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Lecture 08: Map Transformation

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

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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
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