Towards a generalized map algebra: principles and data types Gilberto Câmara Danilo Palomo Ricardo Cartaxo Olga Oliveira

2 Map Algebra 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. A simple example: How to identify all areas without vegetation with slope greater than 15% as high risk. U slope vegetatio n

3 Tomlins Map Algebra [1990] Data type A Map in raster representation Operations LocalFocalZonal Source: Tomlin (1990) n(p) = touchz(p) = inside spatial gl gf n(p) gz z(p)

4 New applications Calculate the deforestation along the rivers and roads

5 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?

6 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.

7 Introduction and motivation The generalized map algebra Compare with the Tomlins proposal Examples Conclusions Outline

8 The generalized map algebra Data type A map in raster or vector representation FunctionSignature getregions map list_region contains map region bool overlaps map region bool insert map region attr_domain map retrieve map region attr_domain new map add map region map remove map region map The map data type

9 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)

10 Nonspatial operation Are the equivalent to Tomlins local operations

11 Nonspatial operators Operators Single argument mathematical functions : log, exp, sin,... Single argument logical function : not. Multiargument functions : sum, product, or,... Instancesm1, m2 : mapmaplist: list_map r : regionreglist : list_region value: attr_domainvaluelist: list_attr_domain func1: func_singlefuncn: func_multiple FunctionSignature single map func_single map map multiple maplist func_multiple map map select map attr_domain comp map map

12 Spatial predicates Directional predicates Source: adapted from Egenhofer and Herring [1991] Source: Winter & Frank [2000] Topological predicates

13 Spatial selection function data selection reference spatial predicate

14 disjoint Application of spatial selection functions data predicate selection reference touch west inside

15 Spatial operation output map data reference output selection spatial predicate composition

16 Spatial operators Operators Multiargument functions : sum, product, or,... Instancesm1, refmap, m2 : map r1, r2 : region v_list: list_attr_domain func: func_multiple pred: spat_pred FunctionSignature spatial map map pred func_multiple map map

17 Motivation The generalized map algebra Comparison with the Tomlins proposal Examples Conclusions Outline

18 Comparison with Tomlins map algebra Focal operation Zonal operation composition output and reference touch composition inside No equivalence composition disjoint / equal west / north /...

19 Motivation The generalized map algebra Comparison with the Tomlins proposal Examples Conclusions Outline

20 lsum := sum (defor touch lsum lsum := sum (defor touch lsum)lsum := sum (defor touch lsum); Find the local sum of regions in a deforestation map Example of spatial operation defor lsum touch 1980 sum:= 230

21 reserves defor Example of spatial operation deforRes := sum (defor inside reserves sum 470 deforRes := sum (defor inside reserves) deforRes := sum (defor inside reserves); Find the sum of deforestation in the native reservations inside deforRes :=

22 roadDesm := mean (defor intersects road roadDesm := mean (defor intersects road) defRoad := mean (defor intersects road); Example of spatial operation road defRoad defor intersects mean 195 := Calculate the mean of the deforestation along the road

23 Motivation The generalized map algebra Comparison with the Tomlins proposal Examples Conclusions Outline

24 Conclusions Our map algebra enables operations that are not directly expressible by Tomlin`s proposal. One of the important result is to show that Tomlins 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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