Algebraic Formalism over Maps

1
Algebraic Formalism over Maps

• João Pedro Cerveira Cordeiro
• Gilberto Câmara
• Ubirajara F. Moura
• Cláudio Barbosa
• Felipe Almeida

GeoInformation Group Image Processing Division
DPI
2
Research Context
This work is intended to contribute toward an
actual formal approach to map algebra.

• Among the references adopted, we emphasize

Tomlin, D., 1990. Geographic Information Systems
and Catographic Modeling. Prentice Hall,
Englewood Cliffs, NJ. Ritter, G.X., Wilson, J.,
Davidson, J. 1990, Image algebra An overview,
Computer Vision, Graphics and Image Processing,
49, 297-331 Takeyama, M. Couclelis, H., 1997.
'Map dynamics integrating cellular automata and
GIS through Geo-Álgebra', International Journal
of Geographical Information Science, 11, 73-91.
We try to accommodate Tomlins classes of
non-local operations into a single paradigm based
only on local operations.
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Motivation and Objectives

• We start by defining a local operator to
  modeling the
• interaction between binary and ordinary maps

• We also formalize the notion of regions from
  which the
• notions of zones and neighborhoods can be
  derived.

• We also show how this approach can help modeling
  some
• spatial variability for neighborhoods.

• A new version of the selecting operator is then
  introduced
• in order to incorporate the weighting of
  location.

• Finally the consistence of the approach proposed
  with the
• principles of geo-algebra is pointed out.

4
Local and Non-local Operations

• Local operation involves different values
  associated to the
• same location, while non-local operations
  firstly consider
• the influence of a set of locations.

• A typical non-local operation consists of three
  basic steps

1. A set of locations is selected 2. A set of
values at selected locations is recorded 3. A
value is summarized from this set and used to
characterize either a single location, or the
whole set of selected locations..

• Actually there is another step regarding the
  weights, or
• multiplicit,y to which different locations must
  be considered.

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Maps, Operators and Expressions
Maps are functions from a spatial domain into an
attribute domain

• A map is an element of VL, the set of functions
• from L, a set of locations, into the set V of
  values

map ? (l, v) m(l) v ? (x, y, v)

• Operations and relations can be Induced from the
• spatial and attribute domains,

A language for expressions is also induced.
(b4 b3) / (b4 b3) veg forest slope gt
30 veg forest and slope gt 30
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Operations involving binary maps
What is the meaning of operating binary and any
other typed maps ?

• A binary map is a function in 0, 1 L .

• We explore the interaction among binary and
  ordinary maps
• by defining a binary operation.

0 0 0 1 1 0 1 1 1 0 0
1 1 1 0 1 1 1 1 0 1 1
0 0 0
Value Null 0 Null Null 1 Value Null

• Its effect corresponds to the selection of
• values at cells marked 1. The value 0
• corresponds to selecting no copies of any value.

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Regions and Boolean expressions
A wide class of sets of locations can be defined
by means of Boolean (logical) operations!

• A type Regions is suggestive here.

Regions regs vegetation forest and slope
lt 30 , vegetation crops and district
second , height gt 1000 or rain low

• The interaction among maps and regions
• is based on the selecting operator.

regs (b4 - b3) / (b4 b3)
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Zones, Neighborhoods and regions
Zones are regions that do not overlap,
neighborhoods are regions given by proximity
relations.

• Types Zones and Neighborhoods can be derived
  from
• Regions.

Zones districts district first, second,
third Zones buffers distance(rivers
main) div 10 Neighborhood close distance()
lt 3

• Combining proximity relations with relations on
  the
• attribute domain allows modeling some spatial
  variability.

Neighborhood near_forest distance() lt 3 and
vegetation forest
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Neighborhoods and regions

• Neighboring locations can be specified by the
  relative
• positioning of locations regarding a focus
  location

• ( img(-1,-1) img(-1,0) img(-1,1)
• img( 0,-1) img( 0, 0) img( 0,1)
• img( 1,-1) img( 1, 0) img( 1,1) ) / 9

1 1 1 1 1 1 1 1 1

• Such specification represents a familiy of
• functions from ZxZ into the binary set 0, 1 .

• Neighborhoods neigh
• (-1,-1,1), (-1,0,1), (-1,1,1),
• (0,-1, 1), (0,0,1), (0,1,1),
• (1,-1, 1), (1,0,1), (1,1,1)

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Neighborhoods and regions

• Boolean expression can also be used to specify
  the
• values

• (-1,-1,slope lt 30), (-1,0, slope lt 20), (-1,1,
  slope lt 10),
• (0,-1, slope lt 20), (0,0, slope lt 20), (0,1,
  slope lt 10),
• (1,-1, slope lt 10), (1,0, slope lt 10), (1,1,
  slope lt 10)

• As for regions in general, Boolean operations
  can be
• used to build new specifications.

new_neigh neigh AND slope lt 30
It is equivalent to write

• (-1,-1,slope lt 30), (-1,0, slope lt 30), (-1,1,
  slope lt 30),
• (0,-1, slope lt 30), (0,0, slope lt 30), (0,1,
  slope lt 30),
• (1,-1, slope lt 30), (1,0, slope lt 30), (1,1,
  slope lt 30)

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Weighted Neighborhoods

• Consider a gradient filtering operation given by
  explicitly
• involving neighboring locations as in a local
  operation.

Sqrt ( ( ( img( 1,-1) 2 img( 1,0) img( 1,1)
) - ( img(-1,-1) 2 img(-1,0)
img(-1,1) ) )2 ( ( img(-1, 1) 2
img(0, 1) img(1, 1) ) - ( img(-1,-1)
2 img(0,-1) img(1,-1) ) )2 )
1 2 1 2 2 1 2 1

• This suggests extending the selecting
• operator so that the weighting of selected
  locations
• can be modeled.

Value Null 0 Null Null
n n copies Null of Value
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Weighted Neighborhoods
Now we can define neighborhoods and express
their interaction with maps

• The gradient filtering operation will involve

Neighborhoods up (-1,-1, 1), (-1, 0,
2), (-1, 1, 1) down (1,-1, 1), ( 1, 0,
2), ( 1, 1, 1) left (-1,-1, 1), (
0,-1, 2), ( 1,-1, 1) right ( 1,-1, 1),
( 0, 1, 2), ( 1, 1, 1)

• The new version of selecting (and weighting)
  operator may
• be used now to model the interactions with
  img.

Sqrt ( ( Sum ( img down) - Sum ( img up ) )2
( Sum ( img right ) - Sum ( img
left ) )2 )
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Summary Functions
This concludes a non-local operation.

• Summarizing is typically done by applying some
• simple statistics to the selected and weighted
  data.

Average (img neigh) Majority (veg ( slope lt
30 AND soil pdz slope gt 30 AND
veg forest ) ) Maximum ( ( ndvi gt 0.5)
heights slope ) Sqrt ( ( Sum ( img down) -
Sum ( img up ) )2 ( Sum ( img
right ) - Sum ( img left ) )2 )
Map1 Map2 Map3 Map4

• Finally, map layers can be assigned to the
  result of
• evaluating algebraic expressions.

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Concluding Remarks

• The approach presented is consistent with
  Takeyamas
• geo-algebra in that regions can be equated to
  influence sets
• and meta-relational maps.

• A formal compromise between language and
  implementation
• may help avoiding some efficiency problems.

• Its suggestive to explore modeling based on
  Cellular
• Automata.by its language counterpart as well.

• Using local operations to modeling not only map
  layers,
• but also the regions used to build them up,
  would also help
• avoiding mixing concepts from spatial and other
  natures.

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Thank You !
GeoInformation Group Image Processing Division
DPI