Hybrid input-output algorithm - Revision history

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	{{More citations needed\|date=April 2010}}		{{More citations needed\|date=April 2010}}

	'''~~Hybrid~~ input-output (HIO) algorithm for phase retrieval''' is a modification of the [[Phase_retrieval#Error_reduction_algorithm\|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 (mathematics)\|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 solution.		The '''hybrid input-output (HIO) algorithm for phase retrieval''' is a modification of the [[Phase_retrieval#Error_reduction_algorithm\|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 (mathematics)\|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 solution.

	Although it has been shown that the method of error reduction converges to a limit (but usually not to the correct or optimal solution) <ref>{{cite journal\|last=Bauschke\|first=Heinz H.\|author2=Combettes, Patrick L. \|author3=Luke, D. Russell \|title=Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization\|journal=Journal of the Optical Society of America A\|date=2002\|volume=19\|issue=7\|pages=1334–45\|doi=10.1364/JOSAA.19.001334\|pmid=12095200\|bibcode=2002JOSAA..19.1334B\|citeseerx=10.1.1.75.1070}}</ref>		Although it has been shown that the method of error reduction converges to a limit (but usually not to the correct or optimal solution) <ref>{{cite journal\|last=Bauschke\|first=Heinz H.\|author2=Combettes, Patrick L. \|author3=Luke, D. Russell \|title=Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization\|journal=Journal of the Optical Society of America A\|date=2002\|volume=19\|issue=7\|pages=1334–45\|doi=10.1364/JOSAA.19.001334\|pmid=12095200\|bibcode=2002JOSAA..19.1334B\|citeseerx=10.1.1.75.1070}}</ref>

MtPenguinMonster: /* top */ Added see also

2024-10-14T01:11:03Z

top: Added see also

← Previous revision		Revision as of 01:11, 14 October 2024
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	there is no limit to how long this process can take. Moreover, the error reduction algorithm will almost certainly find a local minimum instead of the global solution. The HIO differs from error reduction only in one step but this is enough to reduce this problem significantly. Whereas the error reduction approach iteratively improves solutions over time the HIO remodels the previous solution in Fourier space applying negative feedback. By minimizing the [[mean square error]] in Fourier space from the previous solution, the HIO provides a better candidate solution for inverse transforming. Although it is both faster and more powerful than error reduction, the HIO algorithm does have a uniqueness problem.<ref>Miao J, Kirz J, Sayre D, “The oversampling phasing method”, Acta Chryst. (2000), D56, 1312-1315\| https://scripts.iucr.org/cgi-bin/paper?S0907444900008970</ref>		there is no limit to how long this process can take. Moreover, the error reduction algorithm will almost certainly find a local minimum instead of the global solution. The HIO differs from error reduction only in one step but this is enough to reduce this problem significantly. Whereas the error reduction approach iteratively improves solutions over time the HIO remodels the previous solution in Fourier space applying negative feedback. By minimizing the [[mean square error]] in Fourier space from the previous solution, the HIO provides a better candidate solution for inverse transforming. Although it is both faster and more powerful than error reduction, the HIO algorithm does have a uniqueness problem.<ref>Miao J, Kirz J, Sayre D, “The oversampling phasing method”, Acta Chryst. (2000), D56, 1312-1315\| https://scripts.iucr.org/cgi-bin/paper?S0907444900008970</ref>
	Depending on how strong the negative feedback is there can often be more than one solution for any set of diffraction data. Although a problem, it has been shown that many of these possible solutions stem from the fact that HIO allows for mirror images taken in any plane to arise as solutions. In [[crystallography]], the scientist is seldom interested in the atomic coordinates relative to any other reference than the molecule itself and is therefore more than happy with a solution that is upside-down of flipped from the actual image. A downside is that HIO has a tendency to escape both global and local maxima. This problem also depends on the strength of the feedback parameter, and a good solution to this problem is to switch algorithm when the error reaches its minimum. Other methods of phasing a coherent diffraction pattern include [[difference map algorithm]] and "relaxed averaged alternating reflections" or RAAR.<ref>1.Luke Russel D, “Relaxed averaged alternating reflections for diffraction imaging” Inverse problems, (2005) 21, 37-50</ref>		Depending on how strong the negative feedback is there can often be more than one solution for any set of diffraction data. Although a problem, it has been shown that many of these possible solutions stem from the fact that HIO allows for mirror images taken in any plane to arise as solutions. In [[crystallography]], the scientist is seldom interested in the atomic coordinates relative to any other reference than the molecule itself and is therefore more than happy with a solution that is upside-down of flipped from the actual image. A downside is that HIO has a tendency to escape both global and local maxima. This problem also depends on the strength of the feedback parameter, and a good solution to this problem is to switch algorithm when the error reaches its minimum. Other methods of phasing a coherent diffraction pattern include [[difference map algorithm]] and "relaxed averaged alternating reflections" or RAAR.<ref>1.Luke Russel D, “Relaxed averaged alternating reflections for diffraction imaging” Inverse problems, (2005) 21, 37-50</ref>

			== See Also ==
			* [[Phase retrieval]]
			* [[Gerchberg-Saxton algorithm]]

	==References==		==References==

MtPenguinMonster: Adding short description: "Algorithm for phase retrieval"

2024-10-10T03:00:01Z

Adding short description: "Algorithm for phase retrieval"

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			{{Short description\|Algorithm for phase retrieval}}
	{{More citations needed\|date=April 2010}}		{{More citations needed\|date=April 2010}}

MayukhPahari at 07:33, 18 February 2024

2024-02-18T07:33:58Z

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	Although it has been shown that the method of error reduction converges to a limit (but usually not to the correct or optimal solution) <ref>{{cite journal\|last=Bauschke\|first=Heinz H.\|author2=Combettes, Patrick L. \|author3=Luke, D. Russell \|title=Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization\|journal=Journal of the Optical Society of America A\|date=2002\|volume=19\|issue=7\|pages=1334–45\|doi=10.1364/JOSAA.19.001334\|pmid=12095200\|bibcode=2002JOSAA..19.1334B\|citeseerx=10.1.1.75.1070}}</ref>		Although it has been shown that the method of error reduction converges to a limit (but usually not to the correct or optimal solution) <ref>{{cite journal\|last=Bauschke\|first=Heinz H.\|author2=Combettes, Patrick L. \|author3=Luke, D. Russell \|title=Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization\|journal=Journal of the Optical Society of America A\|date=2002\|volume=19\|issue=7\|pages=1334–45\|doi=10.1364/JOSAA.19.001334\|pmid=12095200\|bibcode=2002JOSAA..19.1334B\|citeseerx=10.1.1.75.1070}}</ref>
	<ref>{{cite journal\|last=Fienup\|first=J. R.\|title=Reconstruction of an object from the modulus of its Fourier transform\|journal=Optics Letters\|date=1 July 1978\|volume=3\|issue=1\|pages=27–29\|doi=10.1364/OL.3.000027\|pmid=19684685\|bibcode=1978OptL....3...27F}}</ref>		<ref>{{cite journal\|last=Fienup\|first=J. R.\|title=Reconstruction of an object from the modulus of its Fourier transform\|journal=Optics Letters\|date=1 July 1978\|volume=3\|issue=1\|pages=27–29\|doi=10.1364/OL.3.000027\|pmid=19684685\|bibcode=1978OptL....3...27F}}</ref>
	there is no limit to how long this process can take. Moreover, the error reduction algorithm will almost certainly find a local minimum instead of the global solution. The HIO differs from error reduction only in one step but this is enough to reduce this problem significantly. Whereas the error reduction approach iteratively improves solutions over time the HIO remodels the previous solution in Fourier space applying negative feedback. By minimizing the [[mean square error]] in Fourier space from the previous solution, the HIO provides a better candidate solution for inverse transforming. Although it is both faster and more powerful than error reduction, the HIO algorithm does have a uniqueness problem.<ref>Miao J, Kirz J, Sayre D, “The oversampling phasing method”, Acta Chryst. (2000), D56, 1312-1315</ref>		there is no limit to how long this process can take. Moreover, the error reduction algorithm will almost certainly find a local minimum instead of the global solution. The HIO differs from error reduction only in one step but this is enough to reduce this problem significantly. Whereas the error reduction approach iteratively improves solutions over time the HIO remodels the previous solution in Fourier space applying negative feedback. By minimizing the [[mean square error]] in Fourier space from the previous solution, the HIO provides a better candidate solution for inverse transforming. Although it is both faster and more powerful than error reduction, the HIO algorithm does have a uniqueness problem.<ref>Miao J, Kirz J, Sayre D, “The oversampling phasing method”, Acta Chryst. (2000), D56, 1312-1315\| https://scripts.iucr.org/cgi-bin/paper?S0907444900008970</ref>
	Depending on how strong the negative feedback is there can often be more than one solution for any set of diffraction data. Although a problem, it has been shown that many of these possible solutions stem from the fact that HIO allows for mirror images taken in any plane to arise as solutions. In [[crystallography]], the scientist is seldom interested in the atomic coordinates relative to any other reference than the molecule itself and is therefore more than happy with a solution that is upside-down of flipped from the actual image. A downside is that HIO has a tendency to escape both global and local maxima. This problem also depends on the strength of the feedback parameter, and a good solution to this problem is to switch algorithm when the error reaches its minimum. Other methods of phasing a coherent diffraction pattern include [[difference map algorithm]] and "relaxed averaged alternating reflections" or RAAR.<ref>1.Luke Russel D, “Relaxed averaged alternating reflections for diffraction imaging” Inverse problems, (2005) 21, 37-50</ref>		Depending on how strong the negative feedback is there can often be more than one solution for any set of diffraction data. Although a problem, it has been shown that many of these possible solutions stem from the fact that HIO allows for mirror images taken in any plane to arise as solutions. In [[crystallography]], the scientist is seldom interested in the atomic coordinates relative to any other reference than the molecule itself and is therefore more than happy with a solution that is upside-down of flipped from the actual image. A downside is that HIO has a tendency to escape both global and local maxima. This problem also depends on the strength of the feedback parameter, and a good solution to this problem is to switch algorithm when the error reaches its minimum. Other methods of phasing a coherent diffraction pattern include [[difference map algorithm]] and "relaxed averaged alternating reflections" or RAAR.<ref>1.Luke Russel D, “Relaxed averaged alternating reflections for diffraction imaging” Inverse problems, (2005) 21, 37-50</ref>

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	'''Hybrid input-output (HIO) algorithm for phase retrieval''' is a modification of the [[Phase_retrieval#Error_reduction_algorithm\|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 (mathematics)\|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 solution.		'''Hybrid input-output (HIO) algorithm for phase retrieval''' is a modification of the [[Phase_retrieval#Error_reduction_algorithm\|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 (mathematics)\|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 solution.

	Although it has been shown that the method of error reduction converges to a limit (but usually not to the correct or optimal solution) <ref>{{cite journal\|last=Bauschke\|first=Heinz H.\|author2=Combettes, Patrick L. \|author3=Luke, D. Russell \|title=Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization\|journal=Journal of the Optical Society of America A\|date=2002\|volume=19\|issue=7\|pages=1334–45\|doi=10.1364/JOSAA.19.001334\|pmid=12095200\|bibcode=2002JOSAA..19.1334B}}</ref>		Although it has been shown that the method of error reduction converges to a limit (but usually not to the correct or optimal solution) <ref>{{cite journal\|last=Bauschke\|first=Heinz H.\|author2=Combettes, Patrick L. \|author3=Luke, D. Russell \|title=Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization\|journal=Journal of the Optical Society of America A\|date=2002\|volume=19\|issue=7\|pages=1334–45\|doi=10.1364/JOSAA.19.001334\|pmid=12095200\|bibcode=2002JOSAA..19.1334B\|citeseerx=10.1.1.75.1070}}</ref>
	<ref>{{cite journal\|last=Fienup\|first=J. R.\|title=Reconstruction of an object from the modulus of its Fourier transform\|journal=Optics Letters\|date=1 July 1978\|volume=3\|issue=1\|pages=27–29\|doi=10.1364/OL.3.000027\|pmid=19684685\|bibcode=1978OptL....3...27F}}</ref>		<ref>{{cite journal\|last=Fienup\|first=J. R.\|title=Reconstruction of an object from the modulus of its Fourier transform\|journal=Optics Letters\|date=1 July 1978\|volume=3\|issue=1\|pages=27–29\|doi=10.1364/OL.3.000027\|pmid=19684685\|bibcode=1978OptL....3...27F}}</ref>
	there is no limit to how long this process can take. Moreover, the error reduction algorithm will almost certainly find a local minimum instead of the global solution. The HIO differs from error reduction only in one step but this is enough to reduce this problem significantly. Whereas the error reduction approach iteratively improves solutions over time the HIO remodels the previous solution in Fourier space applying negative feedback. By minimizing the [[mean square error]] in Fourier space from the previous solution, the HIO provides a better candidate solution for inverse transforming. Although it is both faster and more powerful than error reduction, the HIO algorithm does have a uniqueness problem.<ref>Miao J, Kirz J, Sayre D, “The oversampling phasing method”, Acta Chryst. (2000), D56, 1312-1315</ref>		there is no limit to how long this process can take. Moreover, the error reduction algorithm will almost certainly find a local minimum instead of the global solution. The HIO differs from error reduction only in one step but this is enough to reduce this problem significantly. Whereas the error reduction approach iteratively improves solutions over time the HIO remodels the previous solution in Fourier space applying negative feedback. By minimizing the [[mean square error]] in Fourier space from the previous solution, the HIO provides a better candidate solution for inverse transforming. Although it is both faster and more powerful than error reduction, the HIO algorithm does have a uniqueness problem.<ref>Miao J, Kirz J, Sayre D, “The oversampling phasing method”, Acta Chryst. (2000), D56, 1312-1315</ref>

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	Although it has been shown that the method of error reduction converges to a limit (but usually not to the correct or optimal solution) <ref>{{cite journal\|last=Bauschke\|first=Heinz H.\|author2=Combettes, Patrick L. \|author3=Luke, D. Russell \|title=Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization\|journal=Journal of the Optical Society of America A\|date=2002\|volume=19\|issue=7\|pages=1334–45\|doi=10.1364/JOSAA.19.001334\|pmid=12095200\|bibcode=2002JOSAA..19.1334B}}</ref>		Although it has been shown that the method of error reduction converges to a limit (but usually not to the correct or optimal solution) <ref>{{cite journal\|last=Bauschke\|first=Heinz H.\|author2=Combettes, Patrick L. \|author3=Luke, D. Russell \|title=Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization\|journal=Journal of the Optical Society of America A\|date=2002\|volume=19\|issue=7\|pages=1334–45\|doi=10.1364/JOSAA.19.001334\|pmid=12095200\|bibcode=2002JOSAA..19.1334B}}</ref>
	<ref>{{cite journal\|last=Fienup\|first=J. R.\|title=Reconstruction of an object from the modulus of its Fourier transform\|journal=Optics Letters\|date=1 July 1978\|volume=3\|issue=1\|pages=27\|doi=10.1364/OL.3.000027\|pmid=19684685\|bibcode=1978OptL....3...27F}}</ref>		<ref>{{cite journal\|last=Fienup\|first=J. R.\|title=Reconstruction of an object from the modulus of its Fourier transform\|journal=Optics Letters\|date=1 July 1978\|volume=3\|issue=1\|pages=27–29\|doi=10.1364/OL.3.000027\|pmid=19684685\|bibcode=1978OptL....3...27F}}</ref>
	there is no limit to how long this process can take. Moreover, the error reduction algorithm will almost certainly find a local minimum instead of the global solution. The HIO differs from error reduction only in one step but this is enough to reduce this problem significantly. Whereas the error reduction approach iteratively improves solutions over time the HIO remodels the previous solution in Fourier space applying negative feedback. By minimizing the [[mean square error]] in Fourier space from the previous solution, the HIO provides a better candidate solution for inverse transforming. Although it is both faster and more powerful than error reduction, the HIO algorithm does have a uniqueness problem.<ref>Miao J, Kirz J, Sayre D, “The oversampling phasing method”, Acta Chryst. (2000), D56, 1312-1315</ref>		there is no limit to how long this process can take. Moreover, the error reduction algorithm will almost certainly find a local minimum instead of the global solution. The HIO differs from error reduction only in one step but this is enough to reduce this problem significantly. Whereas the error reduction approach iteratively improves solutions over time the HIO remodels the previous solution in Fourier space applying negative feedback. By minimizing the [[mean square error]] in Fourier space from the previous solution, the HIO provides a better candidate solution for inverse transforming. Although it is both faster and more powerful than error reduction, the HIO algorithm does have a uniqueness problem.<ref>Miao J, Kirz J, Sayre D, “The oversampling phasing method”, Acta Chryst. (2000), D56, 1312-1315</ref>
	Depending on how strong the negative feedback is there can often be more than one solution for any set of diffraction data. Although a problem, it has been shown that many of these possible solutions stem from the fact that HIO allows for mirror images taken in any plane to arise as solutions. In [[crystallography]], the scientist is seldom interested in the atomic coordinates relative to any other reference than the molecule itself and is therefore more than happy with a solution that is upside-down of flipped from the actual image. A downside is that HIO has a tendency to escape both global and local maxima. This problem also depends on the strength of the feedback parameter, and a good solution to this problem is to switch algorithm when the error reaches its minimum. Other methods of phasing a coherent diffraction pattern include [[difference map algorithm]] and "relaxed averaged alternating reflections" or RAAR.<ref>1.Luke Russel D, “Relaxed averaged alternating reflections for diffraction imaging” Inverse problems, (2005) 21, 37-50</ref>		Depending on how strong the negative feedback is there can often be more than one solution for any set of diffraction data. Although a problem, it has been shown that many of these possible solutions stem from the fact that HIO allows for mirror images taken in any plane to arise as solutions. In [[crystallography]], the scientist is seldom interested in the atomic coordinates relative to any other reference than the molecule itself and is therefore more than happy with a solution that is upside-down of flipped from the actual image. A downside is that HIO has a tendency to escape both global and local maxima. This problem also depends on the strength of the feedback parameter, and a good solution to this problem is to switch algorithm when the error reaches its minimum. Other methods of phasing a coherent diffraction pattern include [[difference map algorithm]] and "relaxed averaged alternating reflections" or RAAR.<ref>1.Luke Russel D, “Relaxed averaged alternating reflections for diffraction imaging” Inverse problems, (2005) 21, 37-50</ref>

Brycehughes: Brycehughes moved page Hybrid input output (HIO) algorithm for phase retrieval to Hybrid input-output algorithm: common name

2021-01-19T18:16:54Z

Brycehughes moved page Hybrid input output (HIO) algorithm for phase retrieval to Hybrid input-output algorithm: common name

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Chisago: language, links

2021-01-19T13:33:24Z

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	{{More citations needed\|date=April 2010}}		{{More citations needed\|date=April 2010}}

	'''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 ~~inverse~~ 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 (mathematics)]] 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 solution.		'''Hybrid input-output (HIO) algorithm for phase retrieval''' is a modification of the [[Phase_retrieval#Error_reduction_algorithm\|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 (mathematics)\|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 solution.

	Although it has been shown that the method of error reduction converges to a limit (but usually not to the correct or optimal solution) <ref>{{cite journal\|last=Bauschke\|first=Heinz H.\|author2=Combettes, Patrick L. \|author3=Luke, D. Russell \|title=Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization\|journal=Journal of the Optical Society of America A\|date=2002\|volume=19\|issue=7\|pages=1334–45\|doi=10.1364/JOSAA.19.001334\|pmid=12095200\|bibcode=2002JOSAA..19.1334B}}</ref>		Although it has been shown that the method of error reduction converges to a limit (but usually not to the correct or optimal solution) <ref>{{cite journal\|last=Bauschke\|first=Heinz H.\|author2=Combettes, Patrick L. \|author3=Luke, D. Russell \|title=Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization\|journal=Journal of the Optical Society of America A\|date=2002\|volume=19\|issue=7\|pages=1334–45\|doi=10.1364/JOSAA.19.001334\|pmid=12095200\|bibcode=2002JOSAA..19.1334B}}</ref>
	<ref>{{cite journal\|last=Fienup\|first=J. R.\|title=Reconstruction of an object from the modulus of its Fourier transform\|journal=Optics Letters\|date=1 July 1978\|volume=3\|issue=1\|pages=27\|doi=10.1364/OL.3.000027\|pmid=19684685\|bibcode=1978OptL....3...27F}}</ref>		<ref>{{cite journal\|last=Fienup\|first=J. R.\|title=Reconstruction of an object from the modulus of its Fourier transform\|journal=Optics Letters\|date=1 July 1978\|volume=3\|issue=1\|pages=27\|doi=10.1364/OL.3.000027\|pmid=19684685\|bibcode=1978OptL....3...27F}}</ref>
	there is no limit to how long this process can take. Moreover, the error reduction algorithm will almost certainly find a local ~~minima~~ instead of the global. The HIO differs from error reduction only in one step but this is enough to reduce this problem significantly. Whereas the error reduction approach iteratively improves solutions over time the HIO remodels the previous solution in Fourier space applying negative feedback. By minimizing the mean square error in ~~the~~ Fourier space from the previous solution, the HIO provides a better candidate solution for inverse transforming. Although it is both faster and more powerful than error reduction, the HIO algorithm does have a uniqueness problem.<ref>Miao J, Kirz J, Sayre D, “The oversampling phasing method”, Acta Chryst. (2000), D56, 1312-1315</ref>		there is no limit to how long this process can take. Moreover, the error reduction algorithm will almost certainly find a local minimum instead of the global solution. The HIO differs from error reduction only in one step but this is enough to reduce this problem significantly. Whereas the error reduction approach iteratively improves solutions over time the HIO remodels the previous solution in Fourier space applying negative feedback. By minimizing the [[mean square error]] in Fourier space from the previous solution, the HIO provides a better candidate solution for inverse transforming. Although it is both faster and more powerful than error reduction, the HIO algorithm does have a uniqueness problem.<ref>Miao J, Kirz J, Sayre D, “The oversampling phasing method”, Acta Chryst. (2000), D56, 1312-1315</ref>
	Depending on how strong the negative feedback is there can often be more than one solution for any set of diffraction data. Although a problem, it has been shown that many of these possible solutions stem from the fact that HIO allows for mirror images taken in any plane to arise as solutions. In crystallography, the scientist is seldom interested in the atomic coordinates relative to any other reference than the molecule itself and is therefore more than happy with a solution that is upside-down of flipped from the actual image. ~~On the~~ downside, HIO ~~does have~~ a tendency ~~to be able~~ to escape both global and local maxima. This problem also depends on the strength of the feedback parameter, and a good solution to this problem is to switch algorithm when the error reaches its minimum. Other methods of phasing a coherent diffraction pattern include [[difference map algorithm]] and "relaxed averaged alternating reflections" or RAAR.<ref>1.Luke Russel D, “Relaxed averaged alternating reflections for diffraction imaging” Inverse problems, (2005) 21, 37-50</ref>		Depending on how strong the negative feedback is there can often be more than one solution for any set of diffraction data. Although a problem, it has been shown that many of these possible solutions stem from the fact that HIO allows for mirror images taken in any plane to arise as solutions. In [[crystallography]], the scientist is seldom interested in the atomic coordinates relative to any other reference than the molecule itself and is therefore more than happy with a solution that is upside-down of flipped from the actual image. A downside is that HIO has a tendency to escape both global and local maxima. This problem also depends on the strength of the feedback parameter, and a good solution to this problem is to switch algorithm when the error reaches its minimum. Other methods of phasing a coherent diffraction pattern include [[difference map algorithm]] and "relaxed averaged alternating reflections" or RAAR.<ref>1.Luke Russel D, “Relaxed averaged alternating reflections for diffraction imaging” Inverse problems, (2005) 21, 37-50</ref>

	==References==		==References==

John of Reading: /* top */Typo/general fixes, replaced: metod → method (surely?)

2020-12-09T16:45:23Z

top: Typo/general fixes, replaced: metod → method (surely?)

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	{{~~Refimprove~~\|date=April 2010}}		{{More citations needed\|date=April 2010}}

	'''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 inverse 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 (mathematics)]] 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 solution.		'''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 inverse 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 (mathematics)]] 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 solution.
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	Although it has been shown that the method of error reduction converges to a limit (but usually not to the correct or optimal solution) <ref>{{cite journal\|last=Bauschke\|first=Heinz H.\|author2=Combettes, Patrick L. \|author3=Luke, D. Russell \|title=Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization\|journal=Journal of the Optical Society of America A\|date=2002\|volume=19\|issue=7\|pages=1334–45\|doi=10.1364/JOSAA.19.001334\|pmid=12095200\|bibcode=2002JOSAA..19.1334B}}</ref>		Although it has been shown that the method of error reduction converges to a limit (but usually not to the correct or optimal solution) <ref>{{cite journal\|last=Bauschke\|first=Heinz H.\|author2=Combettes, Patrick L. \|author3=Luke, D. Russell \|title=Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization\|journal=Journal of the Optical Society of America A\|date=2002\|volume=19\|issue=7\|pages=1334–45\|doi=10.1364/JOSAA.19.001334\|pmid=12095200\|bibcode=2002JOSAA..19.1334B}}</ref>
	<ref>{{cite journal\|last=Fienup\|first=J. R.\|title=Reconstruction of an object from the modulus of its Fourier transform\|journal=Optics Letters\|date=1 July 1978\|volume=3\|issue=1\|pages=27\|doi=10.1364/OL.3.000027\|pmid=19684685\|bibcode=1978OptL....3...27F}}</ref>		<ref>{{cite journal\|last=Fienup\|first=J. R.\|title=Reconstruction of an object from the modulus of its Fourier transform\|journal=Optics Letters\|date=1 July 1978\|volume=3\|issue=1\|pages=27\|doi=10.1364/OL.3.000027\|pmid=19684685\|bibcode=1978OptL....3...27F}}</ref>
	there is no limit to how long this process can take. Moreover, the error reduction algorithm will almost certainly find a local minima instead of the global. The HIO differs from error reduction only in one step but this is enough to reduce this problem significantly. Whereas the error reduction approach iteratively improves solutions over time the HIO remodels the previous solution in Fourier space applying negative feedback. By minimizing the mean square error in the Fourier space from the previous solution, the HIO provides a better candidate solution for inverse transforming. Although it is both faster and more powerful than error reduction, the HIO algorithm does have a uniqueness problem.<ref>Miao J, Kirz J, Sayre D, “The oversampling phasing ~~metod”~~, Acta Chryst. (2000), D56, 1312-1315</ref>		there is no limit to how long this process can take. Moreover, the error reduction algorithm will almost certainly find a local minima instead of the global. The HIO differs from error reduction only in one step but this is enough to reduce this problem significantly. Whereas the error reduction approach iteratively improves solutions over time the HIO remodels the previous solution in Fourier space applying negative feedback. By minimizing the mean square error in the Fourier space from the previous solution, the HIO provides a better candidate solution for inverse transforming. Although it is both faster and more powerful than error reduction, the HIO algorithm does have a uniqueness problem.<ref>Miao J, Kirz J, Sayre D, “The oversampling phasing method”, Acta Chryst. (2000), D56, 1312-1315</ref>
	Depending on how strong the negative feedback is there can often be more than one solution for any set of diffraction data. Although a problem, it has been shown that many of these possible solutions stem from the fact that HIO allows for mirror images taken in any plane to arise as solutions. In crystallography, the scientist is seldom interested in the atomic coordinates relative to any other reference than the molecule itself and is therefore more than happy with a solution that is upside-down of flipped from the actual image. On the downside, HIO does have a tendency to be able to escape both global and local maxima. This problem also depends on the strength of the feedback parameter, and a good solution to this problem is to switch algorithm when the error reaches its minimum. Other methods of phasing a coherent diffraction pattern include [[difference map algorithm]] and "relaxed averaged alternating reflections" or RAAR.<ref>1.Luke Russel D, “Relaxed averaged alternating reflections for diffraction imaging” Inverse problems, (2005) 21, 37-50</ref>		Depending on how strong the negative feedback is there can often be more than one solution for any set of diffraction data. Although a problem, it has been shown that many of these possible solutions stem from the fact that HIO allows for mirror images taken in any plane to arise as solutions. In crystallography, the scientist is seldom interested in the atomic coordinates relative to any other reference than the molecule itself and is therefore more than happy with a solution that is upside-down of flipped from the actual image. On the downside, HIO does have a tendency to be able to escape both global and local maxima. This problem also depends on the strength of the feedback parameter, and a good solution to this problem is to switch algorithm when the error reaches its minimum. Other methods of phasing a coherent diffraction pattern include [[difference map algorithm]] and "relaxed averaged alternating reflections" or RAAR.<ref>1.Luke Russel D, “Relaxed averaged alternating reflections for diffraction imaging” Inverse problems, (2005) 21, 37-50</ref>

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← Previous revision		Revision as of 03:11, 14 December 2019
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	'''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 inverse 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 (mathematics)]] 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 solution.		'''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 inverse 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 (mathematics)]] 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 solution.

	Although it has been shown that the method of error reduction converges to a limit (but usually not to the correct or optimal solution) <ref>{{cite journal\|last=Bauschke\|first=Heinz H.\|author2=Combettes, Patrick L. \|author3=Luke, D. Russell \|title=Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization\|journal=Journal of the Optical Society of America A\|date=2002\|volume=19\|issue=7\|pages=~~1334~~\|doi=10.1364/JOSAA.19.001334\|bibcode=2002JOSAA..19.1334B}}</ref>		Although it has been shown that the method of error reduction converges to a limit (but usually not to the correct or optimal solution) <ref>{{cite journal\|last=Bauschke\|first=Heinz H.\|author2=Combettes, Patrick L. \|author3=Luke, D. Russell \|title=Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization\|journal=Journal of the Optical Society of America A\|date=2002\|volume=19\|issue=7\|pages=1334–45\|doi=10.1364/JOSAA.19.001334\|pmid=12095200\|bibcode=2002JOSAA..19.1334B}}</ref>
	<ref>{{cite journal\|last=Fienup\|first=J. R.\|title=Reconstruction of an object from the modulus of its Fourier transform\|journal=Optics Letters\|date=1 July 1978\|volume=3\|issue=1\|pages=27\|doi=10.1364/OL.3.000027\|bibcode=1978OptL....3...27F}}</ref>		<ref>{{cite journal\|last=Fienup\|first=J. R.\|title=Reconstruction of an object from the modulus of its Fourier transform\|journal=Optics Letters\|date=1 July 1978\|volume=3\|issue=1\|pages=27\|doi=10.1364/OL.3.000027\|pmid=19684685\|bibcode=1978OptL....3...27F}}</ref>
	there is no limit to how long this process can take. Moreover, the error reduction algorithm will almost certainly find a local minima instead of the global. The HIO differs from error reduction only in one step but this is enough to reduce this problem significantly. Whereas the error reduction approach iteratively improves solutions over time the HIO remodels the previous solution in Fourier space applying negative feedback. By minimizing the mean square error in the Fourier space from the previous solution, the HIO provides a better candidate solution for inverse transforming. Although it is both faster and more powerful than error reduction, the HIO algorithm does have a uniqueness problem.<ref>Miao J, Kirz J, Sayre D, “The oversampling phasing metod”, Acta Chryst. (2000), D56, 1312-1315</ref>		there is no limit to how long this process can take. Moreover, the error reduction algorithm will almost certainly find a local minima instead of the global. The HIO differs from error reduction only in one step but this is enough to reduce this problem significantly. Whereas the error reduction approach iteratively improves solutions over time the HIO remodels the previous solution in Fourier space applying negative feedback. By minimizing the mean square error in the Fourier space from the previous solution, the HIO provides a better candidate solution for inverse transforming. Although it is both faster and more powerful than error reduction, the HIO algorithm does have a uniqueness problem.<ref>Miao J, Kirz J, Sayre D, “The oversampling phasing metod”, Acta Chryst. (2000), D56, 1312-1315</ref>
	Depending on how strong the negative feedback is there can often be more than one solution for any set of diffraction data. Although a problem, it has been shown that many of these possible solutions stem from the fact that HIO allows for mirror images taken in any plane to arise as solutions. In crystallography, the scientist is seldom interested in the atomic coordinates relative to any other reference than the molecule itself and is therefore more than happy with a solution that is upside-down of flipped from the actual image. On the downside, HIO does have a tendency to be able to escape both global and local maxima. This problem also depends on the strength of the feedback parameter, and a good solution to this problem is to switch algorithm when the error reaches its minimum. Other methods of phasing a coherent diffraction pattern include [[difference map algorithm]] and "relaxed averaged alternating reflections" or RAAR.<ref>1.Luke Russel D, “Relaxed averaged alternating reflections for diffraction imaging” Inverse problems, (2005) 21, 37-50</ref>		Depending on how strong the negative feedback is there can often be more than one solution for any set of diffraction data. Although a problem, it has been shown that many of these possible solutions stem from the fact that HIO allows for mirror images taken in any plane to arise as solutions. In crystallography, the scientist is seldom interested in the atomic coordinates relative to any other reference than the molecule itself and is therefore more than happy with a solution that is upside-down of flipped from the actual image. On the downside, HIO does have a tendency to be able to escape both global and local maxima. This problem also depends on the strength of the feedback parameter, and a good solution to this problem is to switch algorithm when the error reaches its minimum. Other methods of phasing a coherent diffraction pattern include [[difference map algorithm]] and "relaxed averaged alternating reflections" or RAAR.<ref>1.Luke Russel D, “Relaxed averaged alternating reflections for diffraction imaging” Inverse problems, (2005) 21, 37-50</ref>

← Previous revision		Revision as of 02:26, 14 October 2024
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	{{More citations needed\|date=April 2010}}		{{More citations needed\|date=April 2010}}

	'''~~Hybrid~~ input-output (HIO) algorithm for phase retrieval''' is a modification of the [[Phase_retrieval#Error_reduction_algorithm\|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 (mathematics)\|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 solution.		The '''hybrid input-output (HIO) algorithm for phase retrieval''' is a modification of the [[Phase_retrieval#Error_reduction_algorithm\|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 (mathematics)\|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 solution.

	Although it has been shown that the method of error reduction converges to a limit (but usually not to the correct or optimal solution) <ref>{{cite journal\|last=Bauschke\|first=Heinz H.\|author2=Combettes, Patrick L. \|author3=Luke, D. Russell \|title=Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization\|journal=Journal of the Optical Society of America A\|date=2002\|volume=19\|issue=7\|pages=1334–45\|doi=10.1364/JOSAA.19.001334\|pmid=12095200\|bibcode=2002JOSAA..19.1334B\|citeseerx=10.1.1.75.1070}}</ref>		Although it has been shown that the method of error reduction converges to a limit (but usually not to the correct or optimal solution) <ref>{{cite journal\|last=Bauschke\|first=Heinz H.\|author2=Combettes, Patrick L. \|author3=Luke, D. Russell \|title=Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization\|journal=Journal of the Optical Society of America A\|date=2002\|volume=19\|issue=7\|pages=1334–45\|doi=10.1364/JOSAA.19.001334\|pmid=12095200\|bibcode=2002JOSAA..19.1334B\|citeseerx=10.1.1.75.1070}}</ref>