Measurement of curvature and twist of a deformed object using digital holography

Measurement of curvature and twist of a deformedobject using digital holography

Wen Chen, Chenggen Quan,* and Cho Jui TayDepartment of Mechanical Engineering, National University of Singapore, 9 Engineering Drive 1, Singapore 117576

*Corresponding author: [email protected]

Received 28 February 2008; revised 23 April 2008; accepted 25 April 2008;posted 25 April 2008 (Doc. ID 93229); published 15 May 2008

Measurement of curvature and twist is an important aspect in the study of object deformation. In recentyears, several methods have been proposed to determine curvature and twist of a deformed object usingdigital shearography. Here we propose a novel method to determine the curvature and twist of a de-formed object using digital holography and a complex phasor. A sine/cosine transformation methodand two-dimensional short time Fourier transform are proposed subsequently to process the wrappedphase maps. It is shown that high-quality phase maps corresponding to curvature and twist can be ob-tained. An experiment is conducted to demonstrate the validity of the proposedmethod. © 2008 OpticalSociety of America

OCIS codes: 090.1995, 090.2880, 120.3940, 120.6650, 100.5070.

1. Introduction

Since flexural and torsional moments are relatedto second-order derivatives of displacement, themeasurement of curvature and twist is an impor-tant aspect in the study of out-of-plane deforma-tion of an object. In the past several decades,several methods [1–11] have been proposed tomeasure the curvature and twist of an object.Hung and Liang [3] described an image-shearingcamera to determine the derivatives of displace-ment and obtained second-order derivativesthrough shifting fringe patterns of first-order deri-vative. Sharma et al. [4] presented a simple three-aperture speckle-shearing arrangement for directmeasurement of curvature of the specimen. Sriramet al. [6] proposed a simple method for measuringthe radius of curvature by using Talbot interfero-metry in a noncollimated light beam. Tay et al. [9]proposed a multiple-image shearography to mea-sure the curvature and twist of a deformed object.In this method, one set of the fringes obtainedbased on the multiple-image shearography repre-sents the second-order displacement derivatives.

Rastogi [10] described an application of electronicspeckle-shearing pattern interferometry for deter-mining the phase maps corresponding to the cur-vature and twist of a deformed specimen. Wanget al. [11] presented a five-aperture shearingand a two-Fourier filtering technique to measurepure curvature and twist distributions of a de-formed object.

Here, we propose a novel method to determine thecurvature and twist of a deformed object using digitalholography (DH). In DH, the object wave can be re-constructed numerically, and the phase can be di-rectly retrieved from reconstructed images orthrough a phase-shifting method [12–14]. Phase dif-ference between the first and second states is ob-tained directly by digital phase subtraction (DPS).In addition, the Fourier transform method [15] canalso be used to extract phase maps. In this study,to improve the quality of phase maps correspondingto second-order displacement derivatives, a complexphasor (CP) approach is employed. A sine/cosinetransformation method and a short-time Fouriertransform (STFT) [16] are proposed subsequentlyto process the wrapped phase maps. An experimentis conducted to demonstrate the validity of the pro-posed method.

0003-6935/08/152874-08$15.00/0© 2008 Optical Society of America

2874 APPLIED OPTICS / Vol. 47, No. 15 / 20 May 2008

2. Theoretical Analysis

A. Hologram Recording and Reconstruction

Figure 1(a) shows a schematic experimental arrange-ment for recording digital holograms. The variablebeam splitter separates a He–Ne laser beam into re-ference and object waves. A beam splitter cube isplaced in front of a CCD camera to combine the objectand reference waves, and a hologram is recorded bythe CCD. As this experimental setup is used, the re-ference wave is considered as a spherical wave. Thepixels of the CCD can be fully used andmuch sharperimages can be obtained. Figure 1(b) shows the struc-ture of a test plate. The aluminum plate is clampedat the edge, and a point load is applied from the rearat the center.Figure 2 shows a coordinate system for numerical

reconstruction of a recorded hologram. As can beseen in Fig. 2, the coordinates of the object, hologram,and reconstruction planes are denoted by ðξ0; η0Þ,ðξ00; η00Þ, and ðx; yÞ, respectively. To obtain well-focusedreconstructed images, one should ensure that the nu-merical reconstruction distance is precisely equal to

the recording distance between the object and theCCD. Since the reference wave is considered as aspherical wave, reconstruction algorithm can be im-plemented as a pure Fourier transform [17]. A pureFourier transform reconstruction algorithm can begiven by

Eðx; yÞ ¼ C exp�iπλL ðx2 þ y2Þ

�I−1½Hðξ00; η00Þ�; ð1Þ

where C denotes a complex constant, L denotes thereconstruction distance, λ is the laser wavelength,I−1 denotes a 2D inverse Fourier transform, andHðξ00; η00Þ denotes the hologram function. A discreterepresentation of Eq. (1) is given by

Eðm;nÞ ¼ C exp�iπλL

�m2

M2Δξ002 þn2

N2Δη002��

XM−1

k¼0

XN−1

l¼0

HðkΔξ00; lΔη00Þ exp�i

�2πmkM

þ 2πnlN

��; ð2Þ

where Eðm;nÞ is a matrix of M ×N points, andΔξ00 and Δη00 denote pixel sizes on the hologram

Fig. 1. (a) Schematic experimental setup: VBS, variable beam splitter; SF, spatial filter; (b) schematic of a plate loaded at center; (c) shiftdirection for numerical determination of curvature ∂2d=∂x2; (d) shift direction for numerical determination of twist ∂2d=∂x∂y.

20 May 2008 / Vol. 47, No. 15 / APPLIED OPTICS 2875

plane. For simplicity, the factors Δx and Δy ofpixel sizes in the reconstruction plane are omitted,and the pixel in the reconstruction plane is denotedas ðm;nÞ.In DH, phase φðm;nÞ and intensity Iðm;nÞ can be

extracted directly from the reconstructed complexamplitude:

φðm;nÞ ¼ arctanIm½Eðm;nÞ�Re½Eðm;nÞ� ; ð3Þ

Iðm;nÞ ¼ jEðm;nÞj2: ð4Þ

To ensure that the camera can resolve the interfer-ence fringe patterns, the spatial frequency shouldcomply with the following requirement:

f ≤2λ sin

�θmax

2

�; ð5Þ

where θmax is the maximum angle between the refer-ence and the object waves. The maximum spatial fre-quency fmax can be calculated by 1=ð2Δξ00Þ: In thisstudy, the maximum angle ½θmax ≈ λ=ð2Δξ00Þ� is ap-proximately 3:9°.

B. Complex Phasor Method

It is well known that the reconstructed image in DHcan be considered as a complex value. The recon-structed image calculated byEq. (2) can be consideredas a complex exponential signal. ACP is calculated bymultiplying Eðm;n; 2Þ at a deformed state by the con-jugate of Eðm;n; 1Þ at an initial state. Without loss ofgenerality, only a given pixel is considered:

Aðm;nÞ ¼ Eðm;n; 2ÞE�ðm;n; 1Þ ¼ aðm;n; 2Þaðm;n; 1Þ× expfi½φðm;n; 2Þ − φðm;n; 1Þ�g

¼ a0ðm;nÞ expfi½Δφðm;nÞ�g; ð6Þ

where �denotes the complex conjugate, theamplitudea0ðm;nÞ ¼ aðm;n; 2Þaðm;n; 1Þ; and the phase differ-ence Δφðm;nÞ ¼ φðm;n; 2Þ − φðm;n; 1Þ.As the CP method is used, filtering algorithms are

engaged to filter the imaginary and the real parts of

the CP. In this study, an average filter is employed.The phase difference in Eq. (6) can be expressed by

Δφðm;nÞ ¼ arctanP

Im½Eðm;n; 2ÞE�ðm;n; 1Þ�=9PRe½Eðm;n; 2ÞE�ðm;n; 1Þ�=9 ;

ð7Þ

where a 3 × 3 window for the average filter is used.A shifted complex amplitude can be easily ob-

tained by shifting the CP map. The complex ampli-tude Γðm;nÞ, corresponding to the first-orderdisplacement derivative, is calculated by multiplyinga shifted CP, A0ðm;nÞ, by the conjugate of the originalCP:

Γðm;nÞ ¼ A0ðm;nÞA�ðm;nÞ ¼ a0ðmþ δx;nÞa0ðm;nÞ× expfi½Δφðmþ δx;nÞ −Δφðm;nÞ�g¼ a00ðm;nÞ expfi½ΔΔφðm;nÞ�g; ð8Þ

where δx denotes a shifting value along the x direc-tion [see Fig. 1(a)], the amplitude a00ðm;nÞ ¼ a0ðmþ δx;nÞa0ðm;nÞ, and ΔΔφðm;nÞ ¼ Δφðmþ δx;nÞ−Δφðm;nÞ.

An average filter is also employed, and the phasedifference in Eq. (8) can be expressed by

ΔΔφðm;nÞ ¼ arctanP

Im½A0ðm;nÞA�ðm;nÞ�=9PRe½A0ðm;nÞA�ðm;nÞ�=9 ; ð9Þ

where a 3 × 3 window for the average filter is used.The complex amplitudes corresponding to the first-order displacement derivative can be further shifted.The complex amplitude Ωðm;nÞ that corresponds tothe second-order displacement derivatives is deter-mined by multiplying a shifted Γ0ðm;nÞ by the conju-gate of the original Γðm;nÞ:

Ωðm;nÞ ¼ Γ0ðm;nÞΓ�ðm;nÞ ¼ a00ðmþ δx;nÞa00ðm;nÞ× expfi½ΔΔφðmþ δx;nÞ −ΔΔφðm;nÞ�g¼ a000ðm;nÞ expðifΔ½ΔΔφðm;nÞ�gÞ; ð10Þ

where a000ðm;nÞ ¼ a00ðmþ δx;nÞa00ðm;nÞ and Δ½ΔΔφðm;nÞ� ¼ ΔΔφðmþ δx;nÞ −ΔΔφðm;nÞ.

An average filter is also engaged, and the phasedifference in Eq. (10) can be expressed by

Δ½ΔΔφðm;nÞ� ¼ arctanP

Im½Γ0ðm;nÞΓ�ðm;nÞ�=9PRe½Γ0ðm;nÞΓ�ðm;nÞ�=9 ;

ð11Þ

where a 3 × 3window for the average filter is used. Itis noteworthy that in Eqs. (6), (8), and (10),Δφðm;nÞ,ΔΔφðm;nÞ, and Δ½ΔΔφðm;nÞ� can be considered asthe phase difference after filtering, and in this studyiterative cycles are employed in the CP method withan average filter.

Fig. 2. Coordinate system for the numerical reconstruction of arecorded hologram.


The phase map corresponding to second-order dis-placement derivatives can be described by

Δ½ΔΔφðm;nÞ� ≈ 4πλ

�∂d∂x

ðmþ δx;nÞ −∂d∂x

ðm;nÞ�δx

≈4πλ∂2d

∂x2δ2x ; ð12Þ

where dðm;nÞ denotes the displacement of an object.It is noted that in Eqs. (6), (8), and (10), although thesame symbol for a pixel is used, it may indicate dif-ferent locations in the reconstructed images or in thephase maps. It is also noteworthy that the phasedifference map should be unwrapped. Similar toEq. (12), analog theoretical expressions can be devel-oped for curvature ∂2d=∂y2 and twist ∂2d=∂x∂y:Figures 1(c) and 1(d) show, respectively, the direc-

tion of shift for the determination of ∂2d=∂x2 and∂2d=∂x∂y. In the measurement of twist, a shift inthe x direction is carried out first followed by a shiftin the y direction. It is noteworthy that only the x − yplane is considered in Figs. 1(c) and 1(d).Using the proposed CP method, the real and ima-

ginary parts of a CP can be filtered. The expressionswith Eqs. (7), (9), and (11) are mainly to show the per-formance of the proposed CP method. It should benoted that not all values are filtered by the ninepoints, thus the Arabic numeral in Eqs. (7), (9),and (11) can be modified correspondingly. It is wellknown that phase itself is not a signal but rathera property of the signal [18]. One advantage of theproposed method is that the shifting process is con-sidered as the CP method and phase manipulationcan be effectively avoided, thus better results canbe expected.As the wrapped phase maps corresponding to sec-

ond-order displacement derivatives are determinednumerically, a conventional sine/cosine transforma-tion algorithm is used to further enhance the qualityof phase maps corresponding to the curvatureor twist.The sine/cosine transformation with an average fil-

ter is expressed by

Δ½ΔΔφðm;nÞ� ¼ arctanP

Sðm;nÞ=9PCðm;nÞ=9 ; ð13Þ

where Sðm;nÞ ¼ sinfΔ½ΔΔφðm;nÞ�g, Cðm;nÞ ¼ cosfΔ½ΔΔφðm;nÞ�g, and Δ½ΔΔφðm;nÞ� denotes the fil-tered wrapped phase difference. A 3 × 3 window forthe average filter is employed. It is also noteworthythat not all values in the wrapped phase map are fil-tered by the nine points. In the sine/cosine transfor-mation method, several iterative cycles are usuallyrequired to achieve satisfactory results. In the firstiterative cycle, the phase values in the wrappedphase map are converted into sine and cosine for-mats, and an average filter is applied. In the seconditerative cycle, an average filter is again applied tothe filtered values obtained in the first cycle. The pro-

cedure is required until the iterative cycles are com-pleted. The final filtered wrapped phase map isobtained by an arctangent operation.

C. Two-Dimensional Short Time Fourier TransformMethod

As the iterative cycle in sine/cosine transformationwith an average filter is large, dense fringe patternsmay be smeared out. Hence, in many cases, the itera-tive cycle should not be too large. After a phase mapcorresponding to second-order displacement deriva-tives is obtained, we introduce a second algorithmbased on two-dimensional STFT which is also knownas windowed Fourier transform [16]. Compared withglobal Fourier transform, STFT has a localizationcharacteristic, thus STFT is able to prevent the pro-pagation of errors more efficiently.

Since exponential signals can be considered as thesource signals of the measurement, the phase map isconverted into an exponential signal before imple-menting the STFT algorithm. The correspondingsine and cosine formats of the wrapped phaseΔ½ΔΔφðx; yÞ� are described by

Sðx; yÞ ¼ sinfΔ½ΔΔφðx; yÞ�g; ð14Þ

Cðx; yÞ ¼ cosfΔ½ΔΔφðx; yÞ�g: ð15Þ

An exponential signal is obtained as follows:

f ðx; yÞ ¼ Cðx; yÞ þ iSðx; yÞ ¼ expðifΔ½ΔΔφðx; yÞ�gÞ:ð16Þ

Two-dimensional STFT of the exponential signalf ðx; yÞ is described by [16]

Sf ðu; v; ξ; ηÞ ¼Z þ∞

−∞

Z þ∞

−∞

f ðx; yÞgðx − u; y − vÞ

× expð−iξx − iηyÞdxdy; ð17Þ

where gðx; yÞ represents a window function. In thisstudy, gðx; yÞ ¼ expð−x2=2σ2x − y2=2σ2yÞ, where σx andσy control the extension of the Gaussian window thatcan provide the smallest Heisenberg box [16]. Notethat, in this study, ðm;nÞ and ðx; yÞ are interchange-ably used as no confusion is raised.

As the signal is converted into the frequency do-main, the noise usually has small coefficients. Coef-ficients smaller than a preset threshold areeliminated, thus the noise is efficiently reduced. Inaddition, STFT is performed over a local area, andthe transform of a signal does not influence pixelsat other positions. The filtered spectrum Sf ðu; v; ξ; ηÞis described by


Sf ðu; v; ξ; ηÞ ¼�Sf ðu; v; ξ; ηÞ if jSf ðu; v; ξ; ηÞj ≥ thrd0 if jSf ðu; v; ξ; ηÞj < thrd ;

ð18Þ

where thrd denotes the preset threshold.After the filtering process, the filtered signal f ðx; yÞ

is obtained by an inverse STFT:

f ðx; yÞ ¼ 1

4π2Z þ∞

−∞

Z þ∞

−∞

Z ηu

ηl

Z ξu

ξlSf ðu; v; ξ; ηÞg

ðx − u; y − vÞ expðiξxþ iηyÞdξdηdudv; ð19Þ

where ξu and ξl denote the upper and lower limits ofξ, and ηu and ηl denote the upper and lower limits of η,respectively. The ranges of ξ and η can be approxi-mately estimated by the analysis of the Fouriertransform of a phase map. Since the STFT spectrumextends infinitely along the ξ and η axes, the esti-mated ranges of ξ and η should be adjusted to focuson the most energy using the selected window size[19]. After the STFT operation, filtered wrappedphase maps are obtained by

Δ½ΔΔφðx; yÞ� ¼ arctanImf ðx; yÞRef ðx; yÞ ; ð20Þ

where Δ½ΔΔφðx; yÞ� denotes the filtered phase map.Figure 3 shows a flow chart for the determination

of curvature and twist of a deformed object using theproposed method. As shown in Fig. 3, the filtering al-gorithm should be simultaneously considered once aphase subtraction operation is carried out. After thewrapped phase maps corresponding to second-orderdisplacement derivatives are filtered by a conven-tional sine/cosine transformation method or a two-dimensional STFT approach, an algorithm isrequired to perform phase unwrapping. In this study,the branch-cut method [18] is employed to do thephase unwrapping.

3. Experiment

As shown in Fig. 1(a), a He–Ne laser beam (wave-length of 632:8nm) is separated into two beams,i.e., reference and object waves. A polarizer is placedin the reference wave path. By adjusting the polari-zer, the intensity of the laser beam in the referencepath can be correspondingly regulated. A CCD arrayis used to record digital holograms at the initial anddeformed states of the object. The CCD has an arrayof 1024 × 1024 pixels, and the size of each pixel is4:65 μm. A circular plate with a central point loadis chosen as a test specimen as shown in Fig. 1(b).The thickness t of the plate at the center is1:5mm, and diameter D of the plate is 50mm. Notethat the point load cannot be too large. White lacqueris sprayed on the surface of the plate to improve re-flectivity from the specimen. The distance betweenthe CCD and the surface of the plate is 92:5 cm.

4. Results and Discussions

In this study, two digital holograms at the initial anddeformed states are recorded by the CCD. Figure 4(a)shows a wrapped phase map of a deformed plate ob-tained by the CPmethod. A 3 × 3 average filter in theCP method is used. The branch-cut unwrapping ap-proach is employed, and a three-dimensional (3D)plot of a continuous phase map that corresponds todisplacement of the plate is shown in Fig. 4(b).

Figure 5(a) shows a wrapped phase map corre-sponding to curvature along the x direction withoutthe use of a filter. It is seen from Fig. 5(a) that thephase map is noisy. Since the wrapped phase mapscontain 2π jumps, phase unwrapping is required toobtain a continuous phase map. However, whenthe wrapped phase map is noisy, the unwrapping op-eration is usually unsuccessful. Hence, two strate-gies are usually adopted to solve the problem. Oneis to improve filter algorithms, while another is to de-velop a more powerful unwrapping algorithm. In thisstudy, the first strategy is considered, and the corre-sponding algorithm is developed to remove the noise.

Fig. 3. Flow chart of measurement of curvature and twist usingthe proposed method.


Figure5(b)showsawrappedphasemapcorrespond-ing to curvature ∂2d=∂x2 using the proposed CP ap-proach. After the phase map [Fig. 5(b)] is obtained, aconventional sine/cosine transformation method isused to further remove the noise. Figure 5(c) showsa filteredphasemapcorresponding to curvaturealongthe x direction using the sine/cosine transformationmethod with a 3 × 3 average filter. The iterative cyclein the filtering algorithm is 19. It can be seen fromFig. 5(c) that a high-quality phase map is obtained.However, it is obvious that some noise still existsat the center. A 3D plot of an unwrapped phase mapfor the wrapped phase map in Fig. 5(c) is shown inFig. 5(d).The proposed method can also be used to deter-

mine curvature along the y direction. Figure 6(a)shows a phase map corresponding to curvature∂2d=∂y2 without filtering. Figure 6(b) shows a phasemap obtained by the CP approach. Figure 6(c) is afiltered wrapped phase map corresponding to curva-ture along the y direction using a sine/cosine trans-formation approach. An average filter with a 3 × 3window is used. The iterative cycle in the filtering

algorithm is 15. A 3D plot of an unwrapped phasemap for the wrapped phase map in Fig. 6(c) is shownin Fig. 6(d). In this study, the value of each shiftingis 30pixels.

Figure 7(a) shows a wrapped phase map corre-sponding to twist ∂2d=∂x∂y without the use of a filter.Figure 7(b) shows a wrapped phase map after filter-ing. The phase map shown in Fig. 7(b) is obtained bydirectly filtering the phase map in Fig. 7(a) by use ofthe conventional sine/cosine transformation ap-proach with a 3 × 3 average filter. The iterative cyclein the filtering algorithm is 25. Figure 7(c) shows awrapped phase map corresponding to twist∂2d=∂x∂y by use of the proposed CP method. Figure 7(d) shows a filtered phase map that is obtained by fil-tering the phase map in Fig. 7(c) by use of the sine/co-sine transformation approach with a 3 × 3 averagefilter. The iterative cycle in the filtering algorithmis also 25. It is seen that a higher-quality phasemap is obtained by use of the CPmethod. A small voidpart in the bottom left corner of Fig. 7(d) might besolved with more appropriate cropping of experimen-tal data. Figure 7(e) shows an unwrapped phase mapand Fig. 7(f) is a 3D plot of the continuous phase mapshown in Fig. 7(e).

In the CP method, the main objective is to find thecenter of gravity of the data. The pixel with the leastamplitude has the weakest influence on the results,while the pixels with a relatively larger amplitudehave more influence on the results. It is noteworthythat the CP cannot be lined up in a definite way. Thelow-pass filter, such as a median filter, will not suc-ceed as the amplitudes of the CP vary within the se-

Fig. 4. (a)Wrapped phasemap obtained by the CPmethod; (b) 3Dplot of a continuous phase map for deformation of a centrallyloaded plate.

Fig. 5. (a) Phase map corresponding to curvature ∂2d=∂x2 with-out filtering; (b) phasemap obtained by use of CP; (c) filtered phasemap by use of sine/cosine transformation for (b); (d) 3D plot of anunwrapped phase map.


lected filter window [20]. It is also noted that in theCP method with an average filter, an appropriate fil-tering window should be determined so as to obtain

satisfactory results. In addition, the CP method ismore suitable for the relatively less-dense fringepatterns.

In two-dimensional STFT, the ranges of ξ and η canbe estimated by the Fourier transform of a phasemap. It is noted that the ranges are adjusted basedon the window size such that the most energy is em-ployed. In this study, the values of both σx and σy areset as 10. Figure 8(a) shows a phase map correspond-ing to curvature along the x direction filtered by theSTFTapproach with a threshold value of 6. As can beseen in Fig. 8(a), the central dashed line in Fig. 5(c)disappears, which indicates that the noise has beenfully eliminated. Figure 8(b) shows a phase map cor-responding to curvature along the y direction filteredby the STFT method with a threshold value of 6. Theresults show higher quality than that in Fig. 6(c).Figure 8(c) shows a phase map corresponding totwist ∂2d=∂x∂y filtered by the STFT method. Thethreshold value is set as 4. It should be noted thatthe phase maps in Figs. 8(a)–8(c) are obtained bySTFT based on the phase maps in Figs. 5(b), 6(b),and 7(c), respectively. It is shown that the two-dimen-sional STFT gives higher-quality phase distributionswhich are much better than the results in Ref. [21].However, how filtering algorithms affect the final re-sults may be further investigated. Since high-quality

Fig. 6. (a) Phase map corresponding to curvature ∂2d=∂y2 with-out filtering; (b) phase map obtained by CP; (c) filtered phase mapfor (b); (d) 3D plot of an unwrapped phase map.

Fig. 7. (a) Phase map corresponding to twist ∂2d=∂x∂ywithout filtering; (b) filtered phase map obtained by use of sine/cosine transforma-tion; (c) phase map obtained by use of CP; (d) filtered phase map obtained by sine/cosine transformation filtering on the phase map in(c); (e) unwrapped phase map; (f) 3D plot of the continuous phase map.


wrapped phase maps are obtained, the phase mapscan also be easily unwrapped.

5. Conclusions

Here, a novel method is proposed to measure the cur-vature and the twist of a deformed object based onDH and CP. It is shown that using the proposedCP method, phase manipulation can be effectivelyavoided, and the real and imaginary parts of theCP can be filtered. The sine/cosine transformationmethod and the two-dimensional STFT algorithmare subsequently employed to process the wrappedphase maps. Experimental results demonstrate thathigh-quality phase maps corresponding to curvatureand twist of a deformed plate can be successfully ob-tained by the proposed method.

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Fig. 8. (a) Phase map corresponding to curvature ∂2d=∂x2 filteredby STFT; (b) phase map corresponding to curvature ∂2d=∂y2

filtered by STFT; (c) phase map corresponding to twist ∂2d=∂x∂yfiltered by STFT.


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Measurement of curvature and twist of a deformed object using digital holography