en>Yobot: WP:CHECKWIKI error #2 fix + general fixes using AWB (9276)

2013-06-21T15:24:00Z

WP:CHECKWIKI error #2 fix + general fixes using AWB (9276)

← Older revision		Revision as of 17:24, 21 June 2013
Line 1:		Line 1:
	~~Alyson Meagher~~ is the ~~name her mothers~~ and ~~fathers gave her but she doesn~~'~~t like when people use her complete name~~. ~~He functions as~~ a ~~bookkeeper~~. ~~Alaska~~ is ~~exactly~~ where I've always ~~been residing~~. The ~~favorite hobby for him and his children~~ is to ~~perform lacross~~ and ~~he would by no means give it up~~.<br><br>~~Here~~ is ~~my web site~~: [http://~~gcjcteam~~.org/~~index~~.~~php?mid~~=~~etc_video&document_srl~~=~~696611&sort_index~~=~~regdate&order_type~~=~~desc good psychic~~]		'''Phase retrieval''' is the process of [[algorithm]]ically finding solutions to the [[phase problem]]. Given a complex signal <math>F(k)</math>, of amplitude <math>\|F (k)\| </math>, and phase <math>\phi(k)</math>:

			::<math>F(k) = \|F(k)\|e^{i \phi(k)} =\int_{-\infty}^{\infty} f(x)\ e^{- 2\pi i u \cdot x}\,dx</math>

			where ''x'' is an ''M''-dimensional spatial coordinate and ''k'' is an ''M''-dimensional spatial frequency coordinate, phase retrieval consists in finding the phase that for a measured amplitude satisfies a set of constraints. Important applications of phase retrieval include [[X-ray crystallography]], [[transmission electron microscopy]] and [[coherent diffractive imaging]], for which <math> M = 2</math>. (Fineup 1982:2759) Uniqueness theorems for both 1-D and 2-D cases of the phase retrieval problem, including the phaseless 1-D inverse scattering problem, were proved by Klibanov and his collaborators (see References).

			== Methods ==

			=== Error reduction algorithm ===
			[[File:GS-diagram.png\|thumb\|right\|400px\|Schematic view of the error reduction algorithm for phase retrieval]]
			The error reduction or [[Gerchberg–Saxton algorithm]] solves the signal equation for <math> f(x) </math> by the iteration of a four-step process. In the first step, an estimate of the object <math> f(x) </math>, <math> g_k(x) </math> undergoes [[Fourier transformation]]:

			::<math>
			G_k(u) = \|G_k(u)\|e^{i \phi_k(u)} = \mathcal{F} (g_k (x))
			</math>

			The experimental value of <math> \|F(u)\| </math>, calculated from the diffraction pattern via the signal equation, is then substituted for <math> \|G_k(u)\| </math>, giving an estimate of the Fourier transformation:

			::<math>
			G'_k(u) = \|F(u)\|e^{i \phi_k(u)}
			</math>

			Next, the estimate of the Fourier transformation <math> G'_k(u) </math> is inverse Fourier transformed:

			::<math>
			g'_k(u) =\|g'_k(x)\|e^{i \theta'_k(x)} = \mathcal{F}^{-1}(G'_k (u))
			</math>

			<math> g'_k (x) </math> then must be changed so that the new estimate of the object, <math> g'_{k+1} (x) </math> satisfies the object constraints. <math> g'_{k+1} (x) </math> is therefore defined piecewise as:
			::<math>
			g'_{k+1} (x) \equiv \begin{cases}
			g'_k(u) & x \notin \gamma \\
			0 & x \in \gamma
			\end{cases}
			</math>
			where <math> \gamma </math> is the domain in which <math> g'_k (x) </math> does not satisfy the object constraints.
			As the calculated image, <math> g'_k (x) </math>, is [[complex number\|complex valued]], <math> \|g'_k(x)\| </math> is replaced by the experimental value of <math> \|f(x)\| </math>, to give the new estimate of the object. (Fienup and Wackerman, 1986:1899)

			::<math>g_{k+1}(x) =\|f(x)\|e^{i \theta'_{k+1}(x)}</math>

			This process is continued until both the Fourier constraint and object constraint are satisfied. Theoretically, the process will always lead to a [[convergence (mathematics)\|convergence]] (Fineup 1982:2761), but the large number of iterations needed to produce a satisfactory image (generally >2000) results in the error-reduction algorithm being unsuitably inefficient for sole use in practical applications.

			=== Hybrid input-output algorithm ===
			The hybrid input-output algorithm is a modification of the error-reduction algorithm - the first three stages are identical. However, <math> g_k(x) </math> no longer acts as an estimate of <math> f(x) </math>, but the input function corresponding to the output function <math> g'_k (x) </math>, which is an estimate of <math> f(x) </math> (Fienup 1982:2762). In the fourth step, when the function <math> g'_k (x) </math> violates the object constraints, the value of <math> g_{k+1}(x) </math> is forced towards zero, but optimally not to zero. The chief advantage of the hybrid input-output algorithm is that the function <math> g_k(x) </math> contains [[feedback\|feedback information]] concerning previous iterations, reducing the probability of stagnation.

			::<math>
			g'_{k+1} (x) \equiv \begin{cases}
			g'_k(u) & x \notin \gamma \\
			g_k (x) - \beta{g'_k(x)} & x \in \gamma
			\end{cases}
			</math>

			Here <math> \beta </math> is a feedback parameter which can take a value between 0 and 1. For most applications, <math> \beta \approx 0.9 </math> gives optimal results.

			=== Shrinkwrap ===
			For a two dimensional phase retrieval problem, there is a [[degeneracy (mathematics)\|degeneracy of solutions]] as <math> f(x) </math> and its conjugate <math> f^*(-x) </math> have the same Fourier modulus. This leads to "image twinning" in which the phase retrieval algorithm stagnates producing an image with features of both the object and its [[complex conjugate\|conjugate]] (Fineup and Wackerman, 1986:1900). The shrinkwrap technique periodically updates the estimate of the support by low-pass ﬁltering the current estimate of the object amplitude (by convolvution with a [[Gaussian function\|Gaussian]]) and applying a threshold, leading to a reduction in the image ambiguity (Marchesini et al., 2003).

			== References ==
			*{{cite journal\|last=Fienup\|first=J.R.\|journal=Applied Optics\|date=1 August 1982\|volume=21\|issue=15\|pages=2758–2769\|url=http://digitus.itk.ppke.hu/~matyi/optika/Phase_Diversity/AO82_PRComparison1.pdf\|title=Phase retrieval algorithms: a comparison.\|bibcode = 1982ApOpt..21.2758F \|doi = 10.1364/AO.21.002758 }}
			*{{cite journal\|author=Fienup, J. R. and Wackerman, C.C. \|title=Phase-retrieval stagnation problems and solutions\|journal=Journal of the Optical Society of America\|date=3 July 1986\|volume=3\|issue=11\|pages=1897–1907\|url=http://www.optics.rochester.edu/workgroups/fienup/PUBLICATIONS/JOSAA86_PRStagnation.pdf}}
			*{{cite journal\|author= Marchesini, S. He, H. Chapman, H. N. Hau-Riege, S. P. Noy, A. Howells, M. R. Weierstall, U. Spence, J.C.H. \|title=X-ray image reconstruction from a diffraction pattern alone \|journal=Phys. Rev. B\|year=2003\|volume=68\|pages=140101(R) \|url=http://arxiv.org/pdf/physics/0306174v2.pdf\|arxiv = cond-mat/0306449 \|bibcode = 2003PhRvB..68b0101C \|doi = 10.1103/PhysRevB.68.020101 }}
			*{{cite journal\|author=Klibanov, M. V.\|journal=Soviet Math. Doklady\|year=1985\|volume=32\|pages=668–670\|title=On uniqueness of the determination of a compactly supported function from the modulus of its Fourier transform }}
			*{{cite journal\|last=Klibanov\|first=M.V.\|journal=Differential Equations\|year=1987\|volume=22\|pages=1232–1240\|title=Determination of a function with compact support from the absolute value of its Fourier transform and an inverse scattering problem }}
			*{{cite journal\|last=Klibanov\|first=M.V.\|journal=Siberian Math J.\|year=1987\|volume=27\|pages=708–719\|title=Inverse scattering problems and restoration of a function from the modulus of its Fourier transform }}
			*{{cite journal\|author=Klibanov, M. V.\|journal=Differential Equations\|year=1989\|volume=25\|pages=520–527\|title=Uniqueness of the determination of distortions of a crystal lattice by the X-ray diffraction in a continuous dynamical model}}
			*{{cite journal\|author=Klibanov, M.V. and Sacks, P.E.\|journal=J. Math. Phys.\|year=1992\|volume=33\|pages=2813--3821\|title=Phaseless inverse scattering and the phase problem in optics }}
			*{{cite journal\|author=Klibanov, M. V., Sacks, P.E.\|journal=J. Comput. Phys.\|year=1994\|volume=112\|pages=273–281\|title=Use of partial knowledge of the potential in the phase problem of inverse scattering }}
			*{{cite journal\|author=Klibanov, M. V., Sacks, P.E., Tikhonravov, A.V.\|journal=Inverse Problems\|year=1995\|volume=11\|pages=1–28\|title=The phase retrieval problem \|bibcode = 1995InvPr..11....1K \|doi = 10.1088/0266-5611/11/1/001 }}
			*{{cite journal\|author=Klibanov, M. V.\|journal=J. Math. Anal. Appl.\|year=2006\|volume=323\|pages=818–843\|title=On the recovery of a 2-D function from the modulus of its Fourier transform }}

			==See also==
			* [[Phase problem]]
			* [[Crystallography]]
			* [[X-ray crystallography]]

			[[Category:Crystallography]]
			[[Category:Mathematical physics]]
			[[Category:Mathematical chemistry]]

en>Avocado: Undid revision 450751075 by 98.180.49.249 (talk) (units seem like a helpful thing to me)

2011-09-16T14:15:52Z

Undid revision 450751075 by 98.180.49.249 (talk) (units seem like a helpful thing to me)

New page

Alyson Meagher is the name her mothers and fathers gave her but she doesn't like when people use her complete name. He functions as a bookkeeper. Alaska is exactly where I've always been residing. The favorite hobby for him and his children is to perform lacross and he would by no means give it up.<br><br>Here is my web site: [http://gcjcteam.org/index.php?mid=etc_video&document_srl=696611&sort_index=regdate&order_type=desc good psychic]

Ballistic limit - Revision history

en>Yobot: WP:CHECKWIKI error #2 fix + general fixes using AWB (9276)

en>Avocado: Undid revision 450751075 by 98.180.49.249 (talk) (units seem like a helpful thing to me)