# Lecture 08: Map Transformation - PowerPoint PPT Presentation

PPT – Lecture 08: Map Transformation PowerPoint presentation | free to download - id: 5ddb19-ZjczM

The Adobe Flash plugin is needed to view this content

Get the plugin now

View by Category
Title:

## Lecture 08: Map Transformation

Description:

### Lecture 08: Map Transformation Geography 128 Analytical and Computer Cartography Spring 2007 Department of Geography University of California, Santa Barbara – PowerPoint PPT presentation

Number of Views:244
Avg rating:3.0/5.0
Slides: 42
Provided by: Magg68
Category:
Tags:
Transcript and Presenter's Notes

Title: Lecture 08: Map Transformation

1
Lecture 08 Map Transformation
• Geography 128
• Analytical and Computer Cartography
• Spring 2007
• Department of Geography
• University of California, Santa Barbara

2
Review of the transformational view of Cartography
• Transformations
• Map scale
• Dimension
• Symbolic content
• Data structures
• Why Transform?
• We may wish to compare maps collected at
different scales.
• We may wish to convert the geometry of the map
base.
• We may wish or need to change the map data
structure.

3
Robinson's Classification
4
Robinson's Classification (cnt.)
• Robinson's Classification was based on dimension
and level of measurement
• Dimension of measurement
• Zero dimensional
• One dimensional
• Two dimensional
• Three dimensional ?
• Level of measurement idea is from Stevens (1946)
• Nominal data assume only existance and type. An
example is a text label on a map.
• Ordinal data assume only ranking. Relations are
like "greater than".
• Interval data have an arbitrary numerical value,
with relative value. Example Elevation.
• Ratio data have an absolute zero and scale.

5
Transformations as Stages in Map Production
• Transformation of level can be shown in making a
choropleth map.
• This transformation is not invertible, but can be
error measured and minimized.

6
David Unwins Extended Classification
• Robinsons idea was extended by David Unwin.
• Unwin separated issue of data from issues of
mapping method, (map type and data type)

7
State Changes and Transformations
• Cartographers are interested in the full set of
state transformation.
• Each map has an optimal path through the set.
• Design cartography primarily concentrates on the
last, or symbolization transformation.
• Four types of transformations shape the mapping
process
• Geocoding (transforming entities to objects
levels, dimension, data structure)
• Map Scale
• Locational Attributes or Map Base
• Symbolization

8
Scale Transformations
• Some transformations "collapse" space e.g. area
to point.
• Map scales of interest to cartography are 11,000
to 1400M.
• Transformations from larger to smaller scale by
the process of generalization.
• At the minimum, generalization involves
simplification, elimination, combination and
displacement.

9
Some Generalization Problems
• Length
• Shape
• Topology

10
Map Generalization and Enhancement
• These steps are conducted under specified and
consistent rules.
• An example is the set of algorithms for point
elimination along a line.
• The inverse of this adds points along a line
enhancement

11
Transformations and Algorithms
• In mathematics, transformations are expressed as
equations.
• Solutions, inversion as so forth are by algebra,
calculus etc.
• In computer science, a set of transformations
defining a process is called an algorithm.
• Any process that can be reduced to a set of steps
can be automated by an algorithm
• data structures transformational algorithms
maps

12
Transformations and Algorithms (cnt.)
13
Transformations of Object Dimension
• The four dimensions of dimension, data can be
represented at any one in one state
• Transformations can move data between states
• Full set of state zero to state one
transformations is then 16 possible
transformations
• Dimensional transformation are only one type
• When dimension collapses to "none" result is a
measurement

14
Map Transformation Algebra
• Transformations map closely onto Matrix algebra
• Almost all spatial data can be placed into an (n
x m) or (n x p) matrix
• Transformations can then be by convolution
(iteration of a matrix over an array OR
• By selecting a small matrix (2 x 2) or (3 x 3)
for multiplication
• Complex transformations can be compounded

15
Transformations as Multiple Steps (Dimensional
Transforms)
16
Map Transformation Algebra (cnt.)
• Matrices have inverses, which reverse effect of
multiplication to yield the identity matrix
• Error creep in when inversion does not result in
identity matrix

17
Map Projection Transformations
• Map projections represent many different types of
transformation
• Perfectly invertible (one-to-one)
• One-to-many
• Many-to-one
• Undefined (non-invertible)
• Imperfectly invertible, e.g. on ellipsoid and
geoid, computational error, rounding etc.
• Some transformations use iterative methods i.e.
algorithms, not formulas

18
Geographic Coordinate Transformation
19
Equatorial Mercator Transformation
20
Planar Geometry vs. Spherical Geometry
• Rule of Sines Distance between points

21
Planar Map Transformations on Points - Length of
a line
• Repetitive application of point-to-point distance
calculation
• For n points, algorithm/formula uses n-1 segments

22
Planar Map Transformations on Points - Centroids
• Multiple point or line or area to be transformed
to single point
• Point can be "real" or representative
• Mean center simple to compute but may fall
outside point cluster or polygon
• Can use point-in-polygon to test for inclusion

23
Planar Map Transformations on Points - Standard
Distance
• Just as centroid is an indication of
representative location, standard distance is
mean dispersion
• Equivalent of standard deviation for an
attribute, mean variation from mean
• Around centroid, makes a "radius" tracing a
circle

24
Planar Map Transformations on Points - Nearest
Neighbor Statistic
• NNS is a single dimensionless scalar that
measures the pattern of a set of point (point-gt
scalar)
• Computes nearest point-to-point separation as a
ratio of expected given the area
• Highly sensitive to the area chosen

25
Planar Map Transformations Based on Lines -
Intersection of two lines
• Absolutely fundamental to many mapping
operations, such as overlay and clipping.
• In raster mode it can be solved by layer overlay.
• In vector mode it must be solved geometrically.
• Lines (2) to point transformation

26
Planar Map Transformations Based on Lines -
Intersection of two lines (cnt.)
• When using this algorithm, a problem exists when
b2 - b1 0 (divide by zero)
• Special case solutions or tests must be used
• These can increase computation time greatly
• Computation time can be reduced by pre-testing,
e.g. based on bounding box.

27
Planar Map Transformations Based on Lines -
Distance from a Point to a Line
28
Planar Map Transformations Based on Areas
• Computing the area of a vector polygon (closed)
• Manually, many methods are used, e.g. cell
counts, point grid.
• For a raster, simply count the interior pixels
• Vector Mode more complex

29
Planar Map Transformations Based on Areas
30
Planar Map Transformations Based on Areas -
Point-in-Polygon
• Again, a basic and fundamental test, used in many
algorithms.
• For raster mode, use overlay.
• For vector mode, many solutions.
• Most commonly used is the Jordan Arc Theorem
• Tests every segment for line intersection.
• Test point selected to be outside polygon.

31
Planar Map Transformations Based on Areas -
Theissen Polygons
• Often called proximal regions or voronoi diagrams
• Often used for contouring terrain, climate,
interpolation, etc

http//en.wiki.mcneel.com/default.aspx/McNeel/Poin
tsetReconstruction.html
32
Affine Transformations
• These are transformation of the fundamental
geometric attributes, i.e. location.
• Influence absolute location, not relative or
topological
• Necessary for many operations, e.g. digitizing,
scanning, geo-registration, and display
• Affine Transformations take place in three steps
(TRS) in order
• Translation
• Rotation
• Scaling

33
Affine Transformations - Translation
• Movement of the origin between coordinate systems

34
Affine Transformations - Rotation
• Rotation of axes by an angle theta

35
Affine Transformations - Scaling
• The numbers along the axes are scaled to
represent the new space scale

36
Affine Transformations
• Possible to use matrix algebra to combine the
whole transformation into one matrix
multiplication.
• Step must then be applied to every point

37
Statistical Space Transformations - Rubber
Sheeting
• Select points in two geometries that match
• Suitable points are targets, e.g. road
intersections, runways etc
• Use least squares transformation to fit image to
map
• Involves tolerance and error distribution
• x y T u v then applied to all pixels
• May require resampling to higher or lower density

38
Statistical Space Transformations - Cartograms
• also known as value-by-area maps and varivalent
projections (Tobler, 1986)
• Deliberate distortion of geometry to new "space"
• Type of non-invertible map projection

39
Symbolization Transformations
• Screen coordinates are often reduced to a
"satndard" device Normalization Transformation
• Standard Device display dimensions are (0,0) to
(1,1)
• World Coordinates-gt Normalized Device Coordinates
gt Device Coordinates

40
Drawing Objects
• Most use model of primitives and attributes
• The Graphical Kernel System (GKS) has six
primives, each has multiple attributes.

41
Next Lecture
• Data Structure Transformation