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

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Fourier transform of vector-valued functions 向量值函数的傅里叶变换
Principles of Biophotonics, Volume 1: Linear systems and the Fourier transform in optics Pub Date : 1900-01-01 DOI: 10.1088/2053-2563/aae121ch10
G. Popescu
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引用次数: 0
Complex signals 复杂的信号
Principles of Biophotonics, Volume 1: Linear systems and the Fourier transform in optics Pub Date : 1900-01-01 DOI: 10.1088/2053-2563/aae121ch7
Gabriel Popescu
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引用次数: 3
3D Fourier transform 三维傅里叶变换
Principles of Biophotonics, Volume 1: Linear systems and the Fourier transform in optics Pub Date : 1900-01-01 DOI: 10.1088/2053-2563/AAE121CH6
G. Popescu
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引用次数: 0
The uncertainty relation 不确定关系
Principles of Biophotonics, Volume 1: Linear systems and the Fourier transform in optics Pub Date : 1900-01-01 DOI: 10.1088/2053-2563/AAE121CH8
G. Popescu
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引用次数: 0
Linear systems with random inputs 具有随机输入的线性系统
Principles of Biophotonics, Volume 1: Linear systems and the Fourier transform in optics Pub Date : 1900-01-01 DOI: 10.1088/2053-2563/AAE121CH9
G. Popescu
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引用次数: 0
Linear systems 线性系统
Principles of Biophotonics, Volume 1: Linear systems and the Fourier transform in optics Pub Date : 1900-01-01 DOI: 10.1088/2053-2563/aae121ch2
Gabriel Popescu
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引用次数: 1
2D Fourier transform 二维傅里叶变换
Principles of Biophotonics, Volume 1: Linear systems and the Fourier transform in optics Pub Date : 1900-01-01 DOI: 10.1088/2053-2563/AAE121CH5
G. Popescu
{"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
Principles of Biophotonics, Volume 1: Linear systems and the Fourier transform in optics Pub Date : 1900-01-01 DOI: 10.1088/2053-2563/AAE121CH3
G. Popescu
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引用次数: 0
1D Fourier transform 一维傅里叶变换
Principles of Biophotonics, Volume 1: Linear systems and the Fourier transform in optics Pub Date : 1900-01-01 DOI: 10.1088/2053-2563/aae121ch4
G. Popescu
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引用次数: 0
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