1. International Journal of Electronics and Communication Engineering & Technology (IJECET),
INTERNATIONAL JOURNAL OF ELECTRONICS AND
ISSN 0976 – 6464(Print), ISSN 0976 – 6472(Online), Special Issue (November, 2013), © IAEME
COMMUNICATION ENGINEERING & TECHNOLOGY (IJECET)
ISSN 0976 – 6464(Print)
ISSN 0976 – 6472(Online)
Special Issue (November, 2013), pp. 210-215
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IJECET
©IAEME
Localization of Free 3D Surfaces by the Mean of Photometric
Stereovision
M Khoudeir, B Bringier
LaboratoireXlim-SIC, UMR-CNRS n°6172
Université de Poitiers
Bvd Marie et Pierre Curie, BP 30179
86962 FuturoscopeCedex, France
majdi.khoudeir@univ-poitiers.fr
ABSTRACT: Within the framework of the analysis of 3D textured surface through image
analysis, we approach here the case of the 3D interfaces (fluid-solid) surfaces for the analysis
of the local variations of their relief. Generally, the interaction between the light and these local
variations of the relief leads to a textured images of these surfaces. Our aim here is to achieve
the feasibility of this measurement through a local relief extraction based on photometric
stereovision. The proposed approach is an original adaptation stereovision based on
photometric model to the case of free surfaces with a high degree of variation and with
Lambertian photometric behaviour .The suggested method is presented and the relevance of
this approach for that kind of surface is tested on particular shape. The results obtained are the
first step of a global study of the displacement of the local variation of this 3D free surface.
KEYWORDS: Free Surface, Image Acquisition, Photometry model, 3D map, Stereovision.
I.
INTRODUCTION
The localization of 3D surfaces and the measurement of their displacements are today a real
challenge. In structure, solid and fluid mechanics, the determination in space and in time of 3D
interfaces (fluid-solid, fluid-fluid) and 3D surfaces (solid or free-surface) is necessary to the
understanding of physical phenomenon and obviously the measurement of the three
components of the displacement is wished. Different techniques to localize in space and in time
the position of free-surface, waves or textured surface have been developed recently by [1] and
[8]. Methods to estimate 3D solid surface have been also quite performed. Fringe projection [7]
or photometric model [2], {3], [4], [5] are applied for different applications and particularly for
solid surface.
Our objective is to adapt our technique of photometric stereovision to the case of free surface
obtained at the fluid-solid interface. At this time of our study, we consider only opaque surface
that photometric properties can be approximated by Lambertian model. To do that, we suggest
to deal with different steps. The first step, which will be presented here consist on extracting
International Conference on Communication Systems (ICCS-2013)
B K Birla Institute of Engineering & Technology (BKBIET), Pilani, India
October 18-20, 2013
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2. International Journal of Electronics and Communication Engineering & Technology (IJECET),
ISSN 0976 – 6464(Print), ISSN 0976 – 6472(Online), Special Issue (November, 2013), © IAEME
the 3D relief of a “static testing surface” in order to put out the relevance of this approach
according to the shape of the surface and to its photometric properties. So, we consider a
corrugated plastic with a wave length representative of fluid surface variation. Then in a
second step, we will extend this approach to the case of dynamic free surface. The proposed
approach is based on a photometric model which is function of the nature of the surface and
the relative positions between the camera, the light source and the surface. This model leads to
a set of three equations including three unknown factors of which two are related to the relief
variations that are the two degrees of freedom describing the surface orientation. The third
one is related to the colorimetric characteristics of the surface and it represents its reflectance.
In a second time, to solve this system of equations, once needs three separate images acquired
using under three different lighting configurations. Then, for the studied surface, we can
extract its local relief. This technique is applied to measure the different positions and
movement of a corrugated plastic. Accuracy of the position and the shape of the considered
surface are presented and compared.
II.
PHOTOMETRIC MODEL BASED METHOD
A. Link between textured surface image and relief
Since surfaces are not specular, Lambertian model [6] was used to characterize their
photometric behaviour and to set the link between grey level, colour information and relief. Let
us consider a rough textured surface composed of Lambertian micro-facets. The surface is
lighted under incidence angle i and observed by camera mounted perpendicularly to the
surface plane (x, y) (Fig. 1). Definitions of the angles shown in the fig. 1 are the following:
i : Incidence angle related to the surface;
 : Incidence angle related to the facets;
 : Angle between the surface normal and the facet normal.

Z

S
i

X



N
i

Y
Fig. 1: Configuration for view shot

In case of Lambertian surface, the image intensity I(x, y) represents the energy received by the
CCD sensor. It is expressed by the following relation:
I x , y  
L x , y 
cos   x , y 
r2
Where I ( x, y) is the Image intensity, L( x, y) the coefficient representative of surface colorimetric
properties,  ( x, y ) the incidence angle related to the facet and r the distance between the
lighting source and the facet.
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B K Birla Institute of Engineering & Technology (BKBIET), Pilani, India
October 18-20, 2013
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3. International Journal of Electronics and Communication Engineering & Technology (IJECET),
ISSN 0976 – 6464(Print), ISSN 0976 – 6472(Online), Special Issue (November, 2013), © IAEME
The distance will be assumed to be constant since relief variations are negligible compared
with the distance surface/lighting source. We try then to express the angle  ( x, y ) as a function
of Cartesian coordinates (x, y, z) of the current point. The following vectors are defined:

Z  [0 0 1]T , vector normal to the surface plane;

S  [sin i cosi
sin i sini
cos i ]T , the unit vector of the light source direction with i the angle


 
between the X axis and the projection of the lighting source vector S in the plane ( X , Y ) ,

N  [ a b c ]  [sin  cos  sin sin  cos ] , the normal to the facet.
Let us consider a point M from the surface. The relief in the neighbourhood of this point can be
considered as a plane defined by the following equation:
P: axb y cz  0

Combining the equation of N with the equation of the plane, we obtain finally:
z
z
cos  sin  cos   sin  sin 
i x
i
i y
i
i
L  x, y 
I  x, y  
r2
1  z x 2  z y 2
The above expression enables us to differentiate different information included in the image
grey level.
B. Method to extract information related to relief
The previous equation is employed to extract the relief information from the image. This
equation comprises three unknowns, two of which are related to relief variations ( z and
x
z
y
) and one to surface colorimetric properties L( x, y) . We have to solve a system with three
equations and three unknowns. To produce the equations, three images are recorded. They
correspond to the same surface lighted under three incidence angles. The incidence angles ()
are low to prevent shadows in the images. Let I1 , I 2 et I 3 be the images shot successively under
the following lighting conditions : 1   and 1  0 ,  2   and 2  2 3 ,  3   and 3   2 3 .
The 3-equation system is the following:
z

cos  
sin 

x
 I 1 x , y   L  x , y 
2
2
1   z  x    z  y 


1 z
3 z

cos  
sin  
sin 


2 x
2 y
 I 2 x , y   L x , y 
2
2

1   z  x    z  y 

1 z
3 z

cos  
sin  
sin 

2 x
2 y
 I 3 x , y   L x , y 
2
2

1   z  x    z  y 


International Conference on Communication Systems (ICCS-2013)
B K Birla Institute of Engineering & Technology (BKBIET), Pilani, India
October 18-20, 2013
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4. International Journal of Electronics and Communication Engineering & Technology (IJECET),
ISSN 0976 – 6464(Print), ISSN 0976 – 6472(Online), Special Issue (November, 2013), © IAEME
Its resolution gives:
2 I 1  I 2  I 3 cos 
 z
 

I 1  I 2  I 3 sin 
 x

3  I 2  I 3 cos 
 z
 
 y
I 1  I 2  I 3 sin 

And then the image of the relief is given through double integral of z according to these
equations.
C. Experimental Apparatus
Experiments are performed on a special device which authorizes the use of the proposed
measurement technique. At this stage, we have chosen to validate and to compare the
measurements the following shape: a corrugated plastic (wave length = 70 mm, amplitude = 20
mm, fig. 2). A blue random white dot pattern has been fixed on each shape. The pattern is
generated by means of a synthetic image generator. The size is 280×200 mm2 yielding an
optical resolution of 100 µm/pixel.
Fig. 2: Corrugated plastic and two-axis translation and one-axis rotation stages
A tri-CCD Sony camera with a 768576 pixel2 resolution is placed along the Z-axis and
recorded the observed images for the different locations of the shape. Three light sources fixed
at 15° with regard to the z-axis are placed at 120° each other. For each location, three
expositions are recorded. The camera and the light sources are placed far enough to have
parallel lines and homogenous illumination.
Three light
Sources
CCD
Camera
Fig. 3: Experimental setup: acquisition and light sources position
Combined movements of translation and rotation have been recorded by the camera. The
range of displacement is included between 0.1 mm to 15 mm in translation and between 1° to
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B K Birla Institute of Engineering & Technology (BKBIET), Pilani, India
October 18-20, 2013
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5. International Journal of Electronics and Communication Engineering & Technology (IJECET),
ISSN 0976 – 6464(Print), ISSN 0976 – 6472(Online), Special Issue (November, 2013), © IAEME
40° in rotation. The precision of the traversing system is about 5 m and about 0.5° for the
rotation stage.
III.
RESULTS
First results have been obtained on the case of the corrugated plastic. The method display with
accuracy the 3D shape of the plastic (Fig. 4). The amplitude and the wave length of the shape
are determined with a good agreement if we compare to the real profile (fig. 5). Evaluation of
the displacements gives the same accuracy. The photometric model seems well adapted to
measure 3-D shapes and the small differences found for the different positions of the plastic
make this technique able to measure the 3D displacement
Fig. 4: 3D shape extraction by photometric stereovision.
Fig. 5: Estimated displacement of a profile section of the 3D surface
IV.
FUTURE WORK
In conclusion, the first results have shown the feasibility of this technique to measure shape
location in space and to follow displacement of this kind of surface. So, we have now to adapt
this approach to the case of real dynamic free surface. For doing that, we are going on one
hand to exploit the spectral properties of coloured image of the surface, and on the other hand
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6. International Journal of Electronics and Communication Engineering & Technology (IJECET),
ISSN 0976 – 6464(Print), ISSN 0976 – 6472(Online), Special Issue (November, 2013), © IAEME
to develop other photometric model in order to take into account surface with specular
behaviour [9].
So, we expect that the use of the RVB images will allow us to extract shape information through
one image acquisition. We hope that at the date of the conference, we’ll be able to present good
results concerning this last step.
REFERENCES
[1] Calluaud D., David L.: Stereoscopic Particle Image Velocimetry measurements of the flow
around a surface mounted block. Experiments in Fluids, Vol 36 n°1, 53-61, 2004.
[2] Dana K., van Ginnken B., Nayar S.K., Koenderink J.J., Reflectance and texture of real world
surfaces ACM Transactions on Graphics, 18(1), p.1-35, 1999.
[3] Khoudeir M., Brochard, J., Benslimane A, Do M.T.,: Estimation to the luminance map for a
Lambertian photometric model: application to the study of road surface roughness. Journal of
Electronic Imaging, Vol 13(3), 515-522, 2004.
[4] McGunnigle G. and. Chantler M.J, Rough surface classification using point statistics from
photometric stereo, Pattern Recognition Letters, n°21, p. 593-604, 2000.
[5] McGunnigle G. and Chantler M.J., Rough surface description using photometric stereo,
Measurement Science and Technology, n°14, p.699-709, 2003.
[6] Oren M. and Nayar S., Generalization of the Lambertian Model and implications for machine
vision International Journal of Computer Vision, n°14, p. 227-251, 1995
[7] Pirodda L.,: Shadow and Projection moiré technique for absolute or relative mapping of
surface shapes. Opt. Eng. 21, 640-9, 1982
[8]WienekeB.,:Stereo-PIV using self-calibration on particle images. Experiments in Fluids
(Online) s00348-005-0962-z, 2005.
[9]Bringier B., Bony A., and Khoudeir M.,Specularity and shadow detection for the multisource
photometric reconstruction of a textured surface.J. Opt Soc Am Sci Vis29(1):11-21 (2012).
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