DE-9IM

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 standard 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" (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, this looks like:
Reading from left-to-right and top-to-bottom, the DE-9IM("a","b") string code is ", the compact representation of .
Spatial predicates.
Spatial predicates are topologically-invariant binary spatial relations based on the DE-9IM. For ease of use "named spatial predicates" have been defined for some common relations.
The "spatial predicate" functions that can be derived (expressed by masks) 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:
Properties.
The spatial predicates have the following properties of binary relations:
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:
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, that is like a casting process between the two different semantic types. Examples:
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, 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, 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: codice_7, codice_8, codice_9, etc. The function codice_10 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 codice_11 or codice_12 or codice_11. It also satisfies codice_14 and codice_15.
When codice_16, the returned DE-9IM code have the semantic of "Intersects(a,b) & Crosses(a,b) & Within(a,b) & CoveredBy(a,b)", that is, returns codice_17 on the boolean expression codice_18.
The use of is faster than direct computing of a set of correspondent predicates. There are cases where using is the only way to compute a complex predicate — see the example of the code codice_19, 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: