Application of the digital shearing method to extract three-component velocity inholographic particle image velocimetry
Identifieur interne : 002207 ( Istex/Corpus ); précédent : 002206; suivant : 002208Application of the digital shearing method to extract three-component velocity inholographic particle image velocimetry
Auteurs : Hui Yang ; Neil Halliwell ; Jeremy CouplandSource :
- Measurement Science and Technology [ 0957-0233 ] ; 2004.
Abstract
We have recently proposed a new method to extract the three-dimensional (3D) velocityvector data from double-exposure holographic particle image velocimetry (HPIV), which wecall the digital shearing method. In contrast to the full 3D correlation, it has been shownthat all three components (3Cs) of particle image displacement can be retrieved using sixtwo-dimensional fast Fourier transform operations and appropriate coordinatetransformations. In this paper we demonstrate the capabilities of this approach on actualHPIV data.The holographic recording method described uses an imaging system to recorda hologram of high numerical aperture using a conventional 35mm film. Theholograms are digitized and particle images are reconstructed numerically. Fromparticle images reconstructed from separate holograms, we illustrate the analysisprocess by computing the 3Cs of particle image displacement in a step-by-stepmanner.
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DOI: 10.1088/0957-0233/15/4/011
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<record><TEI wicri:istexFullTextTei="biblStruct"><teiHeader><fileDesc><titleStmt><title>Application of the digital shearing method to extract three-component velocity inholographic particle image velocimetry</title><author><name sortKey="Yang, Hui" sort="Yang, Hui" uniqKey="Yang H" first="Hui" last="Yang">Hui Yang</name><affiliation><mods:affiliation>Wolfson School of Mechanical and Manufacturing Engineering, Loughborough University, Leicestershire LE11 3TU, UK</mods:affiliation></affiliation><affiliation><mods:affiliation>E-mail:h.yang@lboro.ac.uk</mods:affiliation></affiliation></author><author><name sortKey="Halliwell, Neil" sort="Halliwell, Neil" uniqKey="Halliwell N" first="Neil" last="Halliwell">Neil Halliwell</name><affiliation><mods:affiliation>Wolfson School of Mechanical and Manufacturing Engineering, Loughborough University, Leicestershire LE11 3TU, UK</mods:affiliation></affiliation></author><author><name sortKey="Coupland, Jeremy" sort="Coupland, Jeremy" uniqKey="Coupland J" first="Jeremy" last="Coupland">Jeremy Coupland</name><affiliation><mods:affiliation>Wolfson School of Mechanical and Manufacturing Engineering, Loughborough University, Leicestershire LE11 3TU, UK</mods:affiliation></affiliation></author></titleStmt><publicationStmt><idno type="wicri:source">ISTEX</idno><idno type="RBID">ISTEX:86D152883DF928EC1E7CF7D1150885A17083948A</idno><date when="2004" year="2004">2004</date><idno type="doi">10.1088/0957-0233/15/4/011</idno><idno type="url">https://api.istex.fr/document/86D152883DF928EC1E7CF7D1150885A17083948A/fulltext/pdf</idno><idno type="wicri:Area/Istex/Corpus">002207</idno></publicationStmt><sourceDesc><biblStruct><analytic><title level="a">Application of the digital shearing method to extract three-component velocity inholographic particle image velocimetry</title><author><name sortKey="Yang, Hui" sort="Yang, Hui" uniqKey="Yang H" first="Hui" last="Yang">Hui Yang</name><affiliation><mods:affiliation>Wolfson School of Mechanical and Manufacturing Engineering, Loughborough University, Leicestershire LE11 3TU, UK</mods:affiliation></affiliation><affiliation><mods:affiliation>E-mail:h.yang@lboro.ac.uk</mods:affiliation></affiliation></author><author><name sortKey="Halliwell, Neil" sort="Halliwell, Neil" uniqKey="Halliwell N" first="Neil" last="Halliwell">Neil Halliwell</name><affiliation><mods:affiliation>Wolfson School of Mechanical and Manufacturing Engineering, Loughborough University, Leicestershire LE11 3TU, UK</mods:affiliation></affiliation></author><author><name sortKey="Coupland, Jeremy" sort="Coupland, Jeremy" uniqKey="Coupland J" first="Jeremy" last="Coupland">Jeremy Coupland</name><affiliation><mods:affiliation>Wolfson School of Mechanical and Manufacturing Engineering, Loughborough University, Leicestershire LE11 3TU, UK</mods:affiliation></affiliation></author></analytic><monogr></monogr><series><title level="j">Measurement Science and Technology</title><title level="j" type="abbrev">Meas. 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Technol.</title><idno type="ISSN">0957-0233</idno><idno type="eISSN">1361-6501</idno><imprint><publisher>Institute of Physics Publishing</publisher><date type="published" when="2004">2004</date><biblScope unit="volume">15</biblScope><biblScope unit="issue">4</biblScope><biblScope unit="page" from="694">694</biblScope><biblScope unit="page" to="698">698</biblScope><biblScope unit="production">Printed in the UK</biblScope></imprint><idno type="ISSN">0957-0233</idno></series><idno type="istex">86D152883DF928EC1E7CF7D1150885A17083948A</idno><idno type="DOI">10.1088/0957-0233/15/4/011</idno><idno type="PII">S0957-0233(04)66427-7</idno><idno type="articleID">166427</idno><idno type="articleNumber">011</idno></biblStruct></sourceDesc><seriesStmt><idno type="ISSN">0957-0233</idno></seriesStmt></fileDesc><profileDesc><textClass></textClass><langUsage><language ident="en">en</language></langUsage></profileDesc></teiHeader><front><div type="abstract">We have recently proposed a new method to extract the three-dimensional (3D) velocityvector data from double-exposure holographic particle image velocimetry (HPIV), which wecall the digital shearing method. In contrast to the full 3D correlation, it has been shownthat all three components (3Cs) of particle image displacement can be retrieved using sixtwo-dimensional fast Fourier transform operations and appropriate coordinatetransformations. In this paper we demonstrate the capabilities of this approach on actualHPIV data.The holographic recording method described uses an imaging system to recorda hologram of high numerical aperture using a conventional 35mm film. Theholograms are digitized and particle images are reconstructed numerically. Fromparticle images reconstructed from separate holograms, we illustrate the analysisprocess by computing the 3Cs of particle image displacement in a step-by-stepmanner.</div></front></TEI><istex><corpusName>iop</corpusName><author><json:item><name>Hui Yang</name><affiliations><json:string>Wolfson School of Mechanical and Manufacturing Engineering, Loughborough University, Leicestershire LE11 3TU, UK</json:string><json:string>E-mail:h.yang@lboro.ac.uk</json:string></affiliations></json:item><json:item><name>Neil Halliwell</name><affiliations><json:string>Wolfson School of Mechanical and Manufacturing Engineering, Loughborough University, Leicestershire LE11 3TU, UK</json:string></affiliations></json:item><json:item><name>Jeremy Coupland</name><affiliations><json:string>Wolfson School of Mechanical and Manufacturing Engineering, Loughborough University, Leicestershire LE11 3TU, UK</json:string></affiliations></json:item></author><subject><json:item><lang><json:string>eng</json:string></lang><value>holographic particle image velocimetry (HPIV)</value></json:item><json:item><lang><json:string>eng</json:string></lang><value>digital shearing</value></json:item><json:item><lang><json:string>eng</json:string></lang><value>three-dimension</value></json:item><json:item><lang><json:string>eng</json:string></lang><value>cross-correlation</value></json:item></subject><language><json:string>eng</json:string></language><abstract>We have recently proposed a new method to extract the three-dimensional (3D) velocityvector data from double-exposure holographic particle image velocimetry (HPIV), which wecall the digital shearing method. In contrast to the full 3D correlation, it has been shownthat all three components (3Cs) of particle image displacement can be retrieved using sixtwo-dimensional fast Fourier transform operations and appropriate coordinatetransformations. In this paper we demonstrate the capabilities of this approach on actualHPIV data.The holographic recording method described uses an imaging system to recorda hologram of high numerical aperture using a conventional 35mm film. Theholograms are digitized and particle images are reconstructed numerically. Fromparticle images reconstructed from separate holograms, we illustrate the analysisprocess by computing the 3Cs of particle image displacement in a step-by-stepmanner.</abstract><qualityIndicators><score>4.461</score><pdfVersion>1.4</pdfVersion><pdfPageSize>595 x 841 pts</pdfPageSize><refBibsNative>true</refBibsNative><keywordCount>4</keywordCount><abstractCharCount>920</abstractCharCount><pdfWordCount>2509</pdfWordCount><pdfCharCount>14052</pdfCharCount><pdfPageCount>5</pdfPageCount><abstractWordCount>121</abstractWordCount></qualityIndicators><title>Application of the digital shearing method to extract three-component velocity inholographic particle image velocimetry</title><genre.original><json:string>paper</json:string></genre.original><pii><json:string>S0957-0233(04)66427-7</json:string></pii><genre><json:string>research-article</json:string></genre><host><volume>15</volume><publisherId><json:string>MST</json:string></publisherId><pages><total>5</total><last>698</last><first>694</first></pages><issn><json:string>0957-0233</json:string></issn><issue>4</issue><genre><json:string>journal</json:string></genre><language><json:string>unknown</json:string></language><eissn><json:string>1361-6501</json:string></eissn><title>Measurement Science and Technology</title></host><publicationDate>2004</publicationDate><copyrightDate>2004</copyrightDate><doi><json:string>10.1088/0957-0233/15/4/011</json:string></doi><id>86D152883DF928EC1E7CF7D1150885A17083948A</id><fulltext><json:item><original>true</original><mimetype>application/pdf</mimetype><extension>pdf</extension><uri>https://api.istex.fr/document/86D152883DF928EC1E7CF7D1150885A17083948A/fulltext/pdf</uri></json:item><json:item><original>false</original><mimetype>application/zip</mimetype><extension>zip</extension><uri>https://api.istex.fr/document/86D152883DF928EC1E7CF7D1150885A17083948A/fulltext/zip</uri></json:item><istex:fulltextTEI uri="https://api.istex.fr/document/86D152883DF928EC1E7CF7D1150885A17083948A/fulltext/tei"><teiHeader><fileDesc><titleStmt><title level="a">Application of the digital shearing method to extract three-component velocity inholographic particle image velocimetry</title></titleStmt><publicationStmt><authority>ISTEX</authority><publisher>Institute of Physics Publishing</publisher><availability><p>IOP</p></availability><date>2004</date></publicationStmt><sourceDesc><biblStruct type="inbook"><analytic><title level="a">Application of the digital shearing method to extract three-component velocity inholographic particle image velocimetry</title><author><persName><forename type="first">Hui</forename><surname>Yang</surname></persName><affiliation>Wolfson School of Mechanical and Manufacturing Engineering, Loughborough University, Leicestershire LE11 3TU, UK</affiliation><affiliation>E-mail:h.yang@lboro.ac.uk</affiliation></author><author><persName><forename type="first">Neil</forename><surname>Halliwell</surname></persName><affiliation>Wolfson School of Mechanical and Manufacturing Engineering, Loughborough University, Leicestershire LE11 3TU, UK</affiliation></author><author><persName><forename type="first">Jeremy</forename><surname>Coupland</surname></persName><affiliation>Wolfson School of Mechanical and Manufacturing Engineering, Loughborough University, Leicestershire LE11 3TU, UK</affiliation></author></analytic><monogr><title level="j">Measurement Science and Technology</title><title level="j" type="abbrev">Meas. Sci. Technol.</title><idno type="pISSN">0957-0233</idno><idno type="eISSN">1361-6501</idno><imprint><publisher>Institute of Physics Publishing</publisher><date type="published" when="2004"></date><biblScope unit="volume">15</biblScope><biblScope unit="issue">4</biblScope><biblScope unit="page" from="694">694</biblScope><biblScope unit="page" to="698">698</biblScope><biblScope unit="production">Printed in the UK</biblScope></imprint></monogr><idno type="istex">86D152883DF928EC1E7CF7D1150885A17083948A</idno><idno type="DOI">10.1088/0957-0233/15/4/011</idno><idno type="PII">S0957-0233(04)66427-7</idno><idno type="articleID">166427</idno><idno type="articleNumber">011</idno></biblStruct></sourceDesc></fileDesc><profileDesc><creation><date>2004</date></creation><langUsage><language ident="en">en</language></langUsage><abstract><p>We have recently proposed a new method to extract the three-dimensional (3D) velocityvector data from double-exposure holographic particle image velocimetry (HPIV), which wecall the digital shearing method. In contrast to the full 3D correlation, it has been shownthat all three components (3Cs) of particle image displacement can be retrieved using sixtwo-dimensional fast Fourier transform operations and appropriate coordinatetransformations. In this paper we demonstrate the capabilities of this approach on actualHPIV data.The holographic recording method described uses an imaging system to recorda hologram of high numerical aperture using a conventional 35mm film. Theholograms are digitized and particle images are reconstructed numerically. Fromparticle images reconstructed from separate holograms, we illustrate the analysisprocess by computing the 3Cs of particle image displacement in a step-by-stepmanner.</p></abstract><textClass><keywords scheme="keyword"><list><head>keywords</head><item><term>holographic particle image velocimetry (HPIV)</term></item><item><term>digital shearing</term></item><item><term>three-dimension</term></item><item><term>cross-correlation</term></item></list></keywords></textClass></profileDesc><revisionDesc><change when="2004">Published</change></revisionDesc></teiHeader></istex:fulltextTEI><json:item><original>false</original><mimetype>text/plain</mimetype><extension>txt</extension><uri>https://api.istex.fr/document/86D152883DF928EC1E7CF7D1150885A17083948A/fulltext/txt</uri></json:item></fulltext><metadata><istex:metadataXml wicri:clean="corpus iop not found" wicri:toSee="no header"><istex:xmlDeclaration>version="1.0" encoding="ISO-8859-1"</istex:xmlDeclaration><istex:docType SYSTEM="http://ej.iop.org/dtd/iopv1_4_6.dtd" name="istex:docType"></istex:docType><istex:document><article artid="mst166427"><article-metadata><jnl-data jnlid="mst"><jnl-fullname>Measurement Science and Technology</jnl-fullname><jnl-abbreviation>Meas. Sci. Technol.</jnl-abbreviation><jnl-shortname>MST</jnl-shortname><jnl-issn>0957-0233</jnl-issn><jnl-coden>MSTCEP</jnl-coden><jnl-imprint>Institute of Physics Publishing</jnl-imprint><jnl-web-address>stacks.iop.org/MST</jnl-web-address></jnl-data><volume-data><year-publication>2004</year-publication><volume-number>15</volume-number></volume-data><issue-data><issue-number>4</issue-number><coverdate>April 2004</coverdate></issue-data><article-data><article-type type="paper" sort="regular"></article-type><type-number type="paper" artnum="011">11</type-number><article-number>166427</article-number><first-page>694</first-page><last-page>698</last-page><length>5</length><pii>S0957-0233(04)66427-7</pii><doi>10.1088/0957-0233/15/4/011</doi><copyright>IOP Publishing Ltd</copyright><ccc>0957-0233/04/040694+5$30.00</ccc><printed>Printed in the UK</printed></article-data></article-metadata><header><title-group><title>Application of the digital shearing method to extract three-component velocity in
holographic particle image velocimetry</title><short-title>Application of the digital shearing method to extract three-component velocity in
HPIV
</short-title><ej-title>Application of the digital shearing method to extract three-component velocity in
HPIV
</ej-title></title-group><author-group><author address="mst166427ad1" email="mst166427ea1"><first-names>Hui</first-names><second-name>Yang</second-name></author><author address="mst166427ad1"><first-names>Neil</first-names><second-name>Halliwell</second-name></author><author address="mst166427ad1"><first-names>Jeremy</first-names><second-name>Coupland</second-name></author><short-author-list>H Yang <italic>et al</italic> </short-author-list></author-group><address-group><address id="mst166427ad1" showid="no"><orgname>Wolfson School of Mechanical and Manufacturing Engineering, Loughborough University</orgname>,
Leicestershire LE11 3TU,
<country>UK</country></address><e-address id="mst166427ea1"><email mailto="h.yang@lboro.ac.uk">h.yang@lboro.ac.uk</email></e-address></address-group><history received="18 July 2003" accepted="15 September 2003" online="16 March 2004"></history><abstract-group><abstract><heading>Abstract</heading><p indent="no">We have recently proposed a new method to extract the three-dimensional (3D) velocity
vector data from double-exposure holographic particle image velocimetry (HPIV), which we
call the digital shearing method. In contrast to the full 3D correlation, it has been shown
that all three components (3Cs) of particle image displacement can be retrieved using six
two-dimensional fast Fourier transform operations and appropriate coordinate
transformations. In this paper we demonstrate the capabilities of this approach on actual
HPIV data.</p><p>
The holographic recording method described uses an imaging system to record
a hologram of high numerical aperture using a conventional 35 mm film. The
holograms are digitized and particle images are reconstructed numerically. From
particle images reconstructed from separate holograms, we illustrate the analysis
process by computing the 3Cs of particle image displacement in a step-by-step
manner.</p></abstract></abstract-group><classifications><keywords><keyword>holographic particle image velocimetry (HPIV)</keyword><keyword>digital shearing</keyword><keyword>three-dimension</keyword><keyword>cross-correlation</keyword></keywords></classifications></header><body refstyle="alphabetic"><sec-level1 id="mst166427s1" label="1"><heading>Introduction</heading><p indent="no">Our original work in the field of holographic particle image velocimetry (HPIV)
demonstrated the use of optical methods to correlate holographic recordings
(<cite linkend="mst166427bib3">Coupland and Halliwell 1992</cite>) to provide three-component (3C) velocity
measurements throughout a three-dimensional (3D) fluid flow. We later showed
that the fundamental application of complex correlation analysis to the HPIV
records has many advantages compared to methods that correlate the local image
intensity, including increased tolerance to aberrations caused, for example, by
windows and other imperfections in the optical system (<cite linkend="mst166427bib4">Coupland and
Halliwell 1997</cite>). With the increased power of modern desktop computers we have
considered digital analysis methods that retain the advantages of complex correlation
analysis, but can be implemented at high speed in a robust and more flexible
manner.</p><p>We have recently proposed a high-speed analysis procedure that we have called the digital
shearing method (<cite linkend="mst166427bib6">Yang <italic>et al</italic> 2003</cite>) and have considered its application to
the analysis of holograms recorded and reconstructed by the OCR technique proposed
by <cite linkend="mst166427bib2">Barnhart <italic>et al</italic> (2002)</cite>. In this paper we consider the digital
shearing method as a means to analyse arbitrarily recorded holographic images, and
demonstrate its use in the analysis of digitized holograms recorded on 35 mm
transparencies.
</p></sec-level1><sec-level1 id="mst166427s2" label="2"><heading>Theory</heading><p indent="no">We begin our analysis by considering the propagation of optical fields in 3D space. Since the
holographic recordings of HPIV are generally of high numerical aperture (NA) we prefer to use
a wavevector description of the (scalar) optical field rather than the two-dimensional approach
normally used in Fourier optics (<cite linkend="mst166427bib5">Goodman 1996</cite>). Accordingly, a complex field
<bold>U</bold>(<underline><italic>r</italic></underline>) at a position
vector <underline><italic>r</italic></underline> = (<italic>r</italic><sub><italic>x</italic></sub>,<italic>r</italic><sub><italic>y</italic></sub>,<italic>r</italic><sub><italic>z</italic></sub>)
can be decomposed into its spectrum of plane-wave components
<bold>S</bold>(<underline><italic>k</italic></underline>)
defined by the 3D Fourier transform
<display-eqn id="mst166427eqn1" textype="equation" eqnnum="1"></display-eqn>
where <underline><italic>k</italic></underline> = (<italic>k</italic><sub><italic>x</italic></sub>,<italic>k</italic><sub><italic>y</italic></sub>,<italic>k</italic><sub><italic>z</italic></sub>) and
d <underline><italic>r</italic></underline> conventionally denotes
the scalar quantity d <italic>r</italic><sub><italic>x</italic></sub> d <italic>r</italic><sub><italic>y</italic></sub> d <italic>r</italic><sub><italic>z</italic></sub>. Clearly the field at any point in space can be found by inverse transformation, and is
defined as
<display-eqn id="mst166427eqn2" textype="equation" eqnnum="2"></display-eqn>
Finally, we note that for a monochromatic system,
<display-eqn id="mst166427eqn3" textype="equation" eqnnum="3"></display-eqn>
where λ
is the wavelength. In this way any monochromatic optical field propagating in a
linear isotropic homogeneous (LIH) medium can be decomposed into a plane wave
spectrum in wavevector space that is defined on the surface of a sphere of radius
1/λ. In solid state physics this sphere is usually referred to as the Ewald sphere
(<cite linkend="mst166427bib1">Ashcroft and Mermin 1976</cite>).</p><p>Consider two optical fields, <bold>U</bold><sub>1</sub>(<underline><italic>r</italic></underline>)
and <bold>U</bold><sub>2</sub>(<underline><italic>r</italic></underline>), where
<bold>U</bold><sub>2</sub>(<underline><italic>r</italic></underline>) is an identical but
shifted version of <bold>U</bold><sub>1</sub>(<underline><italic>r</italic></underline>), such that
<display-eqn id="mst166427eqn4" textype="equation" eqnnum="4"></display-eqn>
If <bold>S</bold><sub>1</sub>(<underline><italic>k</italic></underline>)
and <bold>S</bold><sub>2</sub>(<underline><italic>k</italic></underline>)
are the plane wave spectra of these fields, then we have, by the Fourier shift theorem, the
cross-spectral density
<display-eqn id="mst166427eqn5" textype="leqnarray" eqnnum="5"></display-eqn>
where <italic>P</italic><sub>1</sub>(<underline><italic>k</italic></underline>) is the power
spectral density <italic>P</italic><sub>1</sub>(<underline><italic>k</italic></underline>) = |<bold>S</bold><sub>1</sub>(<underline><italic>k</italic></underline>)|<sup>2</sup>. Finally, we have the cross-correlation
<display-eqn id="mst166427eqn6" textype="leqnarray" eqnnum="6"></display-eqn>
where <bold>R</bold><sub>1</sub>(<underline><italic>r</italic></underline>) is the
auto-correlation of the field <bold>U</bold><sub>1</sub>(<underline><italic>r</italic></underline>), δ
is the Dirac delta function and <inline-eqn></inline-eqn> denotes convolution.</p><p>If it is assumed that the fields transmitted, for example, through an aperture in the
reconstructed field are identical but shifted, then the most robust analysis to find the
displacement is to compute the correlation function defined above. Our original work in
this area used optical methods to produce an optical field proportional to the 3D
correlation of the transmitted field (<cite linkend="mst166427bib3">Coupland and Halliwell 1992</cite>). In this
way, to find the particle image displacement we searched the correlation field for the
brightest point, using a voice-coil driven, travelling microscope system. Clearly, the 3D
correlation can be calculated digitally but for a typical fringe pattern digitized to be
256 × 256 pixels resolution it requires approximately 5 Gflops using FFT methods (typically a few
seconds on the fastest current PCs). For this reason we have considered a new method of
analysis.</p><p>The 3D correlation outlined above describes complex HPIV analysis in a concise mathematical
manner; it does, however, give a rather pessimistic impression of the computation required. Since, in
<italic>k</italic>-space, monochromatic fields are represented purely by the two-dimensional surface
of the Ewald sphere, the superposition integral of equation (<eqnref linkend="mst166427eqn2">2</eqnref>) reduces to a
(two-dimensional) surface integral. In this way, a field with no counter-propagating plane
wave components (i.e. forward propagating) is represented by a half sphere in
<italic>k</italic>-space and can be defined unambiguously by the projection of its wavevector
components in a plane. For example a field that is nominally propagating in the
<italic>z</italic> direction is entirely
described by its <italic>k</italic><sub><italic>x</italic></sub>
and <italic>k</italic><sub><italic>y</italic></sub>
components. Upon this basis the more familiar two-dimensional form of Fourier optics is
derived. Since reduction of computing time is of vital importance here, we return to a
two-dimensional representation here.</p><p>Accordingly the cross-spectrum of equation (<eqnref linkend="mst166427eqn5">5</eqnref>) can be written in two-dimensional form as
<display-eqn id="mst166427eqn7" textype="leqnarray" eqnnum="7"></display-eqn>
where
<display-eqn id="mst166427eqn8" textype="equation" eqnnum="8"></display-eqn>
and (neglecting the obliquity factor)
<display-eqn id="mst166427eqn9" textype="equation" eqnnum="9"></display-eqn>
It can be seen from equation (<eqnref linkend="mst166427eqn7">7</eqnref>) that the cross-spectrum of two identical but shifted fields
is the power spectral density of the first exposure modulated by a phase term that is a function
of the displacement. Dividing by the magnitude we can isolate the phase of the cross-spectrum,
&phis;(<italic>k</italic><sub><italic>x</italic></sub>,<italic>k</italic><sub><italic>y</italic></sub>), such that
<display-eqn id="mst166427eqn10" textype="equation" eqnnum="10"></display-eqn>
If the displacement is purely in the
<italic>x</italic> and
<italic>y</italic> directions then
&phis;(<italic>k</italic><sub><italic>x</italic></sub>,<italic>k</italic><sub><italic>y</italic></sub>) increases linearly
with the components <italic>k</italic><sub><italic>x</italic></sub>
and <italic>k</italic><sub><italic>y</italic></sub>. However, if the
displacement is in the <italic>z</italic>
direction then a circular phase map is observed. This phase term can be considered as a
wavefront and its curvature can be measured in a manner that is analogous to a
wavefront shearing interferometer. Accordingly, we calculate the phase difference,
Δ&phis;, between sheared phase terms, given by
<display-eqn id="mst166427eqn11" textype="leqnarray" eqnnum="11"></display-eqn>
If we expand this function using a Taylor series, neglecting third and higher order terms,
we find
<display-eqn id="mst166427eqn12" textype="equation" eqnnum="12"></display-eqn>
where <italic>K</italic><sub><italic>x</italic></sub> = 2Δ<italic>k</italic><sub><italic>x</italic></sub><italic>k</italic><sub><italic>x</italic></sub>/(λ<sup>−2</sup>−Δ<italic>k</italic><sub><italic>x</italic></sub><sup>2</sup>−<italic>k</italic><sub><italic>y</italic></sub><sup>2</sup>)<sup>1/2</sup>. With these coordinate changes, it can be seen that the final distribution is a linear function of
displacement. The rate of change of phase is proportional to the particle image displacement in the
<italic>z</italic>
direction, <italic>s</italic><sub><italic>z</italic></sub>, which can be found by Fourier transformation.</p><p>With reference to equation (<eqnref linkend="mst166427eqn10">10</eqnref>), having found the particle image displacement in the
<italic>z</italic> direction
we subtract 2π<italic>s</italic><sub><italic>z</italic></sub>(λ<sup>−2</sup>−<italic>k</italic><sub><italic>x</italic></sub><sup>2</sup>−<italic>k</italic><sub><italic>y</italic></sub><sup>2</sup>)<sup>1/2</sup>
from &phis;(<italic>k</italic><sub><italic>x</italic></sub>,<italic>k</italic><sub><italic>y</italic></sub>)
to give a final phase distribution that is a linear function of the displacement in the
<italic>x</italic> and <italic>y</italic> directions. Once again the displacement in this direction is found by Fourier
transformation.</p><p>At this point it is worth commenting on the speed of the method. We have
computed the final result using a total of six two-dimensional FFT operations
and these represent the dominant computational overhead. Using an input of
256 × 256 pixel resolution, this requires approximately 27 Mflops and it is estimated that the other
essential computations (coordinate transforms) require approximately 2 Mflops. The digital
shearing method therefore affords a considerable time saving over the full 3D correlation
approach (approximately 5 Gflops).
</p></sec-level1><sec-level1 id="mst166427s3" label="3"><heading>Experiment</heading><p indent="no">In our previous work we have demonstrated the digital shearing method on simulated data
for a range of displacements. We now illustrate the method on HPIV data recorded on
35 mm photographic film. The film used was Kodak Technical Pan 2415 having a maximum
resolution of 320 lpmm (lines per millimetre), and it can be used as a direct replacement for
holographic emulsion provided that the interference image is magnified by about 6
times.</p><p>A forward scatter geometry as shown in figure <figref linkend="mst166427fig1">1</figref> was chosen for reasons of efficiency and simplicity. In essence, an image
plane hologram is formed at the film at a magnification of 6 times using a reversed 50 mm,
F/2 Pentax lens. For the purposes of this experiment the object was a plate of
glass that was sparsely sprayed with a mist of fine particles of approximately
10–50 µm
in diameter and illuminated by a collimated, axial object beam. An off-axis reference beam
that diverges from a (virtual) point that is in the far focal plane of the imaging lens mixes
with the light scattered from the object. In this way the phase curvature introduced by the
imaging process is exactly matched by the curvature of the reference beam such that
straight interference fringes are observed in the film plane in the absence of any
object.</p><figure id="mst166427fig1" parts="single" width="column" position="float" pageposition="top" printstyle="normal" orientation="port"><graphic><graphic-file version="print" format="EPS" scale="100" filename="images/6642701.eps"></graphic-file><graphic-file version="ej" format="JPEG" filename="images/6642701.jpg"></graphic-file></graphic><caption id="mst166427fc1" type="figure" label="Figure 1"><p indent="no">The recording geometry.</p></caption></figure><p>The laser used was a <italic>Q</italic>-switched, diode pumped, doubled Nd:YAG laser with about
150 µJ/pulse. Holograms were taken with a predefined displacement of the slide between exposures. The
35 mm negatives were processed in Kodak D19 developer and initially scanned into a PC.
However, the quality of the scan was not good and the reconstructions presented
below were accomplished by digitizing a small area of the negative using a Nikon
microscope.</p><p>Once recorded, the complex amplitude in the plane of the film at the time of the exposure
can be demodulated numerically in the normal way (multiplying by a linear phase ramp
and low pass filtering). Using the diverging reference beam configuration, the complex
amplitude in the plane of the object is found by scaling this according to the
magnification.</p><p>Digitized holographic images taken from the 35 mm film that correspond to about
1 mm<sup>2</sup>
in the object space are shown in figures <figref linkend="mst166427fig2">2</figref>(a) and (b). High frequency, vertical fringes can be seen in
the first and second exposure images corresponding to a pitch of about
2 µm/cycle
in the object space.</p><figure id="mst166427fig2" parts="single" width="column" position="float" pageposition="top" printstyle="normal" orientation="port"><graphic><graphic-file version="print" format="EPS" scale="110" filename="images/6642702.eps"></graphic-file><graphic-file version="ej" format="JPEG" filename="images/6642702.jpg"></graphic-file></graphic><caption id="mst166427fc2" type="figure" label="Figure 2"><p indent="no">(a) First exposure hologram. (b) Second exposure hologram.</p></caption></figure><figure id="mst166427fig3" parts="single" width="column" position="float" pageposition="top" printstyle="normal" orientation="port"><graphic><graphic-file version="print" format="EPS" scale="106" filename="images/6642703.eps"></graphic-file><graphic-file version="ej" format="JPEG" filename="images/6642703.jpg"></graphic-file></graphic><caption id="mst166427fc3" type="figure" label="Figure 3"><p indent="no">Reconstructed particle images from figures <figref linkend="mst166427fig2">2</figref>(a) and (b).</p></caption></figure><p>The magnitude of the corresponding demodulated images are shown in figures <figref linkend="mst166427fig3">3</figref>(a) and (b). The phase of the cross-spectrum was calculated and is displayed
as a grey-scale phase map in figure <figref linkend="mst166427fig4">4</figref>. Although there is significant noise (due to digitization, we expect), curved
fringes can be seen clearly that appear to be centred toward the bottom right-hand corner
of the image.</p><figure id="mst166427fig4" parts="single" width="column" position="float" pageposition="top" printstyle="normal" orientation="port"><graphic><graphic-file version="print" format="EPS" scale="106" filename="images/6642704.eps"></graphic-file><graphic-file version="ej" format="JPEG" filename="images/6642704.jpg"></graphic-file></graphic><caption id="mst166427fc4" type="figure" label="Figure 4"><p indent="no">The phase map of the cross-spectrum.</p></caption></figure><p>The sheared and coordinate transformed phase map of equation (<eqnref linkend="mst166427eqn12">12</eqnref>) is shown in
figure <figref linkend="mst166427fig5">5</figref>. About one and a half cycles of horizontal fringes can be seen.</p><figure id="mst166427fig5" parts="single" width="column" position="float" pageposition="top" printstyle="normal" orientation="port"><graphic><graphic-file version="print" format="EPS" scale="100" filename="images/6642705.eps"></graphic-file><graphic-file version="ej" format="JPEG" filename="images/6642705.jpg"></graphic-file></graphic><caption id="mst166427fc5" type="figure" label="Figure 5"><p indent="no">The phase map of the coordinate transformed sheared cross-spectrum.</p></caption></figure><p>The displacement in the <italic>z</italic>
direction is found by Fourier transformation of this distribution and is shown in
figure <figref linkend="mst166427fig6">6</figref>. A clear signal peak can be seen.</p><figure id="mst166427fig6" parts="single" width="column" position="float" pageposition="top" printstyle="normal" orientation="port"><graphic><graphic-file version="print" format="EPS" scale="100" filename="images/6642706.eps"></graphic-file><graphic-file version="ej" format="JPEG" filename="images/6642706.jpg"></graphic-file></graphic><caption id="mst166427fc6" type="figure" label="Figure 6"><p indent="no">The modulus of the FFT of the phase distribution illustrated in figure <figref linkend="mst166427fig5">5</figref>.</p></caption></figure><p>Finally, the curvature of the fringes displayed in the phase map of figure <figref linkend="mst166427fig4">4</figref> is removed using the measured
<italic>z</italic>
displacement as shown in figure <figref linkend="mst166427fig6">6</figref>.</p><figure id="mst166427fig7" parts="single" width="column" position="float" pageposition="top" printstyle="normal" orientation="port"><graphic><graphic-file version="print" format="EPS" scale="100" filename="images/6642707.eps"></graphic-file><graphic-file version="ej" format="JPEG" filename="images/6642707.jpg"></graphic-file></graphic><caption id="mst166427fc7" type="figure" label="Figure 7"><p indent="no">The original phase map of the cross-spectrum (figure <figref linkend="mst166427fig4">4</figref>) with curvature subtracted.</p></caption></figure><p>In this case the <italic>x</italic>
and <italic>y</italic>
displacements are found by Fourier transformation. This is shown in figure <figref linkend="mst166427fig8">8</figref> and once again the signal peak can be clearly identified as required.</p><figure id="mst166427fig8" parts="single" width="column" position="float" pageposition="top" printstyle="normal" orientation="port"><graphic><graphic-file version="print" format="EPS" scale="101" filename="images/6642708.eps"></graphic-file><graphic-file version="ej" format="JPEG" filename="images/6642708.jpg"></graphic-file></graphic><caption id="mst166427fc8" type="figure" label="Figure 8"><p indent="no">The modulus of the FFT of the phase distribution illustrated in figure <figref linkend="mst166427fig7">7</figref>.</p></caption></figure></sec-level1><sec-level1 id="mst166427s4" label="4"><heading>Conclusion and discussion</heading><p indent="no">In this paper we have introduced the digital shearing method of HPIV analysis. We have
considered the technique in the most general case, as a means to determine 3D
displacement from holographic reconstructions of the scattered field at the time of each
exposure. The digital shearing method has been illustrated in the analysis of such
recordings made on 35 mm photographic film.</p><p>From our experiments we have found that the digital shearing method is around 250 times
faster than full 3D correlation. It is noted, however, that in theory, the full 3D
correlation is the most robust method to determine the most likely particle image
displacement. This is because the full correlation is an entirely linear operation that
effectively presents a map of all the possible displacements with a magnitude that is
proportional to their relative likelihood. In this way, choosing the largest peak in the
correlation is the (nonlinear) decision step that effectively determines the particle
pairing.</p><p>In contrast, the decision step in the digital shearing method is necessarily made during the procedure to
extract the <italic>z</italic>
displacement. By choosing the largest peak in the FFT of the sheared
wavefront, we are making the decision of the most likely displacement in the
<italic>z</italic>
direction from all the possible alternatives. Since the shear occurs in one direction (the
<italic>x</italic>
direction) the peaks in this distribution occur along a line (the
<italic>x</italic>
axis) and consequently it would be expected that there is a much greater
chance of identifying the wrong peak. However, the process of determining the
<italic>z</italic>
displacement is not entirely linear. An innovative step that we have included in the
digital shearing method is that we shear phase-only representations of an extracted
wavefront. This is clearly a highly nonlinear step and appears to retain the energy in
the dominant spatial frequency or signal whilst spreading the energy of the less
dominant spatial frequencies corresponding to the noise frequencies. From our
experiment this appears to make the digital shearing method similarly robust to full
correlation.
</p></sec-level1></body><back><references><heading>References</heading><reference-list type="alphabetic"><book-ref id="mst166427bib1" author="Ashcroft and Mermin" year-label="1976"><authors><au><second-name>Ashcroft</second-name><first-names>N W</first-names></au><au><second-name>Mermin</second-name><first-names>N D</first-names></au></authors><year>1976</year><book-title>Solid State Physics</book-title><publication><place>Philadelphia, PA</place><publisher>Saunders</publisher></publication><pages>p 101</pages></book-ref><journal-ref id="mst166427bib2" author="Barnhart et al" year-label="2002"><authors><au><second-name>Barnhart</second-name><first-names>D H</first-names></au><au><second-name>Halliwell</second-name><first-names>N A</first-names></au><au><second-name>Coupland</second-name><first-names>J M</first-names></au></authors><year>2002</year><art-title>Object conjugate reconstruction (OCR): a step forward in holographic metrology</art-title><jnl-title>Proc. R. Soc.</jnl-title><part>A</part><volume>458</volume><pages>2083–97</pages></journal-ref><journal-ref id="mst166427bib3" author="Coupland and Halliwell" year-label="1992"><authors><au><second-name>Coupland</second-name><first-names>J M</first-names></au><au><second-name>Halliwell</second-name><first-names>N A</first-names></au></authors><year>1992</year><art-title>Particle image velocimetry: three-dimensional fluid velocity measurements using
holographic recording and optical correlation</art-title><jnl-title>Appl. Opt.</jnl-title><volume>131</volume><pages>1005–7</pages></journal-ref><journal-ref id="mst166427bib4" author="Coupland and Halliwell" year-label="1997"><authors><au><second-name>Coupland</second-name><first-names>J M</first-names></au><au><second-name>Halliwell</second-name><first-names>N A</first-names></au></authors><year>1997</year><art-title>Holographic displacement measurements in fluid and solid mechanics: immunity to
aberrations by optical correlation processing</art-title><jnl-title>Proc. R. Soc.</jnl-title><part>A</part><volume>453</volume><pages>1053–66</pages></journal-ref><book-ref id="mst166427bib5" author="Goodman" year-label="1996"><authors><au><second-name>Goodman</second-name><first-names>J W</first-names></au></authors><year>1996</year><book-title>Introduction to Fourier Optics</book-title><edition>2nd edn</edition><publication><place>New York</place><publisher>McGraw-Hill</publisher></publication></book-ref><journal-ref id="mst166427bib6" author="Yang et al" year-label="2003"><authors><au><second-name>Yang</second-name><first-names>H</first-names></au><au><second-name>Halliwell</second-name><first-names>N</first-names></au><au><second-name>Coupland</second-name><first-names>J</first-names></au></authors><year>2003</year><art-title>Digital shearing method for 3D data extraction in HPIV</art-title><jnl-title>Appl. Opt.</jnl-title><volume>42</volume><misc-text>at press</misc-text></journal-ref></reference-list></references></back></article></istex:document></istex:metadataXml><mods version="3.6"><titleInfo><title>Application of the digital shearing method to extract three-component velocity inholographic particle image velocimetry</title></titleInfo><titleInfo type="abbreviated"><title>Application of the digital shearing method to extract three-component velocity inHPIV</title></titleInfo><titleInfo type="alternative"><title>Application of the digital shearing method to extract three-component velocity in holographic particle image velocimetry</title></titleInfo><name type="personal"><namePart type="given">Hui</namePart><namePart type="family">Yang</namePart><affiliation>Wolfson School of Mechanical and Manufacturing Engineering, Loughborough University, Leicestershire LE11 3TU, UK</affiliation><affiliation>E-mail:h.yang@lboro.ac.uk</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">Neil</namePart><namePart type="family">Halliwell</namePart><affiliation>Wolfson School of Mechanical and Manufacturing Engineering, Loughborough University, Leicestershire LE11 3TU, UK</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">Jeremy</namePart><namePart type="family">Coupland</namePart><affiliation>Wolfson School of Mechanical and Manufacturing Engineering, Loughborough University, Leicestershire LE11 3TU, UK</affiliation><role><roleTerm type="text">author</roleTerm></role></name><typeOfResource>text</typeOfResource><genre type="research-article" displayLabel="paper"></genre><originInfo><publisher>Institute of Physics Publishing</publisher><dateIssued encoding="w3cdtf">2004</dateIssued><copyrightDate encoding="w3cdtf">2004</copyrightDate></originInfo><language><languageTerm type="code" authority="iso639-2b">eng</languageTerm><languageTerm type="code" authority="rfc3066">en</languageTerm></language><physicalDescription><internetMediaType>text/html</internetMediaType><note type="production">Printed in the UK</note></physicalDescription><abstract>We have recently proposed a new method to extract the three-dimensional (3D) velocityvector data from double-exposure holographic particle image velocimetry (HPIV), which wecall the digital shearing method. In contrast to the full 3D correlation, it has been shownthat all three components (3Cs) of particle image displacement can be retrieved using sixtwo-dimensional fast Fourier transform operations and appropriate coordinatetransformations. In this paper we demonstrate the capabilities of this approach on actualHPIV data.The holographic recording method described uses an imaging system to recorda hologram of high numerical aperture using a conventional 35mm film. Theholograms are digitized and particle images are reconstructed numerically. Fromparticle images reconstructed from separate holograms, we illustrate the analysisprocess by computing the 3Cs of particle image displacement in a step-by-stepmanner.</abstract><subject><genre>keywords</genre><topic>holographic particle image velocimetry (HPIV)</topic><topic>digital shearing</topic><topic>three-dimension</topic><topic>cross-correlation</topic></subject><relatedItem type="host"><titleInfo><title>Measurement Science and Technology</title></titleInfo><titleInfo type="abbreviated"><title>Meas. Sci. Technol.</title></titleInfo><genre type="journal">journal</genre><identifier type="ISSN">0957-0233</identifier><identifier type="eISSN">1361-6501</identifier><identifier type="PublisherID">MST</identifier><identifier type="CODEN">MSTCEP</identifier><identifier type="URL">stacks.iop.org/MST</identifier><part><date>2004</date><detail type="volume"><caption>vol.</caption><number>15</number></detail><detail type="issue"><caption>no.</caption><number>4</number></detail><extent unit="pages"><start>694</start><end>698</end><total>5</total></extent></part></relatedItem><identifier type="istex">86D152883DF928EC1E7CF7D1150885A17083948A</identifier><identifier type="DOI">10.1088/0957-0233/15/4/011</identifier><identifier type="PII">S0957-0233(04)66427-7</identifier><identifier type="articleID">166427</identifier><identifier type="articleNumber">011</identifier><accessCondition type="use and reproduction" contentType="copyright">IOP Publishing Ltd</accessCondition><recordInfo><recordContentSource>IOP</recordContentSource><recordOrigin>IOP Publishing Ltd</recordOrigin></recordInfo></mods></metadata><serie></serie></istex></record>Pour manipuler ce document sous Unix (Dilib)
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