Improvement of fringe quality for

phase extraction in double digital

fringe projection

Ubaldo Uribe-López
David Asael Gutiérrez-Hernández
Francisco Javier Casillas-Rodríguez
Miguel Mora-Gonzalez
Jesús Muñoz-Maciel

Ubaldo Uribe-López, David Asael Gutiérrez-Hernández, Francisco Javier Casillas-Rodríguez, Miguel Mora-
Gonzalez, Jesús Muñoz-Maciel, “Improvement of fringe quality for phase extraction in double digital fringe
projection,” Opt. Eng. 58(9), 092605 (2019), doi: 10.1117/1.OE.58.9.092605.

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 Optical Engineering 58(9), 092605 (September 2019)

Improvement of fringe quality for phase extraction in

double digital fringe projection
Ubaldo Uribe-López,a David Asael Gutiérrez-Hernández,b Francisco Javier Casillas-Rodríguez,a,* Miguel Mora-Gonzalez,a
and Jesús Muñoz-Maciela
a
Universidad de Guadalajara, Centro Universitario De Los Lagos, Lagos de Moreno, México
b
Tecnológico Nacional de México, Instituto Tecnológico de León, División de Estudios de Posgrado e Investigación, León, México

Abstract. A procedure to improve the quality of the extracted phase by a passband filter within the spatial
frequency domain is proposed, using the double-digital fringe projection method for obtaining the contour of
the surface of different objects. This method requires the digital projection of two fringe patterns to the
same object to generate an interference pattern containing moiré fringes, which are related to the shape of
the measured object. The proposed method pretends to remove remnant frequencies of the projected fringes
keeping only the moiré pattern. © 2019 Society of Photo-Optical Instrumentation Engineers (SPIE) [DOI: 10.1117/1.OE.58.9.092605]
Keywords: fringe projection; optical phase; frequency filter.
Paper 181709SS received Dec. 1, 2018; accepted for publication Mar. 27, 2019; published online Apr. 15, 2019.

1 Introduction an object to measure, generating an interference pattern cre-

The measurement of the three-dimensional (3-D) shape has ating fringes with a different frequency, which provides
been of great importance in recent years, both in the indus- information of optical phase to obtain the surface shape
trial and scientific fields. Thus, different techniques have of the object.
been developed, including noninvasive ones. These tech- In Fig. 1, the optical configuration used for this work is
niques allow one to obtain surface shape data of objects presented. The two fringe patterns are designed by com-
without any contact that could deform it, increasing the puter and projected over the object overlapping, forming
error of the measurement. a set of fringes with their own frequency, called moiré pat-
Optical noninvasive techniques make it possible to obtain tern, which provides the surface profile of the measured
intensity patterns that later allow us to recover the phase with object.7 A camera is used to capture an image of the
a reduced number of images and thus, obtain the information deformed fringe. Then, the computer processes the resulted
of the measured object. Instead of the high precision, interference fringe pattern to obtain the wrapped and
processing speed, and the simplicity of the optical arrange- unwrapped phase.
ment, these techniques can be susceptible to errors caused by When these fringe patterns are projected simultaneously,
distortions induced by the angle between the camera and the we have the two single-frequency fringes and a third fringe
projector or lighting source, either in phase shifting,1 fringe pattern generated by the superposing of the first two pro-
projection,2,3 or digital holographic,4 being the most jected patterns. So, the resulted interference fringe pattern
common techniques for these purposes. can be expressed as follows:5,8
In this work, we propose an improvement of fringes qual-
ity by means of a bandpass frequency filter, using the double-
digital fringe projection method,5,6 through two digital pro- Iðx; yÞ ¼ aðx; yÞ þ b1 ðx; yÞ cos½ϕ1 ðx; yÞ
EQ-TARGET;temp:intralink-;e001;326;313

jectors and the application of isotropic quadrature transform

(IQT)6 for phase retrieval with a spatial unwrapping algo- þ b2 ðx; yÞ cos½ϕ1 ðx; yÞ þ b3 ðx; yÞ cos½ϕ1 ðx; yÞ;
rithm in objects with different shapes. (1)
In Sec. 2 of this work, the theoretical description of the
proposed technique is presented. It is also described the
experimental setup to be used for the work to be done. To where aðx; yÞ is the average background intensity,
record the intensity patterns, we used a Garmin VIRB XE bn¼1;2;3 ðx; yÞ is the intensity modulation, and ϕn¼1;2;3 ðx; yÞ
camera, with a CMOS sensor, size of 1/2.3″ and a denotes the phases that can be rewritten in function
resolution of 12 MP (3000 × 4000). of carried phases φα;β;γ and initial phases φ1;2;3 as
Section 3 shows the results obtained by applying the pro- follows:
posed technique in some objects, each with different shapes
to measure.
ϕ1 ðx; yÞ ¼ φα ðx; yÞ þ φ1 ðx; yÞ;
EQ-TARGET;temp:intralink-;e002;326;194

2 Experimental Setup and Theoretical Description

This work proposes the overlap of two vertical fringe ϕ2 ðx; yÞ ¼ φβ ðx; yÞ þ φ2 ðx; yÞ;
patterns with the same frequency digitally projected, on ϕ3 ðx; yÞ ¼ φγ ðx; yÞ þ φ3 ðx; yÞ: (2)

*Address all correspondence to Francisco Javier Casillas-Rodríguez, E-mail:

fjcr@culagos.udg.mx 0091-3286/2019/$28.00 © 2019 SPIE

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 Uribe-López et al.: Improvement of fringe quality for phase extraction. . .

1
c ðx; yÞ ¼ bðx; yÞe−iϕðx;yÞ : (6)
2
EQ-TARGET;temp:intralink-;e006;326;741

The intensity Iðx; yÞ of Eq. (4) becomes

Iðx; yÞ ¼ aðx; yÞ þ cðx; yÞ þ c ðx; yÞ:

EQ-TARGET;temp:intralink-;e007;326;710 (7)

Then, the discrete Fourier transform for Iðx; yÞ is

performed:

FfIðx; yÞg ¼ Iðu; vÞ ¼ Aðu; vÞ þ Cðu; vÞ þ C ðu; vÞ:

EQ-TARGET;temp:intralink-;e008;326;657 (8)

Applying the bandpass filter, terms Aðu; vÞ and C ðu; vÞ

are removed, keeping only Cðx; yÞ. Finally, inverse Fourier
transform of the filtered intensity is performed:

F−1 fI FILTERED ðu; vÞg ¼ I FILTERED ðx; yÞ ¼ cðx; yÞ:

EQ-TARGET;temp:intralink-;e009;326;593 (9)

Figure 2 shows the superposition of two simulated fringe

patterns, which generates an interferometric pattern, contains
a moiré fringe pattern, and allows us to observe how we can
get only the moiré fringes by applying the frequency band-
pass filter from the original intensity, removing frequencies
Fig. 1 Optical setup applied for the double digital fringe projection
of the original pattern that overlap.
technique. Once the moiré fringes are filtered, we use phase demodu-
lation from a single interference fringe pattern a function
based on the IQT.10,11 Given the normalized version of a
Then, Eq. (1) can be rewritten in function of Eq. (2): fringe pattern, the corresponding quadrature term can be
obtained as follows:
Iðx; yÞ ¼ aðx; yÞ þ b1 ðx; yÞ cos½φα ðx; yÞ þ φ1 ðx; yÞ
EQ-TARGET;temp:intralink-;e003;63;453

QfI NORMALIZED ðx; yÞg ¼ − sin½ϕðx; yÞ; (10)

þ b2 ðx; yÞ cos½φβ ðx; yÞ þ φ2 ðx; yÞ
EQ-TARGET;temp:intralink-;e010;326;440

þ b3 ðx; yÞ cos½φγ ðx; yÞ þ φ3 ðx; yÞ: where fg is the isotropic quadrature operator. The wrapped
phase of the modulation phase can be obtained from Eq. (4)
However, the digital camera used in the experimental set- as follows:
up has only the possibility of registering one fringe pattern of  
QfI NORMALIZED ðx; yÞg
interference, coming precisely from the superposition of the Wfϕðx; yÞg ¼ arctan − : (11)
I NORMALIZED ðx; yÞ
EQ-TARGET;temp:intralink-;e011;326;377

fringe patterns digitally projected over the object. Then, the

final fringe pattern of interference can be written as follows:
And finally, the wrapped phase Wfϕðx; yÞg is then
EQ-TARGET;temp:intralink-;e004;63;331 Iðx; yÞ ¼ Aðx; yÞ þ Bðx; yÞ cos½ϕðx; yÞ; (4) unwrapped by means of multigrid techniques.12

where Aðx; yÞ is the average intensity, Bðx; yÞ is the ampli- 3 Experiments and Results
tude modulation of fringes, and ϕðx; yÞ is the phase to be According to the procedure described in Sec. 2, Fig. 3 shows
solved for, also known as the wrapped phase. Considering the images of each step to be carried out. First, the interfer-
that ϕðx; yÞ ¼ 2πf 0 x þ φðx; yÞ, where f 0 is the spatial car- ence pattern generated by the two projected fringes captured
rier frequency and φðx; yÞ is the phase modulation of fringes by the camera (a) is obtained. This is filtered by a bandpass
and is related to the shape of the measured object, Eq. (4) can filter in the frequency domain, obtaining the desired moiré
be rewritten as follows: fringes (e). Subsequently, the IQT method is used to obtain
the wrapped phase (g) and finally, the unwrapped phase (h).
EQ-TARGET;temp:intralink-;e005;63;223 Iðx; yÞ ¼ Aðx; yÞ þ Bðx; yÞ cos½2πf 0 x þ φðx; yÞ: (5) In Fig. 4, we can see the surface contour of a foam made
face, measured by the double-digital fringe projection. This
The resulting interference pattern contains the informa- object has some reliefs that are not so easy to characterize
tion about the shape of the measured object but also contains because of the shadows generated by the projection.
unwanted remnants from the carrier projected fringes, as However, this technique provides us important information
shown in Eq. (5). These remnants are removed via the cutoff even in these difficult parts to reach due to the double pro-
frequencies of a bandpass filter in the spatial frequency jection, which handles both sides of the object.
domain.9 To do this, we introduce a pair of complex Other examples of this technique used on other objects
exponential equations cðx; yÞ and its conjugate c  ðx; yÞ with different shapes are shown in Figs. 5 and 6. Figure 5
as follows: provides us the results from a reflective object and even
in this type of surface, managing some parameters like
1 the contrast of the fringes and the brightness of the projec-
cðx; yÞ ¼ bðx; yÞeiϕðx;yÞ ;
2
EQ-TARGET;temp:intralink-;sec2;63;104

tors, we can obtain good results in the shape.

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 Uribe-López et al.: Improvement of fringe quality for phase extraction. . .

Fig. 2 Result of the application of the frequency bandpass filter in the superposition of two simulated
fringe patterns. (a) and (b) Simulated fringe patterns, (c) superposition of the two fringe patterns, and
(d) moiré pattern resulting from the filter application.

Fig. 3 Difference of results in a spherical object, between process without the Fourier filter: (a) original
intensity pattern, (b) Fourier transform of the intensity pattern, (c) wrapped phase, and (d) unwrapped
phase; and process with the Fourier filter: (e) intensity pattern filtered, (f) Fourier transform with bandpass
filter, (g) wrapped phase, and (h) unwrapped phase.

Figure 6 shows another object with different relief scales. decrease in quality of moiré pattern when the spatial fre-
The frequency of the projected fringes generates, after the quency of projected fringes is too low [Fig. 7(a)], making
filter, a moiré pattern (e) with an even lower frequency, the carrier frequency not to disappear completely. On the
which provides the shape of the object roughly, making other hand, the more the carrier spatial frequency increases,
known the importance of increasing the frequency of projec- the greater the number of moiré fringes, and therefore,
tion to obtain a greater definition to the details. increases the information that can be obtained from the
To obtain a quality comparison in the results, in the moiré object. The quality is also subject to the resolution of both
pattern as in the optical phase, fringe patterns with different camera and projector.
spatial frequencies (number of fringes per specific area) were Figure 7 also compares results in wrapped and unwrapped
projected on the same object, in this case a half sphere, mak- phases when the intensity patterns are processed by the
ing different intensity patterns, which were processed and Gaussian filter, removing the carrier frequency (Fig. 7, 3.2
filtered by means of the proposed method. Figure 7 and 4.2), against those that does not (Figs. 7(a)–7(d) 3.1
shows us these differences and we realize that there is a and 4.1).

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Fig. 4 Difference of results in a foam face between process without the Fourier filter: (a) Original intensity
pattern, (b) Fourier transform of the intensity pattern, (c) wrapped phase, and (d) unwrapped phase; and
process with the Fourier filter: (e) intensity pattern filtered, (f) Fourier transform with bandpass filter,
(g) wrapped phase, and (h) unwrapped phase.

Fig. 5 Difference of results in an aluminum can, between process without the Fourier filter: (a) original
intensity pattern, (b) Fourier transform of the intensity pattern, (c) wrapped phase, and (d) unwrapped
phase; and process with the Fourier filter: (e) intensity pattern filtered, (f) Fourier transform with bandpass
filter, (g) wrapped phase, and (h) unwrapped phase.

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Fig. 6 Difference of results in a toy tower, between process without the Fourier filter: (a) original intensity
pattern, (b) Fourier transform of the intensity pattern, (c) wrapped phase, and (d) unwrapped phase; and
process with the Fourier filter: (e) intensity pattern filtered, (f) Fourier transform with bandpass filter,
(g) wrapped phase, and (h) unwrapped phase.

Fig. 7 Comparison with the results of the optical phases of a half sphere subjected to the double fringe
projection, modifying its spatial frequency for each intensity pattern captured. For columns, we
have the next frequencies: (a) 0.12 cycles∕mm, (b) 0.1543 cycles∕mm, (c) 0.2229 cycles∕mm,
(d) 0.2572 cycles∕mm, and (e) 0.3087 cycles∕mm. For rows: (1) double fringe projected patterns,
(2) Gaussian filtered patterns, (3.1) wrapped phase of the original pattern, (3.2) wrapped phase of
the filtered pattern, (4.1) unwrapped phase of the original pattern, and (4.2) unwrapped phase of the
filtered pattern.

Finally, a profile 1-D vector was extracted from 4 Conclusions

unwrapped phases, one for each intensity pattern, and was In this work, we have presented a quality improvement to
compared in a single plot, which can be seen in Fig. 8. obtain the profile surface of objects with different shapes
Here, we can observe a relationship between quality and car- by using a bandpass filter in the frequency domain. This
rier spatial frequency. allows one to get the moiré pattern that results from the

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 Uribe-López et al.: Improvement of fringe quality for phase extraction. . .

Fig. 8 Profile vectors of the unwrapped phases of a half sphere with different spatial frequencies.

superposition of the projected fringes. The proposed method contour and displacement measurements,” Opt. Eng. 55(12), 121719
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a good lighting contrast, even the details in the shape of the membrane by double digital fringe projection,” J. Optoelectron. Adv.
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well as the modification in the object itself. The proposed
method also allows us to evaluate any nonrepeatable tran- Ubaldo Uribe-López is a PhD student and a professor at the
sient events and characterize and extract the mechanics of CULagos from the University of Guadalajara, Lagos de Moreno,
this event with certainty that gives us a view very close to México. He received his master’s degree in optomechatronics from
reality. Optical Research Center (CIO), León, México, and his BSc degree
from the University of La Salle Bajío, León, México. His research inter-
ests include electronic speckle pattern interferometry, digital image
Acknowledgments processing, and optical metrology.

The authors express our gratitude to the Mexican National David Asael Gutiérrez-Hernández received his MSc degree in
Council for Science and Technology (CONACYT), to optics in 2006 from Optical Research Center (CIO), Mexico, and
University Center of Los Lagos (University of Guadalajara), his PhD degree in physics in 2016 from Universidad Autónoma de
Sinaloa (México). He is interested in noninvasive optical techniques
and to the Technological Institute of León for the support for industrial and biomedical applications. He is a senior lecturer at
offered to this project. Instituto Tecnológico de León and is a member of the National
System of Researchers (SNI).

References Francisco Javier Casillas-Rodriguez obtained his BSc degree

in electronics at Aguascalientes Institute of Technology in
1. T. Tahara et al., “Dual-wavelength phase-shifting digital holography
selectively extracting wavelength information from wavelength- Aguascalientes, Mexico, in 1996, and his PhD from the Optical
multiplexed holograms,” Opt. Lett. 40(12), 2810 (2015). Research Center (CIO), Mexico, in 2004. Since 2005, he has been
2. K. Genovese, L. Lamberti, and C. Pappalettere, “Mechanical characteri- with the Universidad de Guadalajara in Lagos de Moreno, Jalisco,
zation of hyper elastic materials with fringe projection and optimization Mexico. His research interests include electronic speckle pattern
techniques,” Opt. Lasers Eng. 44(5), 423–442 (2006). interferometry, fringe analysis, and digital image processing for stress
3. A. L. González-Gómez, J. Meneses-Fonseca, and J. León-Téllez, and vibration analysis.
“Proyección de franjas en metrología óptica facial,” INGE CUC
8(1), 191 (2012). Miguel Mora-Gonzalez is a titular research-professor at the
4. U. Uribe-López, M. Hernández-Montes, and F. Mendoza-Santoyo,
“Fully automated digital holographic interferometer for 360 deg CULagos from the University of Guadalajara, Lagos de Moreno,

Optical Engineering 092605-6 September 2019 • Vol. 58(9)

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 Uribe-López et al.: Improvement of fringe quality for phase extraction. . .

Mexico. He received his BSc degree in electronics from the Jesús Muñoz-Maciel received his electronic engineering title from
Aguascalientes Institute of Technology in Aguascalientes, Mexico, the University of Guadalajara in Mexico in 1995. Then, he joined
in 1996; and his PhD in optics from the University of Guanajuato the Optical Research Center (CIO) in Mexico obtaining his PhD in
(CIO) in Leon, Mexico, in 2003. His current research interests include optics in 2003. Since 2004, he is working at the University of
optical testing, applied optics, optical metrology, liquid crystal dis- Guadalajara in Lagos de Moreno, Mexico, as a research professor.
plays, interferometry, digital image processing, and pattern recogni- His research interest areas include optical metrology, fringe analysis,
tion. He is also a member of SPIE. image processing, and optical tests.

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