3D Surface Acquisition for
FMT Using High-Accuracy
Fringe Projection Profilometry
Seminario ISP
Andrés Marrugo - Feb 2016
2009 IEEE Nuclear Science Symposium Conference Record
M15-6
3D Surface Acquisition for FMT Using HighAccuracy Fringe Projection Profilometry
Juan E. Ortuño, Pedro Guerra, Member, IEEE, George Kontaxakis, Senior Member, IEEE, María J. LedesmaCarbayo, Member, IEEE, and Andrés Santos, Senior Member, IEEE
Abstract–An experimental structured light projection system
which includes a miniaturized projector is described. The system
has been designed to be integrated in a fluorescence molecular
tomography (FMT) prototype in order to reconstruct the surface
of mice and phantom studies. A high-accuracy phase map is
retrieved with phase-shifted sinusoidal fringes. Phase error due
to the nonlinear gamma function of the pico-projector is
calibrated and compensated. Robust phase unwrapping is
performed with an additional Gray-code projection sequence. An
automatic phase-to-height non-linear calibration scheme has
been applied using objects located in the extremes of the field of
view. The accuracy of the proposed method has been tested with
a realistic mouse model and ray-tracing software.
F
I. INTRODUCTION
molecular tomography (FMT) is a
biomedical imaging technique that quantifies the
distribution of fluorescent biomarkers in small animals (e.g.,
mice) [1]. Tomographic reconstruction of fluorophore
distribution at depths of several millimeters to centimeters has
been possible thanks to the emergence of mathematical
models that describe photon propagation in biological tissues
LUORESCENCE
of heights relative to the reference plane. The 3D surface can
then be reconstructed based on triangulation once the system
is properly calibrated.
In this paper, we propose the integration of a fringe pattern
projection system in a compact FMT prototype using
miniaturized digital projectors to acquire the 3D shape of mice
and phantom studies.
II. MATERIALS AND METHODS
The structured light projection system has been designed to
be integrated in a novel FMT CCD camera based system that
works in non-contact geometry [7]. The light emitted by the
fluorophore is captured with a CCD camera located on top of
the scanner and mechanically fixed to the system cabinet in
the way that the focal plane is parallel to the reference plane
where the mouse is placed.
An experimental setup was mounted to test the proposed
code light projection system (Fig. 1) prior to the integration
with the FMT prototype. The assembly consist of a low-cost
CCD camera (Sony DSC-W1) placed in the top of the device
Motivation
Fluorescence Molecular Tomography is an optical imaging
technique which aims at reconstructing the 3D distribution of
fluorescent markers in bio-tissues based on surface
measurements of emitted photons and a model of light
propagation.
[1] M. L. James and S. S. Gambhir, “A Molecular Imaging Primer: Modalities, Imaging Agents, and
Applications,” Physiological Reviews, vol. 92, no. 2, pp. 897–965, Apr. 2012.
Motivation
• FMT has poor spatial resolution.
• To improve resolution the 3D surface of the mouse is
acquired.
•
•
Sub-millimeter precision is desirable for accurate
quantitative measurements.
Authors propose integration of fringe projection
system with FMT prototype.
[1] M. L. James and S. S. Gambhir, “A Molecular Imaging Primer: Modalities, Imaging Agents, and
Applications,” Physiological Reviews, vol. 92, no. 2, pp. 897–965, Apr. 2012.
II. MATERIALS AND METHODS
projector has a minimum focus range of 210 mm, angular
pressed
and
immersed
into
a
chamber
The structured light projection system has been designed to
aperture of 35º and HGVA resolution. The setup has
y diffusive
fluid
This
be integrated matching
in a novel FMT
CCD[3].
camera
based system that
dimensions
of 20×30×30 cm, appropriate to be further
in non-contact
geometry
[7]. The light emitted
by the
nalworks
photon
diffusion
and attenuation,
fluorophore is captured with a CCD camera located on
top of
integrated
in the FMT prototype, replacing the DSC-W1
le the
or scanner
desirable
for
in
vivo
studies.
To
and mechanically fixed to the system cabinet in
camera by the electron-multiplier CCD camera also used to
ns,theFMT
of the
small
usingtofreeway that
focalanimals
plane is parallel
the reference plane
where thehas
mouse
is placed.
schemes
been
proposed [4]. These collect fluorescent emissions.
An experimental setup was mounted to test the proposed
he code
3D surface
of the
mouse
it has
light projection
system
(Fig. and
1) prior
to the integration
with the FMT
prototype. The
assembly consist
millimeter
precision
is desirable
for of a low-cost
CCD camera[5].
(Sony DSC-W1) placed in the top of the device
asurements
and a pico-projector (model PK101, Optoma Technology, Inc)
nge
projection)
3D surface
acquisition
with
DLP® technology
(Texas instruments,
Inc.). The picoprojector
has a minimum
focus range[6].
of 210
plied
for multiple
applications
A mm, angular
of 35º and HGVA resolution. The setup has
3Daperture
measurement
system consists of a
dimensions of 20×30×30 cm, appropriate to be further
rojector,
projects
a series
ofthe DSC-W1
integrated which
in the FMT
prototype,
replacing
camera by
the electron-multiplier
CCD camera
f coded
patterns
onto the object.
The also used to
collect fluorescent emissions.
rted
patterns caused by the variations
Experimental setup
mber 12, 2009. This work was supported in part
ce and by the Spanish Ministry of Science and
2008-06715-C02-02 and CDTEAM-CDTI.
is M. J. Ledesma-Carbayo and A. Santos are
chnology Group, Dpto. Ingeniería Electrónica,
drid, and with the Biomedical Research Center
als and Nanomedicine (CIBER-BBN), Spain
ma, andres@die.upm.es).
omedical Research Center of Bioengineering,
medicine
(CIBER-BBN),
Spain
(email:
Fig. 1. Experimental setup mounted to test the proposed coded light projection
system. The device is composed of a DLP pico-projector and a CCD camera
that collects deformed patterns reflected by the object.
Fig. 1. Experimental setup mounted to test the proposed coded light projection
del fondo continuo y el contraste de las
mación de las franjas
[2], [3], tal y como se
franjas, respectivamente. El término f es la
frecuencia espacial media de las franjas y
observa en la Fig. 2.
proceso físico de codificación de la inmación topográfica se realiza de la si-
Revista INGE CUC,Volumen 8, Número 1, Octubre d
o
El térm
trón d
ción q
franjas
ϕ de la
recons
proyec
la func
Considerando un sistema formador de imágenes no-telecéntrico, el ángulo entre los
ejes de proyección y observación y la influencia de las aberraciones geométricas, la
distribución en intensidad de las imágenes
obtenidas con la cámara CCD sobre el plano de referencia tiene la forma dada por (1).
Para d
versos
vos y
el mét
Shiftin
la fase
lumino
su res
φi con
como
(a)
Fig. 1 Montaje experimental de proyección de franjas
(1)
(b)
Donde Io y A corresponden a la intensidad
La ecu
tra en
del fondo continuo y el contraste de las
ϕ es la fase inicial de las franjas, que cofranjas, respectivamente. El término fo es la rresponde a la deformación inicial sufrida A part
por las franjas. Al ubicar sobre el plano de
las fun
referencia un objeto, la ecuación se modififrecuencia
espacial
media
de
las
franjas
y
[1] A. L. Gonzalez, J. Meneses, and L. León, “Proyección de franjas en metrología óptica y (8).
Fig. 2 Patrón de franjas proyectado sobre (a) un plano de
referencia y (b) mano humana
o
facial,” Revista INGE CUC, vol. 8, no. 1, pp. 191–206, 2012.
ca de acuerdo con (2).
(2)
! g4 − g2 "
calibration #2,(2)is ca
(%xg, 3y −) g1 $&
(3)
h ( x, y ) =
ϕ x, y mod π = arctan #
ϕ ( x, y )
m ( x, y ) + n ( x, y ) ∆(2)
$
A. Fringe pattern projections
Phase error due to the gamma function
DLP
h of
k picox, the
y =
x, y ∆
% g3 − g1 &
is compensated
calibration
[9].
where h(x,y)
is the heightwith
overathe
referencelook-up-table
plane in the (x,y)
In each acquisition, four sinusoidal patterns gi with phase projector
Phase
performed
with an
additionalbetween
Gray-code
point, unwrapping
!" is
phase
difference
the
shifts of 90º are projected:
Phase error due to the gamma function
of the
theisunwrapped
DLP
picoIn
theory,
this
object andof the
reference
and to(m,n)
are the
parameters
sequence
5 bits,
whichplane,
is robust
ambiguities
in surface
projector !is compensated
with a calibration
look-up-table
π "
acquisition
toa [11].
gues
obtained with
aThe
leastsequence
squares[9].
minimization
algorithm
discontinuities.
of
patterns,
including
full
g i ( x, y ) = a ( x, y ) + b ( x, y ) cos # ϕ ( x, y ) + i $ , i = 0,1,2,3 (1)
This
procedure,
from
now
as
%
& performed with
2 is
illuminated
image is
shown in denoted
Fig. 2 in
the case
of aon,RayPhase unwrapping
an calibration
additional
Gray-code
several
measureme
calibration
#1 for of
short,
requires
of the acquisition of at least
tracing
simulation
ain
mouse
model.
sequence
of
5
bits,
which
is
robust
to
ambiguities
surface
where a is the average intensity and b is the modulation in two planar slabs of different height.
The novel autom
each point coordinate
(x,y). Wrapped phase
solved with: of patterns,
discontinuities.
The! issequence
including mapping,
a full denoted
Linear phase-to-height
nowison bas
as
this from
work
#2, case
is calculated
illuminated
2 in the
of a as:
Ray- parameters m and
! g 4 − g 2 " image is shown in Fig.calibration
ϕ ( x, y ) mod π = arctan #
(2)
$
g
g
−
tracing
simulation
of
a
mouse
model. h ( x, y ) = k ( x, y ) ∆ϕ ( x, y )
(4) (x,y
% 3
1 &
variation along
Simulation includes gamma nonlinearity effect of picoprojector, shadows, reflections and albedo differences.
! g4 − g2 "
(
)
Phase error due to the gamma function of the DLP picoprojector is compensated with a calibration look-up-table [9].
Phase unwrapping is performed with an additional Gray-code
sequence of 5 bits, which is robust to ambiguities in surface
discontinuities. The sequence of patterns, including a full
illuminated image is shown in Fig. 2 in the case of a Raytracing simulation of a mouse model.
ϕ ( x, y ) mod π = arctan
∆ϕ #
(
)
(
)
valueone
from
a set of
In theory, this approach only requires
calibration
acquisition to guess the parameter k(x,y)of
. However
in practice
the field
of view
several measurements are performed to increase the accuracy.
The novel automatic phase-to-height mapping presented in
this work is based on calibration #1 and assumes that
parameters m and n in (3) have smooth and quasi-linear
variation along (x,y). This assumption enables estimating their
value from a set of four measurements located in the extremes
Fig. 2. Ray-tracing simulation of mouse model, showing the structured light
of the field of view. This method is denoted as calibration #3.
patterns.
Phase of reference plane in occluded areas (e.g., below the
mouse) is extrapolated from surrounding values using
polynomial splines, so there is no need to perform a dedicated
background acquisition to repeat background acquisitions.
Occluded and shaded zones can be reduced repeating the
sequence of structured pattern projections at two different
pico-projector
orientations.
not occluded or shadowed
Fig. 2. Ray-tracing simulation of mouse model,
showing the
structuredFor
light
pixels, the height is calculated as the mean value of the two
patterns.
results.
Key aspects
✓
Phase error due to gamma function of DLP is compensated with
calibration look-up table.
✓
Phase unwraping perfomed with additional Gray-code sequence
of 5 bits.
✓
Plane of reference in occluded areas (e.g. below mouse) is
extrapolated from surrounding values w/ splines.
✓
Occluded & shaded zones → 2 acquisitions (2 DLP orientations).
✓
Not occluded pixels → mean of 2 acquisitions.
✓
Accuracy tested with simulations (POV ray tracing.)
of Vision™phase
Raytracer
(version
3.6)
softwarethe
[8]. as:
point, !" isPersistence
the
unwrapped
difference
between
π "
!
obtained
with a least squares minimization algorithm [11].
(1)
g ( x, y ) = a (includes
x, y ) + b ( x, ygamma
) cos # ϕ ( xnonlinearity
, y ) + i $ , i = 0,1,2,3
Simulation
effect
of
picoThis calibration procedure, ∆denoted
from now on, as
% (m,n) 2are
& the parameters
object and the
reference
plane,
and
ϕ
x
,
y
(
)
projector, shadows, reflections and albedo differences.
calibration
(3)
h (#1
x, for
y ) =short, requires of the acquisition of at least
obtained withwhere
a least
minimization
a is thesquares
average intensity
and b is the algorithm
modulation in [11].
+
∆
ϕ
m
n
x
y
x
y
x
y
,
,
,
( height.
) ( ) ( )
two planar slabs of different
each
point coordinate
(x,y).
Wrapped phase ! is solved with:
A.
Fringe
pattern
projections
This calibration procedure, denoted from now on, asLinear phase-to-height mapping, denoted from now on as
calibration
#2, is h(x,y)
calculated
as: height over the reference plane in the (x,y)
where
is the
phase
each
acquisition,
patterns gof
i with
g 2sinusoidal
! gfour
"the acquisition
calibration #1ϕIn
for
short,
requires
at
least
4 −of
(2)
y ) mod
( x, of
# g −g $
point, !" is the unwrapped phase difference between the
shifts
90º πare= arctan
projected:
h ( x, y ) = k ( x, y ) ∆ϕ ( x, y )
(4)
%
&
3
1
two planar slabs of different height.
object and the reference plane, and (m,n) are the parameters
πthe" DLP
Linear phase-to-height
denoted
now
Phase error duemapping,
to the gamma
pico-on as
! function offrom
obtained
with a least
this approach
only squares
requires minimization
one calibration algorithm [11].
g i ( x, y ) = a ( x, y ) + b ( x, y ) cos # ϕ ( x, y ) + i $ , i = 0,1,2,3 (1)In theory,
projector is compensated with a calibration
look-up-table
[9]. acquisition
to guess
the parameter
k(x,y) . However
in practice
This
procedure,
from now on, as
%
2 &
calibration
#2,
is
calculated
as:
• Calibration
• calibration
#1
(non-linear)
Calibration
#2denoted
(linear)
Phase unwrapping is
performed
with an additional Gray-code
i
Calibration
several measurements
are for
performed
increase of
thethe
accuracy.
calibration #1
short, torequires
acquisition of at least
tup, a linear empirical
non-automatic
have
been
also
sequence
of the
5 bits,
which intensity
iscalibrations
robust toand
ambiguities
inmodulation
surface
where
a is
average
b is the
in
The novel
automatic
phase-to-height
mapping
presented
in
two planar slabs of different height.
h
k
=
∆
ϕ
(4)
x
,
y
x
,
y
x
,
y
discontinuities.
The sequence
of patterns,
including
a fullwith:
using tested. Nonlinear
[10]
is!calculated
this work Linear
is basedphase-to-height
on calibration #1
and assumes
thatfrom now on as
each
point phase-to-height
coordinate
(x,y). mapping
Wrapped
phase
is solved
mapping,
denoted
illuminated
image
is
shown
in
Fig.
2
in
the
case
of
a
Rayparameters m and n in (3) have smooth and quasi-linear
e [8]. as:
calibration
#2, isassumption
calculated
as: estimating their
tracing
simulation
of a mouse
model.
g
−
g
!
"
variation
along
(x,y).
This
enables
In
theory,
this
approach
only
requires
one
calibration
4
2
picoϕ ( x, y ) mod
π = arctan
(2)
value from a set of four measurements located in the extremes
∆ϕ ( x, y ) #% g3 − g1 $&
h ( x, y ) = k ( x, y ) ∆ϕ ( x, y )
(4)
acquisition
(3) practice
= guess the parameter k(x,y) . However in
h ( x, y )to
of the field of view. This method is denoted as calibration #3.
(
)
(
)
(
)
m ( x, y ) + nare
y ) ∆ϕ ( x, y ) to increase the accuracy.
( x, performed
several measurements
Phase error due
to the gamma function of the DLP piconovel
automatic
phase-to-height
mapping
presented
in
projector
compensated
a calibration
h(x,y)
is theisheight
over thewith
reference
plane inlook-up-table
the (x,y) [9].
phaseThewhere
Phase
unwrapping
is performed
with
Gray-code
point, !"
isbased
the
unwrapped
phase difference
between
the that
this work
is
on calibration
#1 an
andadditional
assumes
sequence
of 5 bits,
which
is (m,n)
robustare
to ambiguities
in surface
object and
the
plane,
and
the
parameters
m
andreference
n in (3)
have
smooth
andparameters
quasi-linear
The sequence
of patterns,
including
obtaineddiscontinuities.
with a least squares
minimization
algorithm
[11]. a full
3 (1)
variation
(x,y). procedure,
This
estimating
This along
calibration
denoted
from
on, of
as atheir
illuminated
imageassumption
is shown
in enables
Fig.
2 innow
the case
Rayvalue calibration
from atracing
set#1offor
four
measurements
located inofthe
extremes
short,
requires
of the
acquisition
at least
simulation
of a mouse
model.
on in two planar slabs of different height.
of the field of view. This method is denoted as calibration #3.
ith:
Linear phase-to-height mapping, denoted from now on as
2. Ray-tracing
simulation
calibrationFig.
#2,
is calculated
as: of mouse model, showing the structured light
patterns.
(2)
In theory, this approach only requires one calibration
acquisition to guess the parameter k(x,y) . However in practice
several measurements are performed to increase the accuracy.
The novel automatic phase-to-height mapping presented in
this work is based on calibration #1 and assumes that
parameters m and n in (3) have smooth and quasi-linear
variation along (x,y). This assumption enables estimating their
•
value from a set of four measurements located in the extremes
of the field of view. This method is denoted as calibration #3.
Calibration #3 (non-linear)
h ( x, y ) = kPhase
(4) the
( x, y )of∆ϕreference
( x, y ) plane in occluded areas (e.g., below
picoe [9].
-code
urface
a full
Ray-
mouse) is extrapolated from surrounding values using
In theory,
this splines,
approach
only
requires
one calibration
polynomial
so there
is no
need to perform
a dedicated
background
acquisition
to repeat
acquisition
to guess the
parameter
k(x,y) background
. However acquisitions.
in practice
Occluded and are
shaded
zones can
be reduced
the
several measurements
performed
to increase
therepeating
accuracy.
sequence of structured pattern projections at two different
The novel
automatic phase-to-height mapping presented in
pico-projector orientations. For not occluded or shadowed
this workpixels,
is based
onis calibration
and value
assumes
the height
calculated as #1
the mean
of the that
two
parameters
m
and
n
in
(3)
have
smooth
and
quasi-linear
results.
Fig. 3. (a) Stepped pyramid-shaped calibration objects situated together with
the object to be measured; (b) detail of unwrapped phase of stepped pyramidal
objects and selected points used to non linear calibration marked in red color.
variation along (x,y). This assumption enables estimating their
B. System calibration
value from
of four simulation
measurements
located
in the extremes
Fig. a2.set
Ray-tracing
of mouse
model, showing
the structured light
The
correspondence
between
phase
values
and
height
over
Stepped
pyramid-shaped
objects were selected
as
patterns.
[10]of the
J. Kofman,
“Comparison
of
linear
and
nonlinear
calibration
methods
for phase-measuring
profilometry,”
Optical
field
of
view.
This
method
is
denoted
as
calibration
#3.
the reference plane is calculated with empirical methods that
calibration objects. The shape of the objects guarantees that it
Engineering,
vol.
no. the
4, p.
043601,
Apr. 2007.
not
need46,
to reference
know
extrinsic
location
and orientation)
Phase
of
plane
in(i.e.,
occluded
areas
(e.g., belowwill
thebe different heights at nearby locations and will be visible
and optical parameters of the pico-projector. In this work we
Jia, Kofman, and English: Comparison of linear and nonlinear calibration methods…
matic diagram of the 3-D shape-measurement system
fringe-projection technique.
Fig. 2 Relationship between the phase of the projected fringe pattern and the height of the object.
Jia, Kofman, and English: Comparison of linear and nonlinear calibration methods…
distribution. The quantity I!x , y" is the intensity detected by
the CCD, and "!t" is the time-varying phase shift.
By applying the phase-shifting algorithm,1 which is the
widely used suitable method, the phase distribution can be
obtained. The minimum number of measurements of the
intensity that are required to reconstruct the unknown phase
distribution is three.1 Here a general case of the N-phase
algorithm is considered, in which the N measurements are
equally spaced over one modulation period. The phase distribution can be expressed as follows2:
ear and nonlinear methods of phase-measuring
and measurement techniques. In this paper, the
lationship between the phase and the height of
urface is formulated by both linear and nonlinns. The nonlinear mapping function is obtained
mple geometrical derivation, and a comparison
ween the two calibration approaches in terms of
nt accuracy. A least-squares method is applied in
and nonlinear calibrations to obtain the systembration parameters. Both numerical simulation
N
ement experiments are carried out to analyze the
f the two calibrations. In Sec. 2, the fringeIi!x,y" sin "i
%
echnique is introduced as background to the fori=1
, i = 1,2,3, . . . ,N,
!2"
!!x,y" = − tan−1 N
eveloped, and the linear and nonlinear formulahe phase-to-height mapping are described,
Ii!x,y" cos "i
%
Fig. 4 methods.
Results of
the real linear and nonlinear
calibration: !a" distribution of K!x , y" of the linear
linear and nonlinear calibration
Seci=1
calibration;
!b,c" distributions of m!x , y" and n!x , y" of the nonlinear calibration.
ents calibration and measurement
experiments,
where the phase shift "i = 2#i / N. The phase distribution
ulation and real, for error analysis of both linear
!!x , y" is wrapped into the range 0 to 2#, due to the periar calibration methods. Section 4 presents the
odic characteristic of the arctangent function. A phase une simulation and real experiments in calibration
15–19
and rms
errors the
over
all runs, using the linear and nonlinear
!"and
= !"the
!x,y"
+ #!x,y",
is necessary
to convert
modulo
wrapping process!19"
!!x,y"
ement
discusses
accuracy
of the linear
[10] J.
Kofman,
“Comparison
and data
nonlinear
methods
phase-measuring
profilometry,”
Optical
2# phase
into theircalibration
natural
range,
whichThe
is afor
continucalibrations.
results
for the linear and
nonlinear methar calibration
methods.
The conclusion
is givenof linear
is calculated
by 043601,
Eqs. ous
!7"Apr.
and2007.
!9" with
where
!"!x , y" vol.
representation
of the phase
map.
Engineering,
46, no. 4, p.
ods,
discussed in Sec. 4, are compared in Fig. 5!a".
known coefficients K!x , y", m!x , y", n!x , y", and h!x , y"; and
e gamma function of the DLP picoIn theory, this approach only requires one cali
with a calibration look-up-table [9]. acquisition to guess the parameter k(x,y) . However in p
ormed with an additional Gray-code several measurements are performed to increase the acc
h is robust to ambiguities in surface
The novel automatic phase-to-height mapping prese
uence of patterns, including a full this work is based on calibration #1 and assum
wn in Fig. 2 in the case of a Ray- parameters m and n in (3) have smooth and quas
ouse model.
variation along (x,y). This assumption enables estimatin
• Calibration #3 assumes nonvalue from a set of four measurements located in the ex
linear parameters change of the field of view. This method is denoted as calibrati
smoothly.
Calibration
•
Estimation of parameters from
4 points in (x,y) w/ Delaunay
Triangulation and polynomial
interpolation.
of mouse model, showing the structured light
ne in occluded areas (e.g., below the
from surrounding values using
intrinsic and extrinsic parameters for the pinhole camera
model using the Zhang method [12]. The procedure employs a
checkerboard patterns placed at different and unknown
orientations. Camera calibration only needs to be calculated
when the camera, fixed on the top of the setup, is moved or
zoomed. Perspective correction does not depend of the picoprojector orientation.
segmentation of pyramid-shaped objects is a fully automatic
Results
process that includes simple but robust histogram-based
segmentation and
operations. A detail of
III.morphological
RESULTS
unwrapped phase of stepped pyramidal objects and selected
points used
in calibration is shown in Fig. 3b.
A. Ray-tracing
models
Interpolation of calibration parameters over the rest of the
Accuracy of the proposed methodology has been tested with
image is performed with a Delaunay triangulation followed by
simulateda smooth
acquisitions
of a phantom
shown in Fig. 4b,
quintic polynomial
interpolation.
composed ofCamera
19 steps
with height
ranges for
fromperspective
1 to 19 mm.
calibration
is needed
correction
(i.e., correction of x-y locations over the reference plane after
height estimation). Camera calibration algorithm obtains
intrinsic and extrinsic parameters for the pinhole camera
model using the Zhang method [12]. The procedure employs a
checkerboard patterns placed at different and unknown
orientations. Camera calibration only needs to be calculated
when the camera, fixed on the top of the setup, is moved or
zoomed. Perspective correction does not depend of the picoprojector orientation.
triangulation (2×105 triangles). The combination of projection
#a and #d was used in order to minimize shaded regions. A
rendering of the 3D surface is shown in Fig 6c.
Fig. 5. Upper plot: Root mean square error (RMSE) in four pico-projector
locations with nonlinear calibration #3; lower plot: RMSE mean of four picoprojector locations using calibrations #1, #2 and #3.
According to plot of Fig. 5, error with calibration #1 is
negligible in the range of 0-20 mm height, so the differences
of height obtained with this method with respect calibrations
#2 and #3 can be considered as an indicator or accuracy of the
Fig 4. (a) PovRay Rendering of mouse model and stepped pyramid-shapped
former methods. Root mean squared difference of measured
calibration objects; (b) phantom usedIII.
to test
the accuracy of algorithms,
RESULTS
heights in the mouse surface model between calibration #1 and
composed of steps of 1 mm height.
#3 and between calibration #1 and #2 can be observed in Fig.
A. Ray-tracing models
Four different
acquisitions
were
done changing
polar
Accuracy
of the proposed
methodology
has been tested
with 6a-b. The mean values are 58 $m and 192 $m, respectively.
acquisitions
of a" phantom
shown infringes:
Fig. 4b,
deflectionsimulated
! and azimuthal
angle
of the projected
of 19"=#25º;
steps withprojection
height ranges
1 to 19
mm.
Projectioncomposed
#a: !=30º,
#b:from
!=30º,
"=165º;
projection #c: !=15º, "=30º; and projection #d: !=25º, "=195º.
Fig. 5. Upper plot: Root mean square error (RMSE) in four pico-projector
Root mean squared error (RMSE) of average measured
locations with nonlinear calibration #3; lower plot: RMSE mean of four picoheights in each step is shown in Fig. 5. The upper plot shows
projector locations using calibrations #1, #2 and #3.
results for each pico-projector position using calibration #3,
According to plot of Fig. 5, error with calibration #1 is
while the lower plot shows the mean value of RMSE over the
negligible in the range of 0-20 mm height, so the differences
four projections with calibrations #1, #2 and #3. Three slabs of
epped pyramid-shapped
ccuracy of algorithms,
e changing polar
projected fringes:
#b: !=30º, "=165º;
#d: !=25º, "=195º.
average measured
e upper plot shows
ing calibration #3,
of RMSE over the
d #3. Three slabs of
calibrations #1 and
alibration #1 in the
for calibration #2,
mum RMSE with
height but linear
of 0.44 mm at 19
g. 4a was obtained
The voxellized data
. The external 3D
projector locations using calibrations #1, #2 and #3.
According to plot of Fig. 5, error with calibration #1 is
negligible in the range of 0-20 mm height, so the differences
of height obtained with this method with respect calibrations
#2 and #3 can be considered as an indicator or accuracy of the
former methods. Root mean squared difference of measured
heights in the mouse surface model between calibration #1 and
#3 and between calibration #1 and #2 can be observed in Fig.
6a-b. The mean values are 58 $m and 192 $m, respectively.
Results
Fig. 6. Mouse surface model: (a) Height difference of calibration #3 with
respect to calibration #1; (b) height difference of calibration #2 with respect to
calibration #1; (c) rendering of result using calibration #3.
B. Experimental setup
Slabs 4, 8, 12 and 16 mm height were used to calibrate the
experimental setup for methods #1 and #2. The acquisitions of
e
f
d
e
,
h
r
9
d
a
D
y
Height
4 mm
8 mm
12 mm
16 mm
RMSE (!m), method #2
256
215
48
193
RMSE (!m), method #3
76
128
113
99
Results
Fig. 6. Mouse surface model: (a) Height difference of calibration #3 with
respect to calibration #1; (b) height difference of calibration #2 with respect to
calibration #1; (c) rendering of result using calibration #3.
Fig.7 shows the reconstructed 3D shape of an agar mouse
phantom and a computer adapter. One pico-projector was used
in these cases so there are shadowed zones. Problematic shiny
surfaces were eliminated in this result.
TABLE 1. HEIGHT ERRORS IN SLABS OF CONSTANT HEIGHT
Height
4 mm
8 mm
12 mm
16 mm
RMSE (!m), method #2
256
215
48
193
RMSE (!m), method #3
76
128
113
99
Fig.7 shows the reconstructed 3D shape of an agar mouse
phantom and a computer adapter. One pico-projector was used
in these cases so there are shadowed zones. Problematic shiny
surfaces were eliminated in this result.
[
[
[
B. Experimental setup
Slabs 4, 8, 12 and 16 mm height were used to calibrate the
experimental setup for methods #1 and #2. The acquisitions of
these slabs were also employed to evaluate the accuracy of
calibration method #3. Table 1 compares the RMSE using
4011calibration #2 and #3.
M
m
m
p
[
[
ACKNOWLEDGMENTS
The authors wish to thank M. Desco, J.J Vaquero a
Aquirre, from the Unidad de Medicina y C[
Experimental, Hospital General Universitario Gre[
Marañon, Madrid, Spain, for its fruitful comments
motivating discussions. We also thank the staff o
mechanical workshop of the ETSIT for manufacturin[
[
phantoms
used in the experiments.
Fig. 7. (a) Full illuminated scene of computer adapter and agar mouse
phantom; (b) height map with one projection set. Shaded surfaces are not
detected and metallic parts have been eliminated to avoid errors; (c) 3D
rendering of reconstructed shapes. REFERENCES
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