Lecture 08: Map Transformation

Geography 128

Analytical and Computer Cartography

Spring 2007

Department of Geography

University of California, Santa Barbara

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.

Robinson's Classification

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.

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.

David Unwin’s Extended Classification

Robinson’s idea was extended by David Unwin.

Unwin separated issue of data from issues of mapping method, (map type and data type)

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

Scale Transformations

Some transformations "collapse" space: e.g. area to point.

Map scales of interest to cartography are 1:1,000 to 1:400M.

Transformations from larger to smaller scale by the process of generalization.

At the minimum, generalization involves simplification, elimination, combination and displacement.

Some Generalization Problems

Length

Shape

Topology

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

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

+ =

Transformations and Algorithms (cnt.)

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

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

Transformations as Multiple Steps (Dimensional Transforms)

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

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

Geographic Coordinate Transformation

Equatorial Mercator Transformation

Planar Geometry vs. Spherical Geometry

Rule of Sines – Distance between points

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

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

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

Planar Map Transformations on Points- Nearest Neighbor Statistic

NNS is a single dimensionless scalar that measures the pattern of a set of point (point-> scalar)

Computes nearest point-to-point separation as a ratio of expected given the area

Highly sensitive to the area chosen

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

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.

Planar Map Transformations Based on Lines- Distance from a Point to a Line

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

Planar Map Transformations Based on Areas

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.

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/PointsetReconstruction.html

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

Affine Transformations- Translation

Movement of the origin between coordinate systems

Affine Transformations- Rotation

Rotation of axes by an angle theta

Affine Transformations- Scaling

The numbers along the axes are scaled to represent the new space scale

Affine Transformations

Possible to use matrix algebra to combine the whole transformation into one matrix multiplication.

Step must then be applied to every point

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

http://tabacco.blog-city.com/red_vs_blue_big_lie_maps__cartograms_of_2004_presidential_el.htm

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

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-> Normalized Device Coordinates > Device Coordinates

Drawing Objects

Most use model of primitives and attributes

The Graphical Kernel System (GKS) has six primives, each has multiple attributes.

Next Lecture

Data Structure Transformation