Holographic particle image velocimetry: signal recovery from under-sampled CCD data
Identifieur interne : 000C43 ( Istex/Corpus ); précédent : 000C42; suivant : 000C44Holographic particle image velocimetry: signal recovery from under-sampled CCD data
Auteurs : J M CouplandSource :
- Measurement Science and Technology [ 0957-0233 ] ; 2004.
Abstract
Holographic particle image velocimetry (HPIV) has now been demonstrated by several research groups as a method to make three-component velocity measurements from a three-dimensional fluid flow field. More recently digital HPIV has become a hot topic with the promise of near-real-time measurements without the often cumbersome optics and wet processing associated with traditional holographic methods. It is clear, however, that CCD cameras have a limited number of pixels and are not capable of resolving more than a small fraction of the interference pattern that is recorded by a typical particulate hologram. In this paper, we consider under-sampling of the interference pattern to reduce the information content and to allow recordings to be made on a CCD sensor. We describe the basic concept of model fitting to under-sampled data and demonstrate signal recovery through computer simulation. A three-dimensional analysis shows that in general, periodic sampling strategies can result in multiple particle images in the reconstruction. It is shown, however, that the significance of these peaks is reduced in the case of high numerical aperture (NA) reconstruction and can be virtually eliminated by dithering the position of sampling apertures.
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DOI: 10.1088/0957-0233/15/4/014
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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="711">711</biblScope><biblScope unit="page" to="717">717</biblScope><biblScope unit="production">Printed in the UK</biblScope></imprint><idno type="ISSN">0957-0233</idno></series><idno type="istex">DF20A8A5F37D37E275101FB40C5E616FA93C7F3C</idno><idno type="DOI">10.1088/0957-0233/15/4/014</idno><idno type="PII">S0957-0233(04)68471-2</idno><idno type="articleID">168471</idno><idno type="articleNumber">014</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">Holographic particle image velocimetry (HPIV) has now been demonstrated by several research groups as a method to make three-component velocity measurements from a three-dimensional fluid flow field. More recently digital HPIV has become a hot topic with the promise of near-real-time measurements without the often cumbersome optics and wet processing associated with traditional holographic methods. It is clear, however, that CCD cameras have a limited number of pixels and are not capable of resolving more than a small fraction of the interference pattern that is recorded by a typical particulate hologram. In this paper, we consider under-sampling of the interference pattern to reduce the information content and to allow recordings to be made on a CCD sensor. We describe the basic concept of model fitting to under-sampled data and demonstrate signal recovery through computer simulation. A three-dimensional analysis shows that in general, periodic sampling strategies can result in multiple particle images in the reconstruction. It is shown, however, that the significance of these peaks is reduced in the case of high numerical aperture (NA) reconstruction and can be virtually eliminated by dithering the position of sampling apertures.</div></front></TEI><istex><corpusName>iop</corpusName><author><json:item><name>J M Coupland</name><affiliations><json:string>Wolfson School of Mechanical and Manufacturing Engineering, Loughborough University, Leicestershire LE11 3TU, UK</json:string><json:string>E-mail:j.m.coupland@lboro.ac.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>under-sampled digital holography</value></json:item></subject><language><json:string>eng</json:string></language><abstract>Holographic particle image velocimetry (HPIV) has now been demonstrated by several research groups as a method to make three-component velocity measurements from a three-dimensional fluid flow field. 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It is shown, however, that the significance of these peaks is reduced in the case of high numerical aperture (NA) reconstruction and can be virtually eliminated by dithering the position of sampling apertures.</abstract><qualityIndicators><score>5.889</score><pdfVersion>1.3</pdfVersion><pdfPageSize>595 x 842 pts (A4)</pdfPageSize><refBibsNative>true</refBibsNative><keywordCount>2</keywordCount><abstractCharCount>1251</abstractCharCount><pdfWordCount>3645</pdfWordCount><pdfCharCount>21324</pdfCharCount><pdfPageCount>7</pdfPageCount><abstractWordCount>187</abstractWordCount></qualityIndicators><title>Holographic particle image velocimetry: signal recovery from under-sampled CCD data</title><genre.original><json:string>paper</json:string></genre.original><pii><json:string>S0957-0233(04)68471-2</json:string></pii><genre><json:string>research-article</json:string></genre><host><volume>15</volume><publisherId><json:string>MST</json:string></publisherId><pages><total>7</total><last>717</last><first>711</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>10.1088/0957-0233/15/4/014</json:string></doi><id>DF20A8A5F37D37E275101FB40C5E616FA93C7F3C</id><fulltext><json:item><original>true</original><mimetype>application/pdf</mimetype><extension>pdf</extension><uri>https://api.istex.fr/document/DF20A8A5F37D37E275101FB40C5E616FA93C7F3C/fulltext/pdf</uri></json:item><json:item><original>false</original><mimetype>application/zip</mimetype><extension>zip</extension><uri>https://api.istex.fr/document/DF20A8A5F37D37E275101FB40C5E616FA93C7F3C/fulltext/zip</uri></json:item><istex:fulltextTEI uri="https://api.istex.fr/document/DF20A8A5F37D37E275101FB40C5E616FA93C7F3C/fulltext/tei"><teiHeader><fileDesc><titleStmt><title level="a">Holographic particle image velocimetry: signal recovery from under-sampled CCD data</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">Holographic particle image velocimetry: signal recovery from under-sampled CCD data</title><author><persName><forename type="first">J M</forename><surname>Coupland</surname></persName><affiliation>Wolfson School of Mechanical and Manufacturing Engineering, Loughborough University, Leicestershire LE11 3TU, UK</affiliation><affiliation>E-mail:j.m.coupland@lboro.ac.uk</affiliation></author></analytic><monogr><title level="j">Measurement Science and Technology</title><title level="j" type="abbrev">Meas. 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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="711">711</biblScope><biblScope unit="page" to="717">717</biblScope><biblScope unit="production">Printed in the UK</biblScope></imprint></monogr><idno type="istex">DF20A8A5F37D37E275101FB40C5E616FA93C7F3C</idno><idno type="DOI">10.1088/0957-0233/15/4/014</idno><idno type="PII">S0957-0233(04)68471-2</idno><idno type="articleID">168471</idno><idno type="articleNumber">014</idno></biblStruct></sourceDesc></fileDesc><profileDesc><creation><date>2004</date></creation><langUsage><language ident="en">en</language></langUsage><abstract><p>Holographic particle image velocimetry (HPIV) has now been demonstrated by several research groups as a method to make three-component velocity measurements from a three-dimensional fluid flow field. More recently digital HPIV has become a hot topic with the promise of near-real-time measurements without the often cumbersome optics and wet processing associated with traditional holographic methods. It is clear, however, that CCD cameras have a limited number of pixels and are not capable of resolving more than a small fraction of the interference pattern that is recorded by a typical particulate hologram. In this paper, we consider under-sampling of the interference pattern to reduce the information content and to allow recordings to be made on a CCD sensor. We describe the basic concept of model fitting to under-sampled data and demonstrate signal recovery through computer simulation. A three-dimensional analysis shows that in general, periodic sampling strategies can result in multiple particle images in the reconstruction. It is shown, however, that the significance of these peaks is reduced in the case of high numerical aperture (NA) reconstruction and can be virtually eliminated by dithering the position of sampling apertures.</p></abstract><textClass><keywords scheme="keyword"><list><head>keywords</head><item><term>holographic particle image velocimetry (HPIV)</term></item><item><term>under-sampled digital holography</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/DF20A8A5F37D37E275101FB40C5E616FA93C7F3C/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="mst168471"><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="014">14</type-number><article-number>168471</article-number><first-page>711</first-page><last-page>717</last-page><length>7</length><pii>S0957-0233(04)68471-2</pii><doi>10.1088/0957-0233/15/4/014</doi><copyright>2004 IOP Publishing Ltd</copyright><ccc>0957-0233/04/040711+07$30.00</ccc><printed>Printed in the UK</printed><features colour="none" mmedia="no" suppdata="no"></features></article-data></article-metadata><header><title-group><title>Holographic particle image velocimetry: signal recovery from under-sampled CCD data</title><short-title>HPIV: signal recovery from under-sampled CCD data</short-title><ej-title>HPIV: signal recovery from under-sampled CCD data</ej-title></title-group><author-group><author address="mst168471ad1" email="mst168471ea1"><first-names>J M</first-names><second-name>Coupland</second-name></author></author-group><address-group><address id="mst168471ad1" showid="no"><orgname>Wolfson School of Mechanical and Manufacturing Engineering, Loughborough University</orgname>, Leicestershire LE11 3TU, <country>UK</country></address><e-address id="mst168471ea1"><email mailto="j.m.coupland@lboro.ac.uk">j.m.coupland@lboro.ac.uk</email></e-address></address-group><history received="4 September 2003" finalform="27 January 2004" online="19 March 2004"></history><abstract-group><abstract><heading>Abstract</heading><p indent="no">Holographic particle image velocimetry (HPIV) has now been demonstrated by several research groups as a method to make three-component velocity measurements from a three-dimensional fluid flow field. More recently digital HPIV has become a hot topic with the promise of near-real-time measurements without the often cumbersome optics and wet processing associated with traditional holographic methods. It is clear, however, that CCD cameras have a limited number of pixels and are not capable of resolving more than a small fraction of the interference pattern that is recorded by a typical particulate hologram. In this paper, we consider under-sampling of the interference pattern to reduce the information content and to allow recordings to be made on a CCD sensor. We describe the basic concept of model fitting to under-sampled data and demonstrate signal recovery through computer simulation. A three-dimensional analysis shows that in general, periodic sampling strategies can result in multiple particle images in the reconstruction. It is shown, however, that the significance of these peaks is reduced in the case of high numerical aperture (NA) reconstruction and can be virtually eliminated by dithering the position of sampling apertures.</p></abstract></abstract-group><classifications><keywords print="yes"><keyword>holographic particle image velocimetry (HPIV)</keyword><keyword>under-sampled digital holography</keyword></keywords></classifications></header><body numbering="bysection"><sec-level1 id="mst168471s1" label="1"><heading>Introduction</heading><p indent="no">Since the mid 1960s it has been recognized that multiple exposure holographic images contain information concerning the three-dimensional displacement of moving objects. In the field of fluid mechanics, pioneering work by Trolinger (<cite linkend="mst168471bib12" show="year">1975</cite>) and Malyak and Thompson (<cite linkend="mst168471bib08" show="year">1984</cite>) demonstrated that, amongst other parameters, velocity could be measured by focusing into a double or multiple pulsed holographic image of a seeded flow using a travelling microscope. Later the correlation methods used in planar particle image velocimetry (PIV) were applied to the analysis of multiple exposure particulate holograms (<cite linkend="mst168471bib05">Coupland and Halliwell 1992</cite>, <cite linkend="mst168471bib02">Barnhart <italic>et al</italic> 1994</cite>) and this approach is now referred to as holographic particle image velocimetry (HPIV).</p><p>HPIV methods and closely related techniques are currently being examined as commercially viable systems for routine three-dimensional fluid flow measurement. Particular emphasis is placed on the ability of the methods to integrate with digital computers and provide data that can be used to validate the models of computational fluid dynamics (CFD). With manufacturers somewhat reticent to produce traditional holographic materials and the corresponding high cost of the materials that are available, the preferred option is to use CCD detectors and computational methods for subsequent reconstruction and analysis. As with the development of planar PIV, this is believed to be an essential step in establishing HPIV for routine use in industry (<cite linkend="mst168471bib07">Hinsch 2002</cite>).</p><p>Recently Onural (<cite linkend="mst168471bib09" show="year">2000</cite>) has considered the recording and reconstruction of holographic images with under-sampled data. This work shows that it is possible to make an exact reconstruction of an object from an under-sampled Fresnel hologram provided that the object has limited spatial extent. In effect, some <italic>a priori</italic> information is used to analyse the recorded data. This concept is discussed further in the following section, with reference to its application in HPIV.</p></sec-level1><sec-level1 id="mst168471s2" label="2"><heading>The concept of model fitting</heading><p indent="no">It has been understood for some time that if some <italic>a priori</italic> information is known about an object then it is possible to use the information present in the image to extract details that would otherwise be lost (<cite linkend="mst168471bib10">Papoulis 1975</cite>). For example, the fact that a given region in space is known to be void of stars can be used to increase the detail in images of surrounding stars above and beyond the Rayleigh resolution limit of the telescope being used. In the case of HPIV, our <italic>a priori</italic> knowledge is that we are recording light scattered by a finite number of particles that essentially behave as point sources. In essence, this allows high numerical aperture recordings with a large field of view to be made using a detector with a low space-bandwidth product.</p><p>Figure <figref linkend="mst168471fig01">1</figref> illustrates the method in its simplest form. In this configuration, a plate consisting of a grid of small sampling apertures is placed close to the object and a CCD camera with reference beam is used to record a coherent image of the plate. The size of sampling apertures depends on the size of the flow field but could be as small as one wavelength such that there is no variation in phase across each aperture.
<figure id="mst168471fig01"><graphic><graphic-file version="print" format="EPS" filename="images/mst168471fig01.eps" width="20.5pc"></graphic-file><graphic-file version="ej" format="JPEG" filename="images/mst168471fig01.jpg"></graphic-file></graphic><caption id="mst168471fc01" label="Figure 1"><p indent="no">Under-sampled recording geometry.</p></caption></figure></p><p>A superposition of an image of the field transmitted by the aperture plate, and a tilted reference beam is recorded by the CCD. An analysis of this system is beyond the scope of this paper, but its purpose is clear: to record the phase and amplitude of the field transmitted by each aperture using a carrier frequency as in conventional holography. In addition, it should be noted that, in practice, at least two CCD pixels per aperture are necessary to do this, and care must be taken to ensure that the aperture stop size and reference beam angle are chosen to suit the pixel spacing (<cite linkend="mst168471bib11">Schnars and Juptner 2002</cite>).</p><p>In general terms, the concept of model fitting is to assume that the measured field can be predicted by a model that is a function of a set of parameters. For the case of HPIV this might be the coordinates of particles that are assumed to behave like ideal point sources. The problem is then to find the parameters that best fit the data. By constructing an error function that represents the difference between the recorded data and that calculated for a set of identical point sources, a best fit can be found and the three-dimensional coordinates of the particles can be estimated.</p><p>It is worth considering how many particles, <italic>N</italic>, can be mapped in this way and this can be estimated as follows. It is clear that each aperture provides two data (phase and amplitude) and each particle has four degrees of freedom (three position variables and phase). If we have <italic>M</italic> apertures, and assume that the problem is well posed, in the best case we could find the position and phase of <italic>N</italic> ⩽ <italic>M</italic>/2 particles (since the number of derived variables must be less than or equal to the number of measurements). It is also clear that at least two CCD pixels are necessary to record the phase and amplitude of the field transmitted by each aperture and we could expect to measure the position of up to 1 million particles with a CCD detector with 2000 × 2000 pixels. It should be noted, however, that this is an upper limit based on the broad assumption that the scattering efficiency of each particle is identical and that there is no extraneous noise.</p><p>Although a model fit of the type discussed above appears to be an intensive task, it should be noted that the operation is exactly equivalent to a decomposition of the (under-sampled) optical field into a set of spherical waves and should be compared with a (Fourier) plane wave decomposition. By choosing the strongest peaks in the spherical wave decomposition, we are effectively finding the set of particle images that best fits the recorded data. In this way, our preliminary investigations suggest that the usual methods of Fourier optics can be applied to the under-sampled data and aliasing effects merely increase background noise as follows.</p></sec-level1><sec-level1 id="mst168471s3" label="3"><heading>Preliminary investigations</heading><p indent="no">The purpose of these investigations was to investigate whether, and in what circumstances, is it possible to reconstruct three-dimensional particle images from an under-sampled recording of the scattered field. Let us assume that the configuration shown in figure <figref linkend="mst168471fig01">1</figref> is capable of measuring the exact phase and amplitude of the incident field at points specified by the position of the sampling apertures. Accordingly, the (under-sampled) optical field, <italic>U<sub>s</sub></italic>(<italic>x<sub>s</sub></italic>, <italic>y<sub>s</sub></italic>) at coordinates (<italic>x<sub>s</sub></italic>, <italic>y<sub>s</sub></italic>) in the plane of the aperture plate, can be written as
<display-eqn id="mst168471eqn01" lines="multiline" eqnnum="1" eqnalign="left"></display-eqn>where (<italic>x<sub>p</sub></italic>, <italic>y<sub>p</sub></italic>, <italic>z<sub>p</sub></italic>) are coordinates of the <italic>N</italic> particles and λ is the wavelength. For a 40 × 40 mm plate, at a wavelength of 500 nm the real part of the optical field calculated at discrete positions with 0.1 mm separation is illustrated in figure <figref linkend="mst168471fig02" override="yes">2(<italic>a</italic>)</figref> for a single particle centrally located 50 mm from the plate and (<italic>b</italic>) for the case of 1000 particles distributed in a 10<sup>3</sup> mm<sup>3</sup> cube centred on the first.
<figure id="mst168471fig02"><graphic><graphic-file version="print" format="EPS" filename="images/mst168471fig02.eps" width="20.5pc"></graphic-file><graphic-file version="ej" format="JPEG" filename="images/mst168471fig02.jpg"></graphic-file></graphic><caption id="mst168471fc02" label="Figure 2"><p indent="no">Under-sampled holograms (dimensions in mm).</p></caption></figure></p><p>It can be seen that figure <figref linkend="mst168471fig02" override="yes">2(<italic>a</italic>)</figref> shows a symmetrical pattern of aliased spatial frequencies while figure <figref linkend="mst168471fig02" override="yes">2(<italic>b</italic>)</figref> appears random. From the (under-sampled) optical field, <italic>U<sub>s</sub></italic>(<italic>x<sub>s</sub></italic>, <italic>y<sub>s</sub></italic>) the optical field in a small region in any plane parallel to the plate can be calculated. In this process a spherical wave is used to demodulate the information in a process that is a mathematical analogue of a lens-less Fourier transform hologram (<cite linkend="mst168471bib06">Goodman 1996</cite>) or the object conjugate reconstruction (OCR) technique proposed by Barnhart <italic>et al</italic> (<cite linkend="mst168471bib03" show="year">2002</cite>). Accordingly, <italic>U<sub>s</sub></italic>(<italic>x<sub>s</sub></italic>, <italic>y<sub>s</sub></italic>) is multiplied by a converging spherical wave, and the optical field, <italic>U<sub>i</sub></italic>(<italic>x<sub>i</sub></italic>, <italic>y<sub>i</sub></italic>), that is parallel to and at a distance <italic>z<sub>i</sub></italic> from the plate, and centred on the point of convergence is found by Fourier transformation such that
<display-eqn id="mst168471eqn02" lines="multiline" eqnnum="2" eqnalign="left" numalign="dropped"></display-eqn>This function was computed for the complex data corresponding to figures <figref linkend="mst168471fig02" override="yes">2(<italic>a</italic>)</figref> and <figref linkend="mst168471fig02" override="yes">(<italic>b</italic>)</figref> and is illustrated in figures <figref linkend="mst168471fig03" override="yes">3(<italic>a</italic>)</figref> and <figref linkend="mst168471fig03" override="yes">(<italic>b</italic>)</figref>, respectively. In figure <figref linkend="mst168471fig03" override="yes">3(<italic>a</italic>)</figref> a single particle is clearly shown at the origin.
<figure id="mst168471fig03"><graphic><graphic-file version="print" format="EPS" filename="images/mst168471fig03.eps" width="17pc"></graphic-file><graphic-file version="ej" format="JPEG" filename="images/mst168471fig03.jpg"></graphic-file></graphic><caption id="mst168471fc03" label="Figure 3"><p indent="no">Numerically reconstructed images.</p></caption></figure></p><p>In figure <figref linkend="mst168471fig03" override="yes">3(<italic>b</italic>)</figref> a set of particle images can be seen. The 1000 particles that are in this image are not totally random but 20 particles have been deliberately arranged to spell the letters ‘L’ and ‘U’ (the particles in the ‘L’ and ‘U’ are separated by 3 µm and ‘U’ is in a plane, 8 µm behind the image plane). Although, there is some evidence of field curvature in the reconstructed image, the ‘L’ is clearly visible and the ‘U’ is noticeably out of focus. Furthermore, it can be seen that none of the randomly positioned particles appear as discrete particles in the reconstruction and their presence merely raises the background noise level.</p><p>The results of these simulations were presented at the <italic>International Workshop on Holographic Metrology in Fluid Mechanics</italic> at Loughborough University (<cite linkend="mst168471bib04">Coupland 2003</cite>). During the discussion period it was pointed out that if the aperture mask was periodic then each particle is reconstructed as a multiplicity of particle images. In this way, it is possible for recordings of particles that were outside the probe region at the time of recording to appear in that region during reconstruction. However, the images shown in figure <figref linkend="mst168471fig03">3</figref> do not show any extraneous particle images and further work was initiated to analyse this as follows.</p></sec-level1><sec-level1 id="mst168471s4" label="4"><heading>Periodic and pseudo-random under-sampling methodologies</heading><p indent="no">To simplify matters mathematically, we now consider an under-sampled Fourier rather than Fresnel hologram as shown in figure <figref linkend="mst168471fig04">4</figref>.
<figure id="mst168471fig04"><graphic><graphic-file version="print" format="EPS" filename="images/mst168471fig04.eps" width="20.5pc"></graphic-file><graphic-file version="ej" format="JPEG" filename="images/mst168471fig04.jpg"></graphic-file></graphic><caption id="mst168471fc04" label="Figure 4"><p indent="no">Under-sampled Fourier transform hologram.</p></caption></figure></p><p>In this case, a lens is used such that the optical field corresponding to each plane wave component is mapped onto a point in the rear focal plane. Since the aim is to analyse the way optical fields are reconstructed in three-dimensional space using a hologram with relatively high numerical aperture (NA), we prefer to use a wave-vector description of the (scalar) optical field rather than the two-dimensional approach normally used in Fourier optics (<cite linkend="mst168471bib06">Goodman 1996</cite>).</p><p>In this way, a complex field <inline-eqn></inline-eqn> defined over a vector space <inline-eqn></inline-eqn> can be reconstructed from its plane wave spectrum <inline-eqn></inline-eqn> defined by the three-dimensional Fourier transform,
<display-eqn id="mst168471eqn03" eqnnum="3" eqnalign="center"></display-eqn>where <inline-eqn></inline-eqn> and d<underline>r</underline> conventionally denotes the scalar quantity d<italic>r<sub>x</sub></italic> d<italic>r<sub>y</sub></italic> d<italic>r<sub>z</sub></italic>. Clearly the field at any point in space can be found by inverse transformation, and is defined as
<display-eqn id="mst168471eqn04" eqnnum="4" eqnalign="center"></display-eqn>Finally we note that for a monochromatic system,
<display-eqn id="mst168471eqn05" eqnnum="5" eqnalign="center"></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 wave-vector 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="mst168471bib01">Ashcroft and Mermin 1976</cite>).</p><p>Since a practical optical system has finite aperture it can only record wave vectors contained within a finite region of the Ewald sphere and, for the case of an under-sampled optical recording system, the plane wave spectrum is sampled at discrete points on the Ewald sphere as shown in figure <figref linkend="mst168471fig05">5</figref>.
<figure id="mst168471fig05"><graphic><graphic-file version="print" format="EPS" filename="images/mst168471fig05.eps" width="18pc"></graphic-file><graphic-file version="ej" format="JPEG" filename="images/mst168471fig05.jpg"></graphic-file></graphic><caption id="mst168471fc05" label="Figure 5"><p indent="no">Sampling in k-space.</p></caption></figure></p><p>Here, the sampling region is centred on the <italic>z</italic>-axis corresponding to the optical axis of the configuration shown in figure <figref linkend="mst168471fig04">4</figref>. Mathematically, the sampled spectrum, <inline-eqn></inline-eqn>, can then be written as
<display-eqn id="mst168471eqn06" eqnnum="6" eqnalign="center"></display-eqn>where <inline-eqn></inline-eqn> represents a sampling function. The field, <inline-eqn></inline-eqn>, reconstructed from this data can then be written as
<display-eqn id="mst168471eqn07" eqnnum="7" eqnalign="center"></display-eqn>where ⊗ and F.T.{ } denote convolution and Fourier transformation, respectively. Clearly, the form of the reconstructed field depends strongly on the Fourier transform of the sampling function. If the spectrum is sampled at points that are uniformly spaced in the <italic>k<sub>x</sub></italic> and <italic>k<sub>y</sub></italic> directions we can write,
<display-eqn id="mst168471eqn08" eqnnum="8" eqnalign="center"></display-eqn>where <italic>a</italic> and <italic>b</italic> are the increments in the <italic>k<sub>x</sub></italic> and <italic>k<sub>y</sub></italic> directions respectively and the comb function, comb( ), is a regular grid of Dirac delta functions defined in the usual manner (<cite linkend="mst168471bib06">Goodman 1996</cite>). In this case it is straightforward to show that,
<display-eqn id="mst168471eqn09" eqnnum="9" eqnalign="center"></display-eqn>Hence, the reconstructed field is replicated, and for the case of a particulate hologram, each particle would be ‘dealt out’ at regular intervals, giving a multitude of particle images as discussed previously.</p><p>We illustrate this effect in two dimensions (<italic>x</italic> and <italic>z</italic>) in figure <figref linkend="mst168471fig06">6</figref>. Here the object is simulated by a point source at the origin <inline-eqn></inline-eqn> such that the plane wave spectrum is of constant amplitude and phase and is sampled at regular intervals in the <italic>k<sub>x</sub></italic> direction. Here the recording corresponds to a simulated NA of approximately 0.5 such that light propagates from left to right. The reconstructed field clearly shows a multiplicity of well-defined images and results in positional ambiguity. The observer is not sure whether the under-sampled hologram is a recording of a single particle or multiple particle images unless it is known <italic>a priori</italic> that the scattering object was restricted to a relatively small region of space. The work by Onural (<cite linkend="mst168471bib09" show="year">2000</cite>) defines the size of this region and shows interestingly that it can be sufficiently far from the origin (in the <italic>x</italic> or <italic>z</italic> direction) that the sampled plane wave spectrum violates the Nyquist limit.
<figure id="mst168471fig06"><graphic><graphic-file version="print" format="EPS" filename="images/mst168471fig06.eps" width="20.5pc"></graphic-file><graphic-file version="ej" format="JPEG" filename="images/mst168471fig06.jpg"></graphic-file></graphic><caption id="mst168471fc06" label="Figure 6"><p indent="no">Result of periodic sampling in k-space (NA = 0.5).</p></caption></figure></p><p>In practice, however, it is actually quite difficult to uniformly sample the wave-vector distribution on the Ewald sphere when there is a large NA. Using the configuration illustrated in figure <figref linkend="mst168471fig04">4</figref>, the geometry dictates that the phase and amplitude at a point (<italic>X</italic>, <italic>Y</italic>) in the focal plane correspond to a wave vector given by
<display-eqn id="mst168471eqn10" lines="block" eqnnum="10" eqnalign="left" numalign="center"></display-eqn>It is clear that these equations are non-linear and a regular grid of apertures in the focal plane is not regular in <italic>k</italic>-space. A simulated two-dimensional reconstruction (again NA = 0.5) using a such a sampling strategy is shown in figure <figref linkend="mst168471fig07">7</figref>.
<figure id="mst168471fig07"><graphic><graphic-file version="print" format="EPS" filename="images/mst168471fig07.eps" width="20.5pc"></graphic-file><graphic-file version="ej" format="JPEG" filename="images/mst168471fig07.jpg"></graphic-file></graphic><caption id="mst168471fc07" label="Figure 7"><p indent="no">Result of periodic sampling in Fourier plane (NA = 0.5).</p></caption></figure></p><p>It is noted that a distinct particle image can be identified at the origin and there are no distinct secondary images. However, the reconstructed particle is not ideal and is accompanied by non-uniform background noise.</p><p>Although the Fresnel configuration of figure <figref linkend="mst168471fig01">1</figref> has not been modelled directly, it is thought that this type of image would be observed in this case and would explain why no spurious images appear in figure <figref linkend="mst168471fig03" override="yes">3(<italic>b</italic>)</figref>. This supposition is credible since, for a sufficiently small region of space at some distance from the aperture plate such that the Fraunhofer condition is satisfied (<cite linkend="mst168471bib06">Goodman 1996</cite>), the optical field observed in the Fresnel region is directly proportional to the spectrum observed in the Fourier plane. It is straightforward to show that for a small region of space at a distance, <italic>f</italic>, along the <italic>z</italic>-axis, the <italic>k</italic>-space mapping of equation (<eqnref linkend="mst168471eqn10">10</eqnref>) applies and the previous observations would be expected. In all other regions appropriate scaling and shift of the <italic>k</italic>-space mapping is necessary, but once again, the sampling in <italic>k</italic>-space is not regular and similar images to that shown in figure <figref linkend="mst168471fig07">7</figref> would be expected once again.</p><p>Returning to the Fourier transform configuration of figure <figref linkend="mst168471fig04">4</figref>, it is also worth noting that as the NA is reduced the mapping from the Fourier plane to <italic>k</italic>-space becomes progressively more linear such that if <italic>X</italic><sup>2</sup> + <italic>Y</italic><sup>2</sup> ≪ <italic>f</italic><sup>2</sup>,
<display-eqn id="mst168471eqn11" eqnnum="11" eqnalign="center"></display-eqn>In essence, this corresponds to the small-angle (paraxial) approximation that is often made in Fourier optics. In this case, periodic sampling in the Fourier plane is equivalent to periodic sampling in <italic>k</italic>-space. Figure <figref linkend="mst168471fig08">8</figref> shows a reconstructed image with a simulated NA of 0.1 and clearly illustrates replicated images once again.
<figure id="mst168471fig08"><graphic><graphic-file version="print" format="EPS" filename="images/mst168471fig08.eps" width="20.5pc"></graphic-file><graphic-file version="ej" format="JPEG" filename="images/mst168471fig08.jpg"></graphic-file></graphic><caption id="mst168471fc08" label="Figure 8"><p indent="no">Result of periodic sampling in Fourier plane (NA = 0.1).</p></caption></figure></p><p>Finally, we return to consider what constitutes the ideal sampling strategy. From equation (<eqnref linkend="mst168471eqn07">7</eqnref>) it can be seen that the best sampling function is one that transforms to be as close to a delta function as possible. If the sampling function comprises randomly positioned points this is approximately true. To re-create this in practice, however, a random mask would need to be created with a corresponding sensor measuring the transmitted field. A more practical approach that appears to work well is to dither the position of the apertures on a regular mask. In the following simulation, the positions of the apertures in uniform mask described have been dithered by a random amount that does not exceed half the average pixel spacing. The resulting reconstruction is shown in figure <figref linkend="mst168471fig09">9</figref>. It can be seen that a single particle image is created and the noise is now more evenly distributed within the reconstructed field.
<figure id="mst168471fig09"><graphic><graphic-file version="print" format="EPS" filename="images/mst168471fig09.eps" width="20.5pc"></graphic-file><graphic-file version="ej" format="JPEG" filename="images/mst168471fig09.jpg"></graphic-file></graphic><caption id="mst168471fc09" label="Figure 9"><p indent="no">Result of dithered sampling in Fourier plane (NA = 0.5).</p></caption></figure></p><p>This strategy works well even at quite small NA. Figure <figref linkend="mst168471fig10">10</figref> shows a reconstruction from similarly dithered apertures with a simulated NA of 0.1 and should be compared with figure <figref linkend="mst168471fig08">8</figref>.
<figure id="mst168471fig10"><graphic><graphic-file version="print" format="EPS" filename="images/mst168471fig10.eps" width="20.5pc"></graphic-file><graphic-file version="ej" format="JPEG" filename="images/mst168471fig10.jpg"></graphic-file></graphic><caption id="mst168471fc10" label="Figure 10"><p indent="no">Result of dithered sampling in Fourier plane (NA = 0.1).</p></caption></figure></p></sec-level1><sec-level1 id="mst168471s5" label="5"><heading>Discussion and conclusions</heading><p indent="no">This paper has introduced the concept of under-sampling as a means to make high NA holographic recordings and reconstructions of seeded fluid flows using a CCD camera. By considering reconstruction to be analogous to a model fitting process, analysis shows that it might be possible to measure the three-dimensional position of a few million particles using typical CCD arrays. Care must be taken, however, to ensure that the sampling array is a-periodic, otherwise each particle will be replicated at regular intervals in the reconstruction. A dithered array of sampling apertures is therefore proposed for this purpose.</p><p>While we have concentrated on theoretical aspects and computer simulations of the under-sampling method in this paper, it is worth commenting at this point on a few of the experimental obstacles that must be overcome before the technique can be implemented in practice.</p><p>The first problem to be considered is the physical construction of the mask. Clearly the mask should be completely opaque with the exception of the sampling apertures and an etched deposit of chrome on glass might be envisaged for this purpose. From the discussions above it is clear that the sampling apertures should be small such that the phase and amplitude of the optical field do not vary appreciably across each aperture. For a high NA Fresnel type hologram similar to that shown in figure <figref linkend="mst168471fig01">1</figref> this could mean sub-wavelength apertures. At this scale, however, the apertures cannot be expected to be perfectly transmitting. For high NA Fourier type holograms such as those shown in figure <figref linkend="mst168471fig04">4</figref>, it is the size of the aperture that restricts the extent of the flow field that can be measured. In this case, the system could be scaled to accommodate any flow field.</p><p>Using a dithered mask of the type discussed, it should be possible to record the phase and amplitude of the light transmitted by each aperture. To analyse the recording, however, it is necessary to ‘demodulate’ the signal in a similar manner to that described by equation (<eqnref linkend="mst168471eqn02">2</eqnref>). In order to do this it is necessary to calculate the phase corresponding to a spherical wave emanating from a point of interest, and this requires knowledge of the physical positions of each aperture. Once again the accuracy required depends on the configuration but could be sub-wavelength, and this could present a problem in practice. One possibility is to record the signal from a monomode fibre physically placed at the position of interest in a manner exactly analogous to the OCR method (<cite linkend="mst168471bib03">Barnhart <italic>et al</italic> 2002</cite>).</p><p>Finally, it is clear that noise, generated for example by scattering from windows and other imperfections, will limit the number of particles that can be identified. If the amplitude of the scattered component of the under-sampled field (the noise) was equivalent to that scattered by 10 000 particles, for example, a proportionate reduction in the maximum number of seeding particles would be expected.</p><p>There are clearly some substantial practical issues to be overcome; however, the simulations presented in this paper suggest that the concept of under-sampling has a sound theoretical basis. The potential to provide data, substantially equivalent to that provided by silver halide-based HPIV but using video technology, justifies further work in this area.</p></sec-level1><acknowledgment><heading>Acknowledgment</heading><p indent="no">The author wishes to gratefully acknowledge valuable discussions on this subject with Wouter Koek (Delft University of Technology).</p></acknowledgment></body><back><references><heading>References</heading><reference-list type="alphabetic"><book-ref id="mst168471bib01" 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="mst168471bib02" author="Barnhart et al" year-label="1994"><authors><au><second-name>Barnhart</second-name><first-names>D H</first-names></au><au><second-name>Adrian</second-name><first-names>R J</first-names></au><au><second-name>Papen</second-name><first-names>G C</first-names></au></authors><year>1994</year><art-title>Phase conjugate holographic system for high resolution particle image velocimetry</art-title><jnl-title>Appl. 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Technol.</jnl-title><volume>13</volume><pages>R85–101</pages></journal-ref><journal-ref id="mst168471bib12" author="Trolinger" year-label="1975"><authors><au><second-name>Trolinger</second-name><first-names>J D</first-names></au></authors><year>1975</year><art-title>Particle field holography</art-title><jnl-title>Opt. Eng.</jnl-title><volume>14</volume><pages>383–92</pages></journal-ref><book-ref id="mst168471bib13" author="Vest" year-label="1979"><authors><au><second-name>Vest</second-name><first-names>C M</first-names></au></authors><year>1979</year><book-title>Holographic Interferometry</book-title><publication><place>New York</place><publisher>Wiley</publisher></publication></book-ref></reference-list></references></back></article></istex:document></istex:metadataXml><mods version="3.6"><titleInfo><title>Holographic particle image velocimetry: signal recovery from under-sampled CCD data</title></titleInfo><titleInfo type="abbreviated"><title>HPIV: signal recovery from under-sampled CCD data</title></titleInfo><titleInfo type="alternative"><title>Holographic particle image velocimetry: signal recovery from under-sampled CCD data</title></titleInfo><name type="personal"><namePart type="given">J M</namePart><namePart type="family">Coupland</namePart><affiliation>Wolfson School of Mechanical and Manufacturing Engineering, Loughborough University, Leicestershire LE11 3TU, UK</affiliation><affiliation>E-mail:j.m.coupland@lboro.ac.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>Holographic particle image velocimetry (HPIV) has now been demonstrated by several research groups as a method to make three-component velocity measurements from a three-dimensional fluid flow field. More recently digital HPIV has become a hot topic with the promise of near-real-time measurements without the often cumbersome optics and wet processing associated with traditional holographic methods. It is clear, however, that CCD cameras have a limited number of pixels and are not capable of resolving more than a small fraction of the interference pattern that is recorded by a typical particulate hologram. In this paper, we consider under-sampling of the interference pattern to reduce the information content and to allow recordings to be made on a CCD sensor. We describe the basic concept of model fitting to under-sampled data and demonstrate signal recovery through computer simulation. A three-dimensional analysis shows that in general, periodic sampling strategies can result in multiple particle images in the reconstruction. It is shown, however, that the significance of these peaks is reduced in the case of high numerical aperture (NA) reconstruction and can be virtually eliminated by dithering the position of sampling apertures.</abstract><subject><genre>keywords</genre><topic>holographic particle image velocimetry (HPIV)</topic><topic>under-sampled digital holography</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>711</start><end>717</end><total>7</total></extent></part></relatedItem><identifier type="istex">DF20A8A5F37D37E275101FB40C5E616FA93C7F3C</identifier><identifier type="DOI">10.1088/0957-0233/15/4/014</identifier><identifier type="PII">S0957-0233(04)68471-2</identifier><identifier type="articleID">168471</identifier><identifier type="articleNumber">014</identifier><accessCondition type="use and reproduction" contentType="copyright">2004 IOP Publishing Ltd</accessCondition><recordInfo><recordContentSource>IOP</recordContentSource><recordOrigin>2004 IOP Publishing Ltd</recordOrigin></recordInfo></mods></metadata><serie></serie></istex></record>Pour manipuler ce document sous Unix (Dilib)
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