you have attended an event. You can pre-book events without paying for a ticket in advance. Here are some of the director's picks.Walk on the Wild SideNot ticketed, FreeJoin storyteller Sarah Law to hear science stories about animals. Along the way you'll meet all sorts of beautiful creatures and discover life cycles and food chains. Best suited to children aged 5-9. Children under 8 must be accompanied by an adult.Introduction to WavesPre-book, PWYDSubjects range from sound waves to gravity waves, and from waves of light to crashing waves on the ocean. Mike Goldsmith explores the fundamental features shared by all waves in the natural world.Science in the FieldNot ticketed, FreeThis storytelling night features a scientist sharing his favourite memories of gathering first-hand data on various field trips. Come along for inspiring and informative stories straight from the scientist's mouth. Join Mark Samuels to find out more in this fun-filled workshop.Festival DinnerPre-book, £25 per personWhether you want to explore more about food, or just fancy a talk over a meal, join us to mark the first science festival in London. Which foods should you eat to trick your brain into thinking that you are full? Find out more from Tom Crawford.（1）In which event can you decide the payment?A. Walk on the Wild SideB. Introduction to WavesC. Science in the FieldD. Festival Dinner（2）Who will talk about experiences of collecting direct data?A.Sarah Law.B.Mike Goldsmith.C.Mark Samuels.D.Tom Crawford.（3）What do the four events have in common?A.Family-based.B.Science-themed.C.Picked by children.D.Filled with adventures.【答案】（1）B（2）C（3）B【解析】【分析】本文是一篇应用文，介绍了Smart Kids收集的在十月份举行的四项以科学会主题的活动，以及各个活动的内容和特色。
2025届湖北省武汉市硚口区高三起点考试七月质量检测英语试卷
2025届湖北省武汉市硚口区高三起点考试七月质量检测英语试卷一、阅读理解Where to eat in MalaysiaMalaysia offers a wide range of food choices, ranging from fine dining to street food. Elsewhere, restaurants and food stalls are readily available.Old China CafeEating here is a little like dining in a museum as the walls are lined with photos and memorabilia of the local Chinese laundryman’s association. Chinese and Asian dishes keep regulars and travellers coming back for more. The menu changes monthly.San PedroThis restaurant serves Portuguese cuisine. Try the baked fish (seabass or red snapper), curry debal chicken (can be quite spicy), fried brinjals and the black pepper crabs, Call ahead to reserve a table (and your fish), because once their fish runs out, they will close for the day.The BungalowFacing the hotel pool and the beach, this restaurant has indoor and outdoor seating. At night, the atmosphere is very romantic. The Chinese menu concentrates on traditional Hainanese dishes, but includes dishes such as fish curry, chicken rice and filet mignon.No. 7 Sup CornerEven if you miss this buffet-style restaurant at the road bend, you will not miss the number of cars parked at the road shoulder or the locals heading to it during lunchtime. It serves Malay dishes. First get your plate of rice, and then take whatever you would like to eat. Present your food plate to the lady at the counter and pay for it. Closed on Fridays and during Ramadan. Come early as there’s always a crowd outside.1．What do Old China Cafe and the Bungalow have in common?A．They are based in hotels.B．They have seasonal set menus.C．They have atmospheric settings.D．They adopt modern cooking methods. 2．Which restaurant serves western cuisine?A．Old China Cafe.B．No.7 Sup Corner.C．The Bungalow.D．San Pedro.3．What do we know about No. 7 Sup Corner?A．It is inconveniently located.B．It is a self-service restaurant.C．It requires a restaurant booking.D．It opens every day except the weekends.Norwegian playwright and author Jon Fosse has just been awarded the 2023 Nobel Prize in Literature for his “innovative plays and prose which give voice to the unsayable”.The Swedish Academy credits Fosse as “one of the most widely performed playwrights in the world”, although the 64-year-old originally made his name as a novelist, beginning with Red, Black in 1983. He has since written many works of prose and poetry. “His literary works, spanning a variety of genres(体裁), comprises about 40 plays and a wealth of novels, poetry collections, essays, children’s books and translations,” said Anders Olsson, chairman of the Nobel Committee for Literature.Jon Fosse draws inspiration from his rural living environment and personal struggles, which have deeply influenced his distinctive writing style. Growing up in a small coastal village, Fosse was immersed in the beauty of nature and the vastness of the sea, which filled him with a sense of solitude and introspection(内省) that is reflected in his works. Fosse’s writing discussed themes of desolateness(孤独，苍凉), longing, and the search for meaning in life. Additionally, his battles with depression have enabled him to explore the depths of human emotions and existential des-pair.“He touches you so deeply when you read his works,” said Anders Olsson. “What is special about him is the closeness in his writing. It touches your deepest feelings-anxieties, insecurities, questions of life and death-which are things that every human being actually faces from the very beginning. In that sense I think he reaches very far and there is a sort of a universal impact of every-thing that he writes-it has appeal to this basic humanity.”Fosse is the first-ever laureate writing in Nynorsk, one of the two official languages of Norwegian, but only used by just 10% of the population. As The Guardian writes, “Many Nynorsk speakers see Fosse ‘as a kind of national hero’ for his championing of the language.” Fosse’s recognition on a global stage promotes the visibility and importance of Nynorsk as a writtenstandard. It will attract more attention to Nynorsk literature, inspire and encourage Nynorsk speakers to continue preserving and promoting their linguistic(语言的) heritage.4．What do we know about Jon Fosse from paragraph 2?A．His most popular work is the play Red, Black.B．He accomplished various genres of literature works.C．Many famous plays are the inspiration for his writing.D．His language skills earned him the Nobel Prize in Literature.5．What does paragraph 3 mainly talk about?A．Jon Fosse’s tough experiences.B．Jon Fosse’s motivation for writing.C．The major themes of Jon Fosse’s works.D．The effect of Jon Fosse’s life on his writing style.6．What is special about Jon Fosse’s works according to Anders Olsson?A．They explore the meaning of life.B．They touch human’s deepest emotions.C．They show the bond between humans and nature.D．They appeal to readers to focus on personal feelings.7．What contribution of Jon Fosse is mentioned in the last paragraph?A．He explains the importance of linguistic diversity.B．He has improved the international status of Norway.C．He promotes the development and preservation of Nynorsk.D．He shapes the basic writing standard of Norwegian literature.It’s one of the most common questions adults ask children: what do you want to be when you grow up? Although childhood is supposed to be fun, kids also discuss important topics regarding their futures. Now, a new study finds that children who set big goals regarding their future status and education often set themselves up for success as they age.The findings, published in the Journal of Personality and Social Psychology, are the first to reveal a connection between life goal development and future success in school or the workplace. Rodica Damian, an associate professor of psychology at the University of Houston, and otherresearchers discovered that as children grow up, their goals naturally begin to change. However, as some childhood goals fall away, other goals related to a family stay strong. These include being close to relatives, building more friendships or finding a romantic partner, and even becoming more involved in your community or helping others.During the study, researchers compared how their goals evolved as children moved from adolescence to adulthood and how a person’s goals impacted their success in school and as an employee later on. Overall, a child’s goals focusing on their education and future status were the most consistent predictors (预测因素) of income in adulthood. Simply put, when a child dreams big about doing well in school or achieving great success as an adult, these goals accurately paint a picture of how successful these children will be. So, the message is simple: dream big and dream of success, kids! Those goals can drive you to success when you grow up!“Our work proves a strong connection between a child’s life goals, educational achievement, and future occupational outcomes. This information is valuable for parents and educators who can use it to encourage children to set ambitious goals. Additionally, it helps develop strategies to support individuals in achieving their goals and reaching their full potential,” Damian concludes. 8．What does the study mainly focus on?A．Why children change their dreams over time.B．Whether childhood goals relate to future success.C．Whether early education determines future career.D．How childhood experiences impact a person’s choice.9．Which of the following goals remains strong as children grow up?A．Connecting with others.B．Achieving academic success.C．Hunting for a high-salary jobD．Improving personal social status.10．What message does the author convey in paragraph 3?A．Dreaming big is potentially associated with success.B．Childhood achievement s greatly affect future careers.C．Educational goals may be more achievable than others.D．Adjusting goals can create new opportunities for success.11．What is the purpose of the last paragraph?A．To explain the purpose of the study.B．To offer suggestions for future studies.C．To stress the significance of the study.D．To point out the limitations of the study.Spiders are master builders, expertly turning silk into complex 3D webs that serve as their home and hunting ground. To gain a deeper understanding of their world, scientists have translated the structure of a spider’s web into music.“The spider lives in an environment of vibrating (振动的) strings,” says Markus Buehler, the project’s principal investigator. “They don’t see very well, so they perceive their surroundings by detecting vibrations, which have different frequencies.” Such vibrations occur, for example, when the spider stretches a strand of silk during construction or when the wind or a trapped fly moves the web.The researchers scanned a natural spider web to capture 2D cross-sections and reconstructed its 3D network using a mathematical model. They assigned different frequencies of sound to strands of the web, creating musical “notes” that they combined in patterns based on the web’s 3D structure to generate music. Then they made a harp-like (像竖琴的) virtual instrument and played the spider web music in several live performances around the world, creating an inspiring harmony of art and science.To gain insights into how spiders build webs, the researchers also scanned a web during construction, transforming each stage into music with different sounds. “The spider’s way of ‘printing’ the web is remarkable because no support material is used, as is often needed in current 3D printing methods,” Buehler says. This knowledge could help develop new 3D printers that work like spiders, enabling them to construct complex structures without using additional materials for support.The team is also interested in learning how to communicate with spiders in their own language. They recorded web vibrations produced when spiders performed different activities, such as building a web, communicating with other spiders or sending signals to admirers. “Now we’re trying to generate signals to basically speak the language of the spider,” Buehler says. “If we expose them to certain patterns of vibrations, can we affect what they do or can we begin to communicate with them? Those are really thrilling ideas, and I believe they could be achieved inthe near future.”12．What do we know about spiders?A．They have good eyesight.B．They adapt to surroundings quickly.C．They spin a web similar to a musical structure.D．They sense the environment through vibrations.13．What is paragraph 3 mainly about?A．Steps to create spider web music.B．Ways to recreate a web’s 3D network.C．Challenges of combining music with science.D．The potential applications of 3D printing methods.14．What do the underlined words “This knowledge” in paragraph 4 refer to?A．3D printers can construct complex structures.B．Spiders’ vibration patterns vary with activities.C．Spiders build webs without using additional materials.D．The web-building process can be translated into music.15．What is Buehler’ attitude towards communicating with spiders?A．Hopeful.B．Doubtful.C．Uncertain.D．Disapproving.The storms in our lives often leave behind gifts if we open our eyes to see them. As the skies clear and you start to emerge from the loss, look for the silver linings. 16 Going through difficulties requires courage and determination. You have to dig deep and find the strength you didn’t know you possessed to overcome the challenges. 17 That self-knowledge and confidence in your abilities will serve you well going forward.Coming out the other side of trouble sometimes gives you a fresh outlook. Priorities shift.18 You have a deeper appreciation for life’s simple moments and clarity about what matters. Use that insight to guide your choices and cherish each day. 19 The old way of doing things no longer works, so you must rebuild and reimagine your life. Though change is hard, it opens up possibilities for new growth and exciting adventures. Look at this as a chance for a fresh start. You can pursue new dreams or make along-wished-for switch in your situation. The future isunwritten.Your struggles and pain have given you understanding for what others maybe going through in their own lives. You now recognize the suffering of people around you and desire to help ease their burdens. Use your experience to be there for friends and family members facing difficulties.20 Your sympathy can make a real difference.A．Dramatic changes often lead to transformation.B．They’re there if you are bent on searching for them.C．Offer a listening ear and share your story with them.D．Storms shape us, but the silver linings make us suffer.E．What seemed important before now pales in comparison.F．Now you know you have the inner power to weather storms.G．This new perspective removed previous doubt about yourself.二、完形填空In my early thirties, I decided to climb Mount Kilimanjaro, the highest mountain in Africa at 5, 895meters above sea level.After months of preparation, I started my great 21 . At the base of the mountain, I met Mik, a local porter, who warmly 22 me. Mik’s job was to carry my equipment up the mountain, set up the tent, and carry everything back down after reaching the 23 .The first day of the climb was 24 the rain forest, across a slippery , muddy ground covered with tree roots and vines (藤蔓). Mik was carrying 30 kilos on his head! By that evening, we made it to 3,000meters. The air was 25 and it was colder. When I arrived at the campsite for the night, my tent was already set up and waiting for me.Day two was much steeper and rockier. I really had to 26 what I was doing. And I felt guilty for the tough 27 Mik had to work in. Yet when I turned to Mik, he said with the biggest smile, “pole, pole,” which means “ 28 , take it easy” in his native language. I 29 back, my burden lighter somehow.Over the next five days, the climb got 30 difficult. The temperatures could 31 from 21℃to below freezing in a few hours. At 5, 700 meters, there’s only half the oxygen 32in each breath compared to at sea level. That leaves many people with severe headache. Yet Mik always had a smile and a positive attitude. It had an enormous impact on me, giving me the 33 to keep going.Even today, when I find myself 34 with anything in life, I just think back to Mik and his smile. A great attitude can bring joy to those around you, or even 35 strangers to the top of a mountain.21．A．performance B．business C．presentation D．adventure 22．A．questioned B．congratulated C．greeted D．invited 23．A．base B．peak C．forest D．border 24．A．outside B．beside C．through D．over 25．A．thinner B．softer C．drier D．sweeter 26．A．focus on B．turn on C．bring up D．break up 27．A．locations B．conditions C．atmosphere D．competition 28．A．stop B．hurry C．relax D．move 29．A．ran B．talked C．smiled D．rode 30．A．slightly B．increasingly C．automatically D．equally 31．A．swing B．circle C．climb D．last 32．A．predictable B．available C．noticeable D．affordable 33．A．impression B．platform C．opportunity D．strength 34．A．cooperating B．struggling C．associating D．communicating 35．A．persuade B．introduce C．motivate D．recommend三、语法填空阅读下面短文，在空白处填入1个适当的单词或括号内单词的正确形式。
扫描印刷图像去网纹方法研究与应用
第１章引言１．１课题背景随着数字多媒体的越来越普及，扫描作为获取数字化信息的一个重要渠道，其应用也变得越来越广泛。
但是几乎所有的报纸、期刊杂志、书籍等印刷品都是采用非连续色阶的半色调（Ｈａｌｆｔｏｎｉｎｇ）印刷技术印制的［１】，这种技术是通过控制印刷网点的大小（调幅加网ＡｌＩＳｃｒｅｅｎｉｎｇ）或者网点的密度（即单位面积内网点的数量，叫做调频加网刚Ｓｃｒｅｅｎｉｎｇ或随机加网）来表现图像的灰度或者色彩［２］。
传统的加网（Ｓｃｒｅｅｎｉｎｇ）技术采用调幅加网，其网格大小是相同的，也就是印刷网点之间的距离是相等的，通过控制网格内墨点的大小来体现图像灰度的级别或色彩浓淡。
决定印刷品质量的参数主要是网格的大小，即相邻网点的距离，距离越小网格越小，印刷品质量越高，通常在表示时采用单位长度内网线的数量来衡量印刷品质量，以Ｉｐｉ（１ｉｎｅｐｅｒｉｎｃｈ）为单位，网线越多（即网格越小）印刷品质量就越好，例如报纸大约为８５１ｐｉ，杂志１３３１ｐｉ，某些高品质的印刷品为１７０１ｐｉ甚至更高。
对于灰度图，一般采用４５。
角（某些印刷品则采用１５。
或７５。
）进行加网，如图ｌ—ｌ所示。
图Ｉ－－Ｉ灰度加网原理示意图（上图为连续色阶的灰度图像，下图为上图加网之后的图像，其中一条网线已标出）而对于彩色图像，则是采用四色（ＣＷＹＫ）印刷，每一色的加网原理与灰度加网相似，为了更好地体现色彩和印刷质量，对四种颜色分别采用不同的角度来加网。
一般青色（Ｃｙａｎ）采用１０５。
即１５。
，品红（Ｍａｇｅｎｔａ）采用７５。
，黄色（Ｙｅｌｌｏｗ）采用９０。
即０。
，黑色（Ｂｌａｃｋ）则用４５。
，如图１—２所示。
图Ｉ－－２四色加网原理示意图摩尔纹（Ｍｏｉｒ６Ｐａｔｔｅｒｎ）是由两个频率的叠加引起的，如图１－－３所示，或者是由于两个频率的角度不同引起的，如图ｌ一４所示，因此对于彩色印刷，有四个角度的频率，所以彩色印刷品本身就含有摩尔纹（梅花状）。
一般来说，两个频率比较接近时，摩尔纹比较明显，两个频率相差较大时，摩尔纹越不明显，当相差８倍以上时，摩尔纹可以忽略［１］。
多贝西小波定义
多贝西小波定义
多贝西小波（Daubechies wavelets）是由比利时数学家Ingrid Daubechies 在1988年提出的一种正交小波基。
它是一种用于信号处理和数据压缩的数学工具，能够将信号在时频域上进行有效的分析和表示。
多贝西小波是一类紧支撑正交小波基，它具有非常好的时域局部化以及频域局部化特性。
多贝西小波的基函数是通过特定的滤波器系数来构造的，这些滤波器系数可以控制小波基函数在时域和频域的特性。
多贝西小波具有一系列重要的性质，如紧支撑性、正交性、可逆性和多分辨率性等。
这些性质使得多贝西小波在信号处理领域中得到了广泛应用，包括图像压缩、图像去噪、图像增强、数据压缩、信号分析等。
多贝西小波的定义是通过一组滤波器系数来实现的，这些滤波器系数决定了多贝西小波的基函数的形状和特性。
根据滤波器系数的不同选择，可以得到不同类型的多贝西小波，如db1、db2、db3等。
每一种多贝西小波都具有特定的频域和时域特性，可以根据具体应用的需求选择合适的多贝西小波。
总而言之，多贝西小波是一种用于信号处理的数学工具，具有良好的时频局部化特性，被广泛应用于图像压缩、信号分析、数据压缩等领域。
小波的几个术语及常见的小波基介绍图文稿
小波的几个术语及常见的小波基介绍集团文件发布号：（9816-UATWW-MWUB-WUNN-INNUL-DQQTY-小波的几个术语及常见的小波基介绍本篇是这段时间学习小波变换的一个收尾，了解一下常见的小波函数，混个脸熟，知道一下常见的几个术语，有个印象即可，这里就当是先作一个备忘录，以后若有需要再深入研究。
一、小波基选择标准小波变换不同于傅里叶变换，根据小波母函数的不同，小波变换的结果也不尽相同。
现实中到底选择使用哪一种小波的标准一般有以下几点：1、支撑长度小波函数Ψ(t)、Ψ(ω)、尺度函数φ(t)和φ(ω)的支撑区间，是当时间或频率趋向于无穷大时，Ψ(t)、Ψ(ω)、φ(t)和φ(ω)从一个有限值收敛到0的长度。
支撑长度越长，一般需要耗费更多的计算时间，且产生更多高幅值的小波系数。
大部分应用选择支撑长度为5~9之间的小波，因为支撑长度太长会产生边界问题，支撑长度太短消失矩太低，不利于信号能量的集中。
这里常常见到“紧支撑”的概念，通俗来讲，对于函数f(x)，如果自变量x在0附近的取值范围内，f(x)能取到值；而在此之外，f(x)取值为0，那么这个函数f(x)就是紧支撑函数，而这个0附近的取值范围就叫做紧支撑集。
总结为一句话就是“除在一个很小的区域外，函数为零，即函数有速降性”。
2、对称性具有对称性的小波，在图像处理中可以很有效地避免相位畸变，因为该小波对应的滤波器具有线性相位的特点。
3、消失矩在实际中，对基本小波往往不仅要求满足容许条件，对还要施加所谓的消失矩（Vanishing Moments）条件，使尽量多的小波系数为零或者产生尽量少的非零小波系数，这样有利于数据压缩和消除噪声。
消失矩越大，就使更多的小波系数为零。
但在一般情况下，消失矩越高，支撑长度也越长。
所以在支撑长度和消失矩上，我们必须要折衷处理。
小波的消失矩的定义为，若其中，Ψ(t)为基本小波，0<=p<N。
则称小波函数具有N阶消失矩。
Wavelet计算信号处理说明书
A. Aldroubi, Oblique and hierarchical multiwavelet bases, To appear in Applied and Camp. Harmonic Analysis, 1997.
J. Allen, Cochlear modeling, IEEE ASSP Magazine, 2:3-29, 1985.
[BMG92] A. Baskurt, I. E. Magnin, and R. Goutte, Adaptive discrete cosine transform coding algorithm for digital mammography, Optical Engineering, 31:1922-1928, Sept. 1992.
A. Aldroubi and M. Unser, Sampling procedures in function spaces and asymptotic equivalence with Shannon's sampling theory, Numer. Funct. Anal. Optimiz., 15:1-21, 1994.
[BT92]
J. J. Benedetto and A. Teolis, An auditory motivated time scale signal representation, IEEE-SP International Symposium on Time-Frequency and Time-Scale analysis, Oct. 1992.
wavelet(小波变换资料)
Sum Results
X1 cos (w) X2 cos (2w) X3 co...s (3w) Xy cos(yw)
Figure 4 depicts the process pictorially: The vectors in the figure just happen to be pointing in a cardinal direction because the strobe frequencies are all multiples of the vector (phasor) rotation rate, but that is not normally the case. Usually the vectors will point in a number of different directions, with a resultant in some direction other than straight up.
For example, in an eight filter bank, a DFT would require 512 computations, while an FFT would only require 56, significantly speeding up processing time.
Fundamental
Third Harmonic Fifth Harmonic
Sum - Approximation of (Square Wave)
Digital Sampling of Waveforms
In order to process a signal digitally, we need to sample the signal frequently enough to create a complete “picture” of the signal. The discrete Fourier transform (DFT) may be used in this regard. Samples are taken at uniform time intervals as shown in Figure 2 and processed.
2024春高中英语Unit12Innovation单元词汇预通关课件北师大版选择性必修第四册
emerge ， decline ， boil ， explode ， regulate ， illustrate ， preserve
4．Profits have ________ as a result of the recent drop in sales.
5 ． You can ________ the temperature in the house by adjusting the thermostat(恒温器).
【答案】4．declined 5．regulate
6．One example will suffice to ________ the point. 7．The sun ________ from behind the clouds. 8．We want to ________ the character of the town while improving the facilities. 【答案】6．illustrate 7．emerged 8．preserve
navy，accuracy，missile，physicist，tube，circuit 7 ． We can predict changes with a surprising degree of ________. 8．A defect was found in the electrical ________. 9 ． These guns will destroy or disable any incoming ________. 【答案】7．accuracy 8．circuit 9．missile
UNIT 12 INNOVATION
单元词汇预通关
一、中英互译 1．n. 轨道 vi. & vt. 沿轨道运行________ 2．adv. 后来，随后________ 3．adv. 有效率地，高效能地________ 【答案】1．2．subsequently 3．efficiently
wavelet as features for objective recognition
image into a hierarchical set of approximations and details. For images analysis we used two-dimensional wavelets and corresponding scaling functions obtained from onedimensional wavelets by tensorial product. The discreet wavelet transform (DWT) of a signal is calculated by passing it through a series of filters (high and low pass filters) and then down-sampled, as we can see from Figure 1. At each level, the signal is decomposed into low and high frequencies, and this decomposition halves the resolution since only half the number of samples are retained to characterize the entire signal. The algorithm retains the even indexed columns respectively rows. Based on this scheme, two-dimensional DWT leads to a decomposition of approximation coefficients at level j in four components: the approximation (CA) at level j+1 and the details (CD) in three orientations (horizontal, vertical, and diagonal). Due to successive downsampling by 2, the signal length must be a power of 2, or a multiple of a power of 2, and the length of the signal determines the maximum levels in which the signal can be decomposed. An example of this wavelet decomposition on level two, applied on a vehicle image can be seen in Figure 2.
小波_基础知识
说明
Z表示整数集合 R表示实数集合 C表示复数集合 Z ＋表示正整数集合 R n 表示n为欧氏空间 内积 x, y
x(t ) y (t )d t
R
常用的距离空间
1.n维欧氏空间R
n
n维向量x ( x1 , x2 , , xn )的全体所组成的集合 . x, y R n , 定义距离 ( x, y ) [ ( xi yi ) ]1/ 2
正如1807年法国的热学工程师 J.B.J.Fourier提出任一函数都能展开成 三角函数的无穷级数的创新概念未能得到 著名数学家grange，place 以及A.M.Legendre的认可一样。幸运的是， 早在七十年代，A.Calderon表示定理的发 现、Hardy空间的原子分解和无条件基的 深入研究为小波变换的诞生做了理论上的 准备，而且J.O.Stromberg还构造了历史 上非常类似于现在的小波基；
1822年Fourier变换,在频域的定位最准确，无任 何时域定位能力。 函数，时域定位完全准确，频域无任何定位能 力 1946年Gabor变换，STFT，窗函数的大小和形状与 时间和频率无关而保持固定不变。不构成正交基。 1982年Burt提出金字塔式图像压缩编码，子带编 码(subband coding),多采样率滤波器组 (multirate sampling filter bank). 1910年Harr提出规范正交基。 1981年Stormberg对Harr系进行改进，证明了小波 函数的存在。 1984年,Morlet提出了连续小波
小波变换如同一台可变焦距的数学显微
镜，改变各种焦距便可探测到被处理信 号中所隐含的奇异点并识别出它的性质， 或分析出非平衡信号所包含的各种成分， 从而可有效地探测并诊断出精密复杂设 备中的疑难故障，是该领域具有明显应 用前景的前沿课题
06小波变换压缩算法
S.Mallat于1988年在构造正交小波基时提出了多分辨率分析 (multiresolution analysis)的概念, 从空间上形象地说明了小波的 多分辨率的特性,提出了正交小波的构造方法和快速算法，叫做 Mallat算法。
38
小波变换实例
一维哈尔小波变换
哈尔函数定义
(x)
1 0
0 x 1 其他
小波变换实例
一维哈尔小波变换
哈尔函数定义
1 0 x 1
(x) 0 其他
基函数
一组线性无关的函数，以用来构造任意给定的 信号
小波变换实例
一维哈尔小波变换
哈尔基函数
最简单的基函数
ij (x) (2 j x i)
DCT 压缩的优点
简单、 便于硬件实现
3
小波变换用于图像压缩的理由
DCT 压缩的缺点
图像是分块处理， 沿块的边界方向相关性被破坏，出现 “blocking
artifacts”
4
傅里叶变换
信号表示
多种方式信号的描述:
例如一个函数表达式，这就是信号的时域表示，
傅里叶变换
1822年,傅里叶提出频率的概念: 通过傅里叶正变换将信号在频 域分解，获得信号的频谱，再通过反变换重建原始信号。
Gabor变换： 时窗函数＝Gauss函数时 时窗函数的Fourier变换仍然是Gauss函数，保证了窗口傅立
叶变换在频域内也有局域化的功能。
窗口傅里叶变换
matlab 小波种类
Matlab小波种类一、引言小波变换是一种基于信号的时频分析方法，通过将信号表示为一组小波基函数的线性组合，可以对信号的时频特性进行精确分析。
Matlab是一种强大的数值计算和数据可视化软件，提供了多种小波种类和相关函数，方便用户进行小波分析研究和应用开发。
本文将介绍Matlab中常用的小波种类，包括Daubechies小波、Symlets小波、Coiflets小波、Haar小波等。
我们将逐个讨论每种小波的特点、使用方法和相关函数，帮助读者更好地理解和应用小波分析。
二、Daubechies小波Daubechies小波是最常用的小波种类之一，由Ingrid Daubechies于1988年提出。
它具有紧支撑、对称性和正交性的特点，适用于信号的精确表示和压缩。
Daubechies小波的主要特点如下：1.紧支撑：Daubechies小波具有有限的非零系数，因此它可以提供信号的紧支撑表示，对于时间和频率局部化特性更好。
2.对称性：Daubechies小波的低通和高通滤波器具有对称性，可以保持信号的平移不变性。
3.正交性：Daubechies小波是正交小波，对于信号的变换和重构过程，可以保持信号能量不变。
在Matlab中，可以使用wavename函数指定Daubechies小波的阶数和名称，例如db1表示Daubechies小波的阶数为1。
三、Symlets小波Symlets小波是Daubechies小波的变种，也具有紧支撑、对称性和正交性的特点。
Symlets小波在时间和频率局部化特性上更加平衡，适用于需要较好时间和频率分辨率的信号分析。
Symlets小波的主要特点如下：1.时间和频率局部化：Symlets小波在时间和频率上更加平衡，可以在需要兼顾时间和频率分辨率的场景中更好地适应。
2.对称性：Symlets小波也具有对称的滤波器，可以保持信号的平移不变性。
3.正交性：Symlets小波是正交小波，能够保持信号的能量不变。
拿破仑也是法兰西科学院的数学院士
[quote]拿破仑•波拿巴（Napoléon Bonaparte），法兰西第一共和国第一执政（1799年- 1804年），法兰西第一帝国及百日王朝的皇帝（1804年- 1814年，1815年）。
也带领考察团出访非洲，计划建造一座横跨欧非的大桥。
但因为仪器的丢失，计划未能实现。
但留下的数据有巨大的价值。
－－－－－－－－－－－－－－－－－－我想起从前看到的关于拿破仑选法兰西科学院院士的经历来了。
可能世人对作为政治家、军事家的拿破仑了解的多，对于他还是法兰西科学院的院士知之甚少了，但他却真是货真价实的院士。
法兰西科学院的院士可不像中国现在的两院院士这么逐年增加，而是有固定名额的，基本上死一个院士才能增补一个院士，所以有很多人虽然在科学上贡献很大，但却因为在世的时候不好，终其一生也不可能获得这样的荣誉，而拿破仑竟然碰到了一个候选院士的机会。
当然，与他竞争的都是真正的科学家，但他们面对这样一个声威日隆的军事天才实在是相形见绌，没听说拿破仑私下做过什么小动作，他还是顺利当选了。
虽然，后来的人发现当年的选举统计数据计算有误(累计票数之和远大于实际投票数)，但拿破仑的当选却真正是符合程序规范的。
这一点值得我们今天的人学习。
－－－－－－－－－－－－－－－－－－－拿破仑与科学家赫赫有名的拿破仑，是举世公认的军事家。
然而，鲜为人知的是，拿破仑真正看重的是科学家。
作为法兰西科学院至高无上的145位院士其中的一名，他是一名货真价实的科学家。
尽管他一生绝大多数时间是军事行动，但他更喜欢与科学为伍——在他的身边总是簇拥着数学家、化学家、天文学家……即使在行军作战时也兴致勃勃跟学者讨论科学问题，甚至出“难题”。
“尊敬的院士们”，拿破仑说：“让我给你们出一道题，不用直尺，仅用圆规，你们能四等分一个圆吗？”正因为拿破仑在数学上的高深造诣，他与11位候选人竞争法兰西科学院院士，并最后当选，成为数学部院士，跻身于科学家行列。
小波重构 python
小波重构 python小波重构是一种信号处理技术，它可以将信号分解成不同频率的小波系数，并通过重构来还原原始信号。
在Python中，可以使用PyWavelets库来实现小波重构。
首先，需要安装PyWavelets库。
可以使用以下命令在终端中安装：```pip install PyWavelets```接下来，可以使用以下代码来实现小波重构：```pythonimport pywtimport numpy as np# 生成测试信号x = np.linspace(0, 1, num=1024)y = np.sin(2 * np.pi * 5 * x) + np.sin(2 * np.pi * 20 * x)# 小波分解coeffs = pywt.wavedec(y, 'db4', level=4)# 将小波系数设置为0，实现降噪coeffs[1:] = (pywt.threshold(i, value=0.5, mode='soft') for i in coeffs[1:])# 小波重构y_reconstructed = pywt.waverec(coeffs, 'db4')# 绘制原始信号和重构信号import matplotlib.pyplot as pltplt.plot(x, y, label='Original signal')plt.plot(x, y_reconstructed, label='Reconstructed signal')plt.legend()plt.show()```在上述代码中，首先生成了一个测试信号，然后使用`pywt.wavedec`函数将信号进行小波分解。
接着，将小波系数设置为0，实现降噪。
最后，使用`pywt.waverec`函数将小波系数重构成原始信号，并绘制出原始信号和重构信号的图像。
pywt.wavedec 用法
PyWavelets（PyWT）是一个用于小波分析的Python库，包括小波变换和滤波器设计。
`wavedec` 是PyWavelets库中的一个函数，用于进行小波分解。
小波分解是一种将信号分解成不同频率组成部分的方法，这些组成部分在时间和频率上都是局部的。
`wavedec`函数可以对手中的数据进行多级小波分解。
`wavedec` 的基本用法如下：```pythonimport pywtcoeffs = pywt.wavedec(data, wavelet, level=None, axis=-1)```其中：* `data` 是你想要进行小波分解的数据。
* `wavelet` 是你选择的小波类型。
PyWavelets库支持许多不同的小波类型，如'haar'，'db2'，'coif1'，'coif2'等等。
* `level` 是你希望进行的分解级别。
如果未指定，`wavedec` 将使用默认级别。
* `axis` 参数定义了分解沿着哪个轴进行。
默认情况下，它沿着数据的最后一个轴进行。
`wavedec` 函数返回一个小波系数（Wavelet Coefficients）列表，这些列表包括每个级别的近似系数和细节系数。
这些系数描述了数据在不同级别和不同频率下的特性。
这些返回的小波系数可以用于重构原始数据。
可以使用`pywt.waverec` 函数来重构数据。
这是一个使用`wavedec` 的简单例子：```pythonimport pywtimport numpy as np# 创建一个简单的信号data = np.sin(2*np.pi*np.linspace(0, 1, num=1024))# 选择小波类型并分解信号coeffs = pywt.wavedec(data, 'haar')# 打印分解的级别print("Decomposition level:", coeffs)```这个例子将打印出信号的分解级别，这里只显示了一个级别，因为'haar' 小波在每个尺度上只有一个系数。
shannon wavelets公式表达
shannon wavelets公式表达Shannon小波是一种在信号处理和数据压缩中经常使用的数学工具。
它基于一种名为Shannon小波函数的函数族，可以将信号分解成不同尺度和频率的成分。
Shannon小波公式是用来计算Shannon小波变换的数学公式。
Shannon小波公式可以表示为：\begin{equation}\psi_{a,b}(t) = \frac{1}{\sqrt{a}} \cdot \psi \left( \frac{t-b}{a} \right)\end{equation}其中，$\psi_{a,b}(t)$表示尺度参数为$a$，位移参数为$b$的Shannon小波函数，$\psi(t)$是标准Shannon小波函数。
Shannon 小波函数的具体形式可以根据需求选择，常见的有Haar小波、Daubechies小波等。
Shannon小波公式的含义是将标准Shannon小波函数在时间和尺度上进行缩放和平移，从而得到不同尺度和频率的小波函数。
尺度参数$a$控制着小波函数的宽度，较大的$a$对应着低频成分，较小的$a$对应着高频成分；位移参数$b$则控制着小波函数在时间轴上的位置。
利用Shannon小波公式，可以将信号分解成不同尺度和频率的成分。
这种分解方法被称为Shannon小波变换。
通过对信号进行Shannon 小波变换，可以得到信号的小波系数，这些小波系数可以用于信号的分析和压缩。
Shannon小波变换可以应用于多个领域。
在信号处理中，它可以用于信号的降噪、边缘检测等任务。
在图像处理中，它可以用于图像的压缩和特征提取。
在数据分析中，它可以用于时间序列的分析和预测。
此外，Shannon小波还可以应用于音频处理、视频处理等领域。
尽管Shannon小波在各个领域都有广泛的应用，但它也存在一些限制。
首先，Shannon小波变换是一种线性变换，对于非线性信号的处理效果可能不理想。
其次，Shannon小波函数的选择对于信号分析的结果有很大影响，不同的小波函数适用于不同类型的信号。
2025高考英语步步高大一轮复习讲义人教版2选择性必修第四册Unit Science Fiction
2025高考英语步步高大一轮复习讲义人教版2选择性必修第四册Ⅰ.阅读单词——会意1．fiction n．小说；虚构的事2．science fiction(informal sci-fi)科幻小说(或影片等)3．bonus n．意外收获；奖金；红利4．ridiculous adj.愚蠢的；荒谬的；荒唐的5．absurd adj.荒谬的；荒唐的6．nail n．指甲；趾甲；钉子v t.(用钉子)钉牢；固定7．suspend v t.悬；挂；暂停；暂缓8．ladder n．梯子；阶梯9．whereas conj.然而；但是；尽管10．rumour n．谣言；传闻11．presume v t.& v i.假定；假设12．fare n．车费；船费；飞机票价13．gramme (NAmE gram)n.克(重量单位)14．venue n．活动场地(如音乐厅、会场等)15．alien n．外星人(生物)；外国人adj.陌生的；外星的；外国的16．inaction n．无行动；不采取措施17．lever n．操纵杆；杠杆18．panel n．控制板；仪表盘；专家咨询组19．inch n．英寸(长度单位，等于2.54厘米)20．grip v t.& v i.紧握；抓紧21．hazy adj.模糊的；朦胧的；困惑的22．puff n．(烟、气等的)一缕；少量；喘息23．jolt n．震动；摇晃；颠簸v t.& v i.(使)震动；摇晃24．flip v t.& v i.(使)快速翻转；(用手指)轻抛25．overstatement n．夸大；夸张Ⅱ.重点单词——记形1．integrity n．诚实正直；完整；完好2．dignity n．庄重；庄严；尊严3．salary n．薪水；薪金4．saleswoman n．女售货员；女推销员5．dismiss v t.让(某人)离开；解散；解雇；消除6．weekly adj.每周的n．周刊7．chairwoman n．女主席；女董事长；女委员长8．flour n．面粉；(谷物磨成的)粉9．salesman n．售货员；推销员10．superior adj.更好的；占优势的；(在级别或重要性上)更高的11．labour n．劳动(者)；体力劳动v i.奋斗；努力工作12．leather n．皮革；[pl.]皮衣；皮外套13．backwards (NAmE backward)ad v.向后；倒着；往回14．niece n．侄女；外甥女15．fetch v t.(去)拿来；(去)请来16．handkerchief n．手帕；纸巾17．lamp n．灯；台灯18．pace n．速度；步伐；节奏v t.& v i.确定速度；调整节奏19．random adj.随机的；不可思议的20．maximum adj.最大极限的n．最大量；最大限度21．stun v t.使震惊；使昏迷Ⅲ.拓展单词——悉变1．appointment n．预约；约会；委任→appoint v t.任命；委派；指定；约定2．guilty adj.内疚的；有罪的；有过失的→guilt n．内疚；罪行；罪过3．declare v t.表明；宣称；公布→declaration n．宣称；声明4．calculate v t.计算；核算；预测→calculator n．计算器→calculation n．计算5．blurred adj.模糊不清的；难以区分的→blur v．(使)变得模糊不清；(使)视线模糊6．division n．分开；分隔；差异；除(法)→divide v．分开；分散；分配；分享7．urge n．强烈的欲望；冲动v t.催促；力劝；大力推荐→urgent adj.紧急的；急迫的8．explode v i.& v t.爆炸；爆破→explosion n．爆炸；(感情)爆发；激增9．mud n．泥；泥浆→muddy adj.泥泞的1．comply /kəm'pla I/v i.遵守；服从comply with遵从；服从2．abundance /ə'bʌndəns/n．大量；丰盛；充裕3．commute /kə'mjuːt/v i.通勤；长途上下班n．每天上班的路程4．pledge /pledʒ/v t.保证给予；正式承诺；发誓n．誓约；捐款承诺5．controversy /'kɒntrəvɜːsi；kən'trɒvəsi/n．争论；争议6．flawless /'flɔːləs/adj.完美的；无瑕的7．whilst /wa I lst/conj.当……时；在……的过程中；尽管8．prestigious /pre'st I dʒəs/adj.有威望的；声誉高的9．collaborate /kə'læbəre I t/v i.合作；协作collaborate with与……合作10．buzz /bʌz/n．喧闹；嘈杂声；嗡嗡声v i.发出嗡嗡声；充满兴奋；闹哄哄Ⅳ.背核心短语1．test out检验；测试2．more like更像是；更接近3．on a．..basis根据；以……的方式(基准)4．pros and cons事物的利与弊；支持与反对5．superior to比……更好；更胜一筹6．take over占上风；取而代之；接管；接手7．conflict with与……冲突或抵触8．turn out关掉；熄灭；在场；使朝外；结果是9．fall away(逐渐)减少；消失10．have an urge to有强烈的欲望做某事Ⅴ.悟经典句式1．On the second morning，Tony brought her breakfast and then asked her whether she needed help dressing.(whether引导的宾语从句)第二天早晨，托尼给她端来了早餐，还问她是否需要帮忙穿衣打扮。
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-Graphicslid4=WT[ li, d4]; lit4=ThresHard[ lid4,1.2]; lin4=WR[ lit4, d4];
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DataPlot[lin4]
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Thresholding
Hard Thresholding
ThresHard[list_List,thresh_]:=Module[{}, Table[ If[ Abs[list[[i]]] < thresh , 0 , list[[i]]] , {i,1,Length[list]}] ]
Soft Thresholding
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lits6=ThresSoft[lid6, 1]; lins6=WR[ lits6, d6]; DataPlot[lins6]
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-Graphics-
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litu6=ThresUniv[lid6]; linu6=WR[ litu6, d6]; DataPlot[linu6]
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-Graphics12
Appl
bb=aa+Table[(Random[]-0.5),{1024}]; DataPlot[bb]
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-Graphicscc=WT[bb,d4]; dd=ThresSoft[cc, 1.3]; ee=WR[dd,d4]; DataPlot[ee]
-Graphics-
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Appl
Different thresholding policies and DAUB #6.
lid6=WT[li, d6]; lith6=ThresHard[lid6, 1.2]; linh6=WR[lith6, d6]; DataPlot[linh6]
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-Graphicsff=WT[bb,d10];
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gg=ThresSoft[ff,0.9]; hh=WR[gg, d10]; DataPlot[hh]
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-Graphicsii=WT[bb,d2]; jj=ThresSoft[ii,1]; kk=WR[jj, d2]
Trimming
Trimm[list_List, lam_]:= Table[ Abs[list[[i]]] Tanh[ lam list[[i]] ], {i,1, Length[list]} ];
Examples
Example from the tutorial, page 6. example={1,0,-3,2,1,0,1,2};
d1={0.7071067811865475, 0.7071067811865475}; d2={0.482962913145, 0.836516303738, 0.224143868042, -0.129409522551}; d4={0.230377813309, 0.714846570553, 0.630880767930, -0.027983769417,-0.187034811719, 0.030841381836 0.032883011667 -0.010597401785}; d6={0.1115407433501098, 0.4946238903984543, 0.7511339080210969, 0.3152503517091975, -0.2262646939654413,-0.1297668675672621,
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DataPlot[er1]
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-GraphicsFunction Sin[x]+Sin[2 x]: Denoising by different wavelet bases and thresholding policies. Plot[ Sin[x] + Sin[2 x], {x, 0, 2 Pi}]
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Appl
lit1=ThresHard[lid1, 1.2]; Short[lit1, 3] {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, <<978>>, 2.08766, 3.70591, -1.67585, 0, 3.73001, 1.59364, -4.72405, -4.12402, 3.28927, 4.3442, -3.79269, 14.0525, 14.5291, 20.6674, 0} lin1=WR[lit1, d1]; DataPlot[lin1]
Appl
DataPlot[example]
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-Graphicsed1=WT[ example, d1] {0.7071067811865475, -3.53553, 0.7071067811865475, -0.707107, 1., -1., -1.41421, 1.41421} et1=ThresHard[ed1, 0.9] {0, -3.53553, 0, 0, 1., -1., -1.41421, 1.41421} er1=WR[et1, d1] {0.5, 0.5, -3., 2., 0.5, 0.5, 1.5, 1.5}
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-Graphicslid1=WT[li, d1]; Short[lid1,3] {-0.191229, -0.342343, -0.288343, -0.475401, -0.510271, 0.0194839, -0.227568, -0.071192, -0.323387, -0.0477633, -0.417172, -0.0400722, -0.0278623, <<10>>7, <<1003>>, 3.28927, 4.3442, -3.79269, 14.0525, 14.5291, 20.6674, 0.0359969}
lid10=WT[ li, d10]; lit10=ThresSoft[ lid10, 1.2]; lin10=WR[lit10, d10]; DataPlot[ lin10 ]
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-Graphicsaa=Table[N[Sign[x] Cos[x]],{x,-1, 511/512, 1/512}]; DataPlot[aa]
Appl
Mathematica Activities
Off[General::spell] Packages needed: <<waves.m <<Statistics`NormalDistribution`
Filters corresponding to Daubechies wavelets
Daubechies family of compactly supported wavelets, DAUB #n, is the orthonormal basis in L^2(R) with properties: (1) support of psi(x) is the interval [0,2 n-1]; (2) 0=int_R psi(x) dx = int_R x psi(x) dx = ... = int_R x^(n-1) psi(x) dx, and (3) psi is in C^{\alpha}, alpha={0.5-epsilon, 0.915, 1.275, 1.596, 1.888, 2.158,...} for n=2,3,4,5,6,... The Haar wavelet is DAUB #1.
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Appl
0.0975016055873234, 0.0275228655303057, -0.0315820393174862, 0.0005538422011614892, 0.004777257510945529, -0.001077301085308485}; d10={0.026670057901, 0.188176800078, 0.527201188932, 0.688459039454, 0.281172343661,-0.249846424327, -0.195946274377, 0.127369340336, 0.093057364604, -0.071394147166,-0.029457536822, 0.033212674059, 0.003606553567,-0.010733175483, 0.001395351747, 0.001992405295,-0.000685856695,-0.000116466855, 0.000093588670,-0.000013264203}; Ploting the data set DataPlot[li_List]:= Plot[ li[[ Floor[ Length[li] x ] + 1 ]], {x,0,0.99999}]