research article three-dimensional identification of

Hindawi Publishing CorporationComputational and Mathematical Methods in MedicineVolume 2013 Article ID 162105 6 pageshttpdxdoiorg1011552013162105

Research ArticleThree-Dimensional Identification of Microorganisms Usinga Digital Holographic Microscope

Ning Wu1 Xiang Wu2 and Tiancai Liang3

1 Shenzhen Key Lab of Wind Power and Smart Grid Harbin Institute of Technology Shenzhen Graduate SchoolShenzhen 518055 China

2 School of Mechanical and Electrical Engineering Harbin Institute of Technology 92 West Dazhi Street Nan Gang DistrictHarbin 150001 China

3 GRG Banking Equipment Co Ltd 9 Kelin Road Science Town Guangzhou 510663 China

Correspondence should be addressed to Xiang Wu xiangwuhiteducn

Received 4 February 2013 Accepted 6 March 2013

Academic Editor Shengyong Chen

Copyright copy 2013 Ning Wu et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper reports a method for three-dimensional (3D) analysis of shift-invariant pattern recognition and applies to holographicimages digitally reconstructed from holographic microscopes It is shown that the sequential application of a 2D filter to the plane-by-plane reconstruction of an optical field is exactly equivalent to the application of amore general filter with a 3D impulse responseWe show that any 3D filters with arbitrary impulse response can be implemented in this wayThis type of processing is applied to thetwo-class problem of distinguishing different types of bacteria It is shown that the proposed technique can be easily implementedusing a modified microscope to develop a powerful and cost-effective system with great potential for biological screening

1 Introduction

In the past high-resolution imaging of three-dimensional(3D) objects or matter suspended in a volume of fluid hasmainly been accomplished using confocal microscopes [1]In recent years however attention has returned to wide-field optical microscopy using coherent illumination andholographic recording techniques that exploit advances indigital imaging and image processing to compute 3D imagesIn contrast with confocal imaging coherent microscopyprovides 3D information from a single recording that canbe processed to obtain imaging modes analogous to darkfield phase or interference contrast as required [2ndash7] Incomparison with incoherent microscopes a coherent instru-ment provides an image that can be focused at a later stageand can be considered as a microscope with an extendeddepth of field For screening purposes the increased depthof field is significant particularly at high magnifications andhigh numerical aperture For example a conventional highmagnification microscope has a depth of field of only afew microns whereas a comparable coherent instrument can

have a depth of field of a few millimetres or so This meansthat around 1000 times the volume of fluid can be screenedfrom the information contained in a single digital recording[8]

The potential of coherent microscopes for automatedbiological screening is clearly dependent on the developmentof robust image or pattern recognition algorithms [9] Inessence the application of pattern recognition techniques tocoherent images is similar to that applied to their incoherentcounterpart The task can be defined as that of highlightingobjects of interest (eg harmful bacteria) from other clutter(eg cell tissue and benign bacteria) This process should beaccomplished regardless of position and orientation of theobjects of interest within the image It can be accomplishedusing variations on correlation processing Linear correlationprocessing has been criticized in the past for its lack ofrotation invariance and its inability to generalize in themanner of neural network classifiers however a cascadeof correlators separated by nonlinear (decision) layers hasconsiderably enhanced performance [5 10] Furthermore wehave shown that this is the architecture a neural network

2 Computational and Mathematical Methods in Medicine

classifier assumes if it is trained to provide a shift-invariantoutput [11 12]

The application of linear correlation processing tothe complex images recorded by a digital phase shiftinginterferometer has recently been demonstrated by Javidiand Tajahuerce [13] Pattern recognition techniques imple-mented using a holographic microscope for the detection ofmicroscale objects has also been considered by Dubois et al[5 14] In these works the 3D sample field was reconstructedplane by plane and image classification was performed bythe application of a 2D correlation filter to each of thereconstructed planes It is noted however that although 2Dcorrelation can be applied independently to different imageplanes it does not take into account the true nature of 3Doptical fields nor that the information in any two planes ofthese fields is in fact highly correlated [15]

In this paper we considered from first principles 3Dshift-invariant pattern recognition applied to optical fieldsreconstructed from digital holographic recordings It willbe shown that the sequential application of a 2D filter toplane-by-plane reconstructions is exactly equivalent to theapplication of a 3D filter to the full 3D reconstruction ofthe optical field However a linear filter designed based onthe plane of focus will not necessarily work for planes outof focus and therefore a 3D nonlinear filtering scheme isintroduced into the optical propagation field The 3D non-linear filter is a system implemented with a general impulseresponse and followed by a nonlinear threshold We willprove with experiment that a 3D nonlinear filtering structurecan significantly improve the classification performance in3D pattern recognition In the experiment we will apply the3D nonlinear filter to 3D images of two types of bacteriarecorded from a holographic microscope and the enhancedclassification performance will be shown

2 Theory

Firstly we define the 3D cross-correlation of complex func-tions 119906(r) and ℎ(r) as

119877 (r) = int+infin

minusinfin

119906 (x) ℎ (x minus r) 119889x (1)

where r is a position vector and 119889x conventionally denotesthe scalar quantity (119889119909 119889119910 119889119911) Assume that119867(k) and 119880(k)are the Fourier transforms of ℎ(r) and 119906(r) respectivelyaccording to the convolution theorem 119877(r) can also bewritten


minusinfin

119880 (k)119867lowast (k) 1198901198952120587ksdotr119889k (2)

where the superscript lowast denotes complex conjugation Forpattern recognition purposes (1) and (2) are equivalent waysto describe the process of correlation filtering defined in spacedomain and frequency domain respectively

It is clear from (1) and (2) that in general 3D correlationfiltering requires 3D integration (in either the space orfrequency domains) However this is not the case whencorrelation filtering is applied to monochromatic optical

fields propagating forward typically the holographic recon-struction of optical fields by digital or optical means Inessence this is because 119880(k) is nonzero only within an areaof a 2D surface and consequently 119906(r) is highly correlated

According to scalar diffraction theory the complexamplitude 119906(r) representing a monochromatic optical fieldpropagation in a uniform dielectric must obey the Helmholtzequation [16] such that

nabla2

119906 (r) + 412058721198962119906 (r) = 0 (3)

where 119896 is a constant Neglecting evanescent waves that occurclose to boundaries and other obstructions it is well knownthat the solutions to this equation are planewaves of the form

119906 (r) = 119860 exp (1198952120587k sdot r) (4)

where 119860 is a complex constant In these equations 119896 andk are the wave number and wave vector respectively andare defined here such that 119896 = |k| = 1120582 where 120582is wavelength In consequence any monochromatic opticalfield propagating a uniform dielectric is described completelyby the superposition of plane waves such that


minusinfin

119880 (k) exp (1198952120587k sdot r) 119889k (5)

where 119880(k) is the spectral density and 119880(k) is the Fouriertransform of 119906(r) such that

119880 (k) = int+infin

minusinfin

119906 (r) exp (minus1198952120587k sdot r) 119889k (6)

It is noted that because 119906(r) consists of plane wavesof single wavelength the values of 119880(k) only exist on aninfinitely thin spherical shell with a radius 119896 = |k| = 1120582 Inconsequence if a general 3D correlation filter with transferfunction 119867(k) is applied to a monochromatic optical field119880(k) then in frequency domain the product 119880(k)119867lowast(k) isalso nonzero only on the spherical shell and consequently willobey the Helmholtz equation If we expand (5) we have

119906 (119903119909 119903119910 119903119911)

=∭

infin

119880(119896119909 119896119910 119896119911) exp (1198952120587 (119896

119909119903119909+ 119896119910119903119910+ 119896119911119903119911)) 120575

times (119896119911plusmn radic

1

1205822minus 1198962

119909minus 1198962

119910)119889119896119909119889119896119910119889119896119911

= ∬

infin

119880(119896119909 119896119910 plusmnradic

1

1205822minus 1198962

119909minus 1198962

119910)

times exp(1198952120587(119896119909119903119909+ 119896119910119903119910

∓119903119911radic1

1205822minus 1198962

119909minus 1198962

119910))119889119896

119909119889119896119910

(7)

The square root in these equations represents light prop-agating through the 119909119910 plane in the positive and negative

Computational and Mathematical Methods in Medicine 3

119911-directions respectively Since most holographic recordingsrecord the flux in only one direction we will consider onlythe positive root According to (7) we can define 119880

119911(119896119909 119896119910)

as the 2D projection of the spectrum onto the plane 119896119911= 0

such that

119880119911(119896119909 119896119910) = 119880(119896

119909 119896119910 radic

1

1205822minus 1198962

119909minus 1198962

119910) (8)

If 119906119885(119903119909 119903119910) represents the optical field in the plane 119903

119911= 119885

we have

119906119885(119903119909 119903119910)

= ∬

infin

119880119885(119896119909 119896119910) exp(1198952120587119885radic 1

1205822minus 1198962

119909minus 1198962

119910)

times exp (1198952120587 (119896119909119903119909+ 119896119910119903119910)) 119889119896119909119889119896119910

(9)

In addition taking the Fourier transform we have

119880119885(119896119909 119896119910)

= exp(minus1198952120587119885radic 1

1205822minus 1198962

119909minus 1198962

119910)

times∬

infin

119906119885(119903119909 119903119910) exp (minus1198952120587 (119896

119909119903119909+ 119896119910119903119910)) 119889119903119909119889119903119910

(10)

Equation (10) allows the spectrum to be calculated from theknowledge of the optical field propagating through a singleplane Equation (9) allows the field in any parallel plane to becalculated

If we consider the application of a general 3D filter tothe reconstruction of a propagating monochromatic field weremember that the product 119880(k)119867lowast(k) only exists on thesurface of a sphere Consequently according to the derivationfrom (7) to (9) we have

119877119885(119903119909 119903119910) = int

+infin

minusinfin

119880119885(119896119909 119896119910)119867lowast

119911(119896119909 119896119910)

times exp(1198952120587119885radic 1

1205822minus 1198962

119909minus 1198962

119910)

times exp (1198952120587 (119903119909119896119909+ 119903119909119896119910)) 119889119896119909119889119896119910

(11)

where119877119885(119903119909 119903119910) is the 3D correlation output in the plane 119903

119911=

119885 and

119867119885(119896119909 119896119910) = 119867(119896

119909 119896119910 radic

1

1205822minus 1198962

119909minus 1198962

119910) (12)

Finally we note that in the space domain the correlation is

119877119885(119903119909 119903119910) = int

+infin

minusinfin

119906119885(119906 V) ℎ

119885(119906 minus 119903

119909 V minus 119903

119910) 119889119906 119889V

(13)

Object beam

Sample

Microscope lens

CCD

Beam splitterReference beam

He-Ne laser Fibre optic probes

120572 (3∘)

Figure 1 Holographic microscope with a coherent laser source

Figure 2 Holographic image with a field of view of 72 times 72120583m(absolute value shown)

where

ℎ119885(119903119909 119903119910) = int

+infin

minusinfin

119867119885(119896119909 119896119910)

times exp (minus1198952120587 (119903119909119896119909+ 119903119909119896119910)) 119889119896119909119889119896119910

(14)

Equation (13) shows that a single plane (119903119911= 119885) of the

3D correlation of a propagating optical field 119906(r) with ageneral impulse response function ℎ(r) can be calculated asa 2D correlation of the field in that plane 119906

119885(119903119909 119903119910) with an

impulse function ℎ119885(119903119909 119903119910) that is defined by (14)

In the recent literature 2D correlation filtering has beenapplied to complex images reconstructed from a digital holo-graphic microscope [14] Practically a digital holographicmicroscope measures the complex amplitude in the planeof focus and the complex amplitude images in the parallelplanes are calculated based on optical propagation theory It isnoted that a linear filter that is designed to performwell in oneplane of focus will not necessarily perform well in anotherand therefore a nonlinear filtering process is required

When the 3D complex amplitude distribution of samplesis reconstructed from the digital holographic recordingcorrelation filters can be applied for pattern recognition


In the field of statistical pattern recognition it is common todescribe a digitized image of any dimension by the orderedvariables in a vector [17] and we adopt this notation hereIn this way the discrete form of a complex 3D image canbe written in vector notation by lexicographically scanningthe 3D image array Thus an 119899-dimensional vector x =

[1199091 1199092 119909

119899]119879 represents a 3D image with 119899 volume ele-

mentsWe define a correlation operator with a filter kernel(or impulse response) h = [ℎ

1 ℎ2 ℎ

119899]119879 is defined as

x =119899

sum

119894=1

ℎlowast

119894minus119899+1119909119894 (15)

where the superscript ldquolowastrdquo denotes the complex conjugate andthe subscript is taken to be modulo 119899 such that

ℎ119899+119886

= ℎ119886 (16)

A nonlinear threshold operator 119879 can be defined in the sameway to operate on the individual components of a vector suchthat

119879x = [1198861199093

1+ 1198871199092

1+ 1198881199091+ 119889 119886119909

3

2+ 1198871199092

2+ 1198881199092

+119889 1198861199093

119899+ 1198871199092

119899+ 119888119909119899+ 119889]

119879

(17)

In general image data from a hologram is a complex-amplitude field however we consider only the intensitydistribution and define a modulus operator that operateson the the output such that

x = [10038161003816100381610038161199091

1003816100381610038161003816

2

10038161003816100381610038161199092

1003816100381610038161003816

2

1003816100381610038161003816119909119899

1003816100381610038161003816

2

] (18)

In this way a 3D nonlinear filter can be expressed as

= 119879119894119894 (19)

where the subscript to each operator denotes the layer inwhich a given operator is applied

Without loss of generality we design the 3D nonlinearfilter to generate a delta function for the objects to berecognized and zero outputs for the patterns to be rejectedFor this purpose we define a matrix set S of 119898 referenceimages such that S = [s

1 s2 s

119898] and the corresponding

output matrix R is given by

R = S (20)

For the optimization of the 3D nonlinear filter a matrix119874 with all the desired outputs intensity images is defined Ingeneral the desired outputs for in-class images will be a zero-valued vector with the first element set to be unit magnitudeand for an out-of-class image the desired output is zero Inorder to train the filter with the desired performance theerror function below is requested to be minimized

119864 =

119899119898

sum

119894=1 119895=1

(119877119894119895minus 119874119894119895)

2

+ 119899

119898

sum

119895=1

(1198771119895minus 1198741119895)

2

(21)

40

35

30

25

20

15

10

5

0

119885po

sitio

ns (120583

)

8070

6050

4030

2010

0

119884 positions (120583) 0 10 20 30 40 50 60 70 80

119883 positions (120583)

7060

5040

3020

10

119884 positions 20 30 40 50 60 70

itions (120583)

Figure 3 3D image of the optical field reconstructed from Figure 2

where119877119894119895and119874

119894119895represent the ith pixel of the jth training and

output image respectivelyThe first term in this expression isthe variance of the actual output from the desired outputThesecond term represents the signal peaks (that for simplicityare defined to be the first term in the output vector) andis given extra weight to ensure that they have the desiredunit magnitude Because (21) is a nonlinear function witha large number of variables it is not possible to find ananalytical solution Hence an iterative method is used inthe minimization process In this case a simulated annealingalgorithm was implemented in the optimization because it ismore likely to reach a global minimum [18]

In the practical implementations of the 3Dnonlinear filterdescribed in this paper we require a filter to identify thepresence of fairly small objects in a relatively large field Inthese cases a relatively small filter kernel is used and thekernel is zero-padded to the same size as the input image Inthe test of this paper the training images are selected to be32 times 32 times 16 elements and we use 16 times 16 elements transferfunction (2D) The filter output the filter kernel and thedesired output images are all zero-padded to a resolution of32 times 32 times 16 elements In this way edge effects in patternrecognition for large images can be avoided

3 Experiment

The objective of the work described in this section was todemonstrate 3D rotationally invariance pattern recognitionbased on digital holographicmicroscopy for the classificationof two species of live bacteria E coli and Pantoea

The digital holographic microscope setup used for thisstudy is illustrated in Figure 1 In this arrangement a He-Nelaser (633 nm) is used as coherent light source and is dividedby a beam splitter and launched into a pair of optical fibresof equal length One fibre supplies the light that forms theobject beam for the holographic recording and is collimatedThe microscope is used in a transmission mode and has anobjective lens with 100x magnification and an oil immersionobjective with an equivalent numerical aperture of NA =125 The object plane is imaged onto a CCD array placed


(a) (b)

Figure 4 Typical bacteria (a) E coli and (b) Pantoea in different rotated orientations

approximately 200mm from the objective It is noted thatbecause the microscope is holographic the object of interestneed not be located in the object plane

The fibre that supplies the reference beam has an opentermination that is arranged to diverge from a point in therear focal plane of the microscope objective In this waythe interference of the light from the reference beam andthe light scattered is recorded at the CCD Phase curvatureintroduced by the imaging process [19] is precisely matchedby the reference curvature and straight interference fringesare observed in the image plane in the absence of anyscattering objects From the analysis in Section 2 we can seethat the interference pattern recorded by the CCD can bedemodulated to give the complex amplitude describing thepropagating field in the object plane For reasons of process-ing efficiency care was taken to adjust the magnification ofthe microscope to match the CCD resolution such that anoptimally sampled (Nyquist) reconstruction is produced

The holographic microscope is implemented with a flowcell that defines an experimental volume The nutrient fluidwith two species of living bacteria E coli and Pantoea issyringed into the flow cell through a pipe Figure 2 showsan image taken from the microscope corresponding tothe absolute value of the complex amplitude in the objectplane In this image the bacteria understood to be E coliare highlighted with circles some out-of-focus bacteria areinvisible on this plane Figure 3 shows a 3D image of the fieldin Figure 2 reconstructed using the method demonstrated inthe above section

In this study a 3D nonlinear filter was trained to highlightlive E coli bacteria floating in the flow cell while thePantoea bacteria will be ignored However the reference setpreparation is one of the most challenging problems forthe identification of the living cells because each of the livebacteria varies in size and shape and appears at randomorientation To recognise the bacteria regardless of theirshapes and orientations adequate representative distortionsof bacteria images must be provided for the 3D nonlinearfilter as reference images

The bacteria images registered as training set can beobtained by directly cropping the cell images from the 3Dreconstructed field or by simulating from the recordedimages For example a selected bacteria image can be rotatedto generate several orientation versions Figure 4(a) showseight absolute value images of a typical rod-shaped E colirotated in steps of 45 degrees Pantoea bacteria have asimilar rod shape but slightly different in size from Ecoli Figure 4(b) shows one of the selected Pantoea in eightdifferent rotated versions

To demonstrate the performance of the 3D nonlinearfilter we train the system to detect E coli bacteria with 42

40

35

30

25

20

15

10

5

0

119885po

sitio

ns (120583

)

8070

6050

4030

2010

0 0 10 20 30 40 50 60 70 80119884 positions (120583) 119883 positions (120583)

Figure 5 3D output for the 3D nonlinear filter trained to recognizeE coli (absolute amplitude value shown)

Figure 6 The projection of the output volume (absolute amplitudevalue shown)

images including 25 E coli and 17 Pantoea images and thefiler is tested with the complex amplitude image in Figure 2Figure 5 shows the 3D image of the 3D filter output Figure 6reports the projection of the output volume onto a planeIt can be seen that most of the E coli bacteria had beenhighlighted by correlation peaks and the Pantoea had beenignored However a small portion of the E coli cannot bedetected this is because the training set with limited numberof reference images does not represent all the distortions andorientations of the bacteria It is expected that classificationrate can be improved if more reference images are includedin the training set


4 Conclusion

This paper describes 3D pattern recognition with a 3D non-linear filter applied to monochromatic optical fields that canbe recorded and reconstructed by holographic microscopesThe 3D extension and formulation of the nonlinear filterconcept has been introduced We have shown with experi-mental data that the 3D nonlinear filtering system providesadditional capability as a means to perform 3D patternrecognition in a shift and rotationally invariant means Wedemonstrate this in practice by applying the 3D nonlinear fil-ter to a holographic recording of the light scattered from twokinds of living bacteria suspended in waterThe experimentaldata demonstrated that the 3D nonlinear filter has good shiftand rotationally invariant property in 3D space

Acknowledgment

Financial support from The Research Fund for the DoctoralProgram of Higher Education (No 20122302120072) to initi-ate this research is gratefully acknowledged

References

[1] M Minsky ldquoMemoir on inventing the confocal scanningmicroscoperdquo Scanning vol 10 no 4 pp 128ndash138 1988

[2] U Schnars and W P O Juptner ldquoDigital recording andnumerical reconstruction of hologramsrdquo Measurement Scienceand Technology vol 13 no 9 pp R85ndashR101 2002

[3] T Zhang and I Yamaguchi ldquoThree-dimensional microscopywith phase-shifting digital holographyrdquo Optics Letters vol 23no 15 pp 1221ndash1223 1998

[4] E Cuche P Marquet and C Depeursinge ldquoSimultaneousamplitude-contrast and quantitative phase-contrastmicroscopyby numerical reconstruction of Fresnel off-axis hologramsrdquoApplied Optics vol 38 no 34 pp 6994ndash7001 1999

[5] F Dubois L Joannes and J C Legros ldquoImproved three-dimensional imaging with a digital holography microscopewith a source of partial spatial coherencerdquo Applied Optics vol38 no 34 pp 7085ndash7094 1999

[6] S Y Chen Y F Li Q Guan and G Xiao ldquoReal-time three-dimensional surface measurement by color encoded light pro-jectionrdquo Applied Physics Letters vol 89 no 11 Article ID 1111083 pages 2006

[7] Z Teng A J Degnan U Sadat et al ldquoCharacterization ofhealing following atherosclerotic carotid plaque rupture inacutely symptomatic patients an exploratory study using invivo cardiovascular magnetic resonancerdquo Journal of Cardiovas-cular Magnetic Resonance vol 13 no 1 article 64 2011

[8] L Lin S Chen Y Shao and Z Gu ldquoPlane-based sampling forray casting algorithm in sequential medical imagesrdquo Computa-tional and Mathematical Methods in Medicine vol 2013 ArticleID 874517 5 pages 2013

[9] Q Guan and B Du ldquoBayes clustering and structural supportvector machines for segmentation of carotid artery plaques inmulti-contrast MRIrdquo Computational and Mathematical Meth-ods in Medicine vol 2012 Article ID 549102 6 pages 2012

[10] F Dubois ldquoNonlinear cascaded correlation processes toimprove the performances of automatic spatial-frequency-selective filters in pattern recognitionrdquo Applied Optics vol 35no 23 pp 4589ndash4597 1996

[11] S Reed and J Coupland ldquoStatistical performance of cascadedlinear shift-invariant processingrdquoApplied Optics vol 39 no 32pp 5949ndash5955 2000

[12] N Wu R D Alcock N A Halliwell and J M CouplandldquoRotationally invariant pattern recognition by use of linear andnonlinear cascaded filtersrdquo Applied Optics vol 44 no 20 pp4315ndash4322 2005

[13] B Javidi and E Tajahuerce ldquoThree-dimensional object recogni-tion by use of digital holographyrdquo Optics Letters vol 25 no 9pp 610ndash612 2000

[14] F Dubois C Minetti O Monnom C Yourassowsky J CLegros and P Kischel ldquoPattern recognition with a digital holo-graphicmicroscopeworking in partially coherent illuminationrdquoApplied Optics vol 41 no 20 pp 4108ndash4119 2002

[15] S Chen and M Zhao ldquoRecent advances in morphological cellimage analysisrdquo Computational and Mathematical Methods inMedicine vol 2012 Article ID 101536 10 pages 2012

[16] A Sommerfeld Partial Differential Equations in Physics Aca-demic Press New York NY USA 1949

[17] K Fukunaga Introduction to Statistical Pattern RecognitionAcademic Press New York NY USA 1972

[18] S Kirkpatrick C D Gelatt and M P Vecchi ldquoOptimization bysimulated annealingrdquo Science vol 220 no 4598 pp 671ndash6801983

[19] J W Goodman Introduction to Fourier Optics McGraw-HillNew York NY USA 1968

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classifier assumes if it is trained to provide a shift-invariantoutput [11 12]

The application of linear correlation processing tothe complex images recorded by a digital phase shiftinginterferometer has recently been demonstrated by Javidiand Tajahuerce [13] Pattern recognition techniques imple-mented using a holographic microscope for the detection ofmicroscale objects has also been considered by Dubois et al[5 14] In these works the 3D sample field was reconstructedplane by plane and image classification was performed bythe application of a 2D correlation filter to each of thereconstructed planes It is noted however that although 2Dcorrelation can be applied independently to different imageplanes it does not take into account the true nature of 3Doptical fields nor that the information in any two planes ofthese fields is in fact highly correlated [15]

In this paper we considered from first principles 3Dshift-invariant pattern recognition applied to optical fieldsreconstructed from digital holographic recordings It willbe shown that the sequential application of a 2D filter toplane-by-plane reconstructions is exactly equivalent to theapplication of a 3D filter to the full 3D reconstruction ofthe optical field However a linear filter designed based onthe plane of focus will not necessarily work for planes outof focus and therefore a 3D nonlinear filtering scheme isintroduced into the optical propagation field The 3D non-linear filter is a system implemented with a general impulseresponse and followed by a nonlinear threshold We willprove with experiment that a 3D nonlinear filtering structurecan significantly improve the classification performance in3D pattern recognition In the experiment we will apply the3D nonlinear filter to 3D images of two types of bacteriarecorded from a holographic microscope and the enhancedclassification performance will be shown

2 Theory

Firstly we define the 3D cross-correlation of complex func-tions 119906(r) and ℎ(r) as


minusinfin

119906 (x) ℎ (x minus r) 119889x (1)

where r is a position vector and 119889x conventionally denotesthe scalar quantity (119889119909 119889119910 119889119911) Assume that119867(k) and 119880(k)are the Fourier transforms of ℎ(r) and 119906(r) respectivelyaccording to the convolution theorem 119877(r) can also bewritten


minusinfin

119880 (k)119867lowast (k) 1198901198952120587ksdotr119889k (2)

where the superscript lowast denotes complex conjugation Forpattern recognition purposes (1) and (2) are equivalent waysto describe the process of correlation filtering defined in spacedomain and frequency domain respectively

It is clear from (1) and (2) that in general 3D correlationfiltering requires 3D integration (in either the space orfrequency domains) However this is not the case whencorrelation filtering is applied to monochromatic optical

fields propagating forward typically the holographic recon-struction of optical fields by digital or optical means Inessence this is because 119880(k) is nonzero only within an areaof a 2D surface and consequently 119906(r) is highly correlated

According to scalar diffraction theory the complexamplitude 119906(r) representing a monochromatic optical fieldpropagation in a uniform dielectric must obey the Helmholtzequation [16] such that

nabla2

119906 (r) + 412058721198962119906 (r) = 0 (3)

where 119896 is a constant Neglecting evanescent waves that occurclose to boundaries and other obstructions it is well knownthat the solutions to this equation are planewaves of the form

119906 (r) = 119860 exp (1198952120587k sdot r) (4)

where 119860 is a complex constant In these equations 119896 andk are the wave number and wave vector respectively andare defined here such that 119896 = |k| = 1120582 where 120582is wavelength In consequence any monochromatic opticalfield propagating a uniform dielectric is described completelyby the superposition of plane waves such that


minusinfin

119880 (k) exp (1198952120587k sdot r) 119889k (5)

where 119880(k) is the spectral density and 119880(k) is the Fouriertransform of 119906(r) such that

119880 (k) = int+infin

minusinfin

119906 (r) exp (minus1198952120587k sdot r) 119889k (6)

It is noted that because 119906(r) consists of plane wavesof single wavelength the values of 119880(k) only exist on aninfinitely thin spherical shell with a radius 119896 = |k| = 1120582 Inconsequence if a general 3D correlation filter with transferfunction 119867(k) is applied to a monochromatic optical field119880(k) then in frequency domain the product 119880(k)119867lowast(k) isalso nonzero only on the spherical shell and consequently willobey the Helmholtz equation If we expand (5) we have

119906 (119903119909 119903119910 119903119911)

=∭

infin

119880(119896119909 119896119910 119896119911) exp (1198952120587 (119896

119909119903119909+ 119896119910119903119910+ 119896119911119903119911)) 120575

times (119896119911plusmn radic

1

1205822minus 1198962

119909minus 1198962

119910)119889119896119909119889119896119910119889119896119911

= ∬

infin

119880(119896119909 119896119910 plusmnradic

1

1205822minus 1198962

119909minus 1198962

119910)

times exp(1198952120587(119896119909119903119909+ 119896119910119903119910

∓119903119911radic1

1205822minus 1198962

119909minus 1198962

119910))119889119896

119909119889119896119910

(7)

The square root in these equations represents light prop-agating through the 119909119910 plane in the positive and negative



119911(119896119909 119896119910)


such that

119880119911(119896119909 119896119910) = 119880(119896

119909 119896119910 radic

1

1205822minus 1198962

119909minus 1198962

119910) (8)


119911= 119885

we have

119906119885(119903119909 119903119910)

= ∬

infin

119880119885(119896119909 119896119910) exp(1198952120587119885radic 1

1205822minus 1198962

119909minus 1198962

119910)

times exp (1198952120587 (119896119909119903119909+ 119896119910119903119910)) 119889119896119909119889119896119910

(9)


119880119885(119896119909 119896119910)

= exp(minus1198952120587119885radic 1

1205822minus 1198962

119909minus 1198962

119910)

times∬

infin

119906119885(119903119909 119903119910) exp (minus1198952120587 (119896

119909119903119909+ 119896119910119903119910)) 119889119903119909119889119903119910

(10)



119877119885(119903119909 119903119910) = int

+infin

minusinfin

119880119885(119896119909 119896119910)119867lowast

119911(119896119909 119896119910)

times exp(1198952120587119885radic 1

1205822minus 1198962

119909minus 1198962

119910)

times exp (1198952120587 (119903119909119896119909+ 119903119909119896119910)) 119889119896119909119889119896119910

(11)


119911=

119885 and

119867119885(119896119909 119896119910) = 119867(119896

119909 119896119910 radic

1

1205822minus 1198962

119909minus 1198962

119910) (12)


119877119885(119903119909 119903119910) = int

+infin

minusinfin

119906119885(119906 V) ℎ

119885(119906 minus 119903

119909 V minus 119903

119910) 119889119906 119889V

(13)

Object beam

Sample

Microscope lens

CCD



120572 (3∘)



where

ℎ119885(119903119909 119903119910) = int

+infin

minusinfin

119867119885(119896119909 119896119910)

times exp (minus1198952120587 (119903119909119896119909+ 119903119909119896119910)) 119889119896119909119889119896119910

(14)



119885(119903119909 119903119910) with an






[1199091 1199092 119909



1 ℎ2 ℎ


x =119899

sum

119894=1

ℎlowast

119894minus119899+1119909119894 (15)


ℎ119899+119886

= ℎ119886 (16)


119879x = [1198861199093

1+ 1198871199092

1+ 1198881199091+ 119889 119886119909

3

2+ 1198871199092

2+ 1198881199092

+119889 1198861199093

119899+ 1198871199092

119899+ 119888119909119899+ 119889]

119879

(17)


x = [10038161003816100381610038161199091

1003816100381610038161003816

2

10038161003816100381610038161199092

1003816100381610038161003816

2

1003816100381610038161003816119909119899

1003816100381610038161003816

2

] (18)


= 119879119894119894 (19)



1 s2 s



R = S (20)


119864 =

119899119898

sum

119894=1 119895=1

(119877119894119895minus 119874119894119895)

2

+ 119899

119898

sum

119895=1

(1198771119895minus 1198741119895)

2

(21)

40

35

30

25

20

15

10

5

0

119885po

sitio

ns (120583

)

8070

6050

4030

2010

0

119884 positions (120583) 0 10 20 30 40 50 60 70 80

119883 positions (120583)

7060

5040

3020

10

119884 positions 20 30 40 50 60 70

itions (120583)


where119877119894119895and119874




3 Experiment




(a) (b)








40

35

30

25

20

15

10

5

0

119885po

sitio

ns (120583

)

8070

6050

4030

2010






4 Conclusion


Acknowledgment


References

























of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of


















119911(119896119909 119896119910)


such that

119880119911(119896119909 119896119910) = 119880(119896

119909 119896119910 radic

1

1205822minus 1198962

119909minus 1198962

119910) (8)


119911= 119885

we have

119906119885(119903119909 119903119910)

= ∬

infin

119880119885(119896119909 119896119910) exp(1198952120587119885radic 1

1205822minus 1198962

119909minus 1198962

119910)

times exp (1198952120587 (119896119909119903119909+ 119896119910119903119910)) 119889119896119909119889119896119910

(9)


119880119885(119896119909 119896119910)

= exp(minus1198952120587119885radic 1

1205822minus 1198962

119909minus 1198962

119910)

times∬

infin

119906119885(119903119909 119903119910) exp (minus1198952120587 (119896

119909119903119909+ 119896119910119903119910)) 119889119903119909119889119903119910

(10)



119877119885(119903119909 119903119910) = int

+infin

minusinfin

119880119885(119896119909 119896119910)119867lowast

119911(119896119909 119896119910)

times exp(1198952120587119885radic 1

1205822minus 1198962

119909minus 1198962

119910)

times exp (1198952120587 (119903119909119896119909+ 119903119909119896119910)) 119889119896119909119889119896119910

(11)


119911=

119885 and

119867119885(119896119909 119896119910) = 119867(119896

119909 119896119910 radic

1

1205822minus 1198962

119909minus 1198962

119910) (12)


119877119885(119903119909 119903119910) = int

+infin

minusinfin

119906119885(119906 V) ℎ

119885(119906 minus 119903

119909 V minus 119903

119910) 119889119906 119889V

(13)

Object beam

Sample

Microscope lens

CCD



120572 (3∘)



where

ℎ119885(119903119909 119903119910) = int

+infin

minusinfin

119867119885(119896119909 119896119910)

times exp (minus1198952120587 (119903119909119896119909+ 119903119909119896119910)) 119889119896119909119889119896119910

(14)



119885(119903119909 119903119910) with an






[1199091 1199092 119909



1 ℎ2 ℎ


x =119899

sum

119894=1

ℎlowast

119894minus119899+1119909119894 (15)


ℎ119899+119886

= ℎ119886 (16)


119879x = [1198861199093

1+ 1198871199092

1+ 1198881199091+ 119889 119886119909

3

2+ 1198871199092

2+ 1198881199092

+119889 1198861199093

119899+ 1198871199092

119899+ 119888119909119899+ 119889]

119879

(17)


x = [10038161003816100381610038161199091

1003816100381610038161003816

2

10038161003816100381610038161199092

1003816100381610038161003816

2

1003816100381610038161003816119909119899

1003816100381610038161003816

2

] (18)


= 119879119894119894 (19)



1 s2 s



R = S (20)


119864 =

119899119898

sum

119894=1 119895=1

(119877119894119895minus 119874119894119895)

2

+ 119899

119898

sum

119895=1

(1198771119895minus 1198741119895)

2

(21)

40

35

30

25

20

15

10

5

0

119885po

sitio

ns (120583

)

8070

6050

4030

2010

0

119884 positions (120583) 0 10 20 30 40 50 60 70 80

119883 positions (120583)

7060

5040

3020

10

119884 positions 20 30 40 50 60 70

itions (120583)


where119877119894119895and119874




3 Experiment




(a) (b)








40

35

30

25

20

15

10

5

0

119885po

sitio

ns (120583

)

8070

6050

4030

2010






4 Conclusion


Acknowledgment


References

























of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of


















[1199091 1199092 119909



1 ℎ2 ℎ


x =119899

sum

119894=1

ℎlowast

119894minus119899+1119909119894 (15)


ℎ119899+119886

= ℎ119886 (16)


119879x = [1198861199093

1+ 1198871199092

1+ 1198881199091+ 119889 119886119909

3

2+ 1198871199092

2+ 1198881199092

+119889 1198861199093

119899+ 1198871199092

119899+ 119888119909119899+ 119889]

119879

(17)


x = [10038161003816100381610038161199091

1003816100381610038161003816

2

10038161003816100381610038161199092

1003816100381610038161003816

2

1003816100381610038161003816119909119899

1003816100381610038161003816

2

] (18)


= 119879119894119894 (19)



1 s2 s



R = S (20)


119864 =

119899119898

sum

119894=1 119895=1

(119877119894119895minus 119874119894119895)

2

+ 119899

119898

sum

119895=1

(1198771119895minus 1198741119895)

2

(21)

40

35

30

25

20

15

10

5

0

119885po

sitio

ns (120583

)

8070

6050

4030

2010

0

119884 positions (120583) 0 10 20 30 40 50 60 70 80

119883 positions (120583)

7060

5040

3020

10

119884 positions 20 30 40 50 60 70

itions (120583)


where119877119894119895and119874




3 Experiment




(a) (b)








40

35

30

25

20

15

10

5

0

119885po

sitio

ns (120583

)

8070

6050

4030

2010






4 Conclusion


Acknowledgment


References

























of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















(a) (b)








40

35

30

25

20

15

10

5

0

119885po

sitio

ns (120583

)

8070

6050

4030

2010






4 Conclusion


Acknowledgment


References

























of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















4 Conclusion


Acknowledgment


References

























of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of





















of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of
















research article three-dimensional identification of

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