From: AAAI-83 Proceedings. Copyright ©1983, AAAI (www.aaai.org). All rights reserved.
ORGANIZATION
PERCEPTUAL
FOR
David
VISUAL
G. Lowe
and Thomas
Computer
Science
Stanford
TJniversity,
is prescnled
showiug
that bottom-up
is usually prerequisite
in terprc tation
of images.
Ihcse
models.
Scvernl
ing which
relations
to the extent
by accident
relations
from
should
which
rcsofutions,
structures
the linking
The
\vhich no single-resolution
Its j>crformance
is demonstrated
such
as the
have
and
on
been
aspect
ment
images
goal
of computer
to prior
label
and
vision
systems
panding
cnvironmcnts
to the image
pcrTormnnce
same
of searching
and
the
of the models
sible mntchcs.
dirncnsiorlal
ing,
far
the
increasing
the
shnpc
from
imagesP-usiilg
However,
considerably.
solved
even
formation,
and
these
of human
stcrco,
to reduce
when
methods
performance
the
given
models
and
fail
in such
simple
the
domains
lhcsc
of
trcat-
regularities
this
paper
of inference
plays
nnd~rlying
prinprcscnt
new seg-
principles.
of irlformation
of image
the problem
and
for many
In
some
and
scurcps
orgailixxtion
simplify
1) A major
reduction
segmentation
related
features
can
of matching
which
provide,
against
in the
-the
search
division
features.
This
hzs
space
of the
world
long
is achieved
image
been
inlo
by
sets
of
as a
recognized
crucial
problem
in image
interpretation.
want to match models against
all possible
of posthree-
of matching
three-dimensional
bottom-up
up”n
valuable
this
a unified
that
form
but
Wit,kin
knowledge:
gcncrality
shnd-
this
;~p-
of detecting
on the irn;lgc
describe
these
[7] was
explicit,
non-accidental.
that
b;v;ed
are three
all of which
models
search
to explain
the
will
motion,
are
Lhe role
methods
There
is one
lhc size of this
problem
full
mentation
is not one of
between
number
based
models
in ex-
problem
describe
but
Recenlly,
They
to
such
general
formulation
on the assumption
image
efforts
lackctl
relations
structure
b>:;cd
in the
many
for the importance
here.
or-
description,
problcrns:
often
in model based recognition,
develop
ciples for this level of interprct>.t,ioll,
tightly-
results
in an excessively
large space
Continued
research
into recovering
from
level
Rather,
correspondences
tcxlurc--promises
md
space
since
domains
1haL two different
data.
for potential
image,
in
with a few well-specified
[Z, 8, It].
The difficulty
unlikely
image
model-
only
general
WC will
visual
and thereby
current
demonstrated
to more
is very
fit the
constituents,
However,
bceu
is to relate
given
of inference
detected
research
of their
them.
have
ambiguity--it
fully
knowledge
interpret
constrained
to compare
vision
reasons
been
developed.
con-
has gone
perceptual
slcctch
of these
fully
topics
or connectivity,
primal
and imposirlg
the same
of
uniformly
texture
have
have
[12] h ave argued
regularities
related
scgmcntstion
and
some
body
of their
segmentalion,
of collinearity
to make
in a form
be applied
phenomena,
inilial
little
boltom-up
n large
knowledge
and
There
of it was never
Tenenbaum
Introduction
image
integrated
wc have
image
into
cm
any
for specific
Marr’s
intended
natural
as
the
index
on this
perception.
plicability.
data.
A major
names
detection
not
effect,ivcJ
describe
we have
research
algorithms
which
paper
we will describe
methods
the significance
of relations
be-
figure/ground
Gestalt
of imaging
could detect.
OR synthetic
under
about
to use
in a way that
before
Previous
and
is a.ble to detect
algorithm
elements
tents.
an algorithm
of segments
algorithm
image
develop
structure
and
to selectively
In this
and cva,luating
ganization,
must be based
significant
be used
knowledge.
tween
of features,
images
it is necessary
to structure
can
to all images
are
are few altcrna-
over a range
to interpret
for detecting
to have arisen
we develop
including
the basis of curvilinearity.
there
detects
in-
world
object
relations
and relations
are invariant
segmentation
at multiple
image
where
that
94305
knowledge,
techniques
of
for dctermin-
distribution
Using these principles
for curve
for accessjug
that they are unlikely
proximity,
which
conditions.
of
functions
be formed:
the surrounding
tives within tilt same
Rinford
California
prior
and
2) three-dimcnsiomrl
descriptions
can only be formed
on properties
three
principlcc J are hypothesized
image
significant
WC describe
and 3) stable
grouping
to the recognition
1) scgmcnlation,
groupings:
tcrpreta,tion,
0.
In order
im&ge features
A BASIS
Department
Stanford,
Abstract
Evidence
AS
RECOGNITION
we do
cornhinations
IlGt
of features
in an image, so good scgmrntat
ion is crucial
for reducing
the combinatorics
of this search.
2) Two-dimcusional
is
in-
cxccllent
inter_:\rrtations,
pnpcrs
[I, 51. For
mu::t
as line
he co!lirlear
viewpoint.
drawings.
255
rrlations
sional
lead
to specific
as we have
c:;amp!c,
collinear
in 3-sp.7cc,
A corollary
of ibis
three-dimcn-
described
barring
is thal
lines
an
thesr
in previous
in tilt
image
accident
image
of
rcla-
tions
are
which
normally
greatly
invariant
simplilies
three-dimensional
3) To
the
objects
extent
different
that
imaging
used,
Not
but
can
be
and
description
For example,
can
under
they
can
of the relations
will
have
be
several
in the
values
collinear
line segments
relative
sizes
I
of world
relative
provides
be used
stable
a body
relation
whose
which
that
tempted
that
the names
by the
gaps,
are
viewpoints,
to access
each
characterized
ments
Note
can
of variation
can be used.
and
to
orientation.
relations
terms
in addition
parameters
image
index
to viewpoint,
of malching
of unknown
these
only
respect
problem
conditions
be used as reliable
knowledge.
with
the
of the
seg-
a viewpoint-invariant
to select
a model
for
at-
a.
matching.
all three
of these
points
assume
that
z/
the relations
found
in the image
are a result
of regularities
in the obThis means
that
any relations
which
jects being viewed.
happen
to arise accidentally
from independent
features
will
only
confuse
the
significant
and
interpretations.
accidental
This
relations
distinction
is a point
I-
between
to which
we
I
will return.
The
importance
of perceptual
recognition:
The
importance
of these
grouping
the
processing
of images
by the
be
demonstrated
by
periment.
In Figure
drawing
for
bottom-up
most
parallelism,
with
identity
of the
object,
difficult
able
to recognize
and
the
of the
ready
been
level
performed---the
task
would
embedded
in a normal
I(b)
segment
location
segments
other
circular
termination
this
segmentation
is in
has
be even harder
scene
nition
times
than
for the
would
would
al-
if the
containing
be expected
first,
this
second
with
3 out
as in l(a)
segment
allow
ma,ny
The
that
the
the
group-
recognizing
lower
it
5 seconds
and with 7 out of 10 subjects
the 60 second time limit.
Presumably,
recognizing
it
if the added
segment
had been placed
at some location
lend itself to perceptual
groupings
the change
which
did not
in recognition
times
would
have
These
figures
human
capability
been
negligible.
can also be used to demonstrate
to make use of top-down
contextual
1935
to limit
the
in-
the
demonstrated
[3], verbal
clues
search
space
in expcrirnents
naming
even
for
forming
performed
vague
a match.
as early
non-visual
as
object
greatly
reduce
the recognition
time.
Subjects
interpret
Figure
l(a) immediately
upon being
told that
it is a bicycle.
esting borderline
where
formation
can
to recognition.
recog-
dramatically
of 10 subjects
As was
classes can
can usually
bottom-up
in recognition,
were
in
center
with
to further
if we assume
figure
only
it to be combined
be coincident
leading
role
with
was placed
grouping.
then
segment,
an important
for
added
in a curvilinear
grouping
play
drawing
The
which
of another
As might
groupings
same
added.
a strategic
within
within
that
unlimit,
1: When opportunities
for bottom-up
grouping
of’image
features
have been removed, as was done for the line drawing of
a bicycle in (a), the drawing is remarkably
difficult to recognize.
The average recognition
time for (a) was over one minute when
the subjects
had no prior knowledge
of the object’s
identity.
When a single line segment was added in (b), which provided
local evidence l’or a curvilinear
grouping,
the recognition
times
were greatly reduced.
formation
is the
with
ings.
Note
were
time
features.
Figure
of this
the
to be rcmark-
a 60 second
45 seconds.
\/
Figure
experiabout
of 10 subjects
object
were
the
b.
endpoint
nothing
proved
out
within
took
that
told
line
we have
In informal
drawing
object
subject
fact
surrounding
a single
the
(e.g.,
l
ex-
a partial
collinearity,
were
--
can
opportunities
eliminated
Nine
in
system
most
symmetry).
this
to recognize.
tenth
spite
that
of signilicant
who
as a stage
visual
constructed
are
and
10 subjects
ably
bicycle
cases
for
psychophysical
a way
segmentation
eliminated
ments
we have
in such
proximity,
operations
human
a straightforward
l(a)
of a bicycle
organization
A demonstration
Thus this figure is on an intereither
bottom-up
or top-down
in-
suddenly
reduce
the search
One can imagine
a series
space and lead
of expcrirnents
this search
space
that would systematically
explore
reduction
in its size created
by different
botlorn-up
and the
or top-
down clues. These figures can also be used to demonstrate
the hunran
eqlrivalcnt
of a back-projection
algorit,hrn [(I] followed
partial
by image-level
matches
cm
matching,
where ccr(.n.in hypothesized
bc used to solve for the position,
oricn-
tation,
and internal
parameters
of the model,
which in turn
lead to accurate
predictions
for further
nlatxhrs
at specific
locations
in the image.
Principles
There
are virtually
be forrncd
bctwccn
principles
can we dcrivc
arc
forming
worth
was rrlcntioncd
and
that
they
than
accidental
for measuring
their
are
rcprcspn
t actual
arisen
by accident.
call arise
positioning
as well as from
by
rxarnining
rounding
to
relation
measures
If there
were
the
tions,
could
this
mcntations.
However,
to be so wide
very
be used
that
weak.
of signi6cancc
tures
are
and
any
We have
with
scale.
Significance
probability
that
is biased
sumption.
But
pendently
since
positioned
respect
to orientation,
discrimination
seems
typically
objects
(leading
location,
operation
major
must
is obviously
tances
that
so the
do not
particular
type
in which
a highly
include
being
tion
to detect
this failure
resources.
limited
tempted
than
can
the
examined.
such
relations,
background
is that
each
It
over
2 shows
to human
vision
targets.
It would
at that
principles
and
but
how
will bc attempted.
scale
above
they
they
at some
scale,
which
will be evaluated
do not specify
There
some
relation
scenes,
groupings
classes
factors
image,
the
only
then
are
the
very
rarely
it is more
likely
is accidental
be possible
to distinguish
the relation
image
measurements).
to devote
than
is useful
easiest
if right-angles
image
from
play
resources
some
accidentals
searching
are
of
instance
it would
Of course,
which
If
structure
(although
for those
and
a role
attempted.
the
from
would
Witkin
to distinguish
be
that
of
to detect
probabilities
in an image
arise
even
in the
therefore
on
irnage-
viewpoints
in viewpoint.
the relation
seldom
all
be pointless
since
prior
should
depend
or other
collinearity
angle
change
relations
do not
over
it would
that
and
arises
An
which
rccogniso
in creating
can perform
images
mented
edge
been
smooth
without
of
still
given
it is also
for properties
common.
these
bottom-up
is the problem
segmentation
a computer
perceptual
of creating
program
processes
appropriately
on
seg-
descriptions.
considerable
The best current
edge operators
which
are then linked
using ncarestinto lists of points.
Although
there
research
into the problem
of fitting
curves
to these lists
exception
these efforts
of points
[9, 10,
have concentrated
111, almost
on a single
pre-selected
resolution
of segmentation
and have attempted
merely to smooth
out noise induced
by the imaging
process.
of
(We
tion
of relations
which
for curve
bottleneck
natural
has
will bc
class
algorithm
detect
“edge points”
neighbor
algorithms
no groupings
for a given
which
arc several
which
accidentals,
But
any
in selecting
from
A significant
of description.
describe
almost
[12] argue
which
when there
presumably
grouping,
then
image
imagingit is only
an example
and
significant
example,
scene
Tenenbaum
of five equally-spaced
of intcrprctlation
candidates
with
less productive
dis-
of the
For
in the
the
accurate-enough
of features
candidates
in
which
position,
in the
in the scene.
at right-angles
typical
significance.
in any absolute
sense,
since groupings
can be atat many different
scales; however,
if there are more
be formed
formed
the
must be aI,1 ributablc
to a lack of computational
This does not mean that groupings
are diameter-
2. few false
The
with
image),
be formed
Figure
a statistically
inde-
relations
light-source
paramct,cr.
common
change
as-
complexity.
false
grouping
purposes
in the
to this
only
too many
significant
sys-
many
all combinations
can
collinear
dots is not apparent
are enough
surrounding
false
be USC~III for
contains
for judging
to test
visual
to independence
computational
rclat,ions
lines
are
for irn:lge
it is present
collinearity
such
important
factor
is the same
that was mrnt ioned earlier---
viewpoint,
because
to the
from
looking
formation
for psychologi-
of scgmcntation
limited
fca-
independence
scale
wort,h
a specific
location,
human
respect
criterion
impossible
in an image,
with
principle
have
and
be
that
arisen
this
a scene
like a reasonable
A second
the
from
this choice.
One
invariance
condition
seems
proportional
It is a matter
direction
of relations
hypothesis
have
to see whether
in any
rela-
computation
to orientation,
would
features.
experimentation
tem
null
by enough competing
candidates
for grouping
within the same
proximity,
as in (b). Th is occurs even though the relation remains
highly significant
in the et
L) atistical
sense and would thcrcfore
likely be of use for recognition.
of seg-
levcll must
our
of five equally-spaced
collinear dots in (a)
spontaneously
by human vision if it is surrounded
is not detected
the
and
images
at this
out
inversely
relation
a set of independent
cal
to the
respect
is then
the
prior
2: The pattern
Figure
any
meaSsures
the significance
to carry
with
that
of features
knowledge
0
sur-
knowledge
respect
independent
the
level
prior
.
it is possible
of common
.
.
0
However,
and
during
chosen
.
*
e
we have
nonrandomness
range
.
or random
likelihood
of
e
0
e
those
for significance
for judging
the
from
0
.
b.
test
distributions
.
.
be Lo distinguish,
These
a significant
expected
scene
image,
of the
can then bc used as the basic
segmentation
process.
regarding
of the
a central
relation
in the
.
As
to the
in the image.
of each
.
a.
only
of viewpoint
structure
is accidental.
significnnce?
All of the relations
of featrtrcs
probabili~;l,ic
which
structures
accidents
accuracy
disLribution
give
given
the
from
relations
useful
must
significant
could
general
Therefore,
process
considered
that
What
struct,urc
align rncn ts.
as possible,
have
those
scgnlcntations
of the segmentation
as accurately
of rclalions
of any image.
for selecting
earlier,
rather
which
number
Lhc elements
CXIcnt
funcLioll
of segmentation
an infinite
use “resolution”
to refer
to the
the smoothed
influence
257
curve
in the
allowable
description.)
context
of curve
segmentatransverse
deviation
from
Although
these
smoothed
results
may
that
the
appear
is because
lower
been
lustrates
the
problem,
other
structures
developed
earlier,
curve
selects
description.
all groupings
These
can
detection
are
lower
3(c).
degree
a wide
significant
human
primitive
curve
-
-
we examine
curva-
arbitrary
that
b.
for
or of constant
to represent
of
range
structures
implementation,
linear
of seg-
the
over
resoluWe have
measures
either
general
when
on the principles
it is possible
of more
in Figure
based
the most
not
3 il-
of collinearity,
which
be splincd
although
segmentation
as in Figure
In our
which
have
Figure
instance
lists
eye,
perform
they
the
in edge-point
and
still
program.
apparent
algorithm,
structure
of resolutions
only
human
can
one
recognized
outlined
nonrandom
curves,
are
arc
a new
mentation
ture.
where
naive
though
by the
to recognize
groupings
the
system
even
described
is adequate
but
to
visual
groupings
and
tion
the
human
resolution
detected
3(b)
reasonable
the
vision
\
smooth
includes
groupings,
the
such
as
spirals.
6.
Measuring
The
is to
the significance
first
task
determine
confuse
in developing
how
each grouping.
linked on the
of a curve
a segmentation
we will
measure
this
nonrandomness
in proximity
of their
from the
would
since
the
point
they
lie, as is shown
have
chosen
would
a point
distance
from
is equal
to
can
looking
the points
for
that
the
considered
point
previously
which
in terms
an overall
Testing
this
angle
between
endpoint,
far and
of
the
points
Since
given
curve.
the
by
away
its
This
curve
in 4(c).
recursively
from
any
of
the likelihood
proximity
these
we
to be how
as shown
calculating
to these
likelihood
together
values
to produce
curve.
all possible
significance
divide the initial
have the highest
of the
can be multiplied
for the
those
points
Therefore,
point
is farthest
points.
they
value
4(b).
of its minimum
considered
are independent,
Given
8 is the
closest
thus
with
as it is to a curve
to 4 or more
point
the distance
of
two.
However,
in linearity
defining
the
be extended
for
4(a) and
nonrandomness
where
from
at at least two
Figure
3: The data in (a) can be segmented
different resolutions
of description,
as shown in (b) and (c). One
instance of collinearity
can only be detected in segmentation
(b)
while the other instance of collinearity
and the parallelism
can
only be detected straightforwardly
in (c).
by
the
be close to the line on which
in Figures
closest
20/n,
the vector
the measurement
to one of the other
is to be as close
the
with
of proximity
close
automatically
to define
unlikely
This
\
of
For example,
if we start
points, we might measure
effects
by being
third
and
algorithm
significance
linearity
by measuring
line joining
the other
confuse
linearity,
the
In this case, since the points were originally
basis of proximity,
we must be careful
not to
of nonrandomness
in linearity.
looking
at a set of only three
significance
one point
segmentation
test
linked list
significance
for
groupings
sets
of points,
of points
values.
we want
to
into segments
which
Previous
methods
of
curve segmentation
have usually
attempted
corners
(tangent
discontinuities)
on a curve,
to search
for
where the curve
F’igure 4: The middle dots in (a) and (b) are both the same
distance from the dashed line joining t,he other two dots.
Yet
the three dots in (b) are much more significant
in terms of their
collinearity
than those in (a), since the middle dot in (a) could
be close to the line merely as a result of its proximity
to the first
endpoint.
Therefore,
we measure the probability
of a l)oint being
within a given distance
from a line in terms of its proximity
to
the clo.;est endpoint defining the line, as shown in (c).
can
bo divided
into different
segments.
Our approach
is
the dual---WC look for segments of the cwve which exhibit
significant
nonrandomness,
continuities
are assigned
ing
segments.
method
In contrast
will fail to assign
and
tangent
to the junctions
to
the
earlier
any segmentation
and
curvature
between
approaches,
where
dis-
neighborour
the curve
258
Figure 5: The small 30 by 45 pixel region of an aerial photograph
shown in (a) was run through
the Marimont
edge detector
to
produce the linked edge points shown in (b). lcigure (c) shows all
the segments at different scales and locations
v;hich were tested
for significance.
After selecting only t’hose sc;mrnts
which were
loca!ly rnaximurn with respect to size of grouping,
and thresholding out those which arc not statistically
significant,
we are left
with the segments shov~n in (d). It is t.hen fairly simple ho form
collinearity
and curvilincarity
relations between t!rese scgrnents
as shown by dotted lines in (e).
Figure 6: The hand-input
curves in (a) have been created to cxhibit significant structure at multiple resolutions.
When these are
given as data to the curve segmentation
algorithm,
it produces
the results shown in (b), which makes these multiple
levels of
structure explicit.
259
appears
to wander
sign multiple
turc
randomly
at diffcrcnt
clearly
of the
allow
and
groupings.
up
groupings
(amounting
scale,
from
that
any
After
of the
than
at
one local
the
given
this
dctcct
each
5(n).
from
data,
shown
tusl
lists.
Figure
apparent.
Given
collinearity
also
he fairly
parallelism,
Figure
which
these
and
by the
local
:;cgmcnts,
lines
straightforward
constant
differing
and
perceptual
image
relations
search
space
pa.per
play
that
must
by
a major
role
be considered
from
the
could
the
for human
in limiting
when
the
matching
of
background
be applied
to
problems
or
segmentation
O., “Inferring surfaces
17 (1981), 205-244.
from images,”
Artifi-
study of a neglected
portion
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46 (1935), 41-75.
in ACRONYM,”
Proceed-
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[7] Marr, David, “Early processing
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PhiloSeries
[lo]
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importance
vision.
range
are the most
(1976),
Workshop
483-524.
Recognition
(New
of
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W.S., and hzriel Rosenfeld,
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of
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Y.,
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demonstrating
organization
pcrccptual
which
Calif-
[ll]
this
accident
im-
of segmenting
over a wide
those
methodology
Pattern
[9] Pavlidis, ‘I‘., Structural
Springer-Verlag,
1977).
5(c).
Summary
We began
preliminary
(Stanford,
groupings.
bottom-up
from
analysis.
R, 275
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other
taken
“Description
and recognition
[S] Nevatia, R., and T.O. Binford,
Artificial Intelligence,
9 (1977).
of curved objects,”
relations
in Figure
to detect
intervals,
the
ma.xima
it is rela-
curvilinearity
dotted
all the
5(d) shows
sclccts
by
same
of other
at groupings
retaining
arisen
This
range
and
arc
methodology
[G] Marirnont,
David, “Segmentation
ings ARPA Image Understanding
ornia, September
1982).
by
to subpixcl
widely
in images
represpect
[5] Lowe, David G. and Thomas Binford,
“The interpretation
of
three-dimensional
structure from image cnrvcs,” Proceedings
UC/U-7
(Vancouver,
Canada, August 1979), 613-628.
in
generated
5(c) shows
the
to have
Psychology,
on
written
points
with
for the parameters
of object models
[4] Lowe, David G., ‘Solving
from image descriptions”
Proceedings
ARPA
Image Understanding Workshop (College Park, MD, April 1980), 121-127.
of an
is shown
data
relations
[3] Leeper,
R., “A
the development
produced.
program
edge
process
as shown
edge
although
to resolution.
them
linked
image
reasoning among 3-D models
[2] Brooks, Rodney A., “Symbolic
and 2-D images,” Artificial Intelligence,
16 (1981).
to
algorithm
image
groupings
correspondence
References
G(b)
ability
have
these
invariant
developed
general
by looking
[l] Binford, Thomas
cial Intelligence,
when
photograph
dotccts
into
are not
to form
proximity,
is also
group-
at different
the
many
We would like to thank Peter Blicher, Chris Goad, David Marimont, Marty Tenenbaurn,
Brian Wandcll,
Andrew Witkin,
and
our other colleagues
for stimulating
cliscussions and suggestions.
This work was supported
under AlU’A
contract
NOO039-82-C0250.
The first author was also supported
by a postgraduate
scholarship
from the Natural Sciences and Engineering
Research
Council of Canada.
different
Figure
arc
prin-
arc
Acknowledgements
which
3.
could
detection
all resolulions,
the thinning
It would
signal
resolutions--results
digit,ized
some
(61, which
values
locations
and
and
will be
algorithm
of an aerial
by an edge
them
data
scales
a wide
There
the algorithm’s
original
significance
between
in Figure
of running
region
results
easy
there
below
at multiple
s h ows
pcrccptual
the
these
and
be detected
algorithms
of the
maximum
exhibits
segmentation
results
The
links
at
respect
a
curve
structures
algorithm
the
specific
distribution.
in action
different
structure
5(b)
and
with
level,
a,nd demonstrates
Marimont
tively
that
of grouping.
algoriLhm
as was
image
after
the
arc locally
if the curve
exhibit
5 shows
groupings
does
through
along
which
of the curve
facility.
accuracy
one
but
grouping,
steps
location
resolutions
which
Figure
David
least
that
these
There
stereo
and
An
viewpoints.
The
data as well as edges
6(a) shows some hand-
much
this
of each
which
maximum
30 by 45 pixel
oil tank
at
have
of its length
It is possible
no single-resolution
a small
will
will
upon
in which
is the
non-
complexity,
were suggested.
based
recognit,ion
in the scene
they
of detecting
demonstrated.
to the extent
plementations
means
and tested
on synthetic
natural
images.
Figure
output
Figure
This
50%
each
in MA~I,ISF’ on a KL-
significant
which
At
since
different
and
An example
structure
to viewpoint,
curve
the curve,
by 50%.
significance
have been
curves
gives
of the
along
problem,
principles
combinatorial
relations
segmentation,
besides
be useful.
resent
implemented
algorithm
resolutions,
by
points
curve
dcvclopcd
problems
would
unlikely
values.
The
drawn
length
curve
covers
segmentations
at different
10 computer
derived
from
differing
of 100 points).
a t,hreshold
at the 0.05 significance
ings are not considered
significant.
This
of
adjacent
at all locations
the
signilicancc
structures
a curve
is executed
only those
in their
full
overlapping
measuring
resolutions
selects
for
three
number
for
was
other
to cover
small
at all scales
ciples,
if we
unifying
structure,
avoiding
for viewpoint-invariant
algorithm
possible
its borders.
procedure
different
every
The
knowledge.
random
looking
struc-
Ilowcvcr,
of only
of the
which
outside
thinning
more
size
segment
attempted
test
it is possible
groupings
groupings
extend
world
will as-
diCfercr;t
a relatively
groupings
given
grouping
to
of error,
groupings
to 6 scales
adjacent
costly
with
the
we examine
with
and
it exhibits
nonrandomncss.
margin
We examine
of two,
to
for
locations
factors
not
bc too
curve
a rcasonal~le
all scales
whcro
resolir Lions.
It would
scgmcnt
at all resolutions,
segmc;ltxtions
using
edge
E. Riseman,
[12] Witkin,
Andrew
P. and .Jay M. Tenenbaum,
“On the role
To appear in Human and Machine
of structure
in vision.”
Vision, Roscnfeld
and Beck, eds. (New York: Academic
Press,
1983).
These
size of the
against
260