Spatial Phase Unwrapping by Mean of a Reliability

Revista Colombiana de Física, vol. 40, No. 1, Abril 2008
Spatial Phase Unwrapping by Mean of a Reliability-Guided Path
J. Garzón1 , C López1, D. Duque1, C. Restrepo1 y N. Álvarez2
1
Grupo de Óptica y Espectroscopía. Centro de Ciencia Básica
Univ. Pontificia Bolivariana. AA 56006. Medellín, Colombia. [email protected]
2
Wizard 3D - Tecnología Interactiva. Envigado, Colombia. wizard-3D.com.
Recibido 22 de Oct. 2007; Aceptado 3 de Mar. 2008; Publicado en línea 15 de Abr. 2008
Resumen
En los sistemas de reconstrucción 3D por luz estructurada o interferométricos, el desenvolvimiento espacial de fase mediante métodos tradicionales constituye un problema difícil de resolver. El ruido inherente a la superficie, la baja visibilidad
de los patrones de franjas, o el método computacional empleado para la extracción del mapa de fase son las principales
causas. Recientes estudios en mapas de fase discontinuos demuestran que es posible llevar a cabo el desenvolvimiento espacial de fase utilizando parámetros de fiabilidad, los cuales contribuyen a reducir la propagación de errores que afectan la
calidad de la corrección de fase. En el presente trabajo, hemos implementado un método para el desenvolvimiento espacial
de mapas de fase utilizando únicamente como parámetro de fiablidad guiado la matriz de modulación de intensidad (IMM)
del patrón de franjas. Antes de calcular la IMM, el mapa de fase es sometido a un conveniente proceso de filtrado. Los
elementos de esta IMM permiten determinar la dirección del desenvolvimiento de fase sobre el mapa de fase filtrado. Esto
quiere decir, que el píxel con el mayor valor de IMM dentro del mapa de dicho parámetro de fiabilidad será corregido en
fase antes que cualquier otro píxel. La eficacia de este método es puesta a prueba en mapas discontinuos, los cuales fueron
obtenidos por medio de técnicas como la transformada de Fourier y el corrimiento de fase.
Palabras claves: modulación de intensidad, corrección espacial, camino de fiabilidad guiado.
Abstract
In the systems of reconstruction 3D of structured light or interferometric, the spatial phase unwrapping by traditional methods constitutes a difficult problem to solve.The inherent noise to the surface, low visibility of the fringes patterns or by
computational method for obtaining the phase are the cause. Recent studies in discontinuous maps show the possibility for
making the spatial phase unwrapping using parameters of reliability, which avoid the propagation of errors affect the quality of the phase unwrapping. In the following work, we have implemented a method for unwrapping of phase spatial maps
taking only as parameter of reliability-guided the intensity modulation matrix (IMM) of the fringes pattern. Before calculating the IMM, the phase map is filtered appropriately. The elements of this IMM are used to identify the direction of phase
unwrapping over the filtered phase map. It means that the pixel with higher parameter value in the parameter map will be
phase unwrapped earlier. The efficiency of the method is proved in discontinuous maps, which were obtained by mean of
Fourier transform and phase shifting.
Key Words: matrix of intensity modulation, spatial unwrapping, reliability-guided path.
© 2008 Revista Colombiana de Física. Todos los derechos reservados.
1. Introducción
final phase of the test object, such as, random noise, local
shadows, under sampling, fringe discontinuities and irregular surface brightness, following an advisable trajectory for
the correction of each pixel present in the phase map, according to the values of the reliability - parameter or pa-
Reliability - guided phase unwrapping algorithm[1] based
on the reliability - guided parameters, allows to obtain a
continuous phase map diminishing significantly all those
factors that introduce errors in pixels of the distribution
150
r col. fís.(c), vool. 40, No. 1, (2008
rev.
8)
rrameters that they indicate what pixels early
e
have to be
c
corrected
and as others later,, taking then an
a advisable orrder
i the process of discontinu
in
uous phase unw
wrapping. Firstt, a
p
parameter
or parameters
p
- set
s is selected for to orient the
p
path
of phase unwrapping. Everything
E
pixxel of the discoont
tinuous
phase map displays a functional value of this par
rameter
or parrameters, reaso
on why pertinnent combinatioons
a due to chooose according to the necessiities of phase unare
u
w
wrapping
in thhe wished appliications. Somee of these param
met
ters
can be: inntensity modullation functionn, the spatial fref
q
quency
inform
mation of the fringe
f
pattern, the phase difffere
ence
between neighboring
n
piixels and somee combinationss of
t
these
parameteers. Intensity modulation
m
funnction commonly
i used for phaase unwrappin
is
ng in noncontacct optical profi
filom
metry.
The vaalue of modullation functionn in the areas of
l
local
shadow and abrupt diiscontinuities is
i lower than the
v
value
in the otther zones, so that
t
it can servve as an excelllent
g
guide
to determ
mine a path off phase unwrappping. For temppor carrier frinnge pattern th
ral
he reliability parameter
p
can be
c
calculated
as:
M(x, y) = I mod(x, y) =
2
N
N
2 ⎡
⎛ 2πn ⎞⎤
⎛ 2πn ⎞⎤ ⎡
⎟⎥
⎟⎥ + ⎢ I n (x, y) cos⎜
⎢ I n (x, y) siin⎜
N ⎣⎢ n=1
⎝ N ⎠⎦⎥
⎝ N ⎠⎦⎥ ⎢⎣ n=1
∑
∑
Fig.1 Fringe
F
pattern projection
p
on an
a object (face)).
2
(1)
And for spatiaal carrier fringee pattern the reeliability param
A
met can be calcuulated as:
ter
M ( x, y ) = I mood ( x, y ) =
[Im
m(c ( x, y ) )]2 + [Re
R (c ( x, y ) )]2
(2
)
Fig.2 Sccheme of reliabillity-guided phasse unwrapping allgorithm of
the objecct(face) and its binary
b
mask
W
With:
1
c( x, y ) = I mod ( x, y ) exp(iφ ( x, y ))
2
(3)
o
from fringe pattern by
c ( x , y ) , whhich could be obtained
Fourier transfo
F
form [2], filterring and inverrse Fourier traansf
form.
2 Reliability-guided phase unwrapping algorithm
2.
a
By means of a fringe pattern
B
n, and the experimental deviice:
system of imaages acquisitio
on adjusted byy a CCD cam
mera
(
(SONY
XC-E
ES50/ES30 XC
C-ST70), an im
mages acquisittion
t
target
(Imaq PXI/PCI-1409),, a SANYO proojector with reesol
lution
of 600x800 and a com
mputer, the frinnge pattern cann be
g
generated
and then deformeed on the test object
o
for the rec
construction
3
3-D.
With the help of camerra CCD, the disd
t
torted
fringe pattern
p
is cap
ptured. The saame proceduree is
f
followed
for thhe fringe patterrn projected onn the plane of the
o
object,
for the case of the profilometry of surfaces
s
by traansf
formed
of Fouurier [3]; requiire only those two frames to be
p
processed
Fig.1.
Intensitty modulation function (IM
MM) is obtaineed from a
fringe pattern
p
by Fourrier transform and
a its values in
i areas of
local shhadow and abrrupt discontinuuities are loweer than the
values in
i other zones, so that it cann be used for spatial
s
unwrappinng algorithm Fig.2, this algorrithm is as folloows:
First, Fourier
F
transforrm method off fringe patternn Fig.3. is
used to obtain the wraapped phase map
m of a face, remember
this veryy important meethod.
Second,, determine the maximum vaalue or startingg point of
phase unwrapping
u
andd mark it in thhe binary maskk as ‘1’ or
‘X’, theen order its 4 neighboring
n
pooints accordingg the IMM
value from
fr
higher to lower one, thhe pixel with maximum
IMM vaalue is put as ‘X’, for examplle 5x5 pixel arrea, Fig.2.
Begin its
i phase unw
wrapping on thhe basis of thhe starting
point[4]] and then marrk this point inn the binary maask as ‘1’,
after orrder the new wrapped
w
pointss and determinne the new
point with
w maximum IMM value again.
a
Repeat each
e
time,
and when all the poinnts of the binarry mask are puut as ‘1’ or
p
unwrappping is finisheed, for ex“white””, means that phase
ample Fig.4
F
and Fig.55.
Unwrappping it makes iteratively, acccording to as it
i is growing the set of pixels thhat conforms thhe vicinity, unntil including the phase
p
map totaally, throwing really
r
the waited for
151
J. Garzón et al.:: Spatial Phase Unnwrapping by Meaan of a Reliability-Guided Path
Fig.3 Fourier transform methodd of fringe patterrn and wrapped phase
p
map.
[1]
[2]
[3]
[4]
Fig.4 Unwrappped phase of a faace after that binnary mask has finnisshed.
Fig.5 Image 3D of unwrapped
u
phasse map.
ttotal correctionn. The conditiion to finish thhe process of unu
w
wrapping
of phase,
p
dictatess the binary mask
m
to it, whhich
b
becomes
a mattrix finally of ones.
o
A
Acknowledgm
ments
The authors accknowledge th
T
he financial suppport of the UniU
v
versidad
Ponttificia Bolivarriana, Medelllín-Colombia, its
C
Centro
de Ciencia Básica an
nd the Centro Integrado paraa el
D
Desarrollo
de la
l Investigació
ón CIDI.
R
Referencias
152
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u
w,” Opt. Eng. 422, pp. 245-261, 2004.
2
alggorithm: a review
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X. Su and W. Chen, Fourier Transform
T
Profiilometry: a
revview, Opt. Eng. 35, pp. 277, 20001.
B.W
Wang, Phase unnwrapping by bloocks, Measurement. No. 25
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