首页 > 最新文献

Principles of Biophotonics, Volume 1: Linear systems and the Fourier transform in optics最新文献

英文 中文
Fourier transform of vector-valued functions 向量值函数的傅里叶变换
G. Popescu
{"title":"Fourier transform of vector-valued functions","authors":"G. Popescu","doi":"10.1088/2053-2563/aae121ch10","DOIUrl":"https://doi.org/10.1088/2053-2563/aae121ch10","url":null,"abstract":"","PeriodicalId":325894,"journal":{"name":"Principles of Biophotonics, Volume 1: Linear systems and the Fourier transform in optics","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129594072","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Complex signals 复杂的信号
Gabriel Popescu
{"title":"Complex signals","authors":"Gabriel Popescu","doi":"10.1088/2053-2563/aae121ch7","DOIUrl":"https://doi.org/10.1088/2053-2563/aae121ch7","url":null,"abstract":"","PeriodicalId":325894,"journal":{"name":"Principles of Biophotonics, Volume 1: Linear systems and the Fourier transform in optics","volume":"61 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132594363","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 3
3D Fourier transform 三维傅里叶变换
G. Popescu
{"title":"3D Fourier transform","authors":"G. Popescu","doi":"10.1088/2053-2563/AAE121CH6","DOIUrl":"https://doi.org/10.1088/2053-2563/AAE121CH6","url":null,"abstract":"","PeriodicalId":325894,"journal":{"name":"Principles of Biophotonics, Volume 1: Linear systems and the Fourier transform in optics","volume":"39 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123537562","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The uncertainty relation 不确定关系
G. Popescu
{"title":"The uncertainty relation","authors":"G. Popescu","doi":"10.1088/2053-2563/AAE121CH8","DOIUrl":"https://doi.org/10.1088/2053-2563/AAE121CH8","url":null,"abstract":"","PeriodicalId":325894,"journal":{"name":"Principles of Biophotonics, Volume 1: Linear systems and the Fourier transform in optics","volume":"101 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123255812","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Linear systems with random inputs 具有随机输入的线性系统
G. Popescu
{"title":"Linear systems with random inputs","authors":"G. Popescu","doi":"10.1088/2053-2563/AAE121CH9","DOIUrl":"https://doi.org/10.1088/2053-2563/AAE121CH9","url":null,"abstract":"","PeriodicalId":325894,"journal":{"name":"Principles of Biophotonics, Volume 1: Linear systems and the Fourier transform in optics","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129659712","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Linear systems 线性系统
Gabriel Popescu
{"title":"Linear systems","authors":"Gabriel Popescu","doi":"10.1088/2053-2563/aae121ch2","DOIUrl":"https://doi.org/10.1088/2053-2563/aae121ch2","url":null,"abstract":"","PeriodicalId":325894,"journal":{"name":"Principles of Biophotonics, Volume 1: Linear systems and the Fourier transform in optics","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127599655","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
2D Fourier transform 二维傅里叶变换
G. Popescu
(a) Consider two independent integer-valued random variables, X and Y. Variable X takes on only the values of the eight integers {1, 2,. .. , 8} and does so with uniform probability. Variable Y may take the value of any positive integer k, with probabilities P {Y = k} = 2 (i) Which random variable has greater uncertainty? Calculate both entropies H(X) and H(Y). [3 marks] (ii) What is the joint entropy H(X, Y) of these random variables, and what is their mutual information I(X; Y)? [2 marks] (b) What is the maximum possible entropy H of an alphabet consisting of N different letters? In such a maximum entropy alphabet, what is the probability of its most likely letter? What is the probability of its least likely letter? Why are fixed length codes inefficient for alphabets whose letters are not equiprobable? Discuss this in relation to Morse Code. [5 marks] (c) Explain why the real-part of a 2D Gabor wavelet has a 2D Fourier transform with two peaks, not just one, as shown in the right panel of the figure below. [4 marks] (d) Show that the set of all Gabor wavelets is closed under convolution, i.e. that the convolution of any two Gabor wavelets is just another Gabor wavelet. [Hint: This property relates to the fact that these wavelets are also closed under multiplication, and that they are also self-Fourier. You may address this question for just 1D wavelets if you wish.] [3 marks] (e) Show that the family of sinc functions used in the Nyquist Sampling Theorem, sinc(x) = sin(λx) λx is closed under convolution. Show further that when two different sinc functions are convolved together, the result is simply whichever one of them had the lower frequency, i.e. the smaller λ.
(a)考虑两个独立的整数值随机变量X和y。变量X只取8个整数{1,2,…, 8},并以均匀概率这样做。变量Y可以取任意正整数k的值,概率P {Y = k} = 2 (i)哪个随机变量的不确定性更大?计算H(X)和H(Y)的熵。[3分](ii)这些随机变量的联合熵H(X, Y)是什么,它们的互信息I(X;Y) ?[2分](b)由N个不同字母组成的字母表的最大可能熵H是多少?在这样一个最大熵字母表中,最有可能出现的字母的概率是多少?最不可能出现的字母的概率是多少?为什么固定长度代码对于字母不是等概率的字母是低效的?讨论这与摩尔斯电码的关系。[5分](c)解释为什么二维Gabor小波的实部具有两个峰的二维傅里叶变换,而不是只有一个,如下图右面板所示。[4标记](d)表明所有Gabor小波的集合在卷积下是封闭的,即任意两个Gabor小波的卷积只是另一个Gabor小波。[提示:这个性质与这些小波在乘法下也是封闭的,它们也是自傅里叶。如果你愿意,你可以只用一维小波来解决这个问题。(e)证明了Nyquist抽样定理中sinc(x) = sin(λx) λx的sinc函数族在卷积下是封闭的。进一步表明,当两个不同的sinc函数卷积在一起时,结果只是其中一个频率较低,即λ较小。
{"title":"2D Fourier transform","authors":"G. Popescu","doi":"10.1088/2053-2563/AAE121CH5","DOIUrl":"https://doi.org/10.1088/2053-2563/AAE121CH5","url":null,"abstract":"(a) Consider two independent integer-valued random variables, X and Y. Variable X takes on only the values of the eight integers {1, 2,. .. , 8} and does so with uniform probability. Variable Y may take the value of any positive integer k, with probabilities P {Y = k} = 2 (i) Which random variable has greater uncertainty? Calculate both entropies H(X) and H(Y). [3 marks] (ii) What is the joint entropy H(X, Y) of these random variables, and what is their mutual information I(X; Y)? [2 marks] (b) What is the maximum possible entropy H of an alphabet consisting of N different letters? In such a maximum entropy alphabet, what is the probability of its most likely letter? What is the probability of its least likely letter? Why are fixed length codes inefficient for alphabets whose letters are not equiprobable? Discuss this in relation to Morse Code. [5 marks] (c) Explain why the real-part of a 2D Gabor wavelet has a 2D Fourier transform with two peaks, not just one, as shown in the right panel of the figure below. [4 marks] (d) Show that the set of all Gabor wavelets is closed under convolution, i.e. that the convolution of any two Gabor wavelets is just another Gabor wavelet. [Hint: This property relates to the fact that these wavelets are also closed under multiplication, and that they are also self-Fourier. You may address this question for just 1D wavelets if you wish.] [3 marks] (e) Show that the family of sinc functions used in the Nyquist Sampling Theorem, sinc(x) = sin(λx) λx is closed under convolution. Show further that when two different sinc functions are convolved together, the result is simply whichever one of them had the lower frequency, i.e. the smaller λ.","PeriodicalId":325894,"journal":{"name":"Principles of Biophotonics, Volume 1: Linear systems and the Fourier transform in optics","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114988632","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Spatial and temporal frequencies 时空频率
G. Popescu
{"title":"Spatial and temporal frequencies","authors":"G. Popescu","doi":"10.1088/2053-2563/AAE121CH3","DOIUrl":"https://doi.org/10.1088/2053-2563/AAE121CH3","url":null,"abstract":"","PeriodicalId":325894,"journal":{"name":"Principles of Biophotonics, Volume 1: Linear systems and the Fourier transform in optics","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133906192","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
1D Fourier transform 一维傅里叶变换
G. Popescu
{"title":"1D Fourier transform","authors":"G. Popescu","doi":"10.1088/2053-2563/aae121ch4","DOIUrl":"https://doi.org/10.1088/2053-2563/aae121ch4","url":null,"abstract":"","PeriodicalId":325894,"journal":{"name":"Principles of Biophotonics, Volume 1: Linear systems and the Fourier transform in optics","volume":"206 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122604283","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
期刊
Principles of Biophotonics, Volume 1: Linear systems and the Fourier transform in optics
全部 Acc. Chem. Res. ACS Applied Bio Materials ACS Appl. Electron. Mater. ACS Appl. Energy Mater. ACS Appl. Mater. Interfaces ACS Appl. Nano Mater. ACS Appl. Polym. Mater. ACS BIOMATER-SCI ENG ACS Catal. ACS Cent. Sci. ACS Chem. Biol. ACS Chemical Health & Safety ACS Chem. Neurosci. ACS Comb. Sci. ACS Earth Space Chem. ACS Energy Lett. ACS Infect. Dis. ACS Macro Lett. ACS Mater. Lett. ACS Med. Chem. Lett. ACS Nano ACS Omega ACS Photonics ACS Sens. ACS Sustainable Chem. Eng. ACS Synth. Biol. Anal. Chem. BIOCHEMISTRY-US Bioconjugate Chem. BIOMACROMOLECULES Chem. Res. Toxicol. Chem. Rev. Chem. Mater. CRYST GROWTH DES ENERG FUEL Environ. Sci. Technol. Environ. Sci. Technol. Lett. Eur. J. Inorg. Chem. IND ENG CHEM RES Inorg. Chem. J. Agric. Food. Chem. J. Chem. Eng. Data J. Chem. Educ. J. Chem. Inf. Model. J. Chem. Theory Comput. J. Med. Chem. J. Nat. Prod. J PROTEOME RES J. Am. Chem. Soc. LANGMUIR MACROMOLECULES Mol. Pharmaceutics Nano Lett. Org. Lett. ORG PROCESS RES DEV ORGANOMETALLICS J. Org. Chem. J. Phys. Chem. J. Phys. Chem. A J. Phys. Chem. B J. Phys. Chem. C J. Phys. Chem. Lett. Analyst Anal. Methods Biomater. Sci. Catal. Sci. Technol. Chem. Commun. Chem. Soc. Rev. CHEM EDUC RES PRACT CRYSTENGCOMM Dalton Trans. Energy Environ. Sci. ENVIRON SCI-NANO ENVIRON SCI-PROC IMP ENVIRON SCI-WAT RES Faraday Discuss. Food Funct. Green Chem. Inorg. Chem. Front. Integr. Biol. J. Anal. At. Spectrom. J. Mater. Chem. A J. Mater. Chem. B J. Mater. Chem. C Lab Chip Mater. Chem. Front. Mater. Horiz. MEDCHEMCOMM Metallomics Mol. Biosyst. Mol. Syst. Des. Eng. Nanoscale Nanoscale Horiz. Nat. Prod. Rep. New J. Chem. Org. Biomol. Chem. Org. Chem. Front. PHOTOCH PHOTOBIO SCI PCCP Polym. Chem.
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
0
微信
客服QQ
Book学术公众号 扫码关注我们
反馈
×
意见反馈
请填写您的意见或建议
请填写您的手机或邮箱
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
现在去查看 取消
×
提示
确定
Book学术官方微信
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术
文献互助 智能选刊 最新文献 互助须知 联系我们:info@booksci.cn
Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。
Copyright © 2023 Book学术 All rights reserved.
ghs 京公网安备 11010802042870号 京ICP备2023020795号-1