Reliable Shadow Area Recovery with Two-Spatial-Frequency

Fringe Projections from Two Projectors

Yang Yang,

Wei Liu,

Yi Ding,

Yate Jiang,

Kai Peng

School of Electronic, Information and Communications, Huazhong University of Science and Technology, 1037 Luoyu

Road, Wuhan, Hubei, China

Wuyi University, Jiangmen, Guangdong, China

Keywords: Fringe projection profilometry, Three-dimensional reconstruction, Fringe analysis, Phase measurement.

Abstract: Shadow always appears on the captured fringe images in traditional fringe projection profilemtry system. To

recover the shadow area, in existing methods, fringe patterns from multiple projectors are employed to calculate

the absolute phase maps of entire object surface. However, if the fringe order errors in the absolute phase maps

were not corrected, significant errors would appear on the reconstructed three-dimensional result. In this paper

we propose a method to recover the entire surface profile with the fringe projections from two projectors. In the

proposed method, the shadow areas are identified, a novel technique is developed to correct the fringe order

errors in the absolute phase maps of the fringe patterns from two projectors, thus the reliability of shadow area

recovery could be improved. This method also can be used for various applications in industries and academics.

The methodology is exemplified with experimental data.

1. INTRODUCTION

Fringe projection profilometry (FPP) is considered

as a promising noncontact measuring method to

obtain three-dimensional profile data in industries

for product surface inspection and visual navigation

for decades (R. Leach,2011)(J. Geng,2011). In FPP, a

set of sinusoidal fringe patterns from micro-mirror

device is generated and projected onto the surface

of measured object and the light signals are captured

by a CCD camera. Due to the nature of structured

light illumination, shadow occurs in the capture

image when the projected illumination signal is

blocked by the shape of measured object. As the

shadow areas of captured image hardly reflect the

modulated light signals associated with object

surface, the captured signals in shadow areas cannot

be directly employed for three-dimensional

reconstruction. To eliminate the effect of shadow

areas in the captured fringe patterns, Lu et al. (L. Lu

et al,2015) developed a method to determine the

shadow areas by mapping the reconstruction depth

result to digital micro-mirror-device plane in pixel-

by-pixel manner.

The shadow areas in captured fringe image are

mainly resulted from the geometry of traditional

fringe projection profilometry system, if the fringe

patterns from different illumination angles could

cover the entire surface and their reflections are

captured by camera, it is possible to recover the

entire profile data. Skydan et al. (O. A. Skydan,2005)

employed the fringe images from multiple angles to

obtain the entire profile 3D reconstruction result.

Cai et al. (Y. Cai et al,2007)

proposed the inverse projected fringe technique

based on multiple projectors to recover the shadow

area more efficiently. Some methods have been

developed to improve the accuracy and robustness

of entire surface acquisition from multiple angles

for entire shape acquisition (W. H. Su,2008)(X.

Liu,2012). However, in these methods, fringe

patterns of only one spatial frequency is used, errors

may occur to the absolute phase maps due to the

complex object surface and shadow areas. Servin et

al. (M. Servin et al,2013) proposed a co-phased

profilometry to convert the phase demodulation of

multiple camera-projector systems into a single

process. In this method, the wrapped phase

extraction can be considered as an extension of the

standard Fourier Transform, thus the reconstructed

result is sensitive to the intensities of background

illumination and projected light signal. The points

of shadow areas generated by each projector may

affect the reconstruction results of the areas without

shadow. Furthermore, the sensitivities of the fringe

patterns from multiple projectors should be the

same to keep the accuracy of measurement, which

is difficult to satisfy in many applications.

Yang, Y., Liu, W., Ding, Y., Jiang, Y. and Peng, K.

Reliable Shadow Area Recovery with Two-Spatial-Frequency Fringe Projections from Two Projectors.

DOI: 10.5220/0008096702430250

In Proceedings of the International Conference on Advances in Computer Technology, Information Science and Communications (CTISC 2019), pages 243-250

ISBN: 978-989-758-357-5

243

This paper proposes a method to improve the

reliability of the 3D profile recovery in shadow

areas with the two projectors in two angles. In

particular, this paper focuses on the case that, for

each projection, the fringe patterns generated by

projector can be reflected by the object surface and

are acquired by the camera. Only parts of the surface

are shadow areas, which are not illuminated by the

fringe pattern of projector.

The proposed method improves the reliability of

the shadow area recovery as follows. The points in

the shadow areas generated by each projector are

identified, the corresponding wrapped phase values

are labeled as invalid. For the points in the areas

illuminated by each projector, the corresponding

wrapped phase maps and absolute phase maps are

calculated with phase-shifting algorithm and two-

spatial-frequency temporal phase unwrapping,

which avoids the influence of the points in shadow

areas. For the possible fringe order errors in the

absolute phase maps around the shadow areas, we

develop a novel fringe order correction technique to

correct these errors. With the proposed method, the

profile of shadow areas generated by each projector

can be recovered. Finally, the complete 3D profile

could be obtained with high reliability.

The rest parts of this paper are organized as

follows. In Section 2 we present the identification

of points in shadow areas. In Section 3, reliable

absolute phase recovery for the fringe patterns

generated by each projector is developed. In Section

4, the method to recover the complete 3D profile is

developed. In Section 5, the experiments are

presented to confirm the method. Section 6

concludes the whole paper.

2. PRINCIPLE OF FPP AND

SHADOW AREA

IDENTIFICATION

2.1 Principle of FPP

Fig.1 shows the system structure diagram of

traditional fringe projection profilometry, which

includes one projector and one camera. Fringe

patterns are generated and projected onto the object

surface and reference plane. Camera acquires these

fringe pattern images for surface profile recovery.

For the area illuminated by projector, the

information of surface profile is modulated into the

deformed fringe pattern, thus it is possible to

reconstruct the surface of object with the captured

fringe images. In existing methods, wrapped phase

extraction should be implemented to obtain the

information of fringe pattern for height distribution

reconstruction. Among various wrapped phase

extraction and phase unwrapping methods, phase

shifting algorithm are the most widely used for the

reliability and the avoidance of the influences from

adjacent pixels, which can eliminate the effects

from the points in shadow area.

Figure 1: System structure diagram of traditional

FPP.

For the N-step phase shifting algorithm, the

fringe patterns captured by camera could be

expressed as

2 ( 1)

( , ) ( , ) ( , )cos(2 ( , ) )

d x y A x y B x y f x x y







   

(1)

where

1,2,3,...,nN

;

( , ) 2x y f x





is the

designed absolute phase map in the projector,

the spatial frequency of the fringe pattern;

( , )A x y

denotes the average intensity,

( , )B x y

denotes the

intensity modulation of the sinusoidal fringe pattern,

( , )xy



is the wrapped phase. When

3N 

( , )A x y

( , )B x y

and

( , )xy



can be calculated as,

( , ) ( , )

A x y d x y





(2)

2 2 2

( , ) [ ( , )sin( )] [ ( , )cos( )]

B x y d x y d x y

N N N









(3)

( , )sin(2 / )

( , ) arctan

( , )cos(2 / )

d x y n N











(4)

Define

( , )

B x y

A x y





(5)

where

( , )xy



denotes the data modulation, it is

always used as the index to evaluate the quality of

the fringe pattern (S. Zhang,2006). Higher

( , )xy



means the better fringe quality.

To recover the surface profile of object, phase

unwrapping should be implemented to obtain the

CTISC 2019 - International Conference on Advances in Computer Technology, Information Science and Communications

244

absolute phase from wrapped ones calculated by

phase-shifting algorithm in Eq.(4) . The relation

between absolute phase and wrapped phase is

expressed as

( , ) 2 ( , ) ( , )x y m x y x y



  

(6)

where

( , )m x y

denotes the fringe order, which is in

integer,

( , )xy

is the absolute phase. When the

absolute phase maps of the fringe patterns on object

( , )

xy

and on reference plane

( , )

xy

are

obtained, the difference of the absolute phase maps

( , )xy

, where

( , ) ( , ) ( , )

x y x y x y    

is used to calculate the height distribution of the

surface profile of object as:

( , )

x y l

h x y







(7)

where

and

are the parameters calibrated

before the measurements.

2.2 Shadow Area Identification

Due to the geometry in traditional fringe projection

profilometry in Fig.1, shadow always occurs in the

captured images. The shadow areas in the captured

images do not reflect the modulated light signal of

the projected fringe patterns. When phase shifting

algorithm or other wrapped phase extraction

algorithm is directly used to the pixels in shadow,

errors will occur in the corresponding wrapped

phase value. These incorrect wrapped phase values

will cause the errors in the reconstruction of the

object. We use Fig.2 to illustrate the reason causing

the shadow in traditional FPP system.

Figure 2: Shadow area formation and absolute.

phase of illuminated area in FPP system.

The field of view of the camera and projector is

12c

E C C

and

12p

E PP

respectively. The light beam

E PS

emitted from the projector reaches point

through point

on the object. Since the light beam

is blocked by the object through point

, shadow

area

PO S

is formed. The camera acquires the

fringe patterns in another direction, so the shadow

area

PO S

will be captured by the camera, which

does not reflect the modulated light signal to the

camera. Thus the pixels in the image acquired by

camera can be divided into two types of area as the

fringe pattern area and shadow area.

To identify the surface profile in shadow areas,

the points in shadow can be recognized by checking

whether the captured fringe patterns are from the

digital micro-mirror device plane (L. Lu,,2015).

Theoretically, if the points of the results calculated

by phase-shifting algorithm were mapped to the

digital micro-mirror device plane, the mapped

points from the shadow area will be out of the

boundary of digital micro-mirror device plane.

Therefore the points in shadow area can be

determined through this mapping. Since the

wrapped phase values in shadow areas are not

reliable, we label these values as invalid to

distinguish from the points in the area illuminated

by projector.

3. RELIABLE ABSOLUTE PHASE

RECOVERY

3.1 Absolute Phase Map Recovery with

Temporal Phase Unwrapping

To calculate the absolute phase and fringe order of

the surface with shadow, we take one section from

point

to point

shown in Fig.2 as an example.

are used to represent the

coordinates of the corresponding points on the

section respectively. For simplicity, we use

()x

and

()mx

to denote absolute phase

( , )xy

and

fringe order

( , )m x y

respectively. In Fig.2, the

areas from point

to point

and from point

1S 

to point

are illuminated with fringe

projections, the area from point

1P 

to point

the shadow. Based on the shadow area identification

in (L. Lu,2015), the unreliable absolute phase values

in shadow from point

1P 

to point

are also

labeled as empty values. For the area illuminated by

fringe projections, the point on the fringe image

captured by camera is one-to-one map to the point

in digital micro-mirror device plane. Therefore,

when the absolute phase and fringe order in the

areas illuminated by fringe projections are correctly

Reliable Shadow Area Recovery with Two-Spatial-Frequency Fringe Projections from Two Projectors

245

recovered, the absolute phase should be

monotonically increasing, the fringe order should be

stepwise increasing. So the absolute phase from

point

to point

and from point

1S 

to point

keeps monotonically increasing, we also have

( 1) ( )SP   

(S. Zhang,2006). Since the fringe

order from point

to point

and from point

1S 

to point

is stepwise increasing, we also

have

( 1) ( )m S m P

(S. Zhang,2006).

In order to recover the absolute phase map of the

surface with shadow, temporal phase unwrapping is

used due to its independence of the wrapped phase

values of adjacent pixels. In temporal phase

unwrapping, the fringe order of absolute phase is

uniquely determined with selected multiple-spatial-

frequency fringe projections(H. O. Saldner,1997)(Liu,

Y. Wang,2010).

However, when the noise in wrapped phase map

exceeds the phase error tolerance bound, errors may

occur in the fringe orders recovered by temporal

phase unwrapping (Y. Ding,2012)(L. Song,2014),

which makes the recovered fringe order value

different from its true value. To illustrate the fringe

order error in the recovered absolute phase, we

show an experiment results shown in Fig.3. In this

experiment, we project two sets of fringes at the two

normalized spatial frequencies 8 and 15 onto the

measured object, the normalized spatial frequency 8

means the image is covered with 8 complete fringe,

the normalized spatial frequency 15 means the

image is covered with 15 complete fringes. We label

the wrapped phase, fringe order and absolute phase

of the points in shadow areas as invalid, which are

indicated as empty in Fig.3(a) and Fig.3(b). The

wrapped phase maps of the area illuminated by

fringe projections are obtained with phase-shifting

algorithm (X. Su,1992), the corresponding fringe

orders are recovered with temporal phase

unwrapping method in (Y. Ding, 2011). The

recovered fringe order and absolute phase and of

section y=690 at the normalized spatial frequency

15 are shown in Fig.3 as follows.

(a)

(b)

Figure 3: The recovered fringe order (a) and the

recovered absolute phase (b) of section y=690 at the

normalized spatial frequency 15.

From Fig. 3, we notice that few points are with

fringe order errors in the area illuminated by fringe

projections, thus it is necessary to correct these

errors for high reliable temporal phase unwrapping.

3.2 Fringe Order Error Correction for

the Absolute Phase Separated by

Shadow Areas

For the surface without shadow, some methods find

and amend the fringe order errors by checking the

numerical difference of the calculated absolute

phase values between neighbor pixels (H.

Wang,2015)(C. Zhang,2015) or designing the filter

window size according to this difference (D.

Zheng,2017). Since the absolute phase and the fringe

order of the area illuminated by fringe projections

are separated into several intervals by shadow areas,

these correction methods cannot be used for the

surface with shadow areas.

To correct the fringe order errors occurring in

the absolute phase map separated by shadow areas,

we have the discussion as follows. According to the

physical meaning of fringe order, the section of

fringe order map is divided into some steps by the

fringe projections, the fringe order values of the

points in each step should be the same. The bounds

of steps can be found by checking the phase

discontinuities in wrapped phase map, the bounds

of shadow areas are also considered as the bounds

of step. The diagram of step division is shown in

Fig.4.

Figure 4: Step division separated by shadow area.

CTISC 2019 - International Conference on Advances in Computer Technology, Information Science and Communications

246

Thus when the fringe order values of the pixels

in one step are not the same, the fringe order error

must occur in this step. Since the noise in temporal

phase unwrapping can be modeled as Gaussian

distribution with the mean as the true fringe order

value (Y. Ding,2017), thus most of the fringe order

values of the pixels in each step recovered by

temporal phase unwrapping will be correct,

experimental results in many references confirm

this inference. Based on this inference, we can find

that in each step, it is high probability that the fringe

order value with the most pixels is the true fringe

order of this step. We consider this fringe order

value as the true fringe order of this step, the fringe

order values of the pixels in this step are all replaced

by this value.

If the fringe order value with the most pixels is

not the true fringe order, this processing can reduce

the random fringe order errors into the same wrong

fringe order value. The fringe order error is in

integer, thus the absolute phase error is in the times



. This kind of fringe order error can be found

by checking the difference of the absolute phase

values in this step with the nearest two steps. Along

the x-axis, one of the nearest two steps is on the right

side of this step, the other is on the left side of this

step. When this error occurs, the difference of the

absolute phase values will exceed pi, the increase

and decrease of absolute phase will appear in pair as

in Fig.3(b).

Therefore, for the surface with shadow, we can

summarize the technique to correct the fringe order

error of the absolute phase separated by the shadow

area as follows:

1. On the fringe order map recovered with

temporal phase unwrapping, divide the section of

fringe order map into several intervals row by row

based on the phase jump and bounds of shadow

areas in wrapped phase as Fig. 4., each interval

corresponds to one step.

2. Count the number of pixels for every fringe

order value in each step. If the fringe order value of

one step keeps the same, the fringe order value is

judged as true fringe order value of this step.

3. For the step with multiple fringe order values,

find the fringe order value with the most pixels. This

fringe order value is judged as the true fringe order

value, the other fringe order values are replaced by

this value.

4. Mark the steps corrected in Step 3. Compare

the absolute phase values on the bounds of these

steps with the absolute phase values on the bounds

of the two nearest steps. In the two nearest steps,

one is on the right side of the corrected step, the

other is on the left side of the corrected step. When

the increase of the absolute phase values on the

bounds of the two nearest steps exceeds



and the

decrease of the absolute phase values on the bounds

of the two nearest steps is less than





, we can

judge the fringe order value of the corrected step as

wrong. If such pair appears, we discard the absolute

phase values in the corresponding step since they

are not reliable. If there is no such pair, it means the

fringe orders are correctly fixed.

By this technique, we can detect and correct the

fringe order errors of the area illuminated by fringe

projections. Thus, the height distribution of such

area can be corrected with high reliability.

4. SHADOW AREA RECOVERY

WITH TWO PROJECTORS

From the discussion in Section 2, we can see that

the shadow area is formed when the illumination

signal emitted by projector is blocked by object

surface. To recover the surface profile in shadow

areas, information of fringe pattern in these areas

should be added. Based on this consideration, we

design a fringe projection profilometry system with

two projectors and one camera, the two projectors

are on the two sides of camera. For the objects with

convex surface profile, two projectors could cover

the entire surface. The diagram of this system is

shown in Fig.5.

(a) (b)

Figure 5: FPP system with two projectors and

one camera.

In Fig.5(a) and Fig.5(b), the shadow area formed

by the left projector can be illuminated by the right

projector, the shadow area formed by the right

projector can be illuminated by the left projector.

The object is illuminated by the fringe patterns

emitted by two projectors sequentially. By this

design, the shadow area from one projector can be

illuminated by another projector. The positions of

two projectors are on the symmetry of camera.

From the discussion in Section 3, we infer that

the possible errors in recovered fringe order values

Reliable Shadow Area Recovery with Two-Spatial-Frequency Fringe Projections from Two Projectors

247

could be corrected by the monotonic property of

absolute phase. Therefore, we design the

projections from the left projector as its true

absolute phase monotonically increasing along the

x-axis, the height distribution of can be obtained.

The projections from the right projector are similar,

the height distribution the area illuminated by the

fringe projection from left projector can be obtained.

The 3D profile of the shadow areas generated by

one projector can be recovered with the reliable

absolute phase of the fringe projections generated

by another projector, the complete three-

dimensional reconstruction result of object can be

obtained. For areas illuminated by two projectors,

we can decide which height value is used for 3D

profile reconstruction based on the quality

evaluation of fringe projection in Eq.(5).

We summarize the procedures to recover the 3D

profile of shadow area as:

1. Fringe patterns are casted onto the object from

two projectors sequentially as Fig.5 and make sure

that the shadow area generated by one projector is

covered by another one. For each projector, two-

spatial-frequency fringe patterns are casted. The

fringe projections from two projectors are designed

as Fig.5(a) and Fig.5(b) respectively.

2. For each projector, the points in the shadow

areas are identified by the method in Section 2.2,

which is proposed in [3], the wrapped phase values

of these points are labeled as empty. The wrapped

phase maps of the areas illuminated by fringe

projections are obtained by phase-shifting

algorithm.

3. For each projector, the absolute phase maps

of the areas illuminated by fringe projections are

obtained with temporal phase unwrapping. The

fringe order errors in the recovered absolute phase

maps are corrected by the novel technique

developed in Section 3.2.

4. The complete 3D profile data are

reconstructed with the corresponding reliable

absolute phase maps of the fringe projections from

two projectors by Eq.(7). For the areas illuminated

by two projectors, we decide which height value is

used based on the quality evaluation of fringe

projection with Eq.(5).

5. EXPERIMENTS

To exemplify the performance of the proposed

method, experiment is implemented. The purpose of

experiment is to show the proposed method could

recover the entire surface correctly. The test object

is a plaster model (Voltaire, the maximal z-axis

direction depth is 130mm), the shadow areas will

occur when these objects are illuminated by fringe

projections. The rich detailed of object surface is

suitable to illustrate the proposed method. Our

experiment system is set up with two projectors and

one camera, the experiment system is shown in

Fig.5. The camera in the experiments is Basler MV-

CS30G with its highest resolution

1296 966

, the

resolution of camera can be adjusted. The projector

is Acer H65010BD.

We use the fringe projection profilometry

system shown in Fig.5 to capture the fringe images

from the two projectors. The captured image is

composed of

648 480

pixels. For each projector,

we project the fringe image of two normalized

spatial frequencies 8 and 15 onto the objects.

The fringe images from two projectors are

shown in Fig.6(a) and Fig.6(c). In these images, we

find some areas do not reflect the modulated light

signal to camera, which are formed as the shadow

area. The wrapped phase maps of the fringe

projections from two projectors are calculated by

phase shifting algorithm for noise reduction. The

shadow points can be found by the method in [3],

thus the wrapped phase values of shadow areas are

labeled as invalid. For the areas without shadow, the

absolute phase maps of these fringe projectors the

two projectors are obtained by the temporal phase

unwrapping developed in [22]. The fringe order

errors in absolute phase maps are corrected by the

technique described in Section 3.2, the corrected

absolute phase maps of the fringe patterns of two

projectors are shown in Fig.6(b) and Fig.6(d). The

absolute phase values in shadow areas are also

labeled as invalid, showing as the white parts in

Fig.6(b) and Fig.6(d) respectively.

With these absolute phase maps, we can obtain

two partial 3D reconstruction results. The complete

3D reconstruction result is obtained with the process

in Section 4, the entire three-dimensional

reconstruction result is shown in Fig.6(e), the unit

of depth data in z-axis direction is mm. We can see

the most parts of shadow areas in Fig.6(a) and

Fig.6(c) are recovered in Fig.6(e), there is no fringe

order error in the complete 3D reconstruction results,

thus the proposed method can recover the shadow

areas with high reliability.

(a)

CTISC 2019 - International Conference on Advances in Computer Technology, Information Science and Communications

248

(b)

(c)

(d)

(e)

Figure 6: (a) Fringe image from lower projector. (b)

Corrected absolute phase map of (a). (c) Fringe

image from higher projector. (d) Corrected absolute

phase map of (c). (e) Complete reconstructed 3D

profile.

6. CONCLUSION

This paper develops a method for entire surface

three-dimensional reconstruction with improved

fringe projection profilometry. To add fringe

information onto the shadow area, two projectors at

symmetrical positions are used to illuminate the

object surface. For the area illuminated by each

projector, points in shadow areas are identified,

phase-shifting algorithm and temporal phase

unwrapping are employed to avoid the influence of

the points in shadow area. The possible fringe order

errors in recovered absolute phase map are

corrected with the mathematical characteristics of

fringe order. Thus the 3D profile of shadow area for

one projector can be reconstructed by the fringe

projections from another projector with high

reliability. The method developed in this paper can

be applied to obtain entire surface profile for

industrial applications.

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