Dimensionally Extended 9-Intersection Model

The Dimensionally Extended 9-Intersection Model (DE-9IM) is a topological model and a standard used to describe the spatial relations of two regions (two geometries in two-dimensions, R²), in geometry, point-set topology, geospatial topology, and fields related to computer spatial analysis. The spatial relations expressed by the model are invariant to rotation, translation and scaling transformations.

The matrix provides an approach for classifying geometry relations. Roughly speaking, with a true/false matrix domain, there are 512 possible 2D topologic relations, that can be grouped into ''binary classification schemes''. The English language contains about 10 schemes (relations), such as "intersects", "touches" and "equals". When testing two geometries against a scheme, the result is a ''spatial predicate'' named by the scheme.

The model was developed by Clementini and others based on the seminal works of Egenhofer and others. It has been used as a basis for standards of '' queries'' and '' assertions'' in geographic information systems (GIS) and spatial databases.

Matrix model

The DE-9IM model is based on a 3×3 intersection matrix with the form:

where is the dimension of the intersection (∩) of the interior (I), boundary (B), and exterior (E) of geometries ''a'' and ''b''.

The terms ''interior'' and ''boundary'' in this article are used in the sense used in algebraic topology and manifold theory, not in the sense used in general topology: for example, the interior of a line segment is the line segment without its endpoints, and its boundary is just the two endpoints (in general topology, the interior of a line segment in the plane is empty and the line segment is its own boundary).

In the notation of topological space operators, the matrix elements can be expressed also as

The dimension of empty sets (∅) are denoted as −1 or (false). The dimension of non-empty sets (¬∅) are denoted with the maximum number of dimensions of the intersection, specifically for points, for lines, for areas. Then, the domain of the model is .

A simplified version of values are obtained mapping the values to (true), so using the boolean domain . The matrix, denoted with operators, can be expressed as

The elements of the matrix can be named as shown below:

Both matrix forms, with dimensional and boolean domains, can be serialized as "''DE-9IM string codes''", which represent them in a single-line string pattern. Since 1999 the ''string codes'' have a standardThe "OpenGIS Simple Features Specification For SQL"
Revision 1.1
was released at May 5, 1999. It was the first international standard to establish the format conventions for ''DE-9IM string codes'', and the names of the "Named Spatial Relationship predicates based on the DE-9IM" (see section with this title). format.

For output checking or pattern analysis, a matrix value (or a string code) can be checked by a " mask": a desired output value with optional asterisk symbols as wildcards — that is, "" indicating output positions that the designer does not care about (free values or "don't-care positions").
The domain of the mask elements is , or for the boolean form.

The simpler models ''4-Intersection'' and ''9-Intersection'' were proposed before DE-9IM for expressing ''spatial relations''M. J. Egenhofer, J. Sharma, and D. Mark (1993)
A Critical Comparison of the 4-Intersection and 9-Intersection Models for Spatial Relations: Formal Analysis
", In

. (and originated the terms ''4IM'' and ''9IM''). They can be used instead of the DE-9IM to optimize computation when input conditions satisfy specific constraints.

Illustration

Visually, for two overlapping polygonal geometries, the result of the function DE_9IM(''a'',''b'') looks like:

This matrix can be serialized. Reading from left-to-right and top-to-bottom, the result is .  So, in a compact representation as string code is ''.

Spatial predicates

Any topological property based on a DE-9IM binary spatial relation is a spatial predicate. For ease of use "named spatial predicates" have been defined for some common relations, which later became standard predicates.
The ''spatial predicate functions'' that can be derived from DE-9IM include:

:Predicates defined with masks of domain :

:Predicates that can be obtained from the above by logical negation or parameter inversion (matrix transposition), as indicated by the last column:

:Predicates that utilize the input dimensions, and are defined with masks of domain :

Notice that:

* The ''topologically equal'' definition does not imply that they have the same points or even that they are of the same class.

* The output of have the information contained in a list of all interpretable predicates about geometries ''a'' and ''b''.

* All predicates are computed by masks. Only ''Crosses'' and ''Overlaps'' have additional conditions about and .

* All mask string codes end with *. This is because ''EE'' is trivially true, and thus provides no useful information.

* The ''Equals'' mask, T*F**FFF*, is the "merge" of ''Contains'' (T*****FF*) and ''Within'' (T*F**F***): .

* The mask T*****FF* occurs in the definition of both ''Contains'' and ''Covers''. ''Covers'' is a more inclusive relation. In particular, unlike ''Contains'' it does not distinguish between points in the boundary and in the interior of geometries. For most situations, ''Covers'' should be used in preference to ''Contains''.

* Similarly, the mask T*F**F*** occurs in the definition of both ''Within'' and ''CoveredBy''. For most situations, ''CoveredBy'' should be used in preference to ''Within''.

* Historically, other terms and other formal approaches have been used to express ''spatial predicates''; for example region connection calculus was introduced in 1992 by Randell, Cohn and Cohn.

Properties

The spatial predicates have the following properties of binary relations:

* Reflexive: Equals, Contains, Covers, CoveredBy, Intersects, Within
* Anti-reflexive: Disjoint
* Symmetric: Equals, Intersects, Crosses, Touches, Overlaps
* Transitive: Equals, Contains, Covers, CoveredBy, Within

Interpretation

The choice of terminology and semantics for the spatial predicates is based on reasonable conventions and the tradition of topological studies.
Relationships such as ''Intersects'', ''Disjoint'', ''Touches'', ''Within'', ''Equals'' (between two geometries ''a'' and ''b'') have an obvious semantic:
; ''Equals'': ''a'' = ''b'' that is (''a'' ∩ ''b'' = ''a'') ∧ (''a'' ∩ ''b'' = ''b'')
; ''Within'': ''a'' ∩ ''b'' = ''a''
; ''Intersects'': ''a'' ∩ ''b'' ≠ ∅
; ''Touches'': (''a'' ∩ ''b'' ≠ ∅) ∧ (''a''^ο ∩ ''b''^ο = ∅)

The predicates ''Contains'' and ''Within'' have subtle aspects to their definition which are contrary to intuition.
For example, a line ''L'' which is completely contained in the boundary of a polygon ''P'' is ''not'' considered to be contained in ''P''. This quirk can be expressed as "Polygons do not contain their boundary". This issue is caused by the final clause of the ''Contains'' definition above: "at least one point of the interior of B lies in the interior of A". For this case, the predicate ''Covers'' has more intuitive semantics (see definition), avoiding boundary considerations.

For better understanding, the dimensionality of inputs can be used as justification for a gradual introduction of semantic complexity:
:

Coverage on possible matrix results

The number of possible results in a boolean ''9IM'' matrix is 2⁹=512, and in a DE-9IM matrix is 3⁹=6561. The percentage of these results that satisfy a specific predicate is determined as following,

On usual applications the geometries intersects ''a priori'', and the other relations are checked.

The composite predicates "''Intersects'' OR ''Disjoint''" and "''Equals'' OR ''Different''" have the sum 100% (always true predicates),
but "''Covers'' OR ''CoveredBy''" have 41%, that is not the sum, because they are not logical complements neither independent relations; idem "''Contains'' OR ''Within''", that have 21%. The sum 25%+12.5%=37.5% is obtained when ignoring overlapping of lines in "''Crosses'' OR ''Overlaps''", because the valid input sets are disjoints.

Queries and assertions

The ''DE-9IM'' offers a full descriptive assertion about the two input geometries. It is a mathematical function that represents a complete set of all possible relations about two entities, like a Truth table, the Three-way comparison, a Karnaugh map or a Venn diagram. Each output value is like a truth table line, that represent relations of specific inputs.

As illustrated above, the output '212101212' resulted from ''DE-9IM''(''a'',''b'') is a complete description of all topologic relations between specific geometries ''a'' and ''b''. It says to us that .

By other hand, if we check predicates like ''Intersects''(''a'',''b'') or ''Touches''(''a'',''b'') — for the same example we have "''Intersects''= and ''Touches''=" — it is an incomplete description of "all topologic relations".
Predicates also do not say any thing about the dimensionality of the geometries (it doesn't matter if ''a'' and ''b'' are lines, areas or points).

This independence of geometry-type and the lack of completeness, on ''predicates'', are useful for general queries about two geometries:
:

For usual applications, the use of ''spatial predicates'' also is justified by being more human-readable than ''DE-9IM'' descriptions: a typical user have better intuition about predicates (than a set of interiors/border/exterior intersections).

Predicates have useful semantic into usual applications, so it is useful the translation of a ''DE-9IM'' description into a list of all associated predicates,Note. Th
Oracle's spatial funcion
do only a partial translation, internally, offering to user a mask for a or-list of predicates to be checked, instead the DE-9IM string. that is like a casting process between the two different semantic types. Examples:

* The string codes "" and "" have the semantic of "''Intersects & Crosses & Overlaps''".
* The string code "" have the semantic of "''Equals''".
* The string codes "", "", "", "", and "" have the semantic of "''Intersects & Touches''".

Standards

The Open Geospatial Consortium (OGC) has standardized the typical spatial predicates (Contains, Crosses, Intersects, Touches, etc.) as boolean functions, and the DE-9IM model,"OpenGIS Implementation Specification for Geographic information - Simple feature access - Part 2: SQL option", OGC, http://www.opengeospatial.org/standards/sfs as a function that returns a string (the DE-9IM code), with domain of , meaning =point, =line, =area, and ="empty set". This DE-9IM string code is a standardized format for data interchange.

The Simple Feature Access (ISO 19125) standard,
Open Geospatial Consortium Inc. (2007), "OpenGIS® Implementation Standard for Geographic
information - Simple feature access - Part 2: SQL option"
OGC document
''06-104r4'' version 1.2.1 (review of 2010-08-04). in the chapter 7.2.8, "SQL routines on type Geometry", recommends as supported routines the ''SQL/MM Spatial'' (ISO 13249-3 Part 3: Spatial) ''ST_Dimension'', ''ST_GeometryType'', ''ST_IsEmpty'', ''ST_IsSimple'', ''ST_Boundary'' for all Geometry Types.
The same standard, consistent with the definitions of relations in "Part 1, Clause 6.1.2.3"
of the SQL/MM, recommends (shall be supported) the function labels: ''ST_Equals'', ''ST_Disjoint'', ''ST_Intersects'', ''ST_Touches'', ''ST_Crosses'', ''ST_Within'', ''ST_Contains'', ''ST_Overlaps'' and ''ST_Relate''.

The DE-9IM in the OGC standards use the following definitions of Interior and Boundary, for the main OGC standard geometry types:

Implementation and practical use

Most spatial databases, such as PostGIS, implements the ''DE-9IM()'' model by the standard functions: ST_Relate, ST_Equals, ST_Intersects, etc. The function ST_Relate(a,b) outputs the standard OGC's ''DE-9IM string code''.

Examples: two geometries, ''a'' and ''b'', that intersects and touches with a point (for instance with and ), can be ST_Relate(a,b)='FF1F0F1F2' or ST_Relate(a,b)='FF10F0102' or ST_Relate(a,b)='FF1F0F1F2'. It also satisfies ST_Intersects(a,b)=true and ST_Touches(a,b)=true.
When ST_Relate(a,b)='0FFFFF212', the returned DE-9IM code have the semantic of "Intersects(a,b) & Crosses(a,b) & Within(a,b) & CoveredBy(a,b)", that is, returns true on the boolean expression ST_Intersects(a,b) AND ST_Crosses(a,b) AND ST_Within(a,b) AND ST_Coveredby(a,b).

The use of is faster than direct computing of a set of correspondent predicates.Chapter 4. Using PostGIS: Data Management and Queries
/ref> There are cases where using is the only way to compute a complex predicate — see the example of the code 0FFFFF0F2,JTS test case of "point A within one of B points", http://www.vividsolutions.com/jts/tests/Run1Case4.html of a point that not "crosses" a multipoint (a object that is a set of points), but predicate ''Crosses'' (when defined by a mask) returns ''true''.

It is usual to overload the by adding a mask parameter,
or use a returned string into the function.
When using , it returns a boolean. Examples:
* ST_Relate(a,b,'*FF*FF212') returns ''true'' when ST_Relate(a,b) is 0FFFFF212 or 01FFFF212, and returns ''false'' when 01FFFF122 or 0FF1FFFFF.
* ST_RelateMatch('0FFFFF212','*FF*FF212') and ST_RelateMatch('01FFFF212','TTF*FF212') are ''true'', ST_RelateMatch('01FFFF122','*FF*FF212') is ''false''.

Synonyms

* "Egenhofer-Matrix" is a synonym for the ''9IM'' 3x3 matrix of boolean domain."Encyclopedia of GIS", S. Shekhar, H. Xiong. .
* "Clementini-Matrix" is a synonym for the DE-9IM 3x3 matrix of domain.
* "Egenhofer operators" and "Clementini operators" are sometimes a reference to matrix elements as ''II'', ''IE'', etc. that can be used in boolean operations. Example: the predicate "''G₁'' contains ''G₂''" can be expressed by "", that can be translated to mask syntax, T*****FF*.
* Predicates "meets" is a synonym for ''touches''; "inside" is a synonym for ''within''
* Oracle's "ANYINTERACT" is a synonym for ''intersects'' and "OVERLAPBDYINTERSECT" is a synonym for ''overlaps''. Its "OVERLAPBDYDISJOINT" does not have a corresponding named predicate.
* In Region connection calculus operators offer some synonyms for predicates: ''disjoint'' is DC (disconnected), ''touches'' is EC (externally connected), ''equals'' is EQ. Other, like ''Overlaps'' as PO (partially overlapping), need context analysis or composition.

See also

References

{{Reflist

External links

Point Set Theory and the DE-9IM Matrix



Matrices
Geometric topology
Geographic data and information
Binary operations
Geometric intersection