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Hybrid Input-output Algorithm
The hybrid input-output (HIO) algorithm for phase retrieval is a modification of the error reduction algorithm for retrieving the phases in coherent diffraction imaging. Determining the phases of a diffraction pattern is crucial since the diffraction pattern of an object is its Fourier transform and in order to properly invert transform the diffraction pattern the phases must be known. Only the amplitude however, can be measured from the intensity of the diffraction pattern and can thus be known experimentally. This fact together with some kind of support constraint can be used in order to iteratively calculate the phases. The HIO algorithm uses negative feedback in Fourier space in order to progressively force the solution to conform to the Fourier domain constraints (support). Unlike the error reduction algorithm which alternately applies Fourier and object constraints the HIO "skips" the object domain step and replaces it with negative feedback acting upon the previous solutio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Phase Retrieval
Phase retrieval is the process of algorithmically finding solutions to the phase problem. Given a complex spectrum F(k), of amplitude , F (k), , and phase \psi(k): ::F(k) = , F(k), e^ =\int_^ f(x)\ e^\,dx where ''x'' is an ''M''-dimensional spatial coordinate and ''k'' is an ''M''-dimensional spatial frequency coordinate. Phase retrieval consists of finding the phase that satisfies a set of constraints for a measured amplitude. Important applications of phase retrieval include X-ray crystallography, transmission electron microscopy and coherent diffractive imaging, for which M = 2. Uniqueness theorems for both 1-D and 2-D cases of the phase retrieval problem, including the phaseless 1-D inverse scattering problem, were proven by Klibanov and his collaborators (see References). Problem formulation Here we consider 1-D discrete Fourier transform (DFT) phase retrieval problem. The DFT of a complex signal f[n] is given by F[k]=\sum_^ f[n] e^=, F[k], \cdot e^ \quad k=0,1, \ldots ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coherent Diffraction Imaging
Coherent diffractive imaging (CDI) a computational microscopy method that reconstructs images from coherent diffraction patterns without the use of lenses. It was first experimentally demonstrated in 1999 by Miao and collaborators using synchrotron X-rays and iterative phase retrieval. CDI has been applied to image structures such as nanotubes, nanocrystals, porous nanocrystalline layers, defects, potentially proteins, and more. In CDI, a highly coherent beam of X-rays, electrons or other wavelike particle or photon is incident on an object. The beam scattered by the object produces a diffraction pattern downstream which is then collected by a detector. This recorded pattern is then used to reconstruct an image via an iterative feedback algorithm. Effectively, the objective lens in a typical microscope is replaced with software to convert from the reciprocal space diffraction pattern into a real space image. The advantage in using no lenses is that the final image is aberratio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fourier Transform
In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the transform is a complex-valued function of frequency. The term ''Fourier transform'' refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made, the output of the operation is sometimes called the frequency domain representation of the original function. The Fourier transform is analogous to decomposing the sound of a musical chord into the intensities of its constituent pitches. Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa, a phenomenon known as the uncertainty principle. The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Support (mathematics)
In mathematics, the support of a real-valued function f is the subset of the function domain of elements that are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smallest closed set containing all points not mapped to zero. This concept is used widely in mathematical analysis. Formulation Suppose that f : X \to \R is a real-valued function whose domain is an arbitrary set X. The of f, written \operatorname(f), is the set of points in X where f is non-zero: \operatorname(f) = \. The support of f is the smallest subset of X with the property that f is zero on the subset's complement. If f(x) = 0 for all but a finite number of points x \in X, then f is said to have . If the set X has an additional structure (for example, a topology), then the support of f is defined in an analogous way as the smallest subset of X of an appropriate type such that f vanishes in an appropriate sense on its complement. The notion of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mean Square Error
In statistics, the mean squared error (MSE) or mean squared deviation (MSD) of an estimator (of a procedure for estimating an unobserved quantity) measures the average of the squares of the errors—that is, the average squared difference between the estimated values and the true value. MSE is a risk function, corresponding to the expected value of the squared error loss. The fact that MSE is almost always strictly positive (and not zero) is because of randomness or because the estimator does not account for information that could produce a more accurate estimate. In machine learning, specifically empirical risk minimization, MSE may refer to the ''empirical'' risk (the average loss on an observed data set), as an estimate of the true MSE (the true risk: the average loss on the actual population distribution). The MSE is a measure of the quality of an estimator. As it is derived from the square of Euclidean distance, it is always a positive value that decreases as the error app ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Crystallography
Crystallography is the branch of science devoted to the study of molecular and crystalline structure and properties. The word ''crystallography'' is derived from the Ancient Greek word (; "clear ice, rock-crystal"), and (; "to write"). In July 2012, the United Nations recognised the importance of the science of crystallography by proclaiming 2014 the International Year of Crystallography.UN announcement "International Year of Crystallography" iycr2014.org. 12 July 2012 Crystallography is a broad topic, and many of its subareas, such as X-ray crystallography, are themselves important scientific topics. Crystallography ranges from the fundamentals of crystal structure to the mathematics of Crystal system, crystal geometry, including those that are Aperiodic crystal, not periodic or quasi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Difference Map Algorithm
Difference commonly refers to: * Difference (philosophy), the set of properties by which items are distinguished * Difference (mathematics), the result of a subtraction Difference, The Difference, Differences or Differently may also refer to: Music * ''Difference'' (album), by Dreamtale, 2005 * ''The Difference'' (album), Pendleton, 2008 * "The Difference" (The Wallflowers song), 1997 * "Differences" (song), by Ginuwine, 2001 * ''Differently'' (album), by Cassie Davis, 2009 ** "Differently" (song), by Cassie Davis, 2009 * "Difference", a song by Benjamin Clementine from the 2022 album ''And I Have Been'' * "The Difference", a song by Matchbox Twenty from the 2002 album ''More Than You Think You Are'' * "The Difference", a song by Westlife from the 2009 album ''Where We Are'' * "The Difference", a song by Nick Jonas from the 2016 album '' Last Year Was Complicated'' * "The Difference", a song by Meek Mill featuring Quavo, from the 2016 mixtape '' DC4'' * "The Difference", a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Phase Retrieval
Phase retrieval is the process of algorithmically finding solutions to the phase problem. Given a complex spectrum F(k), of amplitude , F (k), , and phase \psi(k): ::F(k) = , F(k), e^ =\int_^ f(x)\ e^\,dx where ''x'' is an ''M''-dimensional spatial coordinate and ''k'' is an ''M''-dimensional spatial frequency coordinate. Phase retrieval consists of finding the phase that satisfies a set of constraints for a measured amplitude. Important applications of phase retrieval include X-ray crystallography, transmission electron microscopy and coherent diffractive imaging, for which M = 2. Uniqueness theorems for both 1-D and 2-D cases of the phase retrieval problem, including the phaseless 1-D inverse scattering problem, were proven by Klibanov and his collaborators (see References). Problem formulation Here we consider 1-D discrete Fourier transform (DFT) phase retrieval problem. The DFT of a complex signal f[n] is given by F[k]=\sum_^ f[n] e^=, F[k], \cdot e^ \quad k=0,1, \ldots ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Footnotes
In publishing, a note is a brief text in which the author comments on the subject and themes of the book and names supporting citations. In the editorial production of books and documents, typographically, a note is usually several lines of text at the bottom of the page, at the end of a chapter, at the end of a volume, or a house-style typographic usage throughout the text. Notes are usually identified with superscript numbers or a symbol.''The Oxford Companion to the English Language'' (1992) p. 709. Footnotes are informational notes located at the foot of the thematically relevant page, whilst endnotes are informational notes published at the end of a chapter, the end of a volume, or the conclusion of a multi-volume book. Unlike footnotes, which require manipulating the page design (text-block and page layouts) to accommodate the additional text, endnotes are advantageous to editorial production because the textual inclusion does not alter the design of the publication. H ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
