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Precision Engineering 38 (2014) 512–524

Contents lists available at ScienceDirect

Precision Engineering

jo ur nal homep age: www.elsev ier .com/ locate /prec is ion

ourier transform profilometry employing novel orthogonal ellipticand-pass filtering for accurate 3-D surface reconstruction

iang-Chia Chena,b,∗, Hoang Hong Haic

Department of Mechanical Engineering, National Taiwan University, No. 1, Sec. 4, Roosevelt Rd., Taipei 106, TaiwanSchool of Engineering, the University of South Australia, Building J, Level 1, Mawson Lakes Campus, GPO Box 2471, Adelaide 5001, AustraliaGraduate Institute of Automation Technology, National Taipei University of Technology, No. 1, Sec. 3, Zhong Hsiao East Rd., Taipei 106, Taiwan

r t i c l e i n f o

rticle history:eceived 13 September 2012eceived in revised form6 December 2013ccepted 29 January 2014vailable online 22 February 2014

eywords:utomatic optical inspection (AOI)

n situ measurementourier Transform Profilometry

a b s t r a c t

The article proposes a novel orthogonal elliptic band-pass filtering methodology in Fourier Transform Pro-filometry (FTP) for significant improvement of accurate 3-D measurement surface reconstruction witharbitrary object colors. Compared with phase shifting profilometry (PSP), FTP using fringe projection canachieve a general 3-D surface profilometry more efficiently by employing one-shot imaging. However,a challenging problem commonly encountered by FTP using fringe projection is its unreliable extrac-tion of precise spectral information from the spectral domain especially when the spectral domain iscomplicated to process. Various filtering methods previously proposed in FTP have been proved unsuc-cessful or nonrobust. Thus, a new band-pass filter is developed from an adaptive orthogonal ellipticregion to achieve higher accuracy of 3-D surface reconstruction. A comprehensive theoretical analysisis performed to investigate the physical measurement limits of the proposed method. The experimen-

and-pass filterriangulation Method

tal results obtained confirm that the measurement accuracy of dimension and sphericity can be greatlyenhanced when compared with that achieved by the traditional circular band-pass filter. The proposedmethod is proved to outperform all the other existing FTP band-pass filtering approaches. The maximumdimensional error measured can be controlled within 1.25% of the overall measuring height with varioussurface colors. However, it is also verified that the traditional three-step PSP can achieve slightly bettermeasuring repeatability than the proposed method.

. Introduction

Optical surface profilometry is an important step for 3-D shapeeasurement of a static or dynamic object. It has been widely

sed for machine vision, biotechnology, broadband communica-ions and optoelectronics. One of the current significant issues inptical surface profilometry is to retrieve accurate dimensional andhape information from objects under detection. Active 3-D visionethods generally project structured light patterns (also called

ringes) onto the object surface for 3-D imaging. The most common-D measurement algorithms include phase shifting profilometryPSP) and Fourier Transform Profilometry (FTP). The advantages ofTP over other approaches include its high detection rate and low

ensitivity to environmental disturbance.

In general, PSP [1–5] needs multiple fringe imaging that charac-erizes its non-real-time detection manner, making it not an ideal

∗ Corresponding author at: Department of Mechanical Engineering, Nationalaiwan University, No. 1, Sec. 4, Roosevelt Rd., Taipei 106, Taiwan.el.: +886 2 33662721.

E-mail addresses: lchen@ntu.edu.tw, lcchen@ntut.edu.tw (L.-C. Chen).

141-6359/$ – see front matter © 2014 Elsevier Inc. All rights reserved.ttp://dx.doi.org/10.1016/j.precisioneng.2014.01.006

© 2014 Elsevier Inc. All rights reserved.

candidate for real-time high-speed 3-D detection. Apart from PSP,FTP is one of the core development technologies in 3-D surfaceprofilometry. Its one-shot imaging feature can effectively avoidundesired object shaking or scene disturbance encountered by PSP[6–12]. To acquire 3-D phase maps, FTP records deformed fringeimages and analyzes the shape information by its phase modulationand demodulation process. However, to achieve accurate 3-D pro-file reconstruction wit arbitrary object colors, one of the key issuesessential in FTP is to extract the first-order spectra accurately fromthe spectral domain. Such extraction would require a proper band-pass filter to retain precise spectral information from the first-orderspectra and keep them from mixing with the background spectrum,other spectra of higher order, and noises. Conditions for separat-ing these spectral regions have been investigated by consideringthe minimal distances between the spectral regions [13–16]. Thesingle elliptic band-pass filter [17] was effective in enhancing theaccuracy of band-pass filtering. However, our recent investigationfound that the single elliptic band-pass filter has a fatal drawback of

losing some crucial spectral data in the spectral domain. So far, allthe existing methods were mainly proposed to handle objects witheither single surface color or uniform surface reflectance. None ofthem shows the feasibility of surface measurement with arbitrary

L.-C. Chen, H.H. Hai / Precision Engineering 38 (2014) 512–524 513

propo

sfimfits

pdds3i

2

2

utaoasp

oc

g

object height, and qn = r(x, y)Anein�(x,y).

Fig. 1. System layout of the

urface colors. As a result, the 3-D shape reconstructed using thislter tends to lose the shape edges especially when step-heighteasurement is involved. Thus, up to now, none of the band-pass

lters being developed has effectively addressed the band-pass fil-ering problem in reconstructing accurate object dimension andhapes.

In view of the above, we propose a new orthogonal elliptic band-ass filter (OEBF) to extract more accurately first-order spectralata from the spectral domain. The other five filters previouslyeveloped in the literature are employed to separate the first-orderpectra from the background and higher-order spectra for accurate-D surface reconstruction. The developed filters were compared

n terms of their accuracy in 3-D shape reconstruction.

. Measurement principle

.1. Principle of FTP using a general tilting fringe

The principle layout of the developed 3-D measuring systemsing digital fringe projection is shown in Figs. 1 and 2, in whichhe CCD camera, digital light projector (DLP) and reference planere arranged in a coplanar relationship. The distance between thebjective lens of the projector and that of the CCD camera is defineds d. The reference plane is perpendicular to the projector axis andet at a distance l defined from the objective lens center of therojector.

When a sinusoidal fringe pattern is projected onto the measuredbject, the deformed fringe image observed through a CCD cameraan be expressed as [6,7,17]:

(x, y) = r(x, y)+∞∑

n=−∞Ane2�in(fxx+fyy+�(x,y)) =

+∞∑n=−∞

qne2�in(fxx+fyy) (1)

sed measurement method.

where r(x,y) is a non-uniform distribution of reflectivity on theobject surface, An is the coefficient of Fourier series expansion rep-resenting the deformed fringe, �(x,y) is the phase change due to the

Fig. 2. Optical configuration of the fringe projection measurement.

514 L.-C. Chen, H.H. Hai / Precision Engineering 38 (2014) 512–524

f fund

f

G

wq

fiti

g

we

y

g

a

g

g

i

l

−utt

h

Fig. 3. Changes before and after removal o

By using FFT, the spectrum of g(x,y) can be transformed to therequency domain by 2-D Fourier transform as follows:

(�, �) =∫ +∞

−∞

∫ +∞

−∞g(x, y)e−2�i(�x+�y) dxdy

=+∞∑

n=−∞Qn(� − nfx, � − nfy) (2)

here G(�, �) and Qn(� − nfx, � − nfy) are the spectrum of g(x,y) andn(x,y), respectively.

Because the information to be determined is �(x,y), only therst-order component Q1(� − fx, � − fy) in the spectral domain haso be extracted for Inverse Fourier Transform (IFT). The result of IFTs given by:

(x, y) = F−1[Q1(� − fx, � − fy)] = q1e2�i(fxx+fyy)

= r(x, y)A1e2�i(fxx+fyy)+�(x,y) (3)

here A1 represents the first-order coefficient of Fourier seriesxpansion representing the first-order spectrum, Q1(� − fx, � − fy).

Furthermore, the deformed fringe for the reference plane (h(x,) = 0) can be described as:

0(x, y) = r0(x, y) ·+∞∑

n=−∞Ane2�in(fxx+fyy+�(x,y)) (4)

Similarly, the IFT result of the reference plane can be expresseds:

ˆ0(x, y) = r0(x, y)A1e2�i(fxx+fyy)+�0(x,y) (5)

Combining Eq. (5) with Eq. (7) yields the following equation:

ˆ0(x, y)g∗0(x, y) = r0(x, y)r(x, y)A2

1ei(�(x,y)−�0(x,y)) (6)

As a result, the phase change, ��(x,y), can be obtained from themaginary part of Eq. (7).

og[g(x, y)g∗0(x, y)] = log[r0(x, y)r(x, y)A2

1] + i��(x, y) (7)

In order to obtain the natural phase component ranging from� to �, which corresponds with the height distribution, a phasenwrapping algorithm is applied. Once the phase is unwrapped,he height distribution can be computed from the geometric rela-

ionship expressed as:

(x, y) = ��(x, y)l��(x, y) + 2�f0d

(8)

amental spectra: (a) before and (b) after.

2.2. Strategy for fundamental spectral removal

The background image of the object can be observed through aCCD camera and expressed as:

gb(x, y) = r(x, y) (9)

When Eq. (1) is divided by Eq. (9), the deformed image can befiltered by r(x,y) and becomes:

g(x, y) = gf (x, y)gb(x, y)

=∞∑

n=−∞An exp{i[2�nf0x + n�(x, y)]} (10)

The above reflectance–variance removal operation is to obtainthe continuity of the sinusoidal fringe patterns acquired from var-ious surface-reflected light and to avoid potential phase errorscaused by potential light reflectivity variances of the object surface.

By applying FFT, the spectrum of g(x,y) can be expressed inthe same way as in Eq. (2). To generate a smooth multi-spectrumfringe pattern from the detected deformed image, a normalizationprocess is performed to maximize the dynamic range for all sur-face regions having different surface-reflected light spectra. Thedetected modulation amplitudes of fringe images are normalizedto be a consistent gray-value distribution between 0 and (2n − 1),where n is the number of bits of the CCD. The normalization processfor RGB component can be described as follows:

Ir,n(x, y) = Ir(x, y) − Ir,min(x, y)Ir,max(x, y) − Ir,min(x, y)

· 255

Ig,n(x, y) = Ig(x, y) − Ig,min(x, y)Ig,max(x, y) − Ig,min(x, y)

· 255

Ib,n(x, y) = Ib(x, y) − Ib,min(x, y)Ib,max(x, y) − Ib,min(x, y)

· 255

(11)

where Ir,n(x,y), Ig,n(x,y), and Ib,n(x,y) is the normalized intensityof RGB component, respectively; Ir(x,y), Ig(x,y) and Ib(x,y) are thedetected light intensity of RGB component, respectively, selectedby judging the best Modulation Transfer Function (MTF) of theimage pixel of CCD; Ir,min(x,y), Ig,min(x,y) and Ib,min(x,y) are the mini-mal light intensity of RGB component, respectively; and, Ir,max(x,y),Ig,max(x,y)and Ib,max(x,y) are the maximum light intensity of RGBcomponent, respectively.

In the above process, the pixels with the value of Imaxm are

normalized to 255 and all the other pixels smaller than Imaxm are

normalized to within the range of 0–255, so that the normalized

image has a maximum dynamic range of 255 for achieving the bestdetection resolution. The spectrum distribution scheme can be gen-erally illustrated in Fig. 2. The strategy developed for removing thefundamental spectra is employed to separate the background light

L.-C. Chen, H.H. Hai / Precision Engineering 38 (2014) 512–524 515

ge; (b

dptsfitp

3

demimataibsbddsctobrt

D profiles, the proposed method employs an OEBF that best extractsthe first-order spectral information in band-pass filtering. Fig. 5shows the design of the proposed OEBF where f0A(x,y) and f0B(x,y),

Fig. 4. Fundamental spectral removal process: (a) deformed fringe ima

istribution. As seen in Fig. 3, image normalization is effectivelyerformed to remove the object background light prior to spec-rum filtering. By performing background removal, the undesiredpectral overlapping problem between the fundamental and therst-order spectral regions can be effectively minimized. Fig. 4 illus-rates an example for the developed fundamental spectral removelrocess employed to a multi color step-hight gauge block.

. Design of orthogonal elliptic band-pass filter

Band-pass filters for extracting the first-order spectra have beenescribed and investigated previously [6–18]. The most commonlymployed filter in FTP is the 2-D Hanning filter, which is approxi-ately represented by a circular filtering region. This type of filter

s simple and can be rapidly implemented in computer program-ing. However, in general cases, the first-order spectrum actually

ppears to be a lengthy shape along the X- and Y-axes. It is obvioushat the circular band-pass filter is no longer capable of extractingll first-order spectral regions. From our investigation, the major-ty of the spectrum distribution of a deformed fringe image grabbedy the fringe projection method can be illustrated in Fig. 3(a). Thehape and width of the first order spectrum is mainly influencedy the distribution of the spatial frequency of the deformed fringesetected by the imaging unit. The wider the distribution of theeformed fringe is the more disperse the width of the first orderpectrum becomes. However, by considering the possibly extremeases, such as a spherical surface and a sharp edge, the distribu-ion of the first order spectrum can be reasonably enclosed by an

rthogonal elliptic filter region. Moreover, the separation conditionetween the fundamental and the first (or higher order) spectrumegion is mainly influenced by the fringe density (or pitch), theilting angle of the projected fringe and the sensor geometry. The

) surface-reflected image; and (c) normalized deformed fringe image.

relationship of these parameters is mathematically described inSection 4.

Therefore, it is important to grab as much of the true spectraldata of the first order as possible because all the frequency compo-nents within the region are essentially required for accurate surfacerestoration while the redundant spectral data causing undesiredmeasurement errors should be avoided. To obtain more accurate 3-

Fig. 5. Orthogonal elliptic band-pass filter with background removal.

5 n Engineering 38 (2014) 512–524

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16 L.-C. Chen, H.H. Hai / Precisio

1A(x,y) and f1B(x,y), f2A(x,y) and f0B(x,y) are the highest and low-st points of the background spectrum, the first-order spectrumnd the second-order spectrum, respectively. f0c1(x,y) is the pointaving the same x component as f0R(x,y) and y component as fy1.

2c1(x,y) is the point having the same x component as f2L(x,y) and yomponent as f1y. f0c2(x,y) is the point having the same y components f0A(x,y) and x component as f1x. f2c2(x,y) is the point having theame y component as f2B(x,y) and x component as f1x. P1(f1x, f1y)nd P2(f21x, f2y) representing the spectral peaks of the first-ordernd second-order spectra, respectively, can be effectively identi-ed by locating the positions of the highest intensity within theorresponding spectral data regions.

In the y direction, the semi-major axis (fa1) and the semi-minorxis (fb1) of the data filtering range can be set as:

(x − f1x)2

f 2a1

+ (y − f1y)2

f 2b1

= 1 (12)

ith fa1 < ¯P1f0C1 and fb1 < ¯P1f2C1In the x direction, the semi-major axis (fa2) and the semi-minor

xis (fb2) of the data filtering range can be defined as:

(x − f1x)2

f 2a2

+ (y − f1y)2

f 2b2

= 1 (13)

ith fa2 < ¯P1f0C2 and fb2 < ¯P1f2C2To obtain accurate surface reconstruction, it is important to

etermine the four filtering parameters, fa1, fa2, fb1 and fb2, fornsuring accuracy of profilometry with both dimensional and form-estoration quality. In general, fa1 and fa2 can be set as the full widthf the spectrum since the first-order spectrum usually extends itsength over the entire spectral range.

. Analysis of physical reconstruction limitations of theroposed method

In the following section, the conditions for measuring surfaceradients with the OEBF are discussed. As derived in the previousection, the first spectral component Q1 of the Fourier series haso be determined accurately. To achieve this, the proposed OEBF isequired to separate the first-order spectral data from the higher-rder ones after the background (DC) component has been removedy the proposed background removal strategy.

In general, the instantaneous frequency in the nth-order spec-rum along the x- and y-axes can be calculated, respectively:

fnx = 12�

∂∂x

[2�n(fxx + fyy) + n�(x, y)

]= nfx + n

2�

∂�(x, y)∂x

fny = 12�

∂∂y

[2�n(fxx + fyy) + n�(x, y)

]= nfy + n

2�

∂�(x, y)∂y

(14)

In the spectrum domain, the boundary radius rn for theth-order spectrum (n = 1, 2, . . .,∞) can be shown in Fig. 6. Everypectrum circle has its center (Qn) and a radius (rn) for enclosing allhe essential frequencies required for accurate surface reconstruc-ion in the nth-order spectrum. From the above definition, rn cane expressed as follows:

n =(√

(fnx − nfx)2 + (fny − nfy)2

)max

(15)

By combining with Eq. (14), Eq. (15) can be further expressed:⎛√( )2 ( )2⎞

n = n

2�⎝ ∂�(x, y)

∂x+ ∂�(x, y)

∂y⎠

max

(16)

or n = 1, 2,. . ., ∞.

Fig. 6. Spectral domain obtained by Fourier transform, in which the background(DC), first-order, higher-order spectral components are included.

To separate the first-order spectrum Q1 with the backgrounddata effectively, a separation condition must hold:

fb max + r1 < ¯OQ1 (17)

In the above condition, the background frequency, fbmax, hasbeen minimized and is much smaller than the spatial frequency f0 ofthe projected fringe to avoid undesired spectrum collision betweenfbmax and f1min; and ¯OQ1 is the distance between the zero point andcenter of the first-order frequency.

Hence,

r1 < ¯OQ1 − fb max (18)

By combining Eq. (16) with Eq. (18), the condition can be furthermodeled as:

12�

⎛⎝√(∂�(x, y)

∂x

)2

+(

∂�(x, y)∂y

)2⎞⎠

max

< ¯OQ1 − fb max = f0 − fb max (19)

By assuming the distance from the objective lens center of theprojector to the reference plane to be l � h(x, y) and is the inclinedangle between the projection center and the imaging center, therelationship between the phase change and the object height canbe approximated as:

�(x, y) ≈ ��(x, y) = 2�h(x, y)fxd

l(20)

Therefore, Eq. (19) can be rewritten as:⎛√( ) ( ) ⎞

⎝ ∂h(x, y)∂x

2

+ ∂h(x, y)∂y

2⎠max

<l

d

(f0 − fb max)f0 cos

(21)

L.-C. Chen, H.H. Hai / Precision Eng

Fig. 7. Experimental setup for the developed methodology.

Fig. 8. Measurement results of color standard step-height target: (a) spectral dataobtained using 2D FFT; (b) OEBF; (c) 3-D reconstructed image; (d) cross-sectionalprofile along the horizontal axis; and (e) comparison between the cross sectionswhen the object lies along the horizontal direction and 45◦ directions with the X-axis, respectively. (For interpretation of the references to color in the text, the readeris referred to the web version of the article.)

ineering 38 (2014) 512–524 517

Similarly, for separating the first-order spectral data, Q1, fromthe higher-order spectral data effectively, the following conditioncan be further derived as:

r1 + rn < ¯Q1Qn =√

(nfx − fx)2 + (nfy − fy)2 = (n − 1)f0 (22)

for n = 2,3, . . ., ∞.Substituting rn from Eq. (16), the above equation is further

expressed as:

n + 12�

⎛⎝√(∂�(x, y)

∂x

)2

+(

∂�(x, y)∂y

)2⎞⎠

max

< (n − 1)f0 (23)

By combining Eq. (20), Eq. (23) can be rewritten as:

2�fxd

l

⎛⎝√(∂h(x, y)

∂x

)2

+(

∂h(x, y)∂y

)2⎞⎠

max

<2�(n − 1)

(n + 1)f0 (24)

for n = 2, 3, . . ., ∞.In Eq. (24) since (n−1)

(n+1) , when n = 2, has a minimum equal to 1/3,a stricter condition can be further rewritten as:

2�fxd

l

⎛⎝√(∂h(x, y)

∂x

)2

+(

∂h(x, y)∂y

)2⎞⎠

max

<2�

3f0 (25)

Since fx = f0 cos , the maximum measurable surface gradient canbe given by:⎛⎝√(

∂h(x, y)∂x

)2

+(

∂h(x, y)∂y

)2⎞⎠

max

<l

3d cos

when /= 90◦ (26)

By summarizing Eqs. (21) and (26), the restriction condition forsurface reconstruction with respect to the maximum measurablegradient can be defined as:⎛⎝√(

∂h(x, y)∂x

)2

+(

∂h(x, y)∂y

)2⎞⎠

max

< min(

l

d(f0 − fb max)

f0 cos ,

l

3d cos

)(27)

Therefore, Eq. (27) is employed as the restriction condition ofsurface profilometry by the proposed method. It is clear that thecondition defined in this article is less restricting than the maxi-mum slope l/3d, previously determined by Takeda and Mutoh [6],since the tilting angle provides an extra degree of freedom toincrease space for effective separation of spectral data. Moreover,the proposed background removal strategy has a unique advantagein minimizing any potential spectrum collision between DC and thefirst spectral zone. The analysis indicates that the condition definedhere is more general and can be employed to deal with a greatervariety of detected objects having different surface curvatures, col-ors and reflectivity. According to the curvature of the object surface,the tilting angle of the projected fringe can be adjusted to maximizethe measurable surface gradient. The larger the tilting angle, thehigher the surface curvature can be measured and reconstructed.

However, increase in tilting angle may decrease the detectable sen-sitivity of profile measurement since fx(= f0 cos ) is decreased. Themaximum range of measurement will also reduce fringe sensitivityand accuracy when employing a larger l/d.

518 L.-C. Chen, H.H. Hai / Precision Engineering 38 (2014) 512–524

F lters:

a

aiitt(tppftsffd

ig. 9. Measurement results of pre-calibrated spherical ball using five band-pass filong with their corresponding reconstructed 3-D profiles.

In addition, since the resolution of the imaging sensor forcquiring the deformed fringes plays a vital factor in determin-ng the system measurable resolution, it should also be consideredn the physical limits of surface reconstruction. Again, to obtainhe maximum accuracy of the fringe projection, the frequency ofhe fringe pattern should be higher than a minimum frequencyfproj = (1/2m+1)), where m refers to the number of bits employed byhe digital projector for quantizing the sinusoidal intensity of therojected structured light [19]. According to the Nyquist samplingrinciple, the maximum detectable frequency of the deformedringe pattern should be less than half of the sampling frequency ofhe imaging sensor, fccd (fccd = (1/2s)), where s refers to the pixel

ize of the imaging sensor employed to quantize the deformedringe. Assuming that the Nyquist sampling theorem is satisfied,ccd should be larger than (f1)max, so all the first-order spectra areetectable:

(a) circular, (b) rectangular, (c) hybrid, (d) diamond, (e) single elliptic, and (f) OEBF

fccd > (f1)max (28)

5. Experimental results and analyses

Fig. 7 shows the hardware setup of the experiments. As canbe seen, it comprises a digital micromirror device (DMD) with a1024 × 768 pixel resolution for generating color-encoded fringepatterns, a three-chip color Sony DXC-390 camera with a highspeed of up to 120 frames per second (fps), and a personal com-puter with a Dual Core Intel Pentium D, 3400 MHz with a RAM of2 GB SDRAM. The computer controls the projector and acquires thereflected light that contains deformed fringe images using a suit-

able frame grabber for obtaining the desired characteristics of thecolor pattern.

In the experiment, three pre-calibrated standard step heightsof different colors, namely 3.0 ± 0.001 mm (green), 1.0 ± 0.001 mm

n Engineering 38 (2014) 512–524 519

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scbathtw

(

(

wt

([wp

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T

wcfi

ws

fite

Table 1Measurement errors in diameter and sphericity by using various proposed band-pass filters in FTP.

No. Type ofband-passfilters

Size (pixel) Maximummeasurementerrors indiameter (%)

Sphericity informmeasurement(%)

1 Circular r = 30 7.595 0.0572 Rectangular L1 = 640

L2 = 254.575 0.046

3 Hybridbetween circleandrectangular

r = 30L1 = 25L2 = 640

4.412 0.045

4 Diamond a = 40b = 30

5.172 0.055

5 Single elliptical 2fa = 3002fb = 17

4.356 0.040

using the OEBP over the circular BP. Again, the OEBP also outper-forms the SEBP by two-fold sphericity improvement.

L.-C. Chen, H.H. Hai / Precisio

blue) and 4.0 ± 0.001 mm (red), were employed to evaluate theeasurement accuracy of the developed method. The spectral data

btained by 2-D FFT on the normalized deformed fringe imagere shown in Fig. 8(a). By using the developed OEBF, the first-rder spectral data (shown in Fig. 8(b)) can be extracted from thepectral regions without mixing with the background and otherigher-order spectral regions. The developed band-pass filter canxtract the first-order spectral information from both x- and y-irections of the spectrum. Meanwhile, it also adapts narrow-bandltering to avoid including the negative first-order spectral data

n resolving potential spectral leakage. The reconstructed 3-D mapnd the cross-sectional profile along the Y-axis of the measuredtep-height targets are shown in Fig. 8(c) and (d) and the com-arison between the cross sections along the horizontal and 45◦

irections Fig. 8(e), respectively. Results from a 30-time measuringepeatability test verified that the proposed method had a standardrror of 50.6 �m, indicating that the proposed method can controlell the maximum measured standard deviation within 1.25% of

he overall measurable height range. Again, the developed methods capable of measuring and reconstructing an object with arbitraryurface colors.

Furthermore, to verify the feasibility of the proposed OEBF,urface profilometry on a pre-calibrated CMM spherical ball wasonducted by employing various existing band-pass filters forenchmarking on measurement accuracy in terms of dimensionnd shape reconstruction. In this study, five other most commonypes of band-pass filters in FTP, namely the circular, rectangular,ybrid (circular and rectangular), diamond and single elliptic fil-ers, are selected to perform spectrum separation in comparisonith the proposed OEBF. These filters are defined as follows:

1) The circular band-pass filter is modeled as:

x − h)2 + (y − k)2 = r2 (29)

here (h, k) is the center of the first-order spectral peak, and r ishe radius.

2) The rectangular band-pass filter is defined as:

x2 −(

L1

2

)2]

·[

y2 −(

L2

2

)2]

= 0 (30)

here L1 is the minor axis and L2 is the major axis.The hybrid band-ass filter is defined as the combination of Eqs. (30) and (31).

(x − h)2 + (y − k)2 = r2}U{[

x2 −(

L1

2

)2]

·[

y2 −(

L2

2

)2]

= 0

}(31)

he diamond band-pass filter is defined as:

(x − h)a

+ (y − k)b

= 1 (32)

here a is the long axis and b is the short axis, and (h, k) is theenter of the first-order spectral peak.The single elliptic band-passlter is defined as:

(x − f1x)2

f 2a

+ (y − f1y)2

f 2b

= 1 (33)

here f1x, f1y denote the first-order spectral peaks, fa and fb are theemi-major axis and semi-minor axis, respectively.

Fig. 9 displays the measurement results obtained using theve different filters to compare their reconstruction accuracy inerms of dimensional and spherical errors. The actual sizes of themployed filters are shown in Table 1. To compare the sphericity,

6 Orthogonalelliptical

2fa1 = 2fa2 = 3002fb1 = 2fb2 = 17

1.250 0.018

Figs. 10–12 show the reconstructed 3-D maps and cross sec-tions obtained from the selected three different filters, namely thecircular band-pass, single elliptical band-pass (SEBP) and ortho-gonal elliptical band-pass (OEBP) filters, respectively. As seen inFig. 10, the circular band-pass filter cannot reconstruct the ref-erence spherical ball effectively, showing undesired significantfluctuations in profile generated in the 3-D-reconstructed surface.It is clear to see that the spherical surface measured by the circularband-pass filter has a clear distortion all around the reconstructedsurface and especially poor on the top neighboring surface region.Shown in Fig. 11, it is obvious to see that the SEBP reconstructedsphere is generally much better than the circular filter. However, itis also observable to note that the top neighboring surface regionis not as circular as it should be, in which certain spherical distor-tion exists in there. In comparison to the SEBP and the circular BP,the OEBP reconstructs a sphere with the highest sphericity, shownin Fig. 12. Three-fold sphericity improvement can be achieved by

Fig. 10. 3-D shape and cross section reconstructed by using the circular band-passfilter.

520 L.-C. Chen, H.H. Hai / Precision Engineering 38 (2014) 512–524

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Fig. 13. Measurement result of an arbitrary object colors: (a) original projection

ig. 11. 3-D shape and cross section by using the single elliptical band-pass (SEBP)lter.

It is worthy to note that the spherical surface reconstructedn the above measured example has a very wide range of surfaceurvatures. As seen in Fig. 12, the reconstructed shape is almostymmetric all around the different view angles. It should be notedhat the bottom part cannot be reconstructed completely since theumerical aperture of the imaging objective used is limited andhe part of the measured ball is occluded to the imaging unit. Theesult can indicate that the developed method is capable of recons-ructing the geometric shape of free-form surfaces. In addition, theesult can also indicate that the developed filter is not sensitive tohe object orientation or the fringe projection angle.

The maximum measurement errors in diameter detection were

btained by performing a 30-time repeatability test and evaluatedy least-squares sphericity evaluation. Table 1 shows the maxi-um measurement errors in diameter and sphericity of the ball

alibrated using various filters. As seen, the OEBF can obtain the

ig. 12. 3-D shape and cross section by using the orthogonal elliptical band-passOEBP) filter.

image; (b) deformed fringe image; (c) deformed fringe image after background lightremoval; and, (d) the reconstructed 3-D map.

best measurement result in both ball diameter and sphericity.Improvement in measurement accuracy of more than 6-fold canbe achieved when using the OEBF instead of the circular filter. Ingeneral, the proposed OEBF outperforms the traditional circularfilter with reduction in measurement errors of 2–6 times. No signif-icant differences among the rectangular, hybrid and diamond filterswere observed, and their measurement errors ranged between 4%and 5%. The single elliptic band-pass filter performs better than theother existing filters but much poor than the OEBF.

In order to verify the capability of the proposed FTP method, afree-form toy model with several surface colors shown in Fig. 13(a)was measured using the OEBF shown in Fig. 13(b). With digitalfringe projection, its deformed fringe image was rapidly acquiredby the developed setup and further processed by the backgroundimage removal algorithm for generating the filtered fringe image,shown in Fig. 13(c). By performing the proposed OEBF methodol-ogy, the free-form surface of the measured mask can be successfullyreconstructed, as seen in the reconstructed 3-D map shown inFig. 13(d). Free-form model surface reconstruction was achievedby high-speed 3-D imaging with a frame rate of up to 60 using ahigh-speed CCD with 120 fps.

As indicated and confirmed by the previous studies [13,17],the traditional circular bandpass filter does not generally extractsufficient spectrum information required to reconstruct sharp geo-metric features. Thus, its reconstructed geometry generally tendsto lose sharp corners or edges from the original measured surface.The reconstruction of the corners in a step height target, such as a

gauge block, generally tends to lose the sharp edges and result ina round corner. In contrast to this, the developed OEBF is designedwith an attempt to include all the major spectrum regions required

L.-C. Chen, H.H. Hai / Precision Engineering 38 (2014) 512–524 521

Fig. 14. Spectrum when the object located at 0◦ before and after background image removal: (a) object located at 0◦; (b) spectrum of (a); (c) image after background imageremoval; and (d) spectrum of (c).

Fig. 15. Spectrum when the object located at 45◦ before and after background image removal: (a) object located at 45◦; (b) spectrum of (a); (c) image after background imageremoval; and (d) spectrum of (c).

522 L.-C. Chen, H.H. Hai / Precision Engineering 38 (2014) 512–524

Fig. 16. Spectrum when the object located at 135◦ before and after background image removal: (a) object located at 135◦; (b) spectrum of (a); (c) image after backgroundimage removal; and (d) spectrum of (c).

Fig. 17. 3-D measurement results of color standard step-height target by the OEBF: (a) the front-45 view angle; (b) the side-45 view angle; and (c) the 45 oblique angle.

L.-C. Chen, H.H. Hai / Precision Engineering 38 (2014) 512–524 523

F lar filter: (a) the front-45 view angle; (b) the side-45 view angle; and (c) the 45 obliquea

ts

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ig. 18. 3-D measurement results of color standard step-height target by the circungle.

o reconstruct objects having an arbitrary shape, such as free formurfaces or sharp geometric edges.

As can be seen in Figs. 14–16 , the 3-D spectra of the measuredtep height target orientated at three difference angles: (a) 90◦,b) 45◦ and (c) 135◦ are obtained before and after the backgroundmage removal, respectively, while the structured light pattern isrojected in the same orientation. As can be seen on the shapend size of the spectrum data in these three cases, it can be foundhat the major spectrum shape is of an orthogonal elliptical spec-rum region and its bandwidth size of each elliptical region cane measured as the same size of 300 × 17 pixels in longitudinalnd lateral directions. From this observation, it is clear to see thathe major spectrum region, in terms of shape and size, is main-ained very much consistent in these three cases when the objects rotated from 0◦ to 45◦ and then 135◦. The background (object)pectrum can be seen in the middle of the spectrum and it cane effectively minimized by the proposed strategy for fundamentalpectral removal in the article. Since the orthogonal elliptical band-ass (OEBP) filter is established based on the actual shape and sizef the spectrum data, it do have a better capability than any otherested filters in reconstructing the object phase information for 3-Dhape measurement. This could explain that the step-height tar-et’s reconstruction using the OEBP filter will not affected by theotation of the object.

To test the capability of the developed method in measuringbjects having sharp geometric edges, the same standard stepeight used in Fig. 4 was measured by the circular filter and OEBF

or comparison. As seen in Fig. 17 from three different view anglesthree different view angles: X-, Y- and 45 axes), it is clear toee the sharp corners of the gauge blocks are well reconstructedy the OEBF. In contrast to this, as seen in Figs. 18 and 19, the

Fig. 19. Comparison of cross-section profiles of color standard step-height targetby using the OEBF and circular filter: (a) the horizontal direction and (b) the 45 viewangle.

524 L.-C. Chen, H.H. Hai / Precision Eng

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[profilometry. Opt Eng 1999;38(6):1029–34.

ig. 20. Reconstructed cross-sectional profiles obtained using the developed FTPethod and traditional three-step PSP method.

econstructed results by the circular bandpass filter from three dif-erent view angles have lost the sharp corners and. This clearlyndicates that the advantage of the developed method.

In addition, the same standard step heights used in Fig. 4 werelso measured using the traditional three-step PSP method to ver-fy its measurement accuracy against the proposed approach. Theross-sectional profiles along the Y-axis of the standard step-heightarget shown in Fig. 20 were reconstructed by both methods. As cane seen, the two profiles resemble closely each other but the PSPethod obtains a relatively smoother surface than the one gener-

ted by the developed FTP method. More importantly, results of a0-time repeatability test verified that the proposed method had atandard error of 50.6 �m while that achieved by the three-step PSPethod was 20.6 �m, indicating that the three-step PSP method is

till superior to the developed approach in terms of measurementepeatability. However, using the proposed FTP method, the max-mum measured standard deviation can be controlled well within.25% of the overall measuring height while the number of imagerames required is reduced from 3 to 2. Moreover, it performedell in terms of measurement accuracy of 3-D profilometry when

ompared with the existing FTP methods.

. Conclusions

In this research, band-pass filtering of FTP was studied for bet-er understanding of its design and effectiveness in one-shot FTP-D imaging. To achieve accurate surface profile reconstructionith surface colors, an innovative OEBF was proposed to extract

he first-order spectral data regions for more accurate 3-D profileeconstruction. Experimental results that the measurement accu-acy in terms of dimension and sphericity can be greatly enhancedy several folds when using the proposed OEBF instead of the otherxisting band-pass filters. With the developed method, the maxi-

um measured dimensional error can be controlled within 1.25% of

he overall measuring height. The accuracy of form reconstruction,n terms of ball sphericity, can also be improved consider-bly by using the newly proposed filter. The developed OEBF

[

ineering 38 (2014) 512–524

outperforms the other filters with a 6-fold improvement indimensional measurement accuracy and a 3-fold enhancement insphericity. Nevertheless, the three-step PSP method can still per-form better than the proposed approach in terms of measuringrepeatability and accuracy. Hence, how the PSP method achieves ahigher level of measurement accuracy merits further investigation.

Acknowledgments

The authors would like to thank the National Science CouncilTaiwan, for financially supporting this research under Grant, NSC100-2221-E-002-250-MY3.

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