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. Burigana Mathematical Social Sciences 38 1999 315 –341
Epilogue to their book on ‘‘integration of visual modules’’ state that ‘‘we need a formal theory for combining information from different sources’’ Aloimonos and Shulman,
1989, p. 267. The aim of the present paper is to suggest some simple ideas on this issue, i.e., to draw a minimal conceptual frame for MCP, by defining and relating a few
concepts which appear to be implicit in all its applications. Identifying and making explicit the basic concepts in a rigorous way is an initial and necessary stage for
undertaking formal analysis of the paradigm, its conditions and implications.
The language we adopt for this formalization is elementary set theory, in order to ensure wide generality to the fundamental concepts, and to obtain a system that is
flexible and faithfully applicable to various specifications of MCP more detailed arguments for the set-theoretic approach will be presented in the concluding section.
The main concepts are introduced progressively, starting from variables Section 2, moving to constraints Section 3, systems of constraints Section 4, and pairings
between systems of constraints and valuations of stimulus variables Section 5. Besides definitions of these various concepts, some of their immediate facts and relationships
will be noted, as the result of an initial inquiry into their formal properties. The paper makes use of some simple examples from perceptual science in order to illustrate the
defined concepts, but it offers no fully extended demonstration of the adequacy of the suggested apparatus in dealing with specific research problems on vision. Nevertheless,
in Section 6 we will mention and comment on some key concepts of current approaches to the study of vision which may be expressed, linked and clarified by using terms of the
suggested conceptual system.
2. Variables and valuations
Intuitively, the concept of a variable refers to some kind of results that are partially different or unstable along a series of observations. Equally, the concept of a variable
demands some stability along that series: results must be homogeneous to some extent, and must refer to a common and permanent base, so that they may be properly qualified
as different values of one and the same variable.
These obvious requirements lead to the first components of our theoretical construc- tion. First, a set O of possible observational cases is presumed, each case o [ O being a
closed perceptual episode, which comprises data at separate empirical levels primarily, the S-, I- and P-levels defined in the Introduction. Second, we assume a well-defined set
U of recurrent units i.e., specific objects or events, or selected parts of objects or events; each unit u [ U is presumed to occur at a fixed level of data in all cases in O,
with stable identity but with possibly and partly changing properties. Set U of recurrent units will serve as the required stable base for variable identification.
In relation to the given set O of observational cases, a located variable v is determined by three aspects: type X , as a complete set of admissible values for the
v
variable, state-function s , as a function from O to X , and location l , which is a subset
v v
v
of set U of recurrent units. Type X is here conceived as an abstract and homogeneous
v
inventory of possible properties e.g., continuum of possible reflectances, variety of possible lengths, complete set of admissible distances, etc.. For each case o [ O, the
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corresponding value s o of function s is a point in X , which represents the value
v v
v
taken on by variable v in case o. Location l is the carrier as a system of recurrent units
v
of the properties which constitute the values of variable v. Location l is a singleton
v
u U, if X is a type of attributes or unary properties e.g., a continuum of possible h j
v
reflectances, a pair u, u9 U, if X is a type of binary properties or relations e.g., a h
j
v
complete set of admissible distances, and so on. We presume that the unit or combination of units acting as the location of a given variable, and the properties taken
on as values by that variable, belong to a specific level of data, which remains the same along all cases in O, so that it makes sense to distinguish between S-variables physical
aspects or conditions of the observed realities, I-variables properties in the proximal images of those realities and P-variables contents of the corresponding perceptual
scenes.
Now, besides O and U, a third basic set for our theorization must be introduced. This is a collection V 5 v , . . . , v
of located variables, which are jointly defined in universe h
j
1 m
O of observational cases. Set V is presumed to be exhaustive, in that it contains all variables that are relevant for a fixed perceptual problem; each variable v i 5 1, . . . , m
i
is qualified by some type X , a definite state-function s :O →
X , and a specific location
i i
i
l U. Set V must be imagined as a highly heterogeneous collection which assembles
i
attributive and relational variables, variables of different types, and variables of the same type but with different locations. In particular, set V may contain variables referring to
distinct levels of data, i.e., S-, I- and P-variables. Put M 5 1, 2, . . . , m , which is the h
j index-set of collection V; any subset K 5 i , . . . , i
of M unambiguously determines a h
j
1 k
subset VK 5 v , . . . , v of V; conversely, any subset W 5 v , . . . , v
of V h
j
h j
i 1 ik
i i
1 k
unambiguously specifies a subset M W 5 i , . . . , i of M.
h j
1 k
Any set of pairs y 5 i , y , . . . , i , y , such that y [ X and i ± i
for all
h j
1 1
k k
h i
h h 9
h
1 h ± h9 k, will be referred to as a partial valuation on the set of coordinates M
uy 5 i , . . . , i or, equivalently, for the set of variables V
uy 5 v , . . . , v . In case h
j
h j
1 k
i ik
1
k 5m so that M uy 5 M and V uy 5V , set y will be called a complete valuation. Symbols
Y and X are used here for the set of all partial valuations and, respectively, for its subset of complete valuations; moreover, for any K M, symbol Y
uK denotes the set of all valuations on coordinates in K i.e., Y
uK 5 y [ Y: Muy 5 K , in particular YuM 5 X.
h j
These are the natural reference sets for the combinations of values taken on by sets of variables. In fact, for any o [O, the joint state of a set of variables W 5 v , . . . , v
in
h j
i i
1 k
the observational case o can be described as the set of pairs Wo 5 i , s o, . . . , i ,
h
1 i 1
k
s o , which is a point in Y more specifically, in Y uMuW . Accordingly, set
j
ik
WO 5 Wo: o [ O of the joint states of W in all the observational cases amounts to a h
j subset of Y more specifically, of YM
uW . In particular, for any case o, joint state Vo 5 1, s o, . . . , m, s o of the entire set of variables V is a complete valuation,
h j
1 m
and the exhaustive set VO 5 Vo: o [ O of such joint states is a subset of X. h
j Compatibility between valuations, denoted by symbol
↔ , plays an important role in
the following, and is defined as follows: any two valuations y, y9 [ Y are compatible with each other if coincidence of some of their pairs in the coordinates implies
coincidence of the same pairs in the values or, formally:
9 9
9 9
y ↔
y9 if for all i , y [ y, i , y [ y9, i 5 i implies y 5 y .
h h
h 9 h 9
h h 9
h h 9
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. Burigana Mathematical Social Sciences 38 1999 315 –341
These facts are immediate: y y9 implies y ↔
y9 inclusion between valuations, as sets of pairs, ensures compatibility, if M
uy Muy9 then y ↔
y9 implies y y9, if M uy Muy9 5
5 the two valuations are separate, i.e., they act on disjunct sets of coordinates then y
↔ y9. It is also directly seen that y
↔ y9 iff y y9[Y, i.e., compatibility of two
valuations means that their union is itself a regular valuation. Compatibility is first used in defining some basic operations on sets of valuations. Let
T,Z be subsets of Y. The restriction of T with respect to Z and the disjunction of the two sets are specified by the following formulae:
rT, Z 5 t [ T : t ↔
z for some z [ Z h
j dT, Z 5 y [ Y: y 5 t z for some t [ T, z [ Z.
h j
In other words, restriction rT, Z is the set of valuations in T that are compatible with at least one valuation in Z, and disjunction dT, Z is the set of unions between compatible
valuations in the given sets. If Z 5 y conditioning set Z is a singleton, restriction is h j
denoted simply by rT, y, so that: rT, y 5 t [ T : t
↔ y ;
h j
accordingly, rT, Z 5 rT, y for any Z Y. These are two of various immediate
y [Z
properties of the given concepts: if T T 9 and y y9, then rT, y rT 9, y9
1 if T T 9 and Z Z9, then rT, Z rT 9, Z9.
2 In addition, for any K M and Z Y such that K M z for all z [ Z, the projection of
u
set of valuations Z on set of coordinates K can be considered, which is denoted by pZ, K and is defined as the set of valuations in Y
uK which are compatible with some valuations in Z, i.e.:
pZ, K 5 y [ Y: M uy 5 K, y
↔ z for some z [ Z .
3
h j
It is seen that projection, so conceived, is a special case of restriction, in that pZ, K 5 rY
uK, Z. The last definition in this section regards a correspondence between single partial
valuations or sets of partial valuations and sets of complete valuations. For any y [ Y, the corresponding extension E y is the set of complete valuations which are compatible
with y and more generally, for any Z Y, the corresponding extension EZ is the set of complete valuations which are compatible with at least one valuation in Z, so that
EZ 5
E y. Alternatively, the two concepts may be expressed by using the
y [Z
restriction operation, specifically: E y 5 rX, y
EZ 5 rX, Z . 4
Properties 1 and 2 directly imply that:
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if y y9, then E y E y9 5
if Z Z9, then EZ EZ9. 6
It is also easily seen that: if y
↔ y9, then E y y9 5 E y E y9
7 EZ Z9 5 EZ EZ9
8 EdZ, Z9 5 EZ EZ9.
9 Note that, because of the last two equations, the range of mapping E as a subset of the
power set of X is a lattice with respect to operations of union and intersection.
3. Constraints on variables
