Graphics Reference
In-Depth Information
c
can be estimated, and thus, the normal motion displacement
s
as the projection of
c
onto the surface normal
n
. From that, the velocity
s
of the surface along its normal can be
t
estimated.
Error estimation and compensation: Now assume
p
0
,
p
−
1
, and
p
1
are projector pixel coor-
dinates of
P
0
,
P
−
1
, and
P
1
. As the camera and projector are mounted horizontally, the pro-
jection pattern is invariant vertically, and only the
x
-coordinates are of importance. Hence,
the difference between the points in the projection pattern is
•
p
x
−
p
0
p
0
−
p
1
.
x
=
−
≈
1
As shown earlier, the intensity of an observed pixel in each of the three images depends on
I
0
, amplitude
I
mod
, phase
. In case of a planar surface, uniform, and diffuse,
I
0
and
I
mod
are locally constant on the observed surface. The shift
φ
(
x
,
y
), and shift
θ
θ
is constant. However, as
y
)
changes between the three images at three different
moments in time. At time
t
−
1
,
t
0
, and
t
1
camera observes the intensity as projected by
p
−
1
,
p
0
,
and
p
1
, respectively. By converting
φ
(
x
,
the observed surface is moving, the
N
x
ω
x
being the width of the projection pattern and
N
the number of projected wrapped phrase. The
true intensities are given by
x
into the phase difference we have
θ
=
2
π
;
I
1
(
x
,
y
)
=
I
0
(
x
,
y
)
+
I
mod
(
x
,
y
) cos (
φ
(
x
,
y
)
−
θ
+
θ
)
,
I
2
(
x
,
y
)
=
I
0
(
x
,
y
)
+
I
mod
(
x
,
y
) cos (
φ
(
x
,
y
))
,
and
(1.26)
I
3
(
x
,
y
)
=
I
0
(
x
,
y
)
+
I
mod
(
x
,
y
) cos (
φ
(
x
,
y
)
+
θ
−
θ
)
.
The corrupted shift phase is
θ
−
θ.
The relative phase error
φ
between observed distorted
phase
φ
d
and true phase
φ
t
is
arctan
tan
θ
−
θ
2
g
φ
d
=
,
(1.27)
arctan
tan
2
g
φ
t
=
,
(1.28)
φ
=
φ
d
−
φ
t
,
and
(1.29)
I
1
(
x
,
y
)
−
I
3
(
x
,
y
)
g
=
y
)
.
(1.30)
2
I
2
(
x
,
y
)
−
I
1
(
x
,
y
)
−
I
3
(
x
,
φ
t
can be expressed as Taylor expansion of
φ
d
:
1
2
sin (2
φ
d
)
y
2
O
y
3
,
1
4
sin (4
φ
t
=
φ
d
+
sin (2
φ
d
)
y
−
φ
d
)
−
+
(1.31)
2
tan
(
θ
−
θ
1
,
1
)
. For small motion,
only the first-term of the Taylor expansion is enough. In this case, the undistorted phase
values can be locally approximated to evolve linearly along a scanline of the camera:
φ
t
(
m
)
arctan
tan
2
(
2
y
)
1
=
−
θ
=
θ
−
+
where
y
2
2
tan
(
2
)
=
φ
c
+
φ
m
m
, where
m
is the
x
-coordinate of the pixel. Then a linear least-square
