Towards a generalized
map algebra: principles
and data types
Gilberto Câmara
Danilo Palomo
Ricardo Cartaxo
Olga Oliveira
Map Algebra

A simple example: “How to identify all areas without
vegetation with slope greater than 15% as high risk ”.
slope
U
vegetatio
n


Is an algebraic framework for performing operations on
data stored in a geographical information system (GIS).
Allows the user to model different problems and to obtain
new information from the existing data set.
2
Tomlin’s Map Algebra [1990]

Data type


A Map in raster representation
Operations
Local
Focal
Zonal
gz
gl
gf
z(p)
n(p)
n(p) = touch
Source: Tomlin (1990)
z(p) = inside
spatial
3
New applications
“Calculate the deforestation along the rivers and
roads”
4
Questions

“What is the theoretical foundation for map
algebra?”

“Could this theoretical foundation provide support
for more generic map algebra?”

“How can we define an extended map algebra
suitable
for
both
raster
and
vector
representations?”
5
Response

We take the existing spatial predicates as a
basis for defining a more generic map algebra.
Motivation

To allow the user to model a greater number of
different problems and to obtain a greater
number of new information from the existing data
set.
6
Outline

Introduction and motivation

The generalized map algebra

Compare with the Tomlin’s proposal

Examples

Conclusions
7
The generalized map algebra

Data type
 A map
in raster or vector representation
Function
Signature
getregions map  list_region
contains
map  region  bool
overlaps
map  region  bool
insert
map  region
retrieve
map  region  attr_domain
new
 map
add
map  region  map
remove
map  region  map
 attr_domain  map
The map data type
8
Axioms
(getregions (m)) == G (the extent of the map)
contains (m, r) == true iff r
 G
insert(m,r,val) == error iff contains (m,r) == false
retrieve (m,r) == error iff contains (m,r) == false
retrieve (insert (m, r, val), r) == value
add (m,r) == error iff ((contains (m, r) == true) or
(overlaps (m,r) == true))
remove (m,r) == error iff (contains (m,r) == false)
remove ((add (m, r), r) == add ((remove (m, r), r)
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Nonspatial operation

Are the equivalent to Tomlin’s local operations
10
Nonspatial operators
Instances

m1, m2 : map
r : region
value: attr_domain
list_attr_domain
func1: func_single
maplist: list_map
reglist : list_region
valuelist:
funcn: func_multiple
Function
Signature
single
map  func_single  map  map
multiple
maplist  func_multiple  map  map
select
map  attr_domain  comp  map  map
Operators
 Single
argument mathematical functions: log, exp, sin, ...
 Single argument logical function: not.
 Multiargument functions: sum, product, or, ...
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Spatial predicates
Topological predicates
Source: adapted from Egenhofer
and Herring [1991]
Directional
predicates
Source: Winter & Frank
[2000]
12
Spatial selection function
selection
spatial predicate
data
reference
13
Application of spatial selection functions
selection
predicate
touch
inside
disjoint
west
data
reference
14
Spatial operation
outputmap
output
composition
selection
spatial predicate
data
reference
15
Spatial operators
Instances m1, refmap, m2 : map
v_list: list_attr_domain
pred: spat_pred

r1, r2 : region
func: func_multiple
Function
Signature
spatial
map  map  pred  func_multiple  map  map
Operators
 Multiargument
functions: sum, product, or, ...
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Outline

Motivation

The generalized map algebra

Comparison with the Tomlin’s proposal

Examples

Conclusions
17
Comparison with Tomlin’s map algebra
Focal operation
Zonal operation
No equivalence
composition
composition
composition
inside
disjoint / equal
west / north / ...
touch
output and reference
18
Outline

Motivation

The generalized map algebra

Comparison with the Tomlin’s proposal

Examples

Conclusions
19
Example of spatial operation
Find the local sum of regions in a deforestation map
230
sum
1980
:=
touch
defor
lsum
lsum := sum (defor touch lsum
lsum)
lsum);
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Example of spatial operation
Find the sum of deforestation in the native reservations
470
inside
reserves
sum
:=
defor
deforRes
reserves);
deforRes := :=
sum (defor
sum
inside
(defor reserves
inside
reserves)
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Example of spatial operation
Calculate the mean of the deforestation along the road
195
intersects
road
mean
defor
:=
defRoad
defRoad := mean (defor intersects road
road);
roadDesm
road)
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Outline

Motivation

The generalized map algebra

Comparison with the Tomlin’s proposal

Examples

Conclusions
23
Conclusions

Our map algebra enables operations that are not
directly expressible by Tomlin`s proposal.

One of the important result is to show that
Tomlin’s Map Algebra can be seen as an
application of spatial predicates to coverages.

Further generalizations of the proposed algebra
are possible by involving the full set of spatial
predicates.

The next step in our work is to design, implement
and validate a language that supports the
proposed map algebra for spatial databases.
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