Vision Flash 5
February, 1971
Views on Vision
Eugene C. Freuder
A.
Vision Group
I. Laboratory
M.I.T.
Introduction
in our
introduced
discussion of
the
"object partition
provide a unified theoretical
base for a
we
be expanded upon to
concepts which may
a number of
problem"
wider segment of
our
work in vision research.
This
paper
attempts to
indicate
some of
the
scope and
subtlety of the issues involved.
We discuss semantic, syntactic
and contextual distinctions that
bear on visual
analysis.
We
focus on line predicate analysis, and indicate how various other
aspects
of the information content of
treated
With
the
tame
partition analysis.
view
of
scene
structural
the picture graph can be
approach
This consistency is
analysis
as
an
we
applied
to
used to facilitate a
interlocking
heterarthy
of
knowledge structures,
A
very general discussion in Part
our approach, is
followed in
I of Issues which shape
Part II by
a solidification
and
expansion of the discussion in the cbntext of some specific line
predicates
and the
subproblems of
visual analysis
which they
characterize.
(Familiarity with
Vision
Flash 4,
The
Object
Partition
Problem, is assumed in this paper.)
I,
Model:
Semantic/syntactic
conceptual tool
analysis is gaining currency as a basic
in artificial
intelligence.
However care
must
be
taken if the application is to
We
will
w~11
the
employ
Wleld a net gain In Insight.
mathematical
of
approach
defined
We will not
logicians and model theoHists to such distinctions.
develop
this
Informally,
in
approach
great
an attempt
in
detail
to clarify
but
here,
our
view of
rather,
the scene
analysis problem,
We
start
"structure".
with
The
a
syntactic
language will
"language"
and
consist of
a
senantic
the elements that
form "line drawings" or "plicture graphs", points, line segments,
regions, plus
structure
predicate symbols
will consist
individual
that
picture
discussed
of physical corne s,
various physical predicates.
"Interpretation"
to be
later.
The
edges, faces, and
Our interest lies largely in
maps
graphs,
constructs
into
in
physical
the
the
language,
constructs,
three
dimensional physical scenes.
We
consider
first
Individual
picture
graphs
without
attendant predicate symbols.
We seek
the physical scene
corresponds
graph.
We
physical
to
the
interpretation
Interpretations are
"best"
picture
is
possible.
interpretation,
information about
configuration
may
possible.
the
The
and
nature
be viewed
of
as a
may
decide
that no
More
likely
several
process
obtaining
the
that
of
choosing
furthee
corresponding
problem in
the
specific
physical
determining how
appropriately to apply various predicate symbols to the
picture
graph.
An
Individual
line, for
example,
may be
representing one or several physical edges.
Interpreted as
A predicate symbol
"$" might be applied to the line with the understanding that "$"
Is Interpreted as the physical predicate "is a single edge".
of label
much.
we have seen, in
Howevet,
we generalize
this apprbach
mean
the partition problem paber,
terms,
to work directly in picturd
how convenient It may prove
and
seem to
manufacturing may not
priori this sort
A
below.
More importantly the
predicate concept does not let us forget the semantic content of
our
syntactic
manipulations.
In
particular
the
predicate
approach stimulates the search for basic r lationships that
be
may
used to characterize or Inform the more general questions of
scene analys.is.
We will
focus attention
in this
paper on a
class of predicates we call "line predtiates".
(in this paper we wili take the somewhat inelegant approach
of
using these distinctions where we
need them for clarity and
blurring them where we need to for simplicity.
This approach is
facilitated by the observation that a physical scene, in a given
view, can
indeed be
corresponding
will deal
two
largely
treated
largely isomorphically
dimensional projection.
"line" predicate
with
In
rather
with
the
particular, we
"edge"l
than
predicate terminology.
The
concept of scene plus
view could be expanded on.
think it may be important to recognize that our low level
We
three
dimensional predicates need not concern themselves with "hidden"
views
concerning invariant
view dependent
relations.
relations, but
The equating
may rather describe
of semantic
physical
models with full three dimensional encodings may have needlessly
hindered
and
confused
the
required
fusion
of
syntax
and
semantics
in vision research
work.
At any
rate it will be
understood that we may still use the familiar terminology,
line
at the concave julction of
"the
the two regions", for example,
as an abbreviation of "the edge represented by the line at
the,
etc.".)
Line Predicates:
A
line
predicate
is simply a
assertion involving a line.
use
predicate
vertices
more often
or the regibns of
the picture graph.
oh the
relation
across line boundaries,
vertex junctions.
example,
these
concern the riglons
generally are
predicates to the
Thus the
on either
the
Our predicates
ihht dbscribe
th.
as established at
In the case of the partition predicateS, for
the predicates
line, but we
key prb0ertles
we
of "line drawings."
been focused
will geherally f6tus
of regions
makes an
It may seem tautological that
lines as the basic unit of our analysit
However, attention has
that
various levels
concerned with
the aplication
of
entering a given vertek.
set of lines
of picture
bordering a given
graph elements
are
drawn
together In the line predicate concept.
Some of the properties of lines that are of Interest are:
definition
or membershlp--whether they define or belong to
one or both of the regions they bound, and if one, which;
junction--convex, concave, flat, or illusory;
multiplicity--whether they represent
one physical edge
or
two.
When
the edge Is Illusory we are interested in which plane
Is on top of which, and more generally when an ed.ke defines only
one of the bounding regions
we are concerned with which
region
is "over" or "occluding" the other.
There
are relationships between
Illusory
them obvious.
one edge;
implies
iMplies
flat
two
associated regions belong to
edge Implies that the
one
edges;
thdse properties, some of
the same body, two edies the opposite.
may
properties
be
the
facilitates
perhaps, line
importance
Of equal
operated
with in a
heterarchical
for
predicatds
these
manner which
uniform
required
Interaction
of a
successful vision system.
Analysis:
in brief, any
lIre 0pedlcate
may be
given the
treatment
afforded the "partition predicates" as described in the previous
immediate benefits may may vary from one predicate
The
paper.
to another
Increase
but the
basic
theoretilal advantages
remain,
and
as the aproach Is extended to unify more bf the vision
system analysis.
We begin
predicates)
our treatment
by
establishing a
possible Intetpretations
predicate
from
property.
context
dictate
of
and
of
a
other
base we
knowledge
picture graph.
analysis, or alternative
4
arrive
picture
or rate on a plausibility
a given
at
an
operati
predidate (bt
"complete
From this
have in applying the predicate
of
a llne
set
of
characterizatior'"
of
with
respect
the
cad apply information
structures
to
eliminate,
scale the various choices we
symbols to the various
On
to
this basis
analyses, of a
we can
given scene.
portions
compile ad
We
may
1
n
terms of syntactid rules for manipulating the predicate symbols.
The
procedure will derive, however, from a semantic motivation,
as it seeks to provide a semantic (physical) analysis.
A
complete
characterization
is
Initiated
with
an
enumeration of all possible intepretations of the various vertex
configurations with
predicates).
of the
a given
with the
restrictions;
of
the partition
relation,
for
tend
to reduce
predicate was
example,-and
the
symbols.
predicate concept for posible
corret0onding syntactic resttlcttvi.
type will
set
corresponding predicate
we examine the semantic
equivalence
predicate (or
Syntactically we can list all possible labellings
vertex type
However,
respect to
observed to
so
we
be
an
established a
Symmatrtis within a vettek
number of
different
labelling
possibilities as well.
The
physical domain will very likely be restricted in some
fashion, e.g.
further
concerned
to planar
restrictions
on
in this paper
polyhedral solids.
the
with
predicates.
ady context
This
We
more
may
will
impose
not
be
general than
planar polyhedra, i.e. we are not concerned with curved objects.
In this context we can observe
that any line (segment) must
be
consistently labelled along its length for any of the predicates
we consider.
The line cannot receive one interpretation at one
endpoint, another at the opposite endpoint;
e.g. a line
cannot
change from a concave to a convex junction.
Our "complete characterization" should be undertaken at the
most general possible context level to begin with.
we suggest a
physical domain limited
In our case
to planar polhedra,
with
few
Initial
constraints on the picture graph or Interpretation.
occlusions
"degenerate"
should allow the various
Our most general context
and overlaps of lines and vertlces.
Certain of our
specialists or initial programs may not be able to handle
problems,
but they should certainly
universe
which
our
system
Establishing an optimal
a given approach
of
these
be included in the overall
must
ultimately
understand.
level for the most general context under
Is itself a research decision..
planar polyhedra seems historically
General
views
and technically to be a
good top level context for line drawing analysis.
At this
level,
after the semantic, symmetric and contextual
Simplifications have
'been made,
alternative
labellings
vertex
predicates
and
range
of
we
for
vertex
are left
any
types.
with a
list
given
set
A
complete
of
obtained from applying the
vertices of the
That
cannot accept an overall
vertices
"global
to
receive
This
achieves
combinations.
"one line, one label"
illegal
a
all consistent
criteria, we
labelling that requires any
inconsistencies" as
then
various vertex labellings to the
picture graph in
is we must observe the
line
set of
alternative interpretations of any given picture graph may
be
of
labellings,
and
individual
we
must avoid
in the case of partition
labelling.
complete
characterization
of
property or set of properties in picture graph analysis.
a
given
Often
this much may buy us more than we expecL. The possibilities may
be considerably narrowed down, basic questions such as "is there
any
least
possible Interpretation" will
we
have
a
basis
have been answered.
At the
fo
ing,
and that of our programs.
We
are, of course, generally interested in the question of
which is the "best" or
picture
graph.
"most likely" interpretation of a
We need
to build
on our
given
characterization to
provide an answer to that need.
Cn text:
Contextual Issues arise in several forms in the restriction
or ordering
analysis.
of the
decision space
Often. we
in a
are Interested
Interpretation will be
(or if any
given line
in what the
predicate
most likely
interpretation Is
possible)
under a restricted context.
Further
order
the
context
predicate
reStrictIons may
labelling
analyser may thus be Informed
further narrow
possibilities.
"specialists"
the advice of
the
The ultimate goal,
which can handle any context,
perhaps, on
predicate
by, or alternatively inform,
context analyser of a heterarchical system.
of course, is a system
A
dowh or
employing
the context analyser.
Ih the course of development of such a system, we often need
restrict
the
specialist,
context
analysis,
either
to
produce a
or merely to simplify a problem sufficiently to get
a foothold on It.
appreciate
of
to
the
It is imperative at such times that we fully
nature
restrictions we make.
and
implications
of
the
context
To place this problem In focus we return
briefly to our model theoretic viewpoint.
Restrictions may
semantic
domain, or on
be imposed
on
either the
the interpretation.
for example, that only picture
syntactic
or
We may stipulate,
vertices joining three or
fewer
that only trihedral corners
lines may occur In a picture graph;
vertices are
or that three line picture
domain;
appear in the. physical
may
interpreted as
to be
trihedral
If we are to appreciate
three possibilities are not Identical,
the
of
difficulty
we
accomplishments
understand
must
"general
Huffman's
restrictions.
the
and
problems
our
These
corners.
of
our
of
our
criteria,
for
extent
nature
the
Position"
example, while a nice concept, is misleading in that it does hot
embody all of the restrictions Huffman apperently expects it to.
A common problem of newcomers to the area of partition
is
of restrictions that
the imposition
ahalysis
effectively reduce the
problem to the trivial case where separate objects meet only
occlusions.
T-joint
restriction;
it
restrictions.
quite
reduction need
This
follows
from
quite
not be
natural
at
an explicit
lower
level
Furthermore, exclusive T-joint demarcatlon is a
It happens, however, that it
common physical situation.
is almost
a degenerate
partition
analysis.
case
Unless
as regards
one realizes
thý full
problem
this, there
of
is the
temptation to possibly misuse the principle of generalizatio, in
concluding that the methods used
or
have analogues in the
for example, has
make
in the restricted case
general case.
possibilities in a
it impossible to use In
indication of partition;
extend
The "simple" T-joint,
more general context
that
any straightforward-fashion as an
on the contrary It may prove to be one
of the most misleading and complex vertices to analyze.
Furthermore, we might hote that
restrictions
we make, in order
the nature of thb
context
to achieve va ious results, may
at times be among the most Interesting aspects of our
As
in
other
mathematics, the
required strengths,
considerations
involved
restrictions,
for
realization
of the
an
alternatives, and
-various
can constitute a significant
An appreciation
moreover,
In
theoretical
may be
model
of a cOmprehdnsive analysis,
hypothesis
aspect 6f a result.
context restrictions
adequate
analysis.
required,
or
a model
practical
that can deal
as effectively as possible with both restricted and unrestricted
contexts.
Insofar as they affect possible or likelV labelling choices
for a given predicate, context restrictions can be viewed In
least
two alternative fashions.
area in
which we
question
are to
in order to
changing
assumptive
example
the
possible in a
restrictions, based
the
option of lifting
should
latter case,
on "general
a given
scene, or
they
the
or
lead
to
choice
of
experience", specific
whatever, becomes a
facility.
In
summary, our
define our
these
it, with the
restrictions
In
contextual analysis of
vital
questions, for
physical realization Is
limit or order
such
may formally define the
They may heuristically define our decision space
unsatisfactory results.
context
ask our
of whether any
given context.
They
at
context restrictions
results, and
results
without
we cannot
a clear
help to
properly assess
analysis
of
direct and
or
extend
these contextual
assumptions.
Semantics:
Within any context, whether
It be the most general
context
or
some more restricted one, we use semantic information.
The
nature of the semantic structure Indicates the limits on the set
of possible labelling dhoices in constructing a characterization
of a given
predicate.
Semantic Information
also assists
our
judgment on the relative plausibility of the various choices, in
order
to
inform
our "best
possible
interpretation" decision
procedure.
There
seems
to
be
some
semantic/syntactic distinction
benefit. from
a formal
needless
at
analysis.
confusion
on
the
we
may
this point.
Again
In loglc,
a "theory"
Is a
collection of statements, in a syntactic language, which may
defined
either semantically,
as statements
which are
some semantic domain, or syntactically, as statements
from
-true in
derivable
specified syntactic "axioms" by specified syntacti6 rules.
Often the Interesting "metatheoretical" problem arises,
a
be
whether
semantically defined theory can 6e derived also syntactically
a
The axioms
and
rules
that are
employed
in
a
syntactic
formulation of a semantically defined theory are not, of course,
pulled
out
of
awareness of
a random
the semantic
hat.
They
are chosen
content they
with
will embody
and
a keen
must
Imply.
Similarly
in our work in artificial
intelligence.
We may
seek a "syntactic" procedure for simulating a semantic function,
but there should
must
be
semantically informed
language study,
natural
be no surprise
language
for
may
example,
that our syntactic
to be
reasonable,
the legitimate
be regarded
as
a
operations
In hatural
sentences
of a
semanticaily defined
theory.
to
Some aspects of this theory are particularly
syntactic formalization;
or "syntax".
and
we
generally term "grammar"
This use of the term "syntax" may be
are further
definition
of
precise
model
need be
no
grammar"
these we
hindered
language
by a
"semantics".
that
a
misleading,
correspondingly restricted
However,
theoretic semantic/syntactic
surprise
amenable
reasonable
in
our
more
t6rminology, there
syntactic
"language
must implicitly, or preferably explicitly, ackndwledge
the semantic
generate.
motivation
And
there
for the
need
theory
be
no
semantically informed approach to
simultaheous
syntactic
basis
it is
surprise
attempting
if
to
a successful
language analysis provides a
fbO
"language
grammar"
and
their
own
"language semantics.
'1
Syntactic
momentum.
forebulations
They
may
indeed
sometimes
suggest
generate
new
semantic
insights.
However, we must keep semantic motivations in mind if we are
to
develop a comprehensive and comprehensible syntactic model.
Knowledge Structures:
Semantic
context implications
are certainly
important in
determining plausibilities In our scene analysis.
Knowledge of
many sorts, however, may have a bearing on our decisions for any
particular
problems.
predicate
or
set
of
We require the individual
predicates.
We
face
identification and analysis
of these various aspects of visual knowledge of a scene.
need to employ the heterarchical
structures,
analysis.
in their
twin
interrelation of the
construction, for
a given
And we
knowledge
picture graph
We may be aided in our implementation of the
latter
step if our individual analyses exhibit some common form.
The
partition analysis paper discussed
advantages
of
a
line
predicate,
in some detail the
complete
characterization
approach to the
description and analysis
structure.
noted, in particular, how th6 labelling network
We
of a given
description built in the potentially global
knowledge
Interrelationship of
the picture parts, and the comlete characterization allowed
alternative
aproach
failure
flexibility.
allows us to use
The
predicate
any and all parts
an
labelling
of the network for
information, and both positive and negative Indications.
We are
knowledge
interested in
structures which may be
analysis,
the
predicates, or
amenable
this paper
observation
sets
in the
that
bf predicates,
content of
of
approached with this type of
there
are
which
are worthy
Wd
to an analysis of this sort.
the information
multiplicity
picture
several
of
line
and
believe that much of
graphs can
be
profitably
characterized thus by line predicates.
Certain
sets of
these predicates
units of analysis for various
of
visual
analysis,
may be
useful as basic
specific subproblems In the
e.g. the
partition
several qualities whidh might recommend
problem.
area
There are
a set of predicates
as
basic analytical concepts for dealing with a particular problem.
We would like the predicates to characterize the problem in some
sense.
Ideally
characterize
the
we
would
problem,
like
in
the
the
predicates
sense
interpretations of a
given picture graph,
given
correspond to
problem, will
that
to
all
"fully"
possible
with respect to
the
alternative "labellings" of
the picture graph with the specified predicate symbols, and vice
characterization of the predicate
complete
sense allows
characterization in this
A full 1-1
versa.
the
set to constitute a
complete characterization of the specified problem, In the sense
of all
interpretations
possible
the
of
picture
graph
with
respect to that problem.
usual
The
will carry
over
to
a
particular
the predicate
network analysis
line predicate
advantages of
problem.
characterized
predicate
In
convenient framework on
network is a
which to add the context and knowledge informatioh that we
need
to make our analysis choices.
If
we have chosen our characterizing predicates well, they
will Indeed seem
problem
the
under consideration.
basic
fundamental properties of
to reflect the
of
nature
the
the
This can be most revealing as to
and
problem,
other
relevant
but
secondary or separate con§iderations may then be brought to bear
Isolatl6n and observation of
In a natural and effective model.
criteria of
the basic
their
under
modification
interdependent
understanding,
property, anb
a visual
properties,
and a
restrictions,
context
Is
an
procedural model
observation
ideal
outline
of
and
for
which follows naturally
from our basic theoretical approach.
If indeed many
predicate
of our knowledge
network labelling
conjoining the various
that
much
more
structures have the
crucial process of
.structure, the
structures heterarchically
tractable.
Our
model
same
of
our
may be
made
procedural
implementation of the visual process will be clarified.
II.
Without
high level of
from the (ahem)
descending too far
generality intended for this paper we might indicate some of the
areas in
which
than
other
predicates
line
are studying
we
partition analysis, in order to illustrate and elaborate some of
the issues and concepts discussed here.
Junction:
Huffman has done some work with what we would term the line
concavity,
predicates
convexity
and
(Huffman
"illusory."
uses two predicates which combine the illusory concept
actually
the true definition
with an indication of
These
of the ýdge.)
predicates have great relevance to the partition problem and the
"possible configuration" or "existence" probleh--whether a given
as a physical
any realizable interpretation
picture
graph has
scene.
Their usefulness will be clearer with the inclusion
the
of two coplanar
"flat" predicate (junction
complete
characterization
strongly restricted
predicates
(Huffman
approach
However,
context).
concave, convex,
planes), and a
with a
begins
we believe
flat, illusory,
of
that
the
characterize what
might be termed the "junction problem", and are not
appropriate
as
or possible
basic
units
of
configuration problem.
characterize
analysis
They
for
do
those problems in the
the
not
partition
characterize,
or
fully
manner that we indicated we
would like a set of predicates to characterize a visual analysis
problem.
It should
that
of
be obvious,
from our
any attempt to use the
partition
partition problem
paper,
junction predicates as basic units
analysts
is
unwarranted.
A
complete
characterization could not even be attempted.
The issues Involved in the characterization of the junction
problem
could be
might be
choose
expanded upon
more instructive
the
"possible
at some
for the
length.
purposes of
configuration
However, it
this paper
problem"
for
to
further
discussion.
Configuration:
We
note
"existence"
first
problem
that
is
the
in
"configuration" problem.
recall,
In
"possible
a sense
the case
aspect
or
of
a generAl
of partition
analysts,
we were Interested first In characterizing all possible
(physically realizable) partitions,
one.
an
configuration"
For
the
characterizing
configuration
all possible
then in
problem
choosing the
the
interest
configurations may
beSt
in
lie largely in
determining if any configuration I~ possible, i.e. the existence
problem.
of
course,
essentially
answer is
The question of choosing the "best" configuration is,
of
asks
interest.
for a
generally the
However,
complete
result of
since
this
analysis of
the
question
scene, its
previous decisions
on
the
various aspects of the s:cene analysis.
This
interpretation
become more concrete if
of
of the
"configuration"
we choose an appropriate
problem will
formalization
the configuration problem, e.g6 in terms of line predicates.
Huffman is, of cobrse, not dealing in our
characterization
and
framework,
problem.
existence
wish to
consider
for
appropriate
as a
be proposed
such
labelling
upon
the
We have no argument there, of couase.
We
to
be
would
the
to
approach
characterization
our
bear
predicates
junction
the
whether
another predicate set should
basic analytical
as provided
advice
relevant
junction
designed
several
problem, or whether
configuration
his
presents
clearly
one- among
as
approach
he
characterization, on
by
which
junction knowledge
the
structure might be hung.
Huffmah indicates that the junction predicates are not used
as what
would
we
problem.
In
to
complete
any case, they
less than
even something
respect
term a
the
characterization -of
characterization role
problem.
configuration
hopefully address
nature of the problem.
criteria.
than
proper
choice of
themselves directly
have
to the basic
The junction predicates do not meet this
An attempt to
view them
as characterizing,
rather
relating to, the problem will indicate that there are more
basic predicates,
that
A
with
problem will, as we
characterizing predicates for a given
indicated,
suitable to assume
do not appear
a "full"
this
really
assumed in
seem to
be the
the definition
of convex,
criteria
possible physical
of a
etc;,
realization.
Context:
the basic
issues
involved
should become
clearer
if
we
consider the configuration problem Its most general cOntext.
In
beginning
with the restricted context Huffman deals in, we lose
or obscure some of these issues.
Guzman
distinguishes
pictures
that
have
no
pbssible
physical interpretation as those which could not be produced
by
photographing or projecting any three dimensiondl scene from any
view.
This
definition
seems
desirable
as
a
basic
characterization of an "Impossible" object.
of the standard "optical
Guzman notes that many
illusion"
"impossible" figures do not fall under this definition, however.
The
"Penrose
special two
three
Triangle",
for
example, can
dimensional projection
dimensional
illusions",
or
Constructs.
"Impossible
of at
The
be
least two
study
figures",
obtained
of
different
such
precisely
"optical
formulated,
becomes then a question of which picture graphs have a
Interpretation,
(
given
certain contextual
as a
physical
constraints,
on the
physical domain, the picture domain or the Interpretation.
This
Is the problem that Huffman sets himself, for a context which he
defines.
Under
these
restrictions
circuistances
one
can
interesting aspects of
content
of
a
the
consider
becomes
the study.
reasonable
nature
In
model
of
of
one
a sense
what
the
of
dontext
the
they form
constitutes
most
the
the
"illusion".
Characterization:
We
now
have
an
Idea
of
the
general
nature
of
the
configuration problem to which we wish to apply a line predicate
characterization.
We
begin by seeking
a basic dharacterizing
predicate for the configuration problem, starting at the general
L_·_l~_··~
C
I·1
·
·
·
·
I
context level or arDitrary views or planar
polyhedra,
Context
restrictions may then be adduced in an instructive fashion.
might
tentatively
belongs
propose
the
to the bounddry of
line
following
R1 as defined-by
R2,
We
1
predicate:
or briefly R2
defines R1 along 1, where R1, R2 are faces, bordering an edge 1
in
a given view of the scene, and 1 physically forms a bounding
edge of
R1.
right
might employ
a
"-~-",
predicate symbol
foe
to be interpreted as this predicate, which we refer to
example,
as the
We
"definition" predicate.
angles to a picture line,
The
arrow would
be drawn
at
which Could be labelled with a
single arrow in either direction or with two arrowS
in
directions
Consider the
(abbreviated as a doubld ended arrow)i
opposite
following labelled f.gure:
We do not
predicate
here;
ourselves it
Intend to
Indeed
is the ideal
present a complete
we
have
not
yet
analysis of
fully
this
satisfied
predicate for our purposes.
However,
some further discussion should be instructive.
From our most general semantic context we observe that this
"definition" predicate Is anti-reflexive;
Correspondingly
any label for a line
sides by the same region is impossible.
believe,
encompass all
the truly
RO-RO is
impossible.
which is bordered on both
This criteria will, we
"impossible" objects
in our
general context, and allow us a complete characterization of the
possible configuration problem.
Restricted Context:
As it happens the context considerations here are very much
the partition problem, for example,
the opposite of that in
the
and discover
the problem,
aspects,
not
by
configuration problem we enrich
For the
following sense.
the most
this fashion
in
"human" optical
general
difficult
context,
but by
We may justify enriching the
considering a restricted context.
problem
interesting and
its most
allowing
In
in the
an thterest
to
by appealing
illusions, or by noting the advantageous aspects
of employing and understanding the implications of such
context
restrictions.
In
particular the context restrictions or assumptions that
lead humans to perceive an optical
object"
or
other
assumptions,
system.
experience.
angle
likely
human
They
they
processing
constitute
and
"Impossible
straight
line
overlapping lines, Whatever--
dismissed as weaknesses
More
facilitate
type)--right
absence of colinear
should not be
illusion (of the
are
and
in the human
very
good
heuristics
description
plausibility
perceptual
of
rankings
interpretation decision structure, based in part on a
that
visual
for
the
knowledge
of context plauslbilitids derived from wide visual experience In
a particular environment.
In
summary, strong
very useful
context plausibility
judgments, while
in choosing a "best" Interpretation, may essentially
function as at least a temporary context restriction, which
may
interest
In
determining
been
has
context
based
plausibility
restricted
of context
a study
(Though more interest should therefore be
"Impossible figures".
than
useful
bedeftt from
criteria may well
shown,
Thus our
ilUslons", or "impossible figures".
lead to "optical
in
at times,
context restrictions themselves,
various alternative
the
made to
whith can be
"model"
illusions".)
various human "optical
Consider the simple figure:
This
Is certainly nb Impossible
object in any deep sense.
could easily be looking straight down
at the top of two
bricks, for example, one cube, one L-shaped.
We
nested
If we were looking
at the bases of two pyramids we could shift our viewpoint a fair
amount
part
without changing this projection much.
here
is
in
fact
the
contextual
corresponding
context assumptions and
that
lead
us
a
cube
might
interpretation
correspondingly lead
verifier.
consideration
predicate,)
(This
of
a
to
vision
raises
the
a
junction
as
our
missing
system
obvious
experience,
and
the
context based judgments,
choose
with
The interestihg
to
similar
problem
most
line,
call
plausible
and
should
in
its
line
issues
for
our
and
thý
junction
A complete vision system would, of course, be aware
of the
of an
possibility
Similarly
a
knowledge of
view" to
"overhead
the
to the
lead
restrictions that
on.
back
fall
good
or a vision system, a
Penrose Triangle illusion gives us,
idea of how to physically Yealize the Triangle in a more general
context.
Restricted Characterization:
Very
briefly how, various alternative context restrictions
can be
proposed which
embody
considerable
permit formulation
"defini-tion predicate"
"context
certain
We
observations,
however,
labellings, and to
employ
require
number
rather
us to
catch
as
in the
assure "global
the
donsistency".
other knowledke ttruttures,
semantic
sophisticated
restrict
which
of possible-
configurations, e.g.
f,,lli
tn
the
This enables
labellings.
impossible"
figure.
above
on
restrictions
of principles
possible
We may
also
such the junctioh predicate
labelling network,
Huffman presents several subtle criteria that can be
in
these
roles.
Optimally, if
choice was ideal, any cýiteria
not
our
viewed
characterizing predicate
that implies a picture graph
physically realizable under
is
the given context restrictions
will also imply that no predicate labelling is possible that
is
consistent with the given context.
This
issue
may
partition analysis.
be
clearer in
the
familiar
By applying certain syntactic restrictions,
derived from the general semantic context we set for
to
the partition
cOntext of
predicate labelling procedurd,
"complete characteri-zation"
of
the
partition
ourselves,
we achieved a
problem.
That
meant that for the general context "structure" we had achieved a
1-1
correspondence
betWeen syntactically
allowable labellings
and semantically (physically) possible partitlions.
(Technically
a "complete" and "consistent" characterization.)
We then observed that restricting the context further could
indicate
further
partitions, and
restrictions
on
the
corresponding syntactic
made on the labelling p6ssibillitle.
physically
possible
restrictions could
be
These might take the form
of general principles (such as the "transitive" property usedd in
the general contekt) or
possibilities.
proving that
for
corresponding
could
be
We
any
or
that
all
would
kiiVeh
6-ntext
set of syntactic restrictions
Insure
there
we can continue to
That
is, there
the
this case,
restricted
still
a
1-1
as we
4uestlon
of
have a complete characterizatidn of
configuration problem
since in
was
is the
the partition problem In various restlicted
to
restrictions a
between poSslble syntactic labellings and possible -
physical configurations.
whether
labelli-ng.
did not concern ourselves at that time with
nice (e.g. finite)
found
relationship
|imple elimination of certain
this
contexts.
issue becomes
have noted,
Applied
more vital,
Interest lies
In
the
characterizations, whereas In partition analysis the
general context forms the basis of our problem.
Conclusion
Well clearly
several
We are
concluding
others, as we concluded
this paper
by
beginning
the partition paper by leading
into this discussion.
this
work are
But then, the continuing implications of
Indeed part
"definition" predicate
perhaps,
partly
Patrick
Winston,
as
determination, as
study,
a
on
of the
point of
for example,
formalization of
physically
the partiti6n
this paper.
could
some
associated
Our
be
viewed,
recent
work of
edges
predicate analysis
In
shape
formalizes
the germinal work of Guzman and otheks in partition analysiS.
Hopefully
we
have provided
significance of a nuinber of
some
idea bf
the
scbpe and
the key concepts, obserVatl6ns
methods that are guiding our current vision researdh.
and
Acknowledgements
The theoretical
result
of
a
atmosphere and
happy
the
approach
described
confluence
working
of
a
-
in this
propitious
environment of
the
paper
is a
theoretical
Vision
Group
Vision System.
Among those who contributed to either the atmosphere or the
environment (Ralph Nader excepted):
M.
Dowson
A.
Griffith
A.
Guzman-
B.
Horn
D.
Huffman
J.
Lerman
M.
Minsky
S.
Papert
M.
Rattner
P.
Winston