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Submission
to Siggraph 2001: p
a
pers_0250

1

Sampling
-
Efficient Mapping of Spherical I
m
ages












A
b
stract

Many functions can map images to the sphere for use as enviro
n-
ment maps or spherical panoramas. We develop a new metric that
asymptotically measures how well these maps use a given number
of samples to pr
ovide the greatest worst
-
case frequency content of
the image ev
e
rywhere over the sphere. Since this metric assumes
perfect reconstruction filtering even with highly anisotropic maps,
we define another, conservative measure of sampling efficiency
that pen
alizes anisotropy using the larger singular value of the
mapping’s Jacobian. With these me
t
rics, we compare spherical
maps used previously in computer graphics as well as other ma
p-
pings from cartography, and propose several new, simple
mapping functions (d
ual equidistant and p
o
lar
-
capped maps) that
are significantly more efficient and exhibit less anisotropy. This
is true with respect to either efficiency metric, which we show
agree in the worst case for all but one of the spherical maps pr
e-
sented. Althou
gh we apply the metrics to spherical ma
p
pings,
they are useful for an
a
lyzing texture maps onto any 3D surface.

Additional Keywords:

cartography, environment map, Fourier tran
s
form,
signal processing, singular value, surface parameteriz
a
tion, texture.

1.

I
ntroduction

Spherical images representing the radiance field at a point are
useful to simulate reflections on shiny surfaces (env
i
ronment
maps)
[2]
[7]
[9]
[20]

and to produce arb
i
trary views from a point
(spherical panoramas)
[15]
[5]
. To enhance realism, we expect
more widespread use of such spherical images, including the use
of multip
le spherical and hemispherical images to a
p
proximate
radiance at many points throughout an enviro
n
ment
[11]
.

Rectangular arrays of samples (i.e., texture maps) are ubiquitous
in graphics systems, providing advantages including
loca
l
ity of
reference, simplicity of texel addressing, and ease of filtering for
reconstruction. We omit schemes like spherical wavelets
[17]

which exploit local differences in frequency content, but are more
c
o
m
plicated to implement in hardware. Still, many functions
have been used to map samples from a 2D te
x
ture domain to the
sphere, including the cube map
[7]
[15]
[20]
, Ope
nGL map
[8]
[13]
,
polar coordinate (latitude/longitude) map
[2]
, and dual stere
o-
graphic map
[9]

(called “dual

par
a
bolic” in
[9]
). How should
these mappings be compared? Is there anything be
t
ter?

To answer these questions, we propose the following crit
e
ria:

1.

Sampling efficiency:

the mapping function should su
p
port the
greatest worst
-
c
ase frequency content in the spherical image using
the fewest samples. We prefer a worst
-
case metric assuming blu
r-
ry spots or directions in the ma
p
ping are undesirable, even if they
allow greater fidelity elsewhere. Minimizing texture samples is
importan
t to conserve graphics memory and reduce memory
bandwidth.

2.

Anisotropy
: isotropic filtering methods (e.g., trilinear interp
o-
lation within a MIPMAP)
[21]

penalize maps having significant
anisotropy, since they cause excessive b
lur in the locally stretched
direction. Even hardware having aniso
t
ropic filtering capability
can not tolerate
unbounded

anisotropy, though anisotropy values
up to 2.0 are handled with hardware implement
a
tions such as
Nvidia’s TNT and GeForce
[10]
.

3.

Simplicity of projection function
: texture coordinate gener
a-
tion by the graphics system should be quickly compu
t
able.

4.

Ease of geodesic interpolation
: simple (e.g., linear) interp
o-
l
a
tion over the domain should closely approximate geo
desic
interpol
a
tion over the sphere.

5.

Ease of creation
: ideally, mappings should be suitable for
d
y
namic creation using the rendering hardware. Note though that
maps can be reparameterized using an additional rende
r
ing pass
over a textured, tessellated sph
ere.

Also important is the use of map components, such as the six
faces of the cube map or two hemispheres in the dual stereograp
h-
ic map. With enough components, or “pieces” in a piecewise
map, sampling efficiency can be increased to an asym
p
totic limit.

But more comp
o
nents create more complexity in evaluating the
mapping function, reduced locality of reference, and greater diff
i-
culty in MIPMAP construction. Moreover, eff
i
ciency gains may
be lost through area wasted in packing many small maps into the
te
xture rectangle. For these reasons, only maps having few co
m-
ponents are pract
i
cal.

This paper formally defines sampling efficiency for texture ma
p-
ping functions over regular 2D lattices by appl
y
ing results from
signal pro
c
essing
[14]

and crystallography
[3]

to spatially
-
varying
mappings. We define two metrics; one considering only freque
n-
cy content and another, strictly larger, that takes into account
anisotropy by considering the largest singular
value of the Jac
o
b
i-
an over the domain. Using these metrics, we analyze several
types of spherical maps used previously in computer graphics as
well as other mappings from cartogr
a
phy
[16]
[19]
, and

show that
the two metrics agree in the worst case. We propose several new
ma
p
pings, including the polar
-
capped map and dual equidistant
map, having superior sa
m
pling efficiency and reduced anisotropy.
We also provide theoretical limits on the eff
i
ciency

attainable by
any piecewise
-
differentiable map to the sphere. The results are
Map Name

Sa
m
pling

Req.

Maximum

Anisotropy

Map

Comps.

OpenGL





1

Cube

24

1.73

6

Dual Stere
o
graphic

32

1

2

Lat/Long

19.7



1

Dual Equ
i
distant*

19.7

1.57

2

Low Distortion Equal Area*

19.7

3.
45

1

Polar
-
Capped* (stretch i
n
variant)

14.8

1.41

3

Polar
-
Capped* (conformal)

16.5

1

3

Polar
-
Capped* (hex. reparam.)

13.5

1.73

3

Optimal Isome
t
ric**

12.57

1



Optimal**

10.9

1.73



Figure
1
: Summary of Spherical Map Properties
.

Si ngl e
-
s t a r r ed a r e
new ma ps des c r i bed i n t hi s paper; doubl e
-
s t a r r ed a r e t heor et i c a l l i mi t s
r a t her t han a ct ua l ma ps. Sa mpl i ng r equi r ement i s pr opor t i ona l t o t ext ur e
a r ea r equi r ed f or a des i r ed f r equency c ont ent i n t he wor s t c a s e ( Sec t i on
2.2
). OpenGL’ s i nf i ni t e va l ue mea ns i t s number of s a mpl es gr ows f a s t er
t ha n qua dr at i c a l l y a s f r equenc y c ont ent i nc r ea s es. Sma l l e r a ni s ot r opy
va l ues a r e bet t er and r epr es ent degr ee of l oc a l c onf o
r
ma l i t y ( Sec t i on
2.3
).


Submission to Siggraph 2001: p
a
pers_0250

2

su
m
marized in
Figure
1
. The new maps have simple
-
to
-
evaluate
projection functions that can be implemented using existing
hardware to
reduce storage and bandwidth costs and avoid anis
o-
tropic blur of spher
i
cal images.

2.

Sampling Efficiency and Other Local
Ma
p
ping Properties

The
spherical mapping function
,
, maps
(u,v)

points in the
domain,
D
, to 3D unit
-
length ve
ctors. Existing graphics systems
use rectangular domains and sampling patterns (see Figure 1(a)).
Each
(u,v)

sample from the tabulated spherical image repr
e
sents
the radiance associated with vector
.

Analysis of sampling efficienc
y is first developed for regular la
t-
tices in the plane (Section
2.1
). We then e
x
tend this analysis to
spatially varying mappings by locally projecting the sampling grid
onto the tangent plane at each point and
applying the same planar
analysis (Se
c
tion
2.2
). Section
2.3

presents mathematics for the
analysis of local distortion (aniso
t
ropy).

2.1

Sampling in the Plane

Following

[14]
[6]
, a regular sampling pattern in the plane can be
represented as a
sampling matrix
,
, where

and

are lin
e
arly

independent vectors to two nearest
-
neighbor sample
locations. For example, isotropic rectangular and hexagonal sa
m-
pling have the following sampling m
a
trices:


where


represents the sample spacing. The corresponding sa
m-
ple geometries are
shown in
Figure
2
. Sample locations are
derived using
where
is a vector of integers
(prime denotes transpose).

Denote the continuous signal to be sampled as
. The corr
e-
s
ponding sampled version is
. Taking the
Fourier transform of
and then its inverse, we obtain


where
is in units of radians per unit length. Doing
the same for the discrete si
g
nal,
f
, we get


(
1
)

where
is in units of rad
i
ans. So

.

Substitu
t
ing

yields

.

The double integral in the

plane can be broken into an
infinite sum of integrals each cove
r
ing a square area of
:


Replacing
by
and
by
, simplifying,
and comparing to (1) implies that


where
, and the integer vector
. Thus
the discrete signal’s Fourier transform has a periodicity m
a
trix
related to the original sampling matrix
V

via its inverse transpose.

To eliminate aliasing, we need to bandlimit
such that its
Fourier transform,
, is zero outside a finite region in fr
e-
quency space. The region is chosen so as to have no overlap
between neighboring periodically repeated
tiles

of
. For
the rectangular and he
x
agonal sampling matrix examples, we have


with frequency space tiles shown in
Figure
3
.

For such a bandlimited function, inside the periodic tile contai
n-
ing the origin, called the
baseband
,
B
, the continuous and discrete
Fourier transformed functions are r
e
lated via


implying we can reconstruct such bandlimited functions exactly
from the discrete samples. In fact, the reconstru
c
tion is given by

.

Given a sampling matrix
V
, we find the radius of the largest i
n-
scribed circle in
B
; th
at is, the maximum frequency co
n
tent in all
directions that a circularly bandlimited periodic signal sampled
using
V

can su
p
port. This is calculated by computing

and fin
d
ing half the minimum distance of

Figure
2
: Example Plane Samplings


Figure
3
: Fourier tr
ansform of discrete signals from example sampling
matrices.

B
ot h r ect angul ar and hexagonal s ampl i ng f r om
Fi gur e
2

pr
o-
vide for frequencies up to

in all directions without aliasing

a
freq
uency radius of

is the largest for an inscribed circle in each
periodic tile. But hexagonal sampling requires 13.4% fewer samples to
accomplish this, since the circles are packed more tightly in each hexag
o-
n
al tile, while the s
quare tiles waste space on outside diag
o
nal frequencies.

Submission to Siggraph 2001: p
a
pers_0250

3

the origin to the vector
s
, for int
e
gers
n
1

and
n
2

not
both zero. We thus define the
sampling spectral radius,


, of
V

:


(
2
)

For the rectangular and hexagonal example samplings,
.

It can be shown
[14]

that among all sampling matrices ha
v
ing a
given sampling spectral radius, the most efficient is always is
o-
tropic hexag
onal sampling, in that it covers the greatest area with
the fewest sa
m
ples. Sampling density can be measured by
, for which our example sampling matr
i
ces yield

and
.

2.2

Mapped
Sampling

We are interested in frequency content on the sphere, not the
plane. We therefore locally apply the previous se
c
tion’s analysis
of regular sampling patterns in 2D by approximating the spherical
samples in the neighborhood of
by 2D points projected to
the tangent plane there.

Let

and
. Then an
orthono
r
mal basis for the tangent space is given by


where
represents the diff
e
r-
ential area at
(u,v)
. Letting
,


(
3
)

is an approximatio
n good for small
. Since the Jacobian
J

maps perturbations in any d
o
main direction to the tangent plane, it
represents a “local” sampling matrix. To derive a 2D sampling
matrix, we project it into the plane using the orthonorma
l basis
, a pure rotation which preserves the spectral properties.
Defining
K

as the resulting transformed Jacobian:


For rectangular sampling with spacing

, the
local sampling m
a-
trix

mapped by
S

is t
hen given by
.

We can now analyze the spectral radius determined by
V

at any
point
(u,v)
as if
S

were everywhere equal to the local approxim
a-
tion in (3). By selecting the minimum sampling spectral radius
for any point in
D
, we d
etermine the highest permissible freque
n-
cy,
, in a circularly bandlimited function rectangularly sampled
with spacing

. Su
b
stituting the local sampling matrix into the
definition of sampling spectral radius (2) and simplifying
yields
the
local sampling spectral r
a
dius

:


where
n
1

and
n
2

are integers. We call


the
local sampling spe
c-
tral stretch

of the ma
p
ping. The minimum local sampling spectral
radius over the parameter
d
o
main,
, is then defined via


As expected, for a given mapping function, we can make
arb
i-
trarily large by redu
c
ing

, essentially, by adding more samples.

Mapping efficienci
es can be compared by fixing the sampling
spectral radius and determining the number of samples required to
gene
r
ate that desired frequency content. The number of samples
required to sample the domain
D

is given by


where
A

is t
he area of
D
. Defining the
(spectral)
sampling eff
i-
ciency
of a mapping,
, then the number of
samples required to achieve a minimum spectral radius

is
. Higher efficiency req
uires fewer samples for
a given spectral radius ev
e
rywhere in
D
. We call the reciprocal of
sampling efficiency the
(spectral)

sampling requirement
, d
e
fined


2.3

Local Distortion Anal
y
sis

A mapping function locally transforms an infi
nitesmal circle into
a general ellipse, with eccentricity and rotation that can be dete
r-
mined from the Jacobian of the ma
p
ping. The lengths of the
major and minor axes of this ellipse,

1

and

2
, are given by the
singular values of the J
a
cobian



where
,
, and
. So

1
repr
e
sents the maximum local stretch or length of the longest
vector mapped from the set of unit tangent vectors in the domain,
and

2
the maximum compression or length of the shortest
mapped ve
c
tor. The singular values are related to the differential
area via
. We also define the
an
isotropy

of the mapping,
,
, whose magnitude is useful for measu
r
ing the
severity of artifacts from MIPMAP filtering.

Mapping functions can be categorized by their local properties as
follows:


where

is a constant and th
e prope
r
ties are for all
.

The sampling spectral stretch of a mapping


is closely related to
its largest sing
u
lar value

1
.

because



and since

for integers not both 0. In the case of an
orthogonal parameteriz
a
tion (in which
),


=

1
.

We also have
where


is the surface area of the output
of the mapping function over
D

and
. This is because
isotropic hexagonal sampling is most efficient
[14]
, for which
. For an isometric
map,
.

In practice, graphics hardware properly reco
n
structs maps having
only limited anisotropy, something ignored by the
metric. For
example, the 2D shearing transformation
has
and thus constant sampling requir
e
ment for all
t

(i.e., no
matter how much it shears), while

which increases as t increases. We therefore define another
measure of sampling eff
i
ciency which models

hardware without
any anisotropic reconstruction filtering; i.e., only performing is
o-
tropic reconstruction filtering based on largest stretch. The
stretch sampling requir
e
ment
,
, is defined


Submission to Siggraph 2001: p
a
pers_0250

4

Then
. The smaller singular
value is i
g
nored assuming the source texture has
been properly bandlimited to account for the

worst case stretch (
) by prefiltering more over
the compressed direction corresponding to
.
Optimizing with respect to this metric then r
e-
serves the hardware’s limited aniso
t
ropic filtering
capabi
l
ity

to handle distortions arising from the
perspective transformation and/or reflections
rather than due to the parameterization.
1


3.

Simple Spherical Maps

We analyze the sampling efficiency and anisotr
o-
py of simple projections from cartography
[16]
,
some of which have appeared in computer
grap
h
ics. Where possible, we retain the term
i-
nology of cartography and relate it to that used in
co
m
puter graphics.

3.1

Azimuthal Proje
c
tions

Azimuthal projections transform the sphere into a
tange
nt or intersecting plane such that para
l
lels
(lines of constant latitude on the earth) are pr
o-
jected to circles. Mathematically, these
projections may be mo
d
eled as

(
4
)

where



arbitrarily
reparameterizes the parallel spacing, and

repr
e
sent the
poles.
Figure
5

summarizes the four most important azimuthal
projections from cartography: gn
o
monic, stereographic, Lambert
equal area, and equidistant, which are illu
s
trated in
Figure
4
.

The
gnomonic

map projects the s
phere onto a plane tangent to it,
using a perspective transfo
r
mation that looks directly at the point
of ta
n
gency. It is the projection used in the “cube map” spherical
image
[7]
[15]
, for each of i
ts six faces. The gnomonic map pr
o-
jects great circles on the sphere to straight lines in the map
domain, an advantageous property for texture coordinate interp
o-
lation. Gnomonic maps can also be directly produced using the
perspective projection of the re
ndering hardware. The
stere
o-
graphic

map is a conformal map that also has the property that
circles on the sphere project to circles in the d
o
main.
[9]

proposes
dual stereographic maps, one for each hemisphere, to paramete
r-
ize

environment maps. The
La
m
bert equal area

map is an area
-
preserving map also called the “gazing ball” or OpenGL map



1

Unfortunately, this model does not fully account for difficulties from
anisotropy unless the hardware can isolate the parameterization disto
r-
tion from the “real” (projection) distortion and compensate for pre
-
filtering by a larger kernel in the
direction. This is why we also an
a-
lyze each map’s anisotropy as well as its sampling requirement.

[13]
. Finally, the
equidistant

map is a stretch
-
preserving map that
also pr
e
serves distances to the pole. Al
though it has not been
used in computer graphics, its sampling efficiency exceeds that of
the other azimuthal projections, as we will de
m
onstrate shortly.

The local distortion properties of the four maps can be derived
from (4) using the respective defini
tion of

(
r
)

from the table. To
derive the sa
m
pling efficiency of these maps, first note that the
singular values of the projections are invariant over any circle in
the domain centered at the origin,
. It
can also be shown that the m
etric tensor entries are given by


where
so
d,e



0. Then


Maximizing
over a domain circle

requires minimizing the
denominator ab
ove, since the n
u
merator is invariant over
;
denote this maximum as
. Examining the denominator, which
we d
e
note
,







Original

Equidistant

Stereographic

Gnomonic

Lambert Equal Area

Figure
4
: Azimuthal Projections:

Spherical circles having a radius of 3.5


and arranged at intervals of 12.5


in latitude and 22.5


in long
itude are projected
back into parameter space using the four azimuthal projections. The red circle represents the extent of a hemisphere. For
the gnomonic map, which is unable
to represent the entire hemisphere in a finite domain, a single face from the

cube map is shown in red. Note that greater mapping stretch is indicated by smal
l-
er projected circles in the domain since we are projecting constant size spherical circles back into p
a
rameter space.


Equidistant

Gnomonic

Stereographic

Lambert

Equal Are
a






properties

stretch
-
preserving

great circles
project to lines

conformal, ci
r
cles
project to circles

area
-
preserving

r*

cove
r
s

hemi,sphere

,

,


,

,













1

2







inverse map





Figure
5
: Table of Mathematical Properties of Azimuthal Projections.

We define
. Note that the local properties do not vary as a function of the direction of
the vector
(u,v)
, only as a function of its length
r

or equivale
ntly,

. The maximum larger
singular value,
, and the sampling requirement,
, are taken over the portion of the sphere
from
[0,

],
thus allowing analysis of parts of the sphere from the pole to any parallel, such as
the hemisph
ere (

=

/2)
. The inverse maps implement the texture coordinate generation r
e-
quired by graphics sy
s
tems.

Submission to Siggraph 2001: p
a
pers_0250

5

It can be seen that the minimum

value of
over
, which we
denote
, occurs at
,
, since
d,e



0
and the int
e
ger factors can not both be zero. Thus,

so
; in words,
for any azimuthal projection, the max
i-
mum spe
c
tral stretch over a circle centered at the pole is the same
as the larger singular value anywhere on the circle
. So the spe
c-
tral and stretch sam
pling requir
e
ments agree.

The sampling requirement tabulated in
Figure
5

is then given by
the domain area times the square of the max
i
mum
; i.e.,


While this d
omain is properly a disk in 2D, we take its area as

rather than
to calc
u
late efficiency, assuming the disk
samples must be embedded in a square to allow a practical inde
x-
ing scheme. Because the equidi
stant map is stretch
-
invariant, its
max
i
mum spectral stretch does not depend on

, and thus has
optimal sampling efficiency among all azimuthal projections.


Figure
7

graphs sampling requirements of the various az
imuthal
maps for portions of the sphere up to a hem
i
sphere. A single
gnomonic map can’t encompass the entire hemisphere; its sa
m-
pling requirement is unbounded. The equidi
s
tant map has

while both the stere
o
graphic and equal
area maps

have
. To cover the
sphere, it is more efficient to use two maps each covering a h
e
m
i-
sphere (called a
dual projection
) rather than a single one covering
the entire sphere. The dual equidistant map has sa
m
pling r
e-
quirement
.

The sampling efficiency of a set of gnomonic maps, one for each
face of a polyhedron ci
r
cumscribing the unit sphere, is given by
the surface

area of that polyhedron. This is b
e
cause the maximum
spectral stretch of the gnomonic map is 1 and occurs at the center
(pole) of the projection within each face; the domain area is the
same as the area of the face. The following table lists the sa
m-
plin
g requir
e
ments,
, and maximum anisotropies,
, of gn
o
monic map sets generated from the
platonic solids to cover the sphere. It assumes triangular faces
can be represented without wasting te
x
ture area:

Solid



Components

tetrahedron

41.57

3.00

4

cube

24

1.73

6

octahedron

20.78

1.73

8

dodecahedron

16.65

1.26

12

icosahedron

15.16

1.26

20

Note that the sampling efficiency of dual equidi
s
tant maps is still
better than the octahedral gnomonic map set. By using gnomonic
maps derived from tesse
l
lations of more and more faces, we can
ap
proach but never attain the sampling requirement of an isom
e
t-
ric map of
.

Considering the anisotropy of the azimuthal mappings, the stere
o-
graphic map is best since it is conformal, but the equidistant map
exhibits little aniso
t
ropy for

angles less than about 45

, where it
has
. This can be compared to its still re
a-
sonable anisotropy at the equator,
.
Gnomonic maps have worst
-
case anisotropy at the greatest ang
u-
lar distance from the pole. For
example, the cube map has

at the cube vertices.

3.2

Cylindrical Proje
c
tions

Cylindrical projections transform the sphere to a tangent or inte
r-
secting cylinder such that parallels are pr
o
jected to straight lines,
with the model


where



arbitrarily re
par
-
am
eterizes the parallel spacing, and where

represent
the poles, and

the equator. Three im
portant cylindrical
projections, the plane chart, equal area, and Mercator, are ill
u
s-
trated in
Figure
6

with mathematical properties in
Figure
8
.

The
plane chart

is the sta
ndard latitude/longitude parameteriz
a-
tion of the sphere. It is also stretch
-
preserving. As its name
implies, the
equal area

cylindrical projection preserves area.
Finally the
Mercator
projection is a conformal projection useful
for navigation since it p
rojects
loxodromes

on the sphere, or
curves making a constant angle with the m
e
ridians, into straight
lines. Since all cylindrical projections are orthogonal, their spe
c-
tral stretch is identical to their larger singular value, so the two
sa
m
pling metrics
agree.

To analyze sampling efficiency, we consider the portion of the
sphere from the equator to an angle
. The sampling r
e-
quirement tabulated in
Figure
8

and graphed in
Figure
9

is





(a) Plane Chart

(b) Equal Area

(c) Mercator

Figure
6
: Cylindrical Projections:
Spherical circles are distributed as
defined in
Figure
4
, and proj
ected into the parameter space of the three
cylindrical mappings.
u

values are charted horizontally and
v

values vert
i-
cally. Note that the Mercator projection has an unbounded domain in
v
,
but only the spherical area very close to the poles is greatly st
retched: the
3.5


spherical circles surroun
d
ing the poles map to the horizontal lines
shown at the top and bottom.


Figure
7
: Sampling Requirements of Azimuthal Maps as a Function of
Angular Coverage.

Submission to Siggraph 2001: p
a
pers_0250

6

since th
e domain area is equal to
. The plane
chart is most sampling
-
efficient because it is stretch
-
invariant, just
as the equidistant map is best among azimuthal projections. Co
v-
ering the hemisphere using the plane chart has sampling
requir
ement
, ide
n
tical to that of the
azimuthal equidistant map. The other two maps have unbounded
sampling requirement for hemispherical coverage. The cylindrical
equal area map has very poor sampling eff
i
ciency for any angular
coverage,
while the Mercator projection, at least for angles not
more than about 45

, is only slightly worse than the plane chart,
with the advantage of conforma
l
ity.

At small angles from the equator, the plane chart has little anis
o
t-
ropy. But its anisotropy i
n
creases without bound near the pole
making it a poor choice for MIPMAP texturing. The area
-
preserving map has poo
r anisotropy as well as sampling efficie
n-
cy.

3.3

Low
-
Distortion Equal
-
Area Map

We also present an unusual sphere mapping developed for use in a
stochastic ray tracer. This mapping was designed to project strat
i-
fied and other specialized sampling patterns onto

the sphere and
to project spherical samples into 2D hist
o
gram bins. To solve
those problems, the mapping had to be a bijection b
e
tween the
unit square and the unit sphere, area
-
preserving, and not severely
an
i
sotropic. Its projection,
, is defined by the
composition of three area
-
preserving bije
c
tions. The first is a
mapping from a hemisphere to a disk:
. The
second mapping is from a disk to a half disk:
.

Thus the two halves of a sphere ar
e converted
into half disks, which are joined to form a si
n-
gle disk. The third ma
p
ping is Shirley's area
-
preserving bijection between the disk and the
unit square
[18]
. This leaves us with an i
m
age
of the sphere on the unit s
quare, where the
north and south pole are both mapped to the
center of the square. If we roll the mapping
hal
f
way, the north pole will be on one edge of
the square, the south pole on the o
p
posite edge,
and the interior of the square will be
C
0

cont
i
n-
uous.

See
Figure
11
d for the ma
p
ping applied
to the image of the earth.

Numerical analysis of this mapping shows the
spectral sampling requirement to be
,
about the same as the plane chart a
nd dual
equidistant mappings. But its stretch sampling
requir
e
ment is much worse at
, the
only map described where these metrics dis
a-
g
ree. Its worst
-
case iso
t
ropy is about 0.29.

4.

Polar
-
Capped Maps

Azimuthal and cylindrical projec
tions are
complimentary, in that azimuthal projections tend to have better
sampling efficiency and less anisotropy near the pole, while c
y-
lindrical projections are better near the equator. We can therefore
improve both sampling eff
i
ciency and anisotropy w
ith a three
-
component map set that uses an azimuthal projection for each of
the two poles and a cylindrical projection near the equator, called
a
polar
-
capped map
.

To create an optimal stretch
-
invariant polar
-
capped map, we e
x-
amine the sampling requirement

of a two
-
component map
covering the hemisphere, containing a plane
-
chart projection near
the equator and an azimuthal equidistant pr
o
jection near the pole.
The sa
m
pling requirement of such a map is given by


where


is the angle from the pole where the equidistant proje
c-
tion transitions to the plane chart. We seek


minimizing the
sampling req
uirement, which o
c
curs at

for which the
sampling requirement is
.

To cover the entire sphere, we can therefore continue the equ
a
t
o-
rial map to the southern hemisphere and add a third equidistant
map c
overing the south pole to produce a p
o
lar
-
capped map with
sampling requirement of roughly 14.80 and maximum (worst
-
case) anisotropy of
(occurring at the 45


parallel of the
plane chart). This represents an improvement of 54% ove
r the
dual stere
o
graphic map, 38% over the cube map (but with half as
many map components and much less aniso
t
ropy), and 25% over
the dual equidistant or plane chart maps (but with much less an
i-
sotropy). It is even a slight improvement (2.4%), over the
un
wieldy 20
-
map icosahedral gnomonic set! Moreover, no map
can save more than 26% over its sampling requirement

The domain structure for the stretch
-
invariant polar
-
capped map is

Plane Chart

Equal Area

Mercator





properties

stretch
-
preserving

area
-
preserving

conformal

v co
v
ers

sphere




















inverse map




Figure
8
: Table of Mathematical Properties of Cylindrical Projections.
The maximum sing
u-
lar value and sampling requirement are taken over the part of the sphere from
[0,

];

i.e., from the
equator to the


parallel. Local properties of the cylindrical projections are invariant with respect
to
u
, depending only on
v

or i
n
verting,

.


Figure
9
: Sampling Requirements of Cylind
rical Maps as a Fun
c
tion
of Angular Coverage.


Figure
10
: Stretch
-
Invariant Polar
-
Capped Map:

The rectangle E
represents the plane chart equatorial map, whose vertical resolution
matches that of the two az
i
muthal equidistant polar caps, labeled N and S.
Such a map with sample spacing


produce
s a minimum sampling spectral
r
a
dius of
, for a sampling requirement of
.

Submission to Siggraph 2001: p
a
pers_0250

7

shown in
Figure
10
. Polar
-
capped ma
ps can also be defined that
are conformal (to avoid anisotropy) and hexagonally reparamet
e
r-
ized (to improve sampling efficiency slightly at the expense of
greater aniso
t
ropy)
[22]
.

5.

Results

Figure
11

illustrates five texture maps of the earth with identical
texture area: stretch
-
invariant polar
-
capped (a), gn
o
monic cube
(b), dual stereographic (c), low distortion area preserving (d), and
Lambert equal area (e). Note pa
r
ticularly the size
s of various
features such as the polar ice caps. For example, it can be seen
that the gnomonic cube (
Figure
11
a) shrinks Anar
c
tica, and thus
will sample that area less e
f
fectively than the polar
-
capped map
(
Figure
11
b). The dual stereographic map (
Figure
11
c) reduces
Anarctica still further.

Figure
12

compares the maps using a high
-
frequen
cy test pattern
on the sphere. This pattern is first sampled into te
x
ture maps of
identical area for each of the 5 example maps. We then generate
orthographic views of the sphere textured with each of these r
e-
sults, shown in the top row. We chose a view

where the south
pole has been rotated towards the viewer by 45


so that both polar
and equatorial regions are visible. The row below zooms in on
the south pole of the row above which repr
e
sents a “bad spot”, or
most undersampled region, for each of the m
aps.
2


Our sampling efficiency metric is based on worst
-
case frequency
preservation using the principle that all parts of the sphere must be
sa
m
pled well to avoid visual artifacts. The bottom row of
Figure
12

vali
dates the mathematical ranking of the maps which decrease
in sampling efficiency from left to right. Discriminating between
the low distortion equal area (LDEA) map and the gnomonic cube
is diff
i
cult. This is probably because we applied the spherical
tex
ture using bilinear filtering to simulate typical graphics har
d-
ware, which penalizes the more highly aniso
t
ropic LDEA map.
The top row shows the polar
-
capped map’s sharpness uniformity
over the entire sphere. The other maps have noticeable patches of
bl
urriness near the south pole.

6.

Conclusions and Future Work

Mapping functions used to represent spherical images in graphics
systems are less than optimal in terms of sampling eff
i
ciency.
This paper defines the notion of sampling efficiency and analyzes
exi
sting and new maps in terms of sa
m
pling efficiency and local
anisotropy. We introduce pieces of projections used in carto
g-
r
a
phy to form two
-

and three
-
component maps that have better
sampling efficiency than the best map used previously, with little
or no
anisotropy. Impl
e
mentation involves replacing the texture
coordinate generation function with a simple altern
a
tive.

We believe the sampling efficiency metric can be extended to
Monte Carlo integration for rendering. Stratified stochastic sa
m-
pling seek
s to scatter as few samples as possible to arrive at an
integral estimate with greatest conf
i
dence, and often uses global
mapping functions from simple domains
[1]
[12]
. While area
-
preserving mappings as in
[1]

assure that all samples contribute
equally to the integral estimate, maps with much local stretch, a
typical consequence of area
-
preserving mapping to surfaces with
c
urvature, increase variance (i.e., reduce the effectiveness of str
a
t-
ification). How should these considerations be balanced? A
surface integral of spectral stretch, rather than a si
m
ple maximum,
may be the right comparison metric for mappings used for in
t
e-
grating rather than tabulating functions over su
r
faces.




2

The polar
-
capped map has no bad spot. The other maps have bad spots
at both poles or at the south pole (Lambert).

7.

References

[1]

Arvo, J., “Stratified Sampling of Spherical Triangles,”
Siggraph ’95, August 1995, 437
-
438.

[2]

Blinn, J.F., and M.E. Newell, "Texture and Reflection in
Computer Generated Images",
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ions of the ACM
,
19(10):542
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547, October 1976.

[3]

Brillouin, L.,
Science and Information Theory
, Academic
Press, 105
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111, 1956.

[4]

Cabral, B., M. Olano, and P. Nemec, “Reflection Space Image
Based Rendering,”
Siggraph ’99
, August 1999, 165
-
170.

[5]

Chen, Shenchang,
“QuickTime VR


An Image
-
Based
Approach to Virtual Environment Navigation”,
Si
g
graph ’95
,
August 1995, 29
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38.

[6]

Dudgeon, Dan, and R. Mersereau,
Multidime
n
sional Digital
Signal Processing
, Prentice Hall, 1984.

[7]

Greene, N., "Environment Mapping and other Appli
cations of
World Projections,"
Computer Graphics and Applications
,
6(11):21
-
29, N
o
vember 1986.

[8]

Haeberli, P., and M. Segal, “Texture Mapping as a
Fundamental Drawing Primitive,” in
Fourth Eurographics
Wor
k
shop on Rendering
, June 1993, 259
-
266.

[9]

Heidrich, Wol
fgang, and H.P. Seidel, “Realistic, Har
d
ware
-
Accelerated Shading and Lighting
,” Siggraph ’99
, August
1999, 171
-
178.

[10]

Hüttner, T., and W. Straßer, “Fast Footprint MIPmapping,”
in
Proceedings 1999 Eurographics/Siggraph Workshop on
Graphics Hardware
, Aug. 1999
, 35
-
43.

[11]

Miller, G., M. Halstead, and M. Clifton, “On
-
the
-
fly Texture
Computation for Real
-
Time Surface Shading,”
IEEE
Computer Graphics and Applications
, March 1998, 44
-
58.

[12]

Mitchell, D., “Consequences of Stratified Sampling in
Graphics,”
Siggraph ’96
, Aug
ust 1996, 277
-
280.

[13]

OpenGL Reference Manual
, Addison Wesley, 1992.

[14]

Peterson, D.P., and D. Middleton, “Sampling and
Reconstruction of Wave
-
Number Limited Functions in N
-
Dimensional Euclidean Spaces,”
Information and Control
,
5(1962), 279
-
323.

[15]

Regan, M. and R
. Pose, “Priority Rendering with a Virtual
Reality Address Recalculation Pipeline”,
Si
g
graph ’94
, 155
-
162.

[16]

Robinson, Arthur,
Elements of Cartography
, John Wiley &
Sons, New York, 1960.

[17]

Schröder, P., and W. Sweldens, “Spherical Wavelets:
Efficiently Represe
nting the Sphere,”
Siggraph ’95
, August
1995, 161
-
172.

[18]

Shirley, Peter and Kenneth Chiu, "A Low Distortion Map
B
e
tween Disk and Square", Journal of Graphics Tools, Vol 2,
No 3, pp 45
-
52, 1997.

[19]

Snyder, John P.,
Flattening the Earth: Two Thousand Years
of Map

Projections
, The University of Chicago Press,
London, 1993.

[20]

Voorhies, Douglas, and Jim Foran, “Reflection Vector
Shading Hardware,”
Siggraph ’94
, July 1994, 163
-
166.

[21]

Williams, Lance, “Pyramidal Parametrics,”
Siggraph ’83
,
July 1983, 1
-
11.

[22]

Sampling
-
Efficie
nt Mapping of Spherical Images, TR to be
published, July 2000.

Submission to Siggraph 2001: p
a
pers_0250

8


(a) Stretch
-
invariant Polar Capped



(b) Gnomonic Cube (Cube Map)

(c) Dual Stereographic

(d) Low Distortion Equal A
rea

(e) Lambert Equal Area (OpenGL)

Figure
11
: Texture Maps of the Earth













Original

Polar
-
capped

Gnomonic Cube

Low Distortion

Dual Stereographic

Lambert

Figure
12
: Results on Spherical Test Pattern