Influence of sampling on face measuring system based on composite structured light

doi:10.4236/jbise.2009.28087

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J. Biomedical Science and Engineering, 2009, 2, 606-611

doi: 10.4236/jbise.2009.28087 Published Online December 2009 (http://www.SciRP.org/journal/jbise/ JBiSE

Published Online December 2009 in SciRes. http://www.scirp.org/journal/jbise

Influence of sampling on face measuring system based on

composite structured light

Yang Shen1,2, Hai-Rong Zheng1,2*

1Institute of Biomedical and Health Engineering, Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences,

Shenzhen, China;

2Key Laboratory of Biomedical Informatics and Health Engineering, Chinese Academy of Sciences, Shenzhen, China.

Email: hr.zheng@siat.ac.cn

Received 7 August 2009; revised 4 September 2009; accepted 7 September 2009.

ABSTRACT

Human face can be rebuilt to a three-dimensional (3

D) digital profile based on an optical 3D sensing sys-

tem named Composite Fourier-Transform Profilo-

metry (CFTP) where a composite structured light

will be used. To study the sampling effect during the

digitization process in practical CFTP, the pectinate

function and convolution theorem were introduced to

discuss the potential phase erro rs caused by sampling

the composite pattern along two orthogonal direc-

tions. The selecting criterions of sampling frequencies

are derived and the results indicate that to avoid

spectral aliasing, the sampling frequency along the

phrase variation direction must be at least four times

as the baseband and along the orthogonal direction it

must be at least three times as the larger frequency of

the two carrier frequencies. The practical experiment

of a model face reconstruction verified the theories.

Keywords: Optical 3D Sensing; Composite Structured

Light; Sampling; Spectral Aliasing

1. INTRODUCTION

Structured-light illumination is commonly used as an

active optical 3D sensing technique for automated in-

spection and measuring surface topologies. The Fourier-

Transform Profilometry (FTP) [1,2] is one of the classi-

cal 3D acquisition methods and it has been widely inves-

tigated [3,4,5] because of its advantages of obtaining

data from only one frame and analyzing spectrum in

whole-field as well as high resolution. Recently, an im-

proved FTP method called Composite Fourier-Transform

Profilometry (CFTP) was introduced [6,7]. This novel

method prevents spectral aliasing between zero-frequen-

cy and baseband b y u sing o nly on e gr ating na mely Com-

posite Pattern (CP) that generated by integrating multi-

frame ordinary patterns, so that it allows for real-time

implementations [8].

However, the data in both CFTP and FTP are digitally

sampled to discrete signals during the digitization proc-

ess in practice. The discrete images have periodical Fou-

rier spectrum, and the fundamental spectrum including

the useful information would be overlapped by the adja-

cent periodical weight [9]. Furthermore, th e CP in CFTP

is much more complexity than traditional sine grating,

another kind of spectrum aliasing would be brought in.

In this instance, choosing a proper sampling frequency is

very important for the precise survey.

To study the influence caused by sampling, the know-

ledge of pectinate function and convolution theorem was

employed in this article and the suggestion that how to

select proper sampling frequencies was given. The ex

periment verified the theories, and a beautiful 3D digital

profile of a model face was acquired.

2. METHODS

2.1. CFTP Theory

A Composite Pattern (CP) in CFTP is generated as

shown in Figure 1. The multiframe sine patterns to be

modulated are as follows.

cos(2 )nGcfy n







 (1)

where a constant is used to offset to be non-nega-

tive values, and cnG



is the baseband, represents the y

Figure 1. A composite pattern formed by simple strips.

depth distortion (i.e., phase dimension) direction, re-

presents the phase-shift index from 0 to 1. These signal

Y. Shen et al. / J. Biomedical Science and Engineering 2 (2009) 606-611 607

patterns are multiplied with different carrier frequencies

respectively along the orthogonal direction. Accumulate

all channels such that

(, ){[cos(2)]cos(2)

[cos( 2)]cos(2 )}

xyabcf yfx

cfy fx





 

 

 (2)

where 1

and 2

are carrier frequencies along the

orthogonal direction

, and the projection constants

and are used to make sure the projection intensity of

CP falls into the range of

b(, )

xy [6,7]. Ideally, the

reflected image of the specimen surface can be counted

as follows

(, )(, )(, ){

[cos(2( ,))]cos(2)

[cos(2( ,))]cos(2)}

Pxyarxy brxy

cfyxy fx

cfyxy f



 

 











(3)

where and

(,)rxy (,)



represent the albedo ra-

diation and distorted phase respectively. By means of

2D Fourier-transform and predigestion, the expression

(3) will be translated into (4) shown as follows, where

(,F)



, (,A)



, (,)B



and (,)





represent

the two-dimension Fourier spectrum of (, )

xy ,

, and

(,ar x)ybcr(,x y)1

2(,br x

}



)exp[y j



(, )]x y re-

spectively.

11 1

22 2

(,) (,) (1/2){

[(,)(, )*(, )]

[( ,)(,)*( ,)]

[(,)(,)*(,)]

Bff ff f







 

 





 









(4)

Expression (4) suggests that the two carrier frequen-

cies are evenly distributed and are separated by spectral

frequency of background reflectance. Therefore, a smooth

and flat background had better be selected to minimize

the influence to the carrier spectrums. The distorted im-

age is processed as a set of 1-D signal vectors by

band-pass filters to separate out each channel. Cutoff

frequencies of each band represent the individual pat-

terns like that in traditional



Phase Shift FTP and are

used to retrieve the depth of the measured object based

on the traditional



Phase Shift FTP method [7] as

follows:

0(, )(, )/2hxyxyLf d







 (5)

where and are experimental setup parameters,

represents the reconstructed height.

)

(,hxy

2.2. Influence of Sampling on CFTP

Expression (3) indicates the continuous image, but in

practical experiment it will be digitally sampled to dis-

crete signals by projector and camera, and the discrete

distorted pat te r n is captured as

(, )Sxy

(, )(, )(/,/)

(,)(/)(/)

SxyPxycombxxy y

Pxy combxx combyy





(6)

where is Pectinate Function, (/ ,/ )comb xxyy



and are sampling spacing along phase direction

and orthogonal direction respectively so that

y

1/xfx





and represent the sampling frequency along

the two directions respectively. Here suppose

1/yfy



1/ (max{,})

fymf

fxnff ff





  (7)

where and are multiple units, respectively

represents the multiple relationship between sampling

frequency and the selected experimental setup frequen-

cies along the two orthogonal axes, and both them are

positive numbers. These two introduced parameters en-

able us to calculate the proper sampling frequencies

based on the known baseband and carrier frequencies,

and the selecting criterions of sampling frequencies are

determined as long as and n are definitely.

m n

Eq.4 shows that besides the frequency of background

reflectance (i.e. (,) A



), there are four peak values

along the orthogonal direction, namely 1, 2ff



; in

each peak



there are three peak spectrums along the

phase direction, namely 0, f





. To simplify the

investigation, we will discuss the sampling effects along

the two orthogonal directions respectively.

2.2.1. Sampling Analysis along Pha s e Dire ct i on

Any a peak value of



was selected, e.g.1





, there

are three peak spectrums in the channel along the phase

direction:

1111(,)(,) (,)*(,)

fBfff ff







 

(8)

The discrete spectrums of 1(,)Ff



can be calculated

608 Y. Shen et al. / J. Biomedical Science and Engineering 2 (2009) 606-611

11*

(,)(,) | |()

[(,)

(,)

*(,) ]

fFf ycomby

Bf Nmf

ffNmf





 













 







(9)

where is the convolution operator and is integer.

Eq.9 indicates that the spectrums of

1(,Ff )



repeat

periodically. To against the overlapping of spectrums, we

must make sure



is separated from *



in the same

period, and also make sure (*)



is separated from

*( )



comes from the adja cent periods, in another word,

the sampling fr equency is restricted, there must be at le ast

four sampling dots in one period [9] so that

4 , or 4yfmf fm





 (10)

2.2.2. Sampli n g An al ysi s along the Orthogonal

Direction

Any a peak value of



was selected to study the cor-

responding four peak spectrums of



along the or-

thogonal direction, e.g.







(, )(, )(, )

(,)(,)

fffff







 





  (11)

Here suppose 21

f

comb

. Consider expression (7) and

Pectinate Function along the orthogonal

direction, the discrete spectrums of

(/ )xx

(,)





can be

calculated as (12):

22 12

12 22

(,)(,) ||( )

[ (,)(,)

(,)(,) ]

Ff Ffxcombx

fNnfffNnff

fNnff fNnff













 









 



(12)

Expression (12) indicates that the spectrums of

(,)





repeat periodically with period of . As

shown in Figure 2, the real lines represent the starboard

2nf

(,)





, and the dashed represent the larboard

spectrums of the adjoining period.

Figure 2 indicates that to escape the overlapping of

spectrums, there must be have



1max 2min

2max2 2max

() ()

fnff



 (13)

Imitating the definition of instantaneous frequency in

domain of signal processing [5], we get

1 max1max

2min 2max

2max 2max

()(1/2)|/ |

ff x



 



 



 



(14)

Eq.13 can be m o di f i e d by (5) and (14) such that

Figure 2. Replicated spectrum distribution.



max21 0

max2 0

|/|()(/2 )

|/|(2)(/2 )

hxffL fd

hxnfL fd



 

  (15)

According to (15), we can get

212( )/nff 2f

(16)

If the condition 22



is selected in survey, there

mus t be , so there has

3n223x

nf f.

2.3. Experiment

To support the analysis above, a model face was used as

a tentative test. The projector used was a Panasonic (PT-

P2500) digital projector with resolution of 1024×768.

The image sensor used was a low-aberrance color CCD

camera (Prosilica, EC1350C, made in Canada) with

resolution of 1360×1024 and pixel size of 4.65um×

1.65um, and the maximum frame rate is 18fps. The fo-

cus of the camera lens (KOWA, LM12JCM, made in

Japan) was 12mm. The image board was a 1394 card

(KEC, 1582T, made in Taiwan). The reference plane as

background was a piece of smooth and white board.

Figure 3 illustrates the experimental setup, in which the

geometric parameters were set as =73mm and

=18mm, and the carrier frequencies

and 2

were set as 3/40 line/pixel and 6/40 line/pixel respec-

tively and the baseband



was given as 60/600 line/

pixel. The lens of projector and camera must be at a

same geometric plane surface and here they were setup

coplanar at vertical curve. The horizontal beam con-

tained CP illuminated over against model face, and the

sho o t ing angle of camera was setup as 45 degree w hich is

Y. Shen et al. / J. Biomedical Science and Engineering 2 (2009) 606-611 609

Figure 3. The diagram of the experimental setup.

the optimal angle value [4,5]. Because the specimen was

a model face, the gesture and expression could be with-

out consideration, however, with regard to a real human

face, eyes exposure and shadows caused by gesture must

be considered carefully.

According to the analysis above, to avoid spectral

aliasing, there must be have and . Figure 4

shows the captured distortion composite pattern. Figure

5 indicates the reconstructed profile. From the drawings,

we can find that when sampling frequencies do not sat-

isfy the sampling request, i.e. (a) when

4m3n



and

, the rebuilt errors were big and the details of face

were lost. However, when and , the de-

tails could be retrieved as shown in (b) and (c) with good

resolution.

2n4m3n

3. DISCUSSION

The accurate acquisition of 3D human face appearance

characteristics is very important for designing a facial

contouring surgery, and a good work is based on an ex-

act 3-D face modeling [10]. People hope to find a non-

ontact, rapid, precise way to acquire 3-D digital face

depth, and then based on it to simulate and design an

optimal plan for face surgery by modem technologies

such as computer aided design etc [11].

Figure 4. The captured distortion composite pat-

tern modulated by height of the face model.

(a)

(b)

(c)

Figure 5. The rebuild shape, (a) when m=3, n=2; (b) when

m=4, n=3; (c) m=5, n=5.

610 Y. Shen et al. / J. Biomedical Science and Engineering 2 (2009) 606-611

At present, there are about three types of 3D face

modeling method to extract human face profile: one is

the method based on computer tomography (CT) tech-

nology [12,13,14] and another one is based on passive

optical 3D sensing technique [15,16,17] and the other is

based on active optical 3D sensing technique [18,19,20].

The 3D reconstruction method based on CT technol-

ogy is sensitively to skeleton and is convenient to be

used for craniofacial plastics and oral and maxillofacial

correction of abnormality and such fields, however, 3D

profile of soft tissue is difficult to rebuilt by CT tech-

nology, especially the human face surface features.

The passive optical 3D sensing technique such as ste-

reo vision uses two or more camera systems to capture

the scene in ambient light from different viewpoints and

to determine the height by matching image features of

the corresponding surface features. In this method, a lot

of factors need to be noticed, such as ambient light,

background, vision angle and face gesture, expression

and shading and so on, for they would influence the

measuring accuracy directly. Besides, there always need

to process a mass of data operation like correlation

analysis and matching operation etc. Generally, the pas-

sive optical 3D sensing technique is more often used for

3D object recognition and understan ding. Alo ng with the

development of computing technique, arithmetic speed

is no longer a key limiting factor, and the passive optical

3D sensing technique is widely used in the field of ma-

chine vision.

The active optical 3D sensing technique employs

structured light to illuminate the specimen. The time or

space in structured light will be modulated by height,

and then the 3D information can be extracted from the

observation light by certain unwrapping algorithm [5].

For its feature of non-contact, high resolution and highly

automated, the active optical 3D sensing technique is

used in most 3D sensing systems with the purpose of 3D

surface-shape measurement.

Phase Measuring Profilometry (PMP) is one i mportant

method of active optical 3D sensing technique [5]. In

PMP, sinusoidal fringes and phase shifting technique are

employed to acquire the height information that we

wanted. A flaw of PMP is that it has to capture at least

three continuous modulated phase shifting fringes cor-

responding to a static profile and therefore there will be

some trouble for real-time dynamic measurement, and

during the shooting process a little movement or facial

expression changes of the target human face will poten-

tially bring errors to the demodulated results. By using

fast digital grating projection approach, a series of phase

shifting fringes can be projected and shot within a short

span of time. However, the images photographed by

CCD camera would easily cause drawbacks such as

trailing and distortion etc. due to rapid rotation of the

phase shifting fringes, and then the inaccuracy of meas-

urement will be raised. A one-shot technique, therefore,

becomes a trend [19,20].

Here a novel one-shot approach for 3D human face

profile measurement is introduced. A composite pattern

(CP) is used in place of the series phase shifting fringes

in PMP, and only a single frame of CP is needed to pro-

ject and capture. The CP efficiently combines some

phase shifting fringes and the same number of carrier

gratings, and so that the phase shifting technique can be

also utilized in this approach. This one-shot technique

can avoid some unwanted troubles such as trailing and

distortion etc. that happened in PMP for needing only

one projection and corresponding one capture. Based on

the proposed approach, 3D digital model of real human

face could be acquired more conveniently and exactly.

Here we used this novel method to reconstruct a

model face and acquired a good stereogram under the

proper sampling frequencies which were the focus of our

investigation. Because of the comp lexity of the compos-

ite pattern, another kind of spectrum overlapping would

be brought in by the two modulating gratings during the

digitization process. In this instance, choosing a proper

sampling frequency is very important for the precise

reconstruction. In the paper we discussed the sampling

conditions along two directions and pointed ou t the rules,

and then under the given sampling conditions we ac-

quired a perfect digital 3D face profile.

4. CONCLUSIONS

Composite Fourier Transform Profilometry (CFTP) is an

improved FTP method where a composite structured

light is employed. To study the influence caused by

sampling, the knowledge of pectinate function and con-

volution theorem was used and the suggestion that how

to select proper sampling frequencies was given, that is,

the sampling frequency along the phr ase variation direc-

tion must be at least four times as the baseband and

along the orthogonal direction it must be at least three

times as the larger frequency of the two carrier frequen-

cies.

5. ACKNOWLEDGEMENMTS

This study was supported by the National 973 Basic Research Program

of China (No. 2010C B7 32600).

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