extraction of object position from in-line holograms by using single wavelet coefficient
TRANSCRIPT
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Optics Communications 281 (2008) 1461–1467
Extraction of object position from in-line hologramsby using single wavelet coefficient
Siriwat Soontaranon a, Joewono Widjaja a,*, Toshimitsu Asakura b
a Institute of Science, Suranaree University of Technology, Nakhon Ratchasima 30000, Thailandb Hokkai-Gakuen University, Sapporo, Hokkaido 062-8605, Japan
Received 25 June 2007; received in revised form 31 October 2007; accepted 12 November 2007
Abstract
A novel digital method for tracking position of objects from in-line holograms by using single wavelet coefficient is proposed. In theproposed method, a wavelet transform is used to analyze the holograms. An axial position of the object being studied is determined byusing a real value of a resultant wavelet coefficient appears at a center position of interference fringes. A feasibility of this method isexperimentally verified by analyzing holograms of an optical fiber.� 2007 Elsevier B.V. All rights reserved.
Keywords: Object tracking; In-line holography; Wavelet transform; Digital analysis
1. Introduction
In-line Fraunhofer holography has been found to bevery useful for sizing small particles [1,2]. In in-line particleholography, opaque or semi-transparent particles are illu-minated by a collimated coherent light. By recording aninterference pattern produced between light waves dif-fracted from particles and the one directly transmitted ontoa photographic film, an in-line hologram of the particles isgenerated. The interference pattern in a hologram containsinformation about both a three-dimensional (3-D) spatialposition and a size of the particles which are encoded asa chirp signal and an envelope function, respectively. In aconventional optical reconstruction method, this informa-tion is extracted by illuminating the developed hologramwith the same coherent light. The light transmitted throughthe hologram reconstructs images of the particles at thepositions with the same distances as the recording dis-tances. Since, in general, these distances are not known inadvance, the image planes with the best focus for the par-
0030-4018/$ - see front matter � 2007 Elsevier B.V. All rights reserved.
doi:10.1016/j.optcom.2007.11.027
* Corresponding author. Tel.: +66 44 224194; fax: +66 44 224185.E-mail address: [email protected] (J. Widjaja).
ticles must be investigated by scanning the overall depthalong an optical axis with fine steps. As a consequence,the conventional analyzing method is very tedious and timeconsuming. Therefore, it is not suitable for the human-operator based analysis which is non-repeatable andinaccurate.
To have an automatic analyzing system with high accu-racy, Widjaja proposed the use of a wavelet based corre-lator [3]. In his proposed method, a wavelet transform(WT) is used to enhance edge features of both the imagesof particles reconstructed from the hologram and the imageof a reference particle. By correlating these two edge-enhanced images, the position and the size of particlescan be determined. Although this method is indeed usefulfor analyzing irregular shape of particles, the price paid isthat its optical system becomes complicated.
Use of the WT for extracting the position of particlesfrom digital in-line holograms was proposed by Buraga-Lefebvre et al. [4] and independently by Soontaranonet al. [5,6]. In the former, a diffraction process is regardedas a wavelet transformation with a spherical wave is usedas the analyzing wavelet and an axial distance of the wavepropagation as its dilation. To determine the position of
0 1 2 3 4-1-2-3-4x (mm)
0.6
0.8
1
1.2
1.4
1.6
Inte
nsity
(a.u
.)
Fig. 1. Simulated in-line hologram of a line object.
1462 S. Soontaranon et al. / Optics Communications 281 (2008) 1461–1467
particles, the holograms are digitally wavelet transformedby using a spherical wave-based analyzing wavelet. Theposition of particles is obtained if the resultant WT givesa maximum value. In fact, this approach is equivalent tosearching the in-focus image plane of particles recon-structed from the hologram. However, since the dilationfactor is determined by the axial recording distance, thismethod is useful only for a short axial distance. For alonger distance, an increase of the dilation causes a viola-tion of the admissibility condition of the wavelet.
In the latter [5,6], a Morlet wavelet [7] is used as the ana-lyzing wavelet whose dilation factor is independent uponthe axial distance. By taking the WT of the holograms, aspace-varying frequency of the transmittance of the holo-gram is obtained from resultant wavelet coefficients. Sincethis spatial frequency is inversely proportional to therecording distance, the axial position of the particle withrespect to the recording plane can be measured. For thisreason, our proposed method excels in the point that itobviates the need for searching all depth planes and hasa large far-field number [8]. In this method, the WT outputgives several desired wavelet coefficients. In order to get theaxial distance, the resultant distances calculated from thedesired coefficients are averaged. Since our method extractsthe recording distance from the holograms by using thespace and spatial frequency information, however, theaccuracy of measurement is determined by the spatial res-olution of the used CCD sensor. Furthermore, the mea-surement of this position becomes more difficult when thenumber of particles is high. This is because the interferencefringes produced by other objects distort the fringes beingstudied.
Other method based on Wigner distribution functionhas been proposed as well for extracting 3-D location ofthe object from in-line holograms [9] and depth locationfrom optical scanning holograms [10–13]. However, unlikethe WT, the resolution of the Wigner distribution functionis fixed, because the signal being analyzed is used as a win-dow function [14].
In this work, a novel method for tracking the position ofobjects by extracting the wavelet coefficient obtained at thecenter position of the interference fringes is proposed. It isfound that the dilation value of this wavelet coefficient isdetermined only by the recording distance and the wave-length of the light. In comparison with the previousmethod, this method has several advantages in that firstly,the spatial position of the wavelet coefficients is notrequired. This improves the accuracy of measurement. Sec-ondly, in the wavelet domain the wavelet coefficient at thecenter position of the interference fringes appears at a largedilation value, while those corresponding to high frequencyappear at small dilations. This is because the spatial fre-quency of the fringe at the center position is lower com-pared to those of the other parts. Since they aresignificantly separated in the wavelet domain, this methodis suitable for extracting the axial position from hologramsof multiple objects.
2. In-line Fraunhofer hologram
An amplitude transmittance of the in-line Fraunhoferhologram of a small line object with a radius of a can bemathematically expressed as [2]
IðxÞ ¼ 1� 4affiffiffiffiffikzp cos
px2
kz� p
4
� �sin 2pax
kz2paxkz
" #
þ 4a2
kzsin 2pax
kz2paxkz
" #2
; ð1Þ
where k and z are the wavelength of the illuminating lightand the axial distance between the object and the recordingplane, respectively. The first term of Eq. (1) corresponds tothe directly transmitted light. The second one is the modu-lation of a chirp signal by a sinc function. The recordingdistance z is encoded into the frequency of the chirp signal,while the object size a determines the lobe width of the sincfunction. The third term is a square of the sinc functionwhose amplitude is much smaller than the other terms[15]. Fig. 1 shows a computer plot of Eq. (1) witha = 60 lm, z = 40 cm, and k = 543.5 nm. It is obvious thatthe envelope of chirp signal has a shape of the sinc functionand the frequency of the chirp signal becomes higher as thespatial position increases.
3. Measurement of axial position
In our previous work [5], the interference pattern of thein-line hologram of a small object captured by a CCD sen-sor is analyzed by using the WT. The real value of theresultant wavelet coefficients gives the space-spatial fre-quency variation of the chirp signal which is a functionof the recording distance. The spatial position of thesewavelet coefficients is measured as a relative value to thecenter position of the object. As a consequence, in orderto extract the axial distance, the center position of the
0.6
0.8
1
1.2
1.4
1.6
Inte
nsity
(a.u
.)a
S. Soontaranon et al. / Optics Communications 281 (2008) 1461–1467 1463
object, the dilation and the spatial translation of thedesired wavelet coefficients are required.
In this work, our study shows that the recording dis-tance can be extracted from the dilation of the waveletcoefficient having maximum value which appears alongthe center position of the object. This dilation is determinedby the recording distance and the wavelength of the illumi-nating light which is a known parameter. Thus, by usingthis dilation information, the recording distance of theobject can be measured.
3.1. Wavelet transform
The WT is a mathematical technique which has beenintroduced in signal analysis to overcome the inability ofFourier analysis in providing local frequency spectra. TheWT of a signal pattern s(r) is defined as [7,9]
W ðt; dÞ ¼ 1ffiffiffidp
Z 1
�1g�
r � td
� �sðrÞdr; ð2Þ
where g(r) is the analyzing wavelet function with d and t de-note the dilation (scale) and the translation (shift) parame-ters, respectively. According to Eq. (2), the WT is computedby correlating the analyzed signal with a set of dilated wave-lets whose frequency contents are inversely proportional tothe dilation values. When the signal s(r) has the same fre-quency content as that of the dilated analyzing waveletg(r/d) in the region subtended by g*[(r � t)/d], a correlationpeak is generated in the WT domain. In this work, theMorlet wavelet defined by
gðrÞ ¼ expði2pfgrÞ expð�r2=2Þ ð3Þis employed as the analyzing wavelet. Fig. 2 plots the realvalues of the WT of the hologram shown in Fig. 1. The ver-tical axis is the dilation, while the horizontal axis is thetranslation. The plus and the cross symbols correspond
0 1 2 3 4-1-2-3-4x (mm)
-4
-4.2
-3.6
-3.8
-3.4
-3.2
-3
-4.4
log
(d)
10
Maximum at center positionlog ( )
10 0d
Fig. 2. The resultant WT coefficients with maximum and minimumamplitudes generated from the real value of the WT of the hologramshown in Fig. 1.
to the maximum and the minimum of the wavelet coeffi-cients, respectively. Since the frequency of the chirp signalincreases with respect to the position, the dilation of thewavelet coefficients varies from large to small values. It isworth mentioning that the wavelet coefficient having amaximum value pointed by the arrow sign always appearsat the center position of the interference fringes and isdetermined by a particular dilation d0.
3.2. Determination of the dilation d0
In order to obtain the mathematical expression for thedilation d0, the WT of the hologram around its center posi-tion is first investigated. Since the effect of the envelopemodulation on the chirp signal at this center position isinsignificant, the envelope function is neglected. Fig. 3aand b show the simplified hologram of the line objectshown in Fig. 1, which is generated by omitting the enve-
-4 -3 -2 -1 0 1 2 3 40.4
x (mm)
0 1 2 3 4-1-2-3-4x (mm)
-4
-4.2
-3.6
-3.8
-3.4
-3.2
-3
-4.4
log 10
(d)
b
Fig. 3. (a) Simplified hologram of a line object and (b) its WT.
-4.5 -4 -3.5 -3-6
-4
-2
0
2
4
6
8x 10-3
log10(d)
Ampl
itude
(a.u
.)
-3.3
1st term
2nd term
Fig. 4. Plot of each term in Eq. (6).
1464 S. Soontaranon et al. / Optics Communications 281 (2008) 1461–1467
lope function, and its resultant WT, respectively. The sim-plified hologram can be mathematically written as
uðxÞ ¼ 1� A cosðBx2 � CÞ; ð4Þwhere A ¼ 4a=
ffiffiffiffiffikzp
; B ¼ p=kz and C = p/4. It is obviousthat the signal at the center position of the simplified holo-gram and its corresponding wavelet coefficient are the sameas those of the original hologram. This is because aroundthe center position of the fringes, the amplitude and the fre-quency content of the signal of both holograms are thesame. Therefore the simplification of the hologram canbe justified.
By using the Morlet wavelet as the analyzing wavelet,the WT of the Eq. (4) can be mathematically written as
W ðt; dÞ
¼ffiffiffiffiffiffiffiffi2pd
pexpð�2p2f 2
g Þ � Að1þ 4B2d4Þ�14
ffiffiffiffiffiffipd2
r
� exp4pfgtBd � 2p2f 2
g � 2B2t2d2
1þ 4B2d4
"
þ jBt2 þ 8pfgtB2d3 � 4p2f 2
g Bd2
1þ 4B2d4þ
tan�1 2Bd2� �2
� C
( )#
� Að1þ 4B2d4Þ�14
ffiffiffiffiffiffipd2
rexp
�4pfgtBd � 2p2f 2g � 2B2t2d2
1þ 4B2d4
"
� jBt2 � 8pfgtB2d3 � 4p2f 2
g Bd2
1þ 4B2d4þ tan�1ð2Bd2Þ
2� C
( )#:
ð5ÞIn order to obtain the maximum value of the wavelet coef-ficient at the center position of the interference pattern, Eq.(5) is mathematically evaluated at the translation t = 0.This yields
W ð0; dÞ ¼ffiffiffiffiffiffiffiffi2pdp
expð�2p2f 2g Þ �
Affiffiffiffiffiffiffiffi2pdp
ð1þ 4B2d4Þ14
� exp�2p2f 2
g
1þ 4B2d4
!
� cos4p2f 2
g Bd2
1þ 4B2d4� tan�1ð2Bd2Þ
2þ C
( ): ð6Þ
Plots of the first and the second terms of Eq. (6) as functionof the dilation are shown in Fig. 4 by using the dashed-dot-ted and the solid lines, respectively. The maximum value ofthe wavelet coefficient at the center of the fringe appears atthe dilation log10(d0) = �3.3. In order to determine mathe-matically the dilation d0, the derivative of the wavelet coef-ficient W(0,d) with respect to d is calculated. The value ofthe dilation that causes this resultant derivation is equal tozero gives the dilation d0. However, as illustrated in Fig. 4,the first term appears as a straight line. This is because itsamplitude is in the order of 10�10 which is much smallerthan the second term. By neglecting the first term, Eq. (6)reduces to
W ð0; dÞ ¼ � Affiffiffiffiffiffiffiffi2pdp
1þ 4B2d4� �1
4
exp�2p2f 2
g
1þ 4B2d4
!
� cos4p2f 2
g Bd2
1þ 4B2d4�
tan�1 2Bd2� �2
þ C
( )ð7Þ
which is a multiplication of three terms. A plot of eachterm of Eq. (7) is shown in Fig. 5 where the solid, thedashed-dotted and the dashed lines represent the first,the second and the third terms, respectively. It can be seenthat the first term varies slowly and can be approximatelyconsidered as a constant compared to the second and thethird terms. Thus, by neglecting the first term, Eq. (7)becomes
W ð0; dÞ ¼ � exp�2p2f 2
g
1þ 4B2d4
!
� cos4p2f 2
g Bd2
1þ 4B2d4� tan�1ð2Bd2Þ
2þ C
( ): ð8Þ
The derivative of Eq. (8) with respect to d is
W 0ð0; dÞ
¼ exp�2p2f 2
g
1þ 4B2d4
!8p2f 2
g Bd2 � 32p2f 2g B3d5 � 2Bdð1þ 4B2d4Þ
ð1þ 4B2d4Þ2
" #
� sin4p2f 2
g Bd2
1þ 4B2d4� 1
2tan�1ð2Bd2Þ þ C
( )�
32p2f 2g B2d3
ð1þ 4B2d4Þ2
� exp�2p2f 2
g
1þ 4B2d4
!cos
4p2f 2g Bd2
1þ 4B2d4� 1
2tan�1ð2Bd2Þ þ C
( ):
ð9Þ
The maximum position can be determined when Eq. (9) isequal to zero. This yields
-4.5 -4 -3.5 -3-3.3
Ampl
itude
(a.u
.)
-0.8
-0.48
-0.16
0.16
0.48
0.8
-2
0
2
4
6
8
RHSEq.(12)
LHSEq.(12) Eq.(8)
log10(d)
Fig. 6. Plot of Eq. (8) and the LHS and RHS of Eq. (12).
0 1 2 3 4 5c (a.u.)
-4
-2
0
2
4
6
8
10
Ampl
itude
(a.u
.)
RHS
LHS1.86
Fig. 7. Plot of the LHS and RHS of Eq. (14).
-4.5 -4 -3.5 -3
-1
-0.5
0
0.5
1
Ampl
itude
(a.u
.)
log10(d)
1st term
3rd term
2nd term
Fig. 5. Plot of each term in Eq. (7).
S. Soontaranon et al. / Optics Communications 281 (2008) 1461–1467 1465
tan4p2f 2
g Bd2
1þ 4B2d4� 1
2tan�1ð2Bd2Þ þ C
( )
¼16p2f 2
g Bd2
4p2f 2g � 16p2f 2
g B2d4 � 1� 4B2d4: ð10Þ
Arctan of Eq. (10) is
4p2f 2g Bd2
1þ 4B2d4� 1
2tan�1ð2Bd2Þ þ C þ np
¼ tan�116p2f 2
g Bd2
4p2f 2g � 16p2f 2
g B2d4 � 1� 4B2d4
!; ð11Þ
where n is an integer number which represents the period-icity of the tangent function. By grouping the arctan func-tions, Eq. (11) becomes
4p2f 2g Bd2
1þ 4B2d4þ C þ np
¼ tan�116p2f 2
g Bd2
4p2f 2g � 16p2f 2
g B2d4 � 1� 4B2d4
!
þ 1
2tan�1ð2Bd2Þ: ð12Þ
In order to determine the value of d that gives the desiredmaximum, the right hand side (RHS) and the left handside (LHS) of Eq. (12) are plotted as a function of thedilation for n = �1. They are shown in Fig. 6 as thedash-dot and the dash lines, respectively. The amplitudeof these graphs is represented by the scale on the right-vertical axis. The dilations at the crossing points of thetwo lines are the solution of Eq. (12). In this figure, Eq.(8) is also plotted by using the solid line whose amplitudeis represented by the left-vertical axis. It is clear that forn = �1 there is one crossing point whose dilation givesthe desired maximum wavelet coefficient at the center ofthe interference fringes.
In order to derive mathematically the relationshipbetween this dilation and the axial position of the object,let assume that the solution of Eq. (12) has the form of
d ¼ cffiffiffiBp ð13Þ
with c is a constant. Substitutions of Eq. (13) and n = �1into Eq. (12) gives
4p2f 2g c2
1þ 4c4þ C � p
¼ tan�116p2f 2
g c2
4p2f 2g � 16p2f 2
g c4 � 1� 4c4
( )þ 1
2tan�1ð2c2Þ:
ð14Þ
Fig. 7 shows plots of the RHS and the LHS of Eq. (14) rep-resented by using the dash and the solid lines, respectively,as a function of c. Here, the parameter C = p/4 is used. It is
1466 S. Soontaranon et al. / Optics Communications 281 (2008) 1461–1467
found that there are two crossing points yielding two solu-tions. However, from Fig. 6, the solution of Eq. (12) that isthe high dilation value produces the desired maximum.Thus, c = 1.86 can be regarded as the solution of Eq.(14). By substituting this value of c and the definition ofthe parameter B into Eq. (13), the dilation d0 that givesthe maximum wavelet coefficient along the translationt = 0 are equal to 1:05
ffiffiffiffiffikzp
. Since this value of d0 is ob-tained by using the simplified hologram and several math-ematical approximations done during the derivation, theresultant constant c may contain an error. In order to min-imize this error, the value of c is verified by computing theWT of the holograms I(x) simulated at various recordingdistances. By detecting the dilation d0 from the maximumvalue of the wavelet coefficients along the translationt = 0, the constant c can be calculated by c ¼ d0=
ffiffiffiffiffikzp
.From these computations, the maximum position of d0 isfound to be
d0 ¼ 1:0455ffiffiffiffiffikzp
: ð15Þ
4. Results and discussion
In order to verify our proposed method, the in-line holo-grams of an optical fiber having a diameter of2a = 124.96 lm were experimentally generated. A He–Nelaser having the wavelength of 543.5 nm was employed asthe light source. The holograms were captured by using aCCD sensor HAMAMATSU C5948 having the averagepixel size in horizontal direction of 12.99 lm. The WT ofthe holograms was digitally computed. From the resultantwavelet coefficient obtained at the center position, the axialposition was extracted by using Eq. (15). Fig. 8 shows theerrors in measurements of the axial distance of the opticalfiber from the holograms. These errors are the percentagedifferences between the axial distances measured by using
Single wavelet coefficientPrevious method
0
0.5
1
1.5
2
erro
r (%
)
z (cm)12 14 16 18 20
Fig. 8. Errors in measurement of z from the optically generatedholograms of the optical fiber by using the WT.
our proposed method and the reading of millimeter scalesengraved on an optical rail onto which the optical fiberand the CCD sensor are installed. The circle and the crosssigns correspond to the errors obtained from the new pro-posed and the previous method [5], respectively. Althoughthe proposed method use only single wavelet coefficient atthe center position of the object for extracting the record-ing distance, its measurement results show that the errorsare generally smaller and fluctuate less compared to thoseextracted from the multiple wavelet coefficients obtainedfrom the previous method. Therefore, the proposedmethod has an advantage in that the measurement of therecording distance can be obtained with higher accuracybecause it does not depend on the spatial resolution ofthe CCD sensor which is used to record holograms. As aresult, our proposed method is useful for extraction ofthe axial distance from the in-line holograms of multipleobjects where the overlapping of multiple interferencefringes occurs.
5. Conclusion
We have proposed and verified experimentally anovel method for tracking the position of objects fromthe in-line holograms by using single wavelet coefficientobtained at the center position of the interferencefringes. The main advantage of the proposed methodis that the measurement of the axial distance is notdetermined by the spatial position of the wavelet coeffi-cients. The experimental results show that the proposedmethod could extract the desired information with highaccuracy.
Acknowledgement
The financial support from the Thailand Research Fundthrough the Royal Golden Jubilee scholarship (Grant No.PHD/0188/2542) is acknowledged.
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