From: AAAI-82 Proceedings. Copyright ©1982, AAAI (www.aaai.org). All rights reserved.
DETERMINING
SURFACE
Paul
Amaranth
Academic
Computer
Services
Oakland
University
Rochester,
TYPE
FROM SURFACE
Intelli.gent
Department
and
MI
ABSTRACT
By exploiting
the relationship
which
exists
between
the Gaussian
image and
gradient
space,
can
be
utilized
in
the
two
representations
conjunction
to facilitate
the
interpretation
of
set
of surface
normals.
The projection
of the
a
can
provide
clear
Gaussian
image onto
a plane
quickly
surface.
I
yield
general
knowledge
is
but only
the binary
image.
This
binary
image
a discreet
version
of the space
we term
the
Unit
Gradient
Space
(UGS).
Analysis
using
the
UGS
quickly
technique
for
provides
a
promising
information
about
the surfaces
acquiring
basic
under
consideration.
The
ion
recogni
t i on [5]
and segmentat
ion [3] us i ng these
however,
the process
of
normals.
In
general,
interpreting
this
local
information
to infer
the
underlying
surface
type
is not well
understood.
Given
a
the
question:
This
addresses
paw
collection
of surface
normals
generated
from
a
what
can be determined
single,
smooth
surface,
variation
of
the
about
the surface?
Using
a
for
gradient
space,
we
present
an
approach
simple
is
purposely
is to quickly
the surface.
surfaces.
Our
simple
and limited
obtain
some
basic
If
surface
translated
spatial
normals
image
The
and
Gaussian
is
looks
mode 1
observer
centered.
with
the
Y axis
pointing
projected
The
to the
right.
onto
the
image at
pointing
Space
viewer
entirely
along
up
and
scene
is
the
Z axis
the
X
axis
orthogonally
Z=l.
image coordinate
system
defined,
With
the
the
system
of
coordinate
we now consider
the
is closely
The
UGS
spaces.
gradient
various
and
the
gaussian
space
to
gradient
related
If
the equation
for
a smooth
surface
is
sphere.
then
the observer
directed
of the
form z=f (x,y),
normals
to
that
surface
are
Cp, q, -11 = C x/ 2,
given
by:
y/ z, -11
eq.
1
(Fig.
1) Gradient
space
consists
of
the
points
[p,q]
each
of
which
represents
a parti;ilar
a
surface
orientation.
The gaussian
sphere
Each point
on its
surface
sphere
of unit
radius.
the orientation
of the plane
tangent
represents
to the sphere
at that
point.
The gaussian
sphere
space
can be defined
as tangent
to the gradient
Kender
gradient
space
origin.
c71
the
at
various
illustrates
the
relationships
among the
very
spaces
I I
Image
I I I
I ntroduct
University
Ml 48202
of
There
are
many methods
in low-level
machine
orientation
vision
which
derive
the 3-D surface
coordinate
point
in an image.
Optical
at
each
texture
gradient
and
flow
Cl981,
c71,
[2,6] are all
techniques
used
photometric
stereo
vectors.
generate
these
surface
norma 1
to
for
object
have
been
suggested
Methods
distinguishing
representation
as our object
knowledge
of
William
Jaynes
Systems
Laboratory
of Computer
Science
Wayne State
Detroit,
48063
traces
which
the underlying
NORMALS
If
clearly.
the
gaussian
sphere
and
Representation
information
generated
is
from
ignored
and
an
image
all
are
to
a
common
origin,
the gaussian
image
is formed
[4,8].
This
image can be seen
as
the
locus
of points
formed
by
the
intersection
of
the
unit
normals
with
a unit
sphere
called
the Gaussian
sphere.
Any rotation
in
the
scene
causes
a corresponding
rotation
of the Gaussian
sphere
about
its
origin.
Histograming
the X
and
Y
components
of the unit
normals
produces
a 2-D
approximation
of the Gaussian
image
cal led
the
Gauss ian
[3]. At the present
histogram
time,
we
do not utilize
the cell
counts
of the histogram,
Fig.
1.
The
ob server
conica
1 and
dir
a
ected
CYl indric
norma
al sur
S
ace.
of
a
the gradient
space
sys tern
coord i nate
conditions
hold:
1) The gradient
plane
Z=-1
2)
The
the
are
embedded
that
such
space
is
within
the
defined
the scene
following
by
intermediate
genera
consequently,
p and q axes
are
aligned
same units
as the x and
with
and
y axes
sl /
CP.
IlCP9
q9
Pl
=
Discussion
concerned
with
surfaces,
not
entire
is
The
embedding
gradient
space,
i nto
space
system.
the
= 0
eq. 3. 4
UGS for
a
right
circular
cone
1 /
/
(b’
+
(b2
+
l),
1)
eq.
5,
6
given
by
1 /
(1
/
q12>
-
1
eq.
7
In real
scenes,
it
is unlikely
that
objects
wi 11
be
so
conveniently
oriented.
As
consequence,
the
UGS represenation
of either
cone or a cylinder
in some arbitrary
orientation
Y
a
a
may
describe
a
section
of
an
ellipse.
Consequently,
the
various
surf aces
not
are
distinquishable.
Another
possibility
for
standard
orientation
is
suggested
by
the
fol lowing
observation.
If
a
singly-curved
surface
is oriented
such that
a plane
tangent
to
the
surface
is parallel
to the
image plane,
the
UGS curve
of that
surface
will
pass
through
the
origin.
The
point
on
the curve
at the origin
corresponds
to
the
rul ing
on
the
surface
determined
by the
tangent
plane.
Conversely,
if
the UGS curve
of a singly-curved
surface
passes
through
the
origin,
then
there
exists
a ruling
on the
surface
with
a
tangent
plane
which
is
parallel
to
the
image plane.
Thus,
a standard
orientation
can be defined
as one in which
some
ruling
of
the surface
is parallel
to the
image
plane.
For a cylinder,
this
equivalent
to
is
setting
the axis
of the surface
parallel
to the
image plane
and
results
in
a
straight
1 ine
Unit Gradient 1
I
Space
2.
in
b =
;
Fig.
oneand,
where
b is the
ratio
of height
to the radius
of
the base.
These
equations
show that
in the given
orientation,
conical
and
cylindrical
surfaces
give
rise
to straight
line
segments
in the
UGS.
I ntui tively,
it
is
easy
to
see
that
the
projection
of an arc
of a sphere
onto
a
plane
perpendicular
to
the plane
defined
by that
arc
will
result
in a line
segment.
The UGS line
for
a
cyl i nder
will
pass
through
the origin,
while
that
of the cone cannot.
Further,
the
ratio
of
the
height
to the
radius
of the base
of a cone
objects
[5]
and,
at this
point,
we wil 1 narrow
single
our
perspective
to
encompass
only
restriction,
we will
surfaces.
As
a
further
surfaces
consider
only
conical
and
cylindrical
cross
section.
We do this
for
with
a
circular
two reasons.
First,
the form of
their
Gaussian
images
straightforward
and
easy
to
are
great
visualize.
Cylinders
lie
along
arcs
of
cones
form arcs
on lesser
circles.
circles
and
they
are
singly
curved
surfaces
and
Secondly,
therefore
representative
of a class
of surfaces
Gradient :s
Space
,
ql
= b*sin(t)
ql =
We are
the
a
[lo]
2
Pl
IV
sin(t),
The
curve
is given
by
-1111
eq.
in
in
Given
a curve
in the UGS, we want
to
know
if
we
can
determine
whether
it
indicates
a
cylinder
or a cone and what,
if any,,
parameters
of
the surface
can be determined.
First,
let
us
assume
that
the axis
of
the
cone
or
cyl inder
generating
the
surface
is parallel
to the
image
plane.
Since
a rotation
about
the Z axis
results
in a corresponding
rotation
in the UGS,
we
can
assume
without
loss
of generality
that
the axis
is parallel
to the Y axis.
It can be shown
that
a right
circular
cylinder
results
in points
on a
curve
in the
UGS given
parametrically
by
be
seen as an orthogonal
projection
of the
same
is
a
hemisphere
onto
the Z=-1
plane.
Hence,
it
of gradient
space
and consists
version
bounded
of the points
=
also
result
space
UGS.
between
any
of
The center
of the Gaussian
sphere
is at
the origin
number
of
interesting
properties
become
then
a
current
For
the
understandable.
readi ly
the most
important
result
however,
discussion,
various
the
is that
the
rotational
coupling
of
to
figure
2.
Refer
obv i ous .
becomes
spaces
Gradient
space
then
corresponds
to
a
central
one
hemisphere
of the gaussian
project
ion
of
clearly
sphere
onto
the Z=-1
plane.
The UGS can
q13
falling
In particular,
and
singly-curved
parameter
the
3)
CPl9
in
complexity;
planar
surfaces.
surface
must
curve
in
gradient
1
of the Gaussian
sphere,
and the unit
gradient
coordinate
the
scene
56
through
a
cone
Thus we
the
origin
will
have
of
still
a fast
the
UGS.
The
UGS curve
for
lie
on an ellipse,
however.
method
of distinguishing
the
symmetrical
aligning
the
surfaces,
projection
surface
the
with
ql
this
has
y
of the
axis
of
the
the
axis
effect
of
of
the
UGS.
surfaces.
cylindrical
conical
and
At
this
point,
is
also
surfaces
are
easily
distinguished.
It
recover
the
ratio
of height
to the
possible
to
angle
at
radius
of the base
in the
form of the
In this
orientation,
the
apex
of the cone.
the
in
distance
between
the endpoints
of the
curve
the UGS is a measure
of the angle
of the apex
of
the original
conical
solid.
For a circular
cone,
this
angle
is equal
to:
In this
standard
orientation,
equation
7
wi 11
not
necessarily
be
valid.
It
is still
possible
to recover
the ratio
of height
to
the
radius
of
the base,
however,
by exploiting
the
fact
that
symmetries
in a surface
are
reflected
symmetries
in the Gaussian
image.
The center
by
of the arc
in the Gaussian
image will
correspond
to a ruling
down the center
of the
cylinder
or
cone.
The
following
procedure
is based
on this
observation.
2 I
acos(
dist
/
2 )
This
yields
only
an approximate
value,
because
of the
digitization
problem
later.
V
method
8
however,
mentioned
Method
VI
The
eq.
standardizing
the
object
to
the observer
consists
of
position
relative
two parts.
The first
part
rotates
the
Gaussian
image
so that
the center
of the arc
lies
on the
the center
of the arc
is at the
Z axis;
that
is,
origin
of the UGS. This
effectively
rotates
the
the
plane
tangent
to
the
surface
so
that
approximate
middle
of the surface
is parallel
to
the
image plane.
The
second
part
performs
a
rotation
about
the Z axis
so that
the UGS curve
For
is symmetrical
with
respect
to the ql axis.
Resu 1 ts
of
analysis
was
generated
by
a
Data
for
the
parameters
of
an
given
program
which,
produced
the
normals
for
the visible
object,
surface.
The surface
normals
were
produced
by
analytically
determining
the orientation
of each
surface
patches
as sampled
through
a 30
of
the
space.
A variety
of
image
by 30 grid
in the
surfaces
at
various
conical
cylindrical
and
The
.resulting
manufactured.
orientations
were
of normals
were
processed
using
the method
sets
described
above.
In all
cases,
the surfaces
were
standard
brought
either
to,
or very
close
to,
a
Figures
3 and 4 show typical
orientation.
-------------------------,----------------------*-0
I
I
A.
i
--------~---“---q--~---*-3--L--p--~*-~-- S--5--~-‘-,-SI
B,
Fig.
3.
Gaussian
histograms
for
a cylinder
(a) an unstandardized
pos i t ion and
(b) the
standard
position.
Fig.
in
57
4.
Gaussian
histograms
for
a cone
(a) an unstandardized
position
(b) the
standard
position.
in
and
results
for
The
cone.
simi
circular
cylinder
and
a
and
between
conical
For
apparent.
are
readily
used
to
equation
8 was
a circular
differences
cylindrical
conical
surfaces
surfaces,
determine
illustrated
the
angle
of
in
table
the
apex.
As
can
1.
Results
are
be
the
seen,
used
of
the
surface.
Table
Apex angles
circular
cones
using
equation
in
to
surf aces
distinct
aid
in
at
traces
different
. This
orientations
fact
could
be
[3].
segmentation
Finally,
there
is
much
more
which
has yet
to be exploited.
Flatter
yield
a
more
condensed
image
in
parameters
of the elliptical
curve
reflect
on
the
curvature
of
the
surface.
The counts
in the cells
of
histogram
concern
the
relative
size
of curvature
of a surface.
These
are
for
further
investigation.
percent
of
computed
values
are within
thirteen
actual
values
calculated
analytically.
Note
the
values
the
robustness
of the method;
consistent
starting
the
regard
1 ess
of
obtained
are
orientation
lar
result
information
surf aces
the
UGS. The
in
the
UGS
generating
the Gaussian
and/or
rate
all
matters
1.
calculated
for
various
at different
orientations,
8. All
values
in degrees.
REFERENCES
Cl1
Actual
Angle
Z
Rotation
X
Rotation
53
Computed
Angle
61
20
45
-45
2:
67
90
90
1;
-45
0
0
0
110
110
127
127
viI
0
45
0
-20
-20
;:
::
Concluding
Dl
59
76
76
96
96
:i
Our
of
method
of the
adjusting
112
112
Remarks
c41
Horn,
B.K.P.
representations
A.I.
Lab.,
M.I.T.,
c51
lkeuchi,
using
IJCAI-81
“Sequ i ns
and
Quillssurf ace
topography.
for
A.I.
Memo 536
(1979).
K.
“Recognition
of
the
Extended
Gaussian
(19811,
pp. 595-600.
3-D
Image.”
”
Objects
Proc.
C61
“Determining
lkeuchi,
K.
Surface
Orientations
of Specular
Surfaces
by Using
the Photometric
Stereo
Method.”
IEEE Trans.
m,
3 (1981). pp. 661-669.
171
Kender,
J.
Computer
University,
L81
standard
initial
it may
in
a
Prazdny,
K.
From
Map
Cybernetics,
c91
equivalent
to the standard
defined
above.
A variety
of
approaches
were
used to counter
this
difficulty,
it
has not been completely
eliminated.
This
but
the
problem
as
does
not appear
to be a
major
orientation
and
ideal
difference
between
the
that
obtained
is only
a few percent.
“Shape
from
Science,
Texture.”
Dept.
Carnegie-Mellon
~~~-~~-81-102
“Egomotion
Optical
36
of
(1981).
and
Relative
Depth
Biological
Flow.”
(1981) pp. 87-102.
“Us i ng
D.A.
A. I.
Lab.,
Smith,
Images.”
Enhanced
M.I.T.,
A.I.
Spherical
Memo 530
(1979) [IO]
Woodham,
R.J.
“Reflectance
Analyzing
Surface
for
Castings.”
A. I.
Lab.,
(1978)
singly-curved
When dealing
with
it may not be necessary
for
the
input
only
one surface.
Different
types
of
R.
“Three-Dimensional
The Gaussian
Image and
Proc.
PRIP-81
(1981).
symmetrical
in the
the
however,
resulting
quite
m-328.-
Dane,
C. and Bajcsy,
Segmentation
Using
Spatial
Information.”
resolution.
the
PP.
c31
134
134
makes
use
of
surfaces
apparent
nature
Depend i ng
on
orientation.
the
surface,
orientation
of
asymmetr i cal ,
appear
somewhat
that
is not
final
orientation
Coleman,
E.N.
and Jain,
R.
“Obtaining
3Shape of Textured
and Specular
Dimensional
Surfaces
Using
Four
Source
Photometry.”
Computer
Graphics
and Image Processing
18
(1982),
The UGS image
is formed
by
quantizing
and
[3]
histograming
the normals.
As Dane and Bajcsy
in
1 ies
have
pointed
out,
a
major
problem
determine
quantization
effects.
Resolution
will
whether
a curve
in the UGS is seen
as a point,
a
smooth
curve
or as disjoint
points.
Dealing
with
allows
the
time,
however,
one
surface
at
a
possibility
Clocksin,
W.F.
“Determining
the Orientation
Flow.”
of
Surfaces
from
Optical
Proc.
AISB/GI,
Hamburg
(1978),
pp.
93-102.
surfaces,
to contain
surfaces
or
56
.
Map
Defects
M. I .T.,
Techniques
in
Metal
A I -TR-457