图像处理（图像处理、波形处理） - 版本历史

Li.qun：/* 27. Fourier Shell Correlation, FSC */

2020-03-23T06:31:10Z

‎27. Fourier Shell Correlation, FSC

Li.qun：/* 27. Fourier Shell Correlation, FSC */

2020-03-23T06:30:16Z

‎27. Fourier Shell Correlation, FSC

Li.qun：/* 27. Fourier Shell Correlation, FSC */

2020-03-23T06:25:51Z

‎27. Fourier Shell Correlation, FSC

Li.qun：/* 27. Fourier Shell Correlation, FSC */

2020-03-23T06:24:11Z

‎27. Fourier Shell Correlation, FSC

Li.qun：/* 27. Fourier Shell Correlation, FSC */

2020-03-23T06:22:09Z

‎27. Fourier Shell Correlation, FSC

2020年3月23日 (一) 06:19 Li.qun

2020-03-23T06:19:59Z

Li.qun：/* 18-3．inverse Fourier transform */

2019-11-01T07:07:11Z

‎18-3．inverse Fourier transform

Li.qun：/* 18-1. Fourier transform */

2019-11-01T07:00:12Z

‎18-1. Fourier transform

Li.qun：/* 18-1. Fourier transform */

2019-11-01T06:58:41Z

‎18-1. Fourier transform

Li.qun：/* 18-1. Fourier transform */

2019-11-01T06:56:00Z

‎18-1. Fourier transform

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 ==27. Fourier Shell Correlation, FSC==
 Fourier Shell Correlation (FSC) is a measure of reliability of the three-dimensional (3D) structures of biological macromolecules obtained by “single particle analysis”. FSC is expressed as a function of spatial frequency by the following equation (normalized cross correlation).
 [[文件:Fourier-shell-correlation 01.jpg]]

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 ==27. Fourier Shell Correlation, FSC==
 Fourier Shell Correlation (FSC) is a measure of reliability of the three-dimensional (3D) structures of biological macromolecules obtained by “single particle analysis”. FSC is expressed as a function of spatial frequency by the following equation (normalized cross correlation).
 Here, F1(k) and F2(k)are 3D Fourier transforms of the two structures reconstructed from arbitrary two independent sets of TEM images obtained. F2(k)* is a complex conjugate of F2(k), and Σk,ΔkF(k) means summing up F(k) over a small range Δk around k.
 FSC (k) takes a value between +1 and –1 for every spatial frequency k. The value of FSC (k) close to +1 means that the two reconstructed structures agree well, indicating a high degree of reliability of the obtained structure. If FSC does not decrease monotonically with spatial frequency k at high frequencies, the reliability of the FSC function is considered to be low.

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   (Proofread by Specially Appointed Professor Keichi Namba, Osaka University)
 [[文件:Fourier-shell-correlation 02.JPG]]      [[文件:Fourier-shell-correlation 03.JPG]]
 Fig1. 3D density map of mouse apoferritin obtained by single particle analysis and its FSC. (a) 3D density map. (b) An FSC function calculated from the two 3D density maps. The resolution was determined to be 0.153 nm from a spatial frequency of 6.5 nm-1 at which FSC = 0.143.

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 In single particle analysis, it is common to define the spatial resolution of the obtained structure to be the reciprocal of the spatial frequency at which FSC = 0.143. (The value “0.143” is a value adopted based on statistical discussion about the correlation of random noise and on referring the resolution defined for X-ray crystallography.) The resolution for apoferritin was determined to be 0.153 nm from a spatial frequency of 6.5 nm-1 at which FSC takes a value of 0.143.
   (Proofread by Specially Appointed Professor Keichi Namba, Osaka University)
-[[文件:Fourier-shell-correlation 03.JPG]]
+[[文件:Fourier-shell-correlation 02.JPG]]      [[文件:Fourier-shell-correlation 03.JPG]]
 Fig1. 3D density map of mouse apoferritin obtained by single particle analysis and its FSC. (a) 3D density map. (b) An FSC function calculated from the two 3D density maps. The resolution was determined to be 0.153 nm from a spatial frequency of 6.5 nm-1 at which FSC = 0.143.

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 In single particle analysis, it is common to define the spatial resolution of the obtained structure to be the reciprocal of the spatial frequency at which FSC = 0.143. (The value “0.143” is a value adopted based on statistical discussion about the correlation of random noise and on referring the resolution defined for X-ray crystallography.) The resolution for apoferritin was determined to be 0.153 nm from a spatial frequency of 6.5 nm-1 at which FSC takes a value of 0.143.
   (Proofread by Specially Appointed Professor Keichi Namba, Osaka University)
+[[文件:Fourier-shell-correlation 03.JPG]]
 Fig1. 3D density map of mouse apoferritin obtained by single particle analysis and its FSC. (a) 3D density map. (b) An FSC function calculated from the two 3D density maps. The resolution was determined to be 0.153 nm from a spatial frequency of 6.5 nm-1 at which FSC = 0.143.

←上一版本		2020年3月23日 (一) 06:19的版本
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	<pre>Related term		<pre>Related term
	curve fitting, Gaussian function, Lorentzian function</pre>		curve fitting, Gaussian function, Lorentzian function</pre>
		+	==27. Fourier Shell Correlation, FSC==
		+	Fourier Shell Correlation (FSC) is a measure of reliability of the three-dimensional (3D) structures of biological macromolecules obtained by “single particle analysis”. FSC is expressed as a function of spatial frequency by the following equation (normalized cross correlation).
		+	Here, F1(k) and F2(k)are 3D Fourier transforms of the two structures reconstructed from arbitrary two independent sets of TEM images obtained. F2(k)* is a complex conjugate of F2(k), and Σk,ΔkF(k) means summing up F(k) over a small range Δk around k.
		+	FSC (k) takes a value between +1 and –1 for every spatial frequency k. The value of FSC (k) close to +1 means that the two reconstructed structures agree well, indicating a high degree of reliability of the obtained structure. If FSC does not decrease monotonically with spatial frequency k at high frequencies, the reliability of the FSC function is considered to be low.
		+	The following example shows the FSC and spatial resolution for mouse apoferritin.
		+	First, a data set of TEM images acquired for single particle analysis is randomly divided into two groups, and two independent 3D structures are obtained by image-processing analysis for each group. FSC is calculated using the two obtained 3D structures.
		+	Fig. 1(a) shows a 3D density map of apoferritin reconstructed from 120,295 particles extracted from 840 cryo-TEM images. Fig. 1(b) is the FSC calculated for this 3D density map. The values of FSC at low frequencies are close to +1 and damped with increasing frequency monotonically towards 0. This behavior confirms high reliability of the FSC function.
		+	In single particle analysis, it is common to define the spatial resolution of the obtained structure to be the reciprocal of the spatial frequency at which FSC = 0.143. (The value “0.143” is a value adopted based on statistical discussion about the correlation of random noise and on referring the resolution defined for X-ray crystallography.) The resolution for apoferritin was determined to be 0.153 nm from a spatial frequency of 6.5 nm-1 at which FSC takes a value of 0.143.
		+	(Proofread by Specially Appointed Professor Keichi Namba, Osaka University)
		+
		+	Fig1. 3D density map of mouse apoferritin obtained by single particle analysis and its FSC. (a) 3D density map. (b) An FSC function calculated from the two 3D density maps. The resolution was determined to be 0.153 nm from a spatial frequency of 6.5 nm-1 at which FSC = 0.143.

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 Young fringe</pre>
 ===18-3．inverse Fourier transform===
-Fourier transform is a method that transforms a function of certain variables into the function of the variables conjugate to the certain variables. On the other hand, "inverse Fourier transform" is a method that transforms the Fourier-transformed function into the function of the original variable. For example, when a crystal potential as a function of position is Fourier-transformed, crystal structure factors are obtained as a function of wavenumber. Next, when the crystal structure factors are inverse-Fourier-transformed, the crystal potential as the function of position is obtained. In a TEM, Fourier transform and inverse Fourier transform of the specimen crystal are automatically executed, so that the diffraction pattern and structure image are obtained at the back focal plane and the image plane, respectively.
+Fourier transform is a method that transforms a function of certain variables into the function of the variables conjugate to the certain variables. On the other hand, "inverse Fourier transform" is a method that transforms the Fourier-transformed function into a function of the original variable. For example, when a crystal potential as a function of position is Fourier-transformed, crystal structure factors are obtained as a function of wavenumber. Next, when the crystal structure factors are inverse-Fourier-transformed, the crystal potential as the function of position is obtained. In TEM imaging, Fourier transform and inverse Fourier transform of the specimen are automatically executed, so that the diffraction pattern and structure image are obtained at the back focal plane and the image plane, respectively.
 <pre>Related term
 Fourier transform</pre>
 ==19. Fourier mask==
 The "Fourier mask" is a mask inserted on the diffraction plane to remove the noise of a lattice image or a structure image, allowing only diffraction spots to pass and background noise between the diffraction spots to be cut.

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 A method that transforms a function of certain variables into a function of the variables conjugate to the certain variables. For example, "Fourier transform" is used for light or sound as a function of time to express it as a function of frequency (spectral analysis), or for scattered waves from an object or a crystal potential as a function of position to transform it into a diffraction pattern or crystal structure factors as a function of wavenumber.
 The imaging lens system of a TEM can be considered as a system in which scattered waves from an object or a crystal potential are Fourier transformed into a diffraction pattern and subsequently inverse Fourier transformed to an image of the object, where the assumption of Fraunhofer diffraction is applied.
 [[文件:Fourier transform-3.jpg]]
 Here, ψ(r) stands for the scattered waves from the specimen, being a function of the real space r, and Ψ(k) for the diffracted waves formed on the back focal plane of the imaging lens, being a function of the reciprocal space k [1/m]. The first equation expresses Fourier transform. That is, the scattered waves ψ(r) produced by the specimen are transformed to a diffraction pattern Ψ(k). The second equation expresses inverse Fourier transform. That is, the diffraction pattern Ψ(k) is transformed to a specimen’s image ψ(r).
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 Fig. 1 TEM imaging by a lens (image and diffraction pattern)
 The imaging process of a transmission electron microscope can be considered as a series of Fourier transform from scattered waves from a specimen to a diffraction pattern and of inverse Fourier transform of the diffraction pattern to an image of the specimen. ψ(r) expresses scattered waves of the specimen, being the function of position r, and Ψ(k) expresses diffracted waves formed on the back focal plane of the imaging lens, being a function of the wave number k [1/m]. Φ(x)∝ψ(x/M) expresses the scattered waves on the imaging plane, which is an image of the specimen, being the function of position r. It is noted that Φ(x) is equal to ψ(r) shown in the second equation, except for a magnification M and image inversion caused by the lens.
 Fig. 2 (a) Diffraction pattern (Fourier transform pattern) of an Au crystal. (b) Crystal lattice image of the Au crystal and its FFT pattern. Accelerating voltage: 300 kV.

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 ==18. Fourier transform ==
 ===18-1. Fourier transform===
-A method that transforms a function of certain variables into the function of the variables conjugate to the certain variables. For example, "Fourier transform" is used for light or sound as a function of time to express it as the function of frequency (spectral analysis) or for an object or a crystal potential to transform it into crystal structure factors as a function of wavenumber or a diffraction pattern.
+A method that transforms a function of certain variables into a function of the variables conjugate to the certain variables. For example, "Fourier transform" is used for light or sound as a function of time to express it as a function of frequency (spectral analysis), or for scattered waves from an object or a crystal potential as a function of position to transform it into a diffraction pattern or crystal structure factors as a function of wavenumber.
 <pre>Related term
 inverse Fourier transform, Fourier synthesis</pre>
 ===18-2．fast Fourier transform===
 A computing method to decrease the calculation time of a Fourier transform. Since "fast Fourier transform (FFT)" is executed using a computer, calculations are performed by a discrete finite sum, instead by an integration. It greatly decreases the computation time by dividing the calculation into certain sub-groups and by changing the order of the calculation. In a discrete Fourier transform with a period of N, the number of operation is normally proportional to N2 but it is decreased to N log N in FFT. This method is used for analysis of high-resolution images.