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becomes taller and its shape approaches a circle. It includes more of the scene (only
those parts located directly above and below the observer are omitted) but with more
distortions, especially at the top and bottom.
As in the cylindrical panoramic projection, horizontal lines are projected on the
band as sinusoids, but we now show that even vertical lines, which in the cylindrical
projection are projected straight, now become curved. Figure 4.27b shows the barrel
from above (i.e., looking in the
y
direction). A long vertical line (parallel to the
y
axis)
is shown, and we assume that a general point on this line is projected to a point
v
on
the barrel. After the barrel is unrolled, the
y
coordinate of point
v
varies in the range
[
Rθ,
+
Rθ
]. The
x
coordinate depends on the
y
coordinate and equals the radius of
the barrel at height
y
times the angle
φ
. The radius of the barrel at height
y
is easily
−
seen to be
R
cos(
y/R
), so point
v
is located on the band at position
φR
cos(
y/R
)
,y
,
where
−Rθ ≤ y ≤
+
Rθ
. This position varies from
φR
cos(
−θ
)
, −Rθ
to (
φR,
0) to
φR
cos(
θ
)
,Rθ
when
y
varies from
Rθ
to0to
Rθ
. The projection of the vertical line
onthebandisthereforethethickcurveshowninFigure4.27c. Itiseasytoseethat
the closer
θ
is to
π/
2 (or 180
◦
), the smaller cos
θ
is and the more curved (distorted) the
projection.
Given an arbitrary point
P
=(
x, y, z
), it is relatively easy to calculate the
xy
coordinates of its projection on the band. Figure 4.27b shows the situation on the
xz
plane and makes it clear that the
x
coordinate of the projected point on the band is
the arc
Rφ
.Sincetan
φ
=
x/z
,wegetthe
x
coordinate as
R
arctan(
x/z
). Similarly,
Figure 4.27a shows that the
y
coordinate of the projected point on the band is the
arc
Rα
or
R
arctan(
y/z
). Thus, the projected point has band coordinates (
Rφ, Rα
)or
−
R
arctan(
x/z
)
,R
arctan(
y/z
)
.Both
φ
and
α
canvaryintheinterval[
−
π,
+
π
], so the
projected
x
coordinate varies in [
πR,
+
πR
]. The projected
y
coordinate varies in the
same interval, but it is clear from the figure that any point
P
for which
−
|
α
|
is greater
than
is projected outside the barrel (i.e., on one of the sphere parts that have been
removed) and should consequently be rejected.
The
IPIX Wizard
software [IPIX 05] can create a spherical panorama from two
scanned fisheye photographs.
To some people, spherical panoramas may seem less interesting (and perhaps also
less useful) than cylindrical panoramas, as the following 1998 quotation, from David
Palermo, a virtual-reality professional, suggests: “Our market is not craving [sphere-
shaped panoramas] right now. You can convey a sense of place without looking at the
sky or floor.”
|
θ
|
For me it remains an open question whether [this work] pertains to the realm
of mathematics or to that of art.
—M.C.Escher
4.6.1 Curvilinear Perspective
However, Figure 4.28 (courtesy of Dick Termes) suggests that it is possible to create
full spherical panoramas that show everything an observer sees in front of him and
behind him, while also maintaining their artistic value in spite of the many vertical
and horizontal distortions. The reader should especially note that the few vertical and
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