Slides_Chapter6
lecture6-slides
2
3
4
5
Information Sets, cont’ d
1
Every node in the same information set must have the same number of choices. If not, the decision maker can tell where he is from looking at the number of choices he has. We also assume that players remember their own choices and do forget information that they know previously. This means that (1) every node in the same information set must involve the decision maker choosing the same choices in the past, and (2) if a player can distinguish between two nodes in the past, then the successors of that two nodes must not be in the same information set of the same player. Examples.
Extensive Form Games (Overview)
1 2
There are two ways to describe any strategic situations. Normal-form games describe a game by its strategies— complete contingent plans of how to react in each possible scenario. Extensive-form games also explicitly describe the game is played through time, including details about who moves …rst, who moves second, and so on. In games with multiple moves/stages/steps, …nding Nash equilibria could be hard. There are also typically many Nash equilibria. Looking at the extensive form simpli…es the process of …nding Nash equilibria and also rules out equilibria that are unreaonable.
Powerpoint_Slides_to_Chapter_01 WITH NOTES
11. Understand how this book is organized
5
DEFINITION OF MANAGEMENT
Its Nature and Purpose
Management is the process of designing and maintaining an environment in which individuals, working together in groups, efficiently accomplish selected aims.
24
The Systems Model of Management
globalization
5. Explain the concepts of productivity, effectiveness, and efficiency
6. Describe the evoபைடு நூலகம்ution of management and some recent
contributions to management thought
9
Time Spent in Carrying Out Managerial Functions
10
Managerial Skills and the Organizational Hierarchy
The four skills for administrators are: Technical Human Conceptual Design skills
Productivity implies effectiveness and efficiency in individual and organizational performance Effectiveness is the achievement of objectives Efficiency is the achievement of the ends with the least amount of resources (time, money, etc.)
《跨文化交际语篇分析法》详解
“There is a taxi at the door.” →different way of being “at” the door→ the taxi at some distance at the door in a roadway or a driveway The difference lies in what we know about men and taxis and how they wait “at” doors.
a. walking down the street and stop a stranger to ask —— “It’s 2:00” —— “Thank you”
b. in a elementary school classroom teacher “ What time is it?” —— “It’s 2:00” —— “Very good, Frankie.” c. A husband and a wife are at dinner in the home of friends wife: “ What time is it ?” =“ Don’t you think it is time we were leaving?”
《与陌生人交际——跨文化交流方法》William
B. Gudykunst /Young Yun Kim，2007， 上海外语教育出版社
4. Dynamics of Intercultural Communication 《跨文化交际动力》
Carley H. Dodd，2006年 上海外语教育出版社
2. Sentence-level ambiguity in language
The meaning in sentence resides not in the sentence alone but in the situation in which it is used as well. Example: “What time is it?”
slides_chap07
Query Languages
There are a number of techniques to enhance the usefulness of the queries Some examples are the expansion of a word to the set of its synonyms or the use of a thesaurus Some words which are very frequent and do not carry meaning (called stopwords) may be removed We refer to words that can be used to match query terms as keywords
Queries: Languages & Properties, Modern Information Retrieval, Addison Wesley, 2010 – p. 10
Word Queries
The result of word queries is the set of documents containing at least one of the words of the query Further, the resulting documents are ranked according to the degree of similarity with respect to the query To support ranking, two common statistics on word occurrences inside texts are commonly used
Additionally, it may be required that the exact positions in which a word occurs in the text should be provided This might be useful for highlighting word occurrences in snippets, for instance, during the display of results
chapter06TheGlobalStockMarket(国际投资,英文版)
Many stock markets also trade other financial
• Offer little or no investment advice
• No ‘hand-holding’ services provided
• Little opportunity for churning
• Lower commissions
– Range from $20 to $50
12
Trading on Margin
When opening a new account with a
A variety of different stock markets exist
– For instance, Germany’s market is over 400 years old while Tanzania’s began in 1998
– Macedonia has only two companies trading on its stock exchange while India has 5,840
instruments
Consolidations are merging stock markets and
technology is pulling national markets together
Francis & Ibbotson
Chapter 6: The Global Stock Market
Chapter6-Feature
Harder still?
NASA Mars Rover images
Answer below (look for tiny colored squares…)
NASA Mars Rover images with SIFT feature matches Figure by Noah Snavely
Quantity: many features can be generated for even small objects
Efficiency: close to real-time performance
Extensibility: can easily be extended to wide range of differing feature types, with each adding robustness
directions
1 and 2 are small; E is almost constant
in all directions
“Flat” region
“Edge” 1 >> 2 1
27
Harris Detector: Mathematics
Measure of corner response:
2
2
w( x, y) uI
x, y
x
u , v
I x2 w( x, y ) x, y IxI y
I x I y u 2 Iy v
21
Harris Detector: Mathematics
For small shifts [u,v] we have a bilinear approximation:
Slides
概率模型
– Queuing Theory – Monte Carlo – Markov：HMM，MDP，POMDP，CPOMDP
Algorithm
出版社：机械工业出版社原书名： Introduction to Algorithms(Second Edition) 原出版社： The MIT Press 作者： (美)Thomas H.Cormen Charles E.Leiserson 美 L.Rivest Clifford Stein 丛书名： 国外优秀信息科学与技术系列教学用书 出版社：高等教育出版社
合理假设
工行银行经理韦小宝为考虑设置一种新的单队列排队系统， 需要对现有系统进行分析。现有系统中有5个服务点，当顾 客走进银行，他们可能选择5个服务点中任一个。在繁忙期 间，两位顾客到达的平均间隔时间是3分钟，为一位顾客服 务的平均时间为2.5分钟。请建立银行服务模型。
1. 2. 假定服务时间T服从平均值为2.5的指数分布（即参数 =0.4）；或 假定服务 时间T服从平均值为2.5的正态分布，应设定适当的标准差值. 将顾客的来到看成泊松流，即顾客到达的间隔时间相互独立，都服从参数为 1/3的指数分布。
工行银行经理韦小宝为考虑设置一种新的单队列排队系统， 需要对现有系统进行分析。现有系统中有5个服务点，当顾 客走进银行，他们可能选择5个服务点中任一个。在繁忙期 间，两位顾客到达的平均间隔时间是3分钟，为一位顾客服 务的平均时间为2.5分钟。请建立银行服务模型。 排队论模型 （Queuing Theory）X/Y/Z/A/B/C X—顾客相继到达的间隔时间的分布； Y—服务时间的分布；
出国？读研？就业？
出国、读研
– 目标？ – 手段
给自己足够的时间思考自己的方向
slides01-01
© 2006 Pearson/Prentice Hall
Page 1.16
1.5 Systems Approach
The Element of a System • Activities and objects
Pfleeger and Atlee, Software Engineering: Theory and Practice
© 2006 Pearson/Prentice Hall
Page 1.15
1.5 Systems Approach
• Identify activities and objects • Define the system boundary
Solving Problems (continued) • Method: refers to a formal procedure • Tool: an instrument or automated system for accomplishing something in a better way • Procedure: a combination of tools and techniques to produce a product • Paradigm: philosophy or approach for building a product
Pfleeger and Atlee, Software Engineering: Theory and Practice
© 2006 Pearson/Prentice Hall
Page 1.8
1.2 How Successful Have We Been?
LectureSlides.ppt
© 2007 Pearson Addison-Wesley. All rights reserved
0-8
Figure 6.3 A function for checkbook balancing constructed from simpler functions
© 2007 Pearson Addison-Wesley. All rights reserved
0-26
Figure 6.18 Two distinct parse trees for the statement if B1 then if B2 then S1 else S2
Figure 6.19 An object-oriented approach to the translation process
0-6
Third Generation Language
• Uses high-level primitives
– Similar to our pseudocode in Chapter 5
• Machine independent (mostly) • Examples: FORTRAN, COBOL • Each primitive corresponds to a sequence of
0-12
Figure 6.5 A two-dimensional array with two rows and nine columns
© 2007 Pearson Addison-Wesley. All rights reserved
0-13
Figure 6.6 The conceptual structure of the heterogeneous array Employee
国外项目管理课件Project Management Chapter 1 Slides_Project Management Intro
Execution – the actual “work” of the project is performed.
Termination – project is transferred to the customer, resources reassigned, project is closed out.
6.Understand and explain the project life cycle, its stages, and the activities that typically occur at each stage in the project.
7.Understand the concept of project “success,” including various definitions of success, as well as the alternative models of success.
1-9
WHY ARE PROJECTS IMPORTANT?
1. Shortened product life cycles 2. Narrow product launch windows 3. Increasingly complex and technical products 4. Emergence of global markets 5. An economic period marked by low inflation
PROJECT
New process or product One objective One-shot-limited life More heterogeneous Integrated system efforts Greater uncertainty Outside of line organization Violates established practice Upsets status quo
aug_09_ch_6_rev
• In essence, people assign probabilities to favourable/unfavorable outcomes that are just too high/low given historical experience or reasoned analysis. • Excessive optimism and miscalibration can go hand in hand.(being overconfident in more than one sense)
• Both men and women expected their portfolios to outperform the market – but gap greater for men
• Also evidence that highly educated, highincome people are more overconfident.
– For example, gamblers may think that they can control the dice or the cards。例如，赌徒可能认为他们
能够控制骰子或者卡片。
6
Excessive optimism 过度乐观
• Present when people’s predictions about the future are unrealistically optimistic. 当人们的预测是不切实际的乐观时就表
3
Calibration-based overconfidence cont.
• Or use confidence interval approach.或使用置信区间方法 • Suppose individuals are asked to construct 90% confidence intervals (e.g., height of Mount Everest). • A percentage of individuals usually less than 90% usually comes up with intervals that bracket true answer. 包含真实答案的区间常常小于人们提出的90%。 • This also suggests miscalibration-based overconfidence. 这也暗示了基于刻度错误的过度自信。
chapter06[1]
![chapter06[1]](https://img.taocdn.com/s3/m/bca469f34693daef5ef73d0b.png)
Francis & Ibbotson
Chapter 6: The Global Stock Market
4
Brokerage Services
Brokers
– Sales people (earning a commission) employed by dealers – Have no money invested in the dealer’s security inventory – Help create markets by buying and selling from employer’s inventory
The Global Stock Market
Chapter 6
Slides by: Pamela L. Hall, Western Washington University
Francis & Ibbotson Chapter 6: The Global Stock Market
1
Background
11
Types of Trading Orders
Scale order
– Requires buying or selling part of the order at each price as market prices change
Fill or kill (FOK) order
Cumbersome and not all brokers accept them
5
Brokerage Services
Types of stock brokerage services
– Full-service
Examples Take buy and sell orders –Merrill Lynch Extend margin credit to customers –Goldman Sachs Hold clients’ securities in safe keeping –PaineWebber Collect cash dividends –Morgan Stanley Dean Witter Provide free investment research –Salomon Smith Barney Perform ‘hand-holding’ services
项目管理Project management课件Chapter 6 Slides_Project Teams
PROJECT TEAM BUILDING, CONFLICT, ANDNEGOTIATIONChapter 6CHAPTER 6LEARNING OBJECTIVESAfter completing this chapter, you should be able to: 1.Understand the steps involved in project teambuilding.2.Know the characteristics of effective projectteams and why teams fail.3.Know the stages in the development of groups.4.Describe how to achieve cross‐functionalcooperation in teams.CHAPTER 6LEARNING OBJECTIVESAfter completing this chapter, you should be able to: 5.See the advantages and challenges of projectteams.6.Understand the nature of conflict and evaluateresponse method.7.Understand the importance of negotiation skillsin project management.PMBOK CORE CONCEPTSProject Management Body of Knowledge (PMBoK) covered in this chapter includes:1.Plan Human Resource Management (PMBoK9.1)2.Acquire Project Team (PMBoK 9.2)3.Develop Project Team (PMBoK 9.3)4.Manage Project Team (PMBoK 9.4)Basic Stepsin Assembling a Project Team FIGURE 6.2组建项目团队的基本步骤项目绩效冲突震荡，间断平衡，效益提升CONFLICT RESOLUTIONMediate –diffusion/confrontationArbitrate –judgmentControl –cool down periodAccept –unmanageableEliminate –transferConflict is often evidence of progress!SUMMARY1.Understand the steps involved in project teambuilding.2.Know the characteristics of effective projectteams and why teams fail.3.Know the stages in the development of groups.4.Describe how to achieve cross‐functionalcooperation in teams.SUMMARY5.See the advantages and challenges of projectteams.6.Understand the nature of conflict and evaluateresponse method.7.Understand the importance of negotiation skillsin project management.。
Slides_Lecture2
Parentheses define sub-computations-complete these to get values before evaluating larger expression
- (2+3)*4 - 5*4 - 20
Operator precedence:
- In the absence of parentheses (within which expressions are first reduced), operators are executed left to right, first using **, then * and /, and then + and -
© 2014 中北大学软件学院
Page 1-12
Python
Some simple examples
>>> 3 + 5
8 >>> 3.14 * 20
62.8
>>> (2 + 3)*4 20
>>> 2 + 3*4
14
© 2014 中北大学软件学院
Page 1-13
Python
Performing simple operations
terms –
高级语言使用更多抽象的形式转换矩阵，计算函数 In a compiled language, those abstractions are converted back into low level instructions, then executed 编译类语言，那些抽象形式的语句就会先转换为低级指令，然后再 执行。
© 2014 中北大学软件学院 Page 1-6
slides_chapter_19
Content
Does the content of critical pages match what is supposed to be there? Do key phrases exist continually in highly-changeable pages? Do critical pages maintain quality content from version to version? What about dynamically generated HTML pages?
5
WebApp Design Goals
Consistency Content should be constructed consistently Graphic design (aesthetics) should present a consistent look across all parts of the WebApp Architectural design should establish templates that lead to a consistent hypermedia structure Interface design should define consistent modes of interaction, navigation and content display Navigation mechanisms should be used consistently across all WebApp elements
Hale Waihona Puke technology8
WebApp Interface Design
Where am I? The interface should
无线与移动网络
network infrastructure
第六章: 无线和移动网络
6-8
无线网络的元素
自组织模式 没有基站 节点能够传输到链 路范围内的其他节点 节点自己组织成为 一个网络：在它们 之间路由
第六章: 无线和移动网络
6-9
第 6 章 大 纲：
6.1 概述 6.2 无线链路和网络的特征
CDMA
6.3 IEEE 802.11 无线LANs (“wi-fi”) 6.4 蜂窝因特网接入
体系结构\标准(e.g., GSM)
无线特性
6.5 6.6 6.7 6.8
原理: 编址和路由到移动用户 移动IP 蜂窝网络中管理移动性 移动性对高层协议的影响
移动性
第六章: 无线和移动网络
6-10
6.2 无线链路和网络的特征
与有线网络不同的是 : 衰减的信号强度:无线电信号在穿过物体传播时会衰 减（路径损耗） 来自其他源的干扰: 标准化的无线网络频率（如 2.4GHz)是与其他设备共享的（如电话）或者受其他 设备干扰（如马达） 多路径传播: 电磁波受物体和地面反射，导致到达目 的地的时间有些不一样 无线链路上的通信更加困难（即使是点对点链路）
space
信号衰减: B, A 能够彼此听到 B, C 能够彼此听到 A, C 不能听到它们在B的相 互干扰
第六章: 无线和移动网络 6-12
码分多址 (CDMA)
用于几种无线广播信道标准 (蜂窝, 卫星等) 唯一“编码”分配给每个用户;即代码集的划分 所有用户共享同样的频率, 但是每个有自己的“码片”序列（ 即编码）用来编码数据 码片序列：M位0,1二进制数，即一个比特时隙划分成M个 微时隙。 数学上的表示: 数据1表示为+1, 数据0表示为-1） M位0,1二进制数的码片序列以矢量形式表示，其中1:+1，0:-1 编码的信息 = (原始数据) X (码片序列) 解码: 编码信息和码片序列的内积 允许多个用户“共存”, 并且同时最小干扰地传输－线性叠加
slides（pdf）-natevw
[and yes just consistently use sel/d/i/j by default]
"Towards Reusable Charts"
article by Mike Bostock
/mike/chart/
!
(c.f. https:///natevw/5979039 for a boilerplate-reducing helper)
!
sel.append('span').text(function (d) { return d[0].toUpperCase()+d.slice(1);
});
Why not web apps?
There are only two hard things in Computer Science: cache invalidation and naming things.
return d[0].toUpperCase()+d.slice(1); }); } </script>
Magic Ink
/MagicInk/
Three types of apps
/MagicInk/#what_is_software_for
What is D3?
[JavaScript library by Mike Bostock et al.]
Байду номын сангаас
D3 applies Data to DOMs
var arr = ["eggs", "bacon", "coffee"], sel = d3.selectAll("li").data(arr);
presentation_用语大全(word文档)
Chapter I Beginning the Speech Text (3)I. Opening Remarks开场： (3)Sample Opening Remarks (3)Expressing thanks to the Chairperson 向主持人致谢 (3)Forms of Address and Greetings对听众的称呼 (3)Expressing Pleasure and Honor 向听众致意 (3)Others 细节，如确认话筒音量 (3)Reference to the Audience 与听众呼应 (3)II. Introducing the Subject and the outline of the Presentation引入话题 (4)Background Information (4)Topic 4Outlining (4)Purpose/Objective (4)Comprehensive Samples (4)Expressions on Other Occasions (5)Checking the Microphone (5)Chapter II Developing the Speech Text (6)I. Announcing the Beginning of the Speech Text (6)II. Shifting to the Next Main Point (6)III. Resuming the Topic (7)IV. Introducing the Supporting Materials (7)V. Repairing a Slip of Tone (7)VI. Expressions Concerning Audio-Visual Aids (8)VI. Explaining the Contents on the Slides (8)VII. Demanding to Show the Next Slide (8)VIII. Returning to the Previous Slide (9)IX. Correcting a Mistake (9)X. Indicating the Speed of Running Slides (9)Chapter III Ending the Speech Text (9)I. Signaling the Beginning of the End Part (9)II. Summarizing (10)III. Concluding (10)IV. Quoting (10)V. Closing (10)VI. Samples (11)[ Sample 1 ] (11)[ Sample 2 ] ' (11)[ Sample 3 ] (11)[ Sample 4 ] (11)[ Sample 5 ] (11)Chapter IV Asking and Answering Questions (11)I. Inviting Questions (11)II. Raising Questions (12)Signaling Your Intention to Ask a Question (12)Expressing Your Attitude (12)Asking the Specific Question (12)III. Response to Questions (12)Asking for Repetition (12)Welcoming the Question (13)Repeating or Paraphrasing the Question (13)Responding to Difficult or Challenging Question (13)Responding to Improper Question (14)Responding to the Questions You Do Not Know (14)Referring Back to the Questioner (14)Chapter V Talking During the Break .............................................................. 错误！未定义书签。
Coursera课程下载和存档计划[转载]
![Coursera课程下载和存档计划[转载]](https://img.taocdn.com/s3/m/9b699dfbe109581b6bd97f19227916888486b911.png)
Coursera课程下载和存档计划[转载]上周三收到平台的群发邮件，⼤意是Coursera将在6⽉30号彻底关闭旧的课程平台，全⾯升级到新的课程平台上，⼀些旧的课程资源（课程视频、课程资料）将不再保存，如果你之前学习过相关的课程，或者有⼼仪的课程，Coursera建议你将这些课程资源下载下来备份。
说实话，⾃从这⼀两年逐渐进⾏“商业升级”以后，我已经很少在这个平台上学习公开课了，反⽽是的⼀些课程更吸引我，特别是课程质量，后者显得更⽤⼼很多。
不过作为最早的MOOC平台Coursera，曾经诞⽣了很多经典课程，要是这些课程真的随Coursera平台的切换⽽丢失，实在可惜。
这⾥曾经整理过⼀批“”，很多来⾃于⼤家的贡献和分享，不过这也是两三年前的事情，⼀些课程⽹盘资源已经失效，这封邮件促使我开始检查这些⽹盘资源，特别是来⾃Coursera平台的课程资源。
之前有些课程资源没有下载或者没有⽹盘资源，以为只要有Coursera账号，就可以随时登陆上去在线观看就可以了，也没有下载的欲望，现在不同了，例如斯坦福⼤学Dan Jurafsky和Christopher Manning的，例如⼀直没有开课却可以preview观看的⼤⽜Pedro Domingos的，下载和备份是必须的。
⼯欲善其事，必先利其器，针对Coursera的下载⼯具有很多，包括⼀些浏览器插件，不过这⾥推荐的是Python下载⼯具, 简称coursera-dl。
这个神器早在⼏年前我就⽤过，印象深刻，这次重拾，依然⾮常⽅便给⼒。
最简单的安装⽅法是“pip install coursera”，可参考github上该项⽬的安装说明。
下⾯以Mac OS系统为例简单说明⼀下基于virtualenv的安装使⽤⽅法，该⽅法对ubuntu这样的linux系统应该有效，windows 下没有测试，未知。
⾸先从github上获取代码，git clone或者直接下载zip源码⽂件均可：Cloning into ‘coursera-dl’…remote: Counting objects: 3357, done.remote: Compressing objects: 100% (14/14), done.remote: Total 3357 (delta 6), reused 0 (delta 0), pack-reused 3343Receiving objects: 100% (3357/3357), 1.39 MiB | 75 KiB/s, done.Resolving deltas: 100% (1852/1852), done.cd coursera-dl/virtualenv my-courseraNew python executable in /Users/xxxxxx/project/mooc/test/coursera-dl/my-coursera/bin/pythonInstalling setuptools, pip, wheel…done.source my-coursera/bin/activatepip install -r requirements.txtCollecting beautifulsoup4>=4.1.3 (from -r requirements.txt (line 1))…..Installing collected packages: beautifulsoup4, six, html5lib, requests, urllib3, pyasn1, keyringSuccessfully installed beautifulsoup4-4.4.1 html5lib-1.0b8 keyring-9.0 pyasn1-0.1.9 requests-2.10.0 six-1.10.0 urllib3-1.16安装完毕，以下是coursera-dl的详细⽤法:General: coursera-dl -u -p modelthinking-004Multiple classes: coursera-dl -u -p saas historyofrock1-001 algo-2012-002Filter by section name: coursera-dl -u -p -sf “Chapter_Four” crypto-004Filter by lecture name: coursera-dl -u -p -lf “3.1_” ml-2012-002Download only ppt files: coursera-dl -u -p -f “ppt” qcomp-2012-001Use a ~/.netrc file: coursera-dl -n — matrix-001Get the preview classes: coursera-dl -n -b ni-001Specify download path: coursera-dl -n –path=C:\Coursera\Classes\ comnetworks-002Display help: coursera-dl –helpMaintain a list of classes in a dir:Initialize: mkdir -p CURRENT/{class1,class2,..classN}Update: coursera-dl -n –path CURRENT `\ls CURRENT`我们以Coursera上密歇根⼤学的“”课程为例，在旧的课程课程主页“”，⾸先需要加⼊（Enroll）该课程的⼀个班次，⽬前只有2015年10⽉到12⽉开过⼀轮课，加⼊该轮课程后，进⼊到课程详细页⾯，可以看到⽹页链接类似这个形式：对于Coursera Downloader来说，主要需要的就是这个”nlpintro-001″课程班次短链接，然后就可以尝试下载了，这⾥⽤ –path指定了课程下载路径：coursera-dl -u ⽤户邮箱 -p ⽤户密码 --path=../../coursera_backup/ nlpintro-001然后就开始了下载历程。
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Recurrence relation
Legendre moments
Legendre and geometric moments
Chebyshev polynomials
First kind
Second kind
Chebyshev polynomials on <-1,1>
http://zoi.utia.cas.cz/moment_invariants
The slides accompanying the book
J. Flusser, T. Suk, and B. Zitová
Moments and Moment Invariants in Pattern Recognition
- special initial values:
Gegenbauer polynomials
Jacobi polynomials
Further generalization, parameters α and β
Laguerre and Hermite polynomials
• Infinite interval of orthogonality • Suitable for particular applications only
Copyright notice
The slides can be used freely for non-profit education provided that the source is appropriately cited. Please report any usage on a regular basis (namely in university courses) to the authors. For commercial usage ask the authors for permission. The slides containing animations are not appropriate to print. © Jan Flusser, Tomas Suk, and Barbara Zitová, 2009
Zernike rotation invariants
Phason
Normalization to rotation
Pseudo - Zernike polynomials
need not be even → redundancy
Orthogonal Fourier-Mellin moments
Moments orthogonal on a square
is a system of 1D orthogonal polynomials
Common 1D orthogonal polynomials
• • • • • • Legendre Chebyshev Gegenbauer Jacobi (generalized) Laguerre Hermite <-1,1> <-1,1> <-1,1> <-1,1> or <0,1> <0,∞) (-∞,∞)
• Image reconstruction from OG moments on a disk (Zernike)
Image reconstruction from Legendre moments
Continuous orthogonality
Image reconstruction from Zernike moments
Discrete Chebyshev polynomials
Krawtchouk polynomials
Krawtchouk polynomials
Weighted Krawtchouk polynomials
p=0.5
p=0.2
Dual-Hahn and Racah polynomials
First kind
Second kind
Chebyshev polynomials on <-2,2>
First kind
Second kind
Chebyshev polynomials in 2D
Chebyshev polynomials – orthogonality
Continuous
Orthogonal FM moments – radial part
Orthogonal FM moments – the basis functions
Jacobi-Fourier and Chebyshev-Fourier moments
Combination: radial part – 1D orthogonal polynomials angular part – traditional harmonic function
• • • • • • Zernike Pseudo-Zernike Orthogonal Fourier-Mellin Jacobi-Fourier Chebyshev-Fourier Radial harmonic Fourier
Zernike polynomials
Definition
Orthogonality
Better for polar raster
Image reconstruction from discrete Chebyshev moments
– precise reconstruction
Illustrates discrimination power of the moments
Reconstruction of large images
Contents 1. Introduction to moments
2. Invariants to translation, rotation and scaling
3. Affine moment invariants 4. Implicit invariants to elastic transformations
Discrete Chebyshev moments – precise up to 1075th order
Limitation is imposed by floating-point underflow
Orthogonal moments in 3D
Ω - Rectangular cuboid
Nonuniform lattice
They were adapted such that s is a traditional coordinate in a discrete image.
Moments orthogonal on a disk
Radial part Angular part
Moments orthogonal on a disk
They can be normalized to discrete orthogonality
Chebyshev moments
Gegenbauer polynomials
Generalization of both Legendre and Chebyshev polynomials – parameter λ
12 x 12 13 x 13
Image reconstruction
• Reconstruction from geometric moments via FT
Image reconstruction via Fourier transform
Image reconstruction
• Image reconstruction from OG moments on a square
Motivation for using OG moments • Stable calculation by recurrent relations • Easier and stable image reconstruction
Numerical stability
How to avoid numerical problems with high dynamic range of geometric moments?
Standard powers
Orthogonal polynomials
Calculation using recurrent relations
Two kinds of orthogonality
• Moments (polynomials) orthogonal on a unit square • Moments (polynomials) orthogonal on a unit disk
Zernike polynomials – radial part in 1D
Zernike polynomials – radial part in 2D
Zernike polynomials
Zernike moments
Mapping of Cartesian coordinates x,y to polar coordinates r,φ:
Ω - Cylinder
Discrete OG polynomials on a square
Discrete variable
Discrete orthogonality
Discrete 1D OG polynomials
• • • • • • Discrete Chebyshev Discrete Laguerre Krawtchouk Hahn Dual-Hahn Racah
• Whole image is mapped inside the unit disk • Translation and scaling invariance