EFFICIENT IMPLEMENTATION
TECHNIQUES FOR
TOPOLOGICAL PREDICATES
ON COMPLEX SPATIAL
OBJECTS
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9-Intersection Matrix Characterization method.
Optimized method for Predicate Verification
(matrix thinning)
Optimized method for Predicate Determination
(Minimum cost Decision Tree)
Implementation, Testing, Approach Assessment and
Performance study.
9-Intersection Matrix Characterization
Method
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Uniquely characterize each element (matrix predicate)
of the 3x3 matrix of the 9-intersection matrix by means
of Topological feature Vectors vF and vG
Each predicate is specified as the logical conjunction of
the characterizations of the nine matrix predicates
Since the feature vectors are different for each type
combination, characterization of each matrix predicate
is different for each type combination.
9th matrix predicate always yields true
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Predicate Verification: Based on the predicate to be
verified and the feature vectors we perform the
predicate verification in a loop. After computation of
value of the matrix predicate i (1<i<8) we compare it
with the corresponding value of the matrix predicate
p(i) of p and proceed (i+1)if its true, stop if its false.
Predicate Determination: First we evaluate the 9IM
characterizations of the matrix(8 predicates) as m. And
m is then compared against the matrices pi of all n
(1<i<n) topological predicates.
For both proper and dimension-refined
predicates(additional characterizations).
Type combination dependent 9Intersection Matrix Characterization.
•Each characterization can be performed in constant time and its
correctness can be shown by a simple proof.
•We will discuss Characterizations for all type Combinations
F-Point G-Point (Lemma 1)
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Since the boundary is empty for a point object 3x3
matrix can be reduced to 2x2 matrix.
Overlap
F-Point G-Line (Lemma 2)
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Since boundary of a point object is empty, this
matrix can be reduced to a 2x3 matrix
F-Point G-Region (Lemma 3)
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Is Reduced to 2x3 matrix as the points’ boundary is
an empty object.
F-Line G-Line (Lemma 4)
F-Line G-Region (Lemma 5)
F-Region G-Region (Lemma 6)
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Lemma 6: Translation of each segment class into a
Boolean matrix predicate expression
F-Region G-Region
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Lemma 7: translation of some matrix predicates into
segment classes.
F-Region G-Region
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Lemma 8: some Implications between Matrix predicates.
(ii) & (iii) If the boundary of a complex region intersects
the exterior of the other complex region, both its
interior and its exterior intersect the exterior of the
other region.
(iv) & (v) intersecting the interior.
Theorem
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Uses previous lemmas and characterizes the 9-IM
for region/region
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Lemma 7(i)
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Lemma 6(v)
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Lemma 6(iii)& 6(iv)
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Lemma 8(iv) & 8(v)
The above two correspond to lemma 6(iv) & 6(iii) respectively
The forward implication corresponds to Lemma 7(ii) & 7(iii) respectively
Backward implication require Lemma 6(i), (ii) and lemma 8(ii),(iii) (0/1)(1/0) of F[G]
And lemma 6(iv),(vi) (1/1) of F and G
and lemma 6(iv)(iii) and lemma 8(v)(iv) (1/2)(2/1) of G[F]
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Either they share a common meeting (intersection point) or there are two half
segments of F and G whose segment components are equal(0/2)(2/0)(1/1).
Backward: 6(v) (0/2)(2/0) and lemma 6(vi) (1/1) and 8(i) bound_poi_shared
The above correspond to Lemma 6(i) and 6(ii) resp
Matrix Characterization
Benefits
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Systematically developed and not an adhoc
approach.
Has Formal and sound Foundation. (we can be sure
about the correctness of the matrix predicates)
Independent of number of topological
predicates.(only requires a constant number of
evaluations for matrix predicate characterizations).
The correctness of the implementation has been
proved.
Evaluation of Dimension-Refined
Topological Predicates.
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Replacement of the topological invariant of emptiness and non-emptiness by the
topological invariant of the dimension of an intersection leads to the 9-intersection
dimension matrix.
The dimension extended model considers the dimension of the intersections and
combines it with topological predicates for spatial data-types.
These dimension predicates are refinements of the topological predicates on spatial
data-types.
All type combinations that involve points cannot be dimension-refined.
This is only possible if one-dimensional components of a spatial object like interior
of a line object or boundary of a region object are involved.
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We distinguish the different dimensions of the maximal
connected components of a given point set in the twodimensional space and define a dimension type.
Union of dimensions of its maximal connected
components.
1st symbol is used to represent the undefined dimension
of an empty point set.
Only In the case of two 1-Dimensional components their
intersection may lead to components of different
dimensions. (interiors of Line, Interior of line and
boundary of region, boundaries of region objects)
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F,G – {Line, Region}
If p yields false then pd is false.
If p yields true then we use the dimension
characterizations that are given in the table.
•Dimension characterizations are different for different type combination, they
are unique.
• Verification:Look up the type combination and the dimension of interest and
evaluate its corresponding as the result for the dimension refined predicate.
•Determination: apply 9IMC for matching topological predicate p and then
we consecutively evaluate the 4 dimension characterizations of the type
combination. If the result is empty then pd coincides with p else we obtain
either pd=0D-p , pd=01D-p, pd=1D-p.
Optimized evaluation methods