Coordinate Transformations and Geometric Procedures

Objectives of lecture:

1. Geometric Issues
   Registration
   Projections
   Generalization
2. Building Larger Coverages
   Rubber Sheet
   Zipper

Geometric Issues

If you didn't figure it out by now, GIS runs on coordinates. If you are weak on coordinate reference systems, check out Peter Dana's lecture notes (Geographer's Craft Project).

Registration

Registration connects known points on the map image to their intended location. This requires knowing the location of some set of points (tics in some parlance) in both coordinate systems. The selection of tic points is not as simple as it sounds. The distribution of tic points should be fairly uniform across your study area. If tic marks are always along roads, some areas might not be as carefully transformed. Errors in the sources might correlate with the landscape, creating bias and undiscovered troubles in analysis. Also, hidden in most software: a 'fit' computed between measurements.

Kinds of registration equations:

{Geometric components: rotate, translate, scale} [Notation: x,y IN; XY out]

Equation 10-1: Similarity transformation

X = A + Cx + Dy
Y = B - Dx + Cy

where C = [scale factor] times cosine ( [angle of rotation] )
D = [scale factor] times sine ( [rotation angle] )
A & B are offsets for the center of rotation in output coordinates {Translation}

Equation 10-2: Affine transformation

{ because maps may not have the same scale in X & Y}

X = A + Cx + Dy
Y = B - Ex + Fy

Many more: Helmert transformation (can deal with radial distortion as from a camera lens)

Kinds of fitting:

• Least Squares: Standard regression equations minimize the sum of squared deviations (measured at the tic points, of course). This procedure is the Best Linear Unbiased Estimator (BLUE) or most efficient in using the information available, BUT only if the error assumptions ("normal" error distribution, etc.) apply. See Blunders.
• Robust Statistics: various fitting procedures (such as Least Median Squares) are more robust to violations of the standard assumptions.

Geodetic Control:

the real source for registration points. Measured by geodetic surveying, often used to 'control' surveys to construct maps. (most features on map are indirectly measured on the earth, connected by geodetic control)
About National Geodetic Reference System; Peter Dana's lecture notes on Geodetic Datums;

Representation of coordinates:

watch out for roundoff. Computers may only have 6 digits unless you use the much slower 'double precision'.

Projections

Map may not match projection of coordinate system [See: Spatial Reference Systems]. Do not expect the above equations to make the projection change along with the transformation. The geometry of the ellipsoid and projection matter.

In USA: Feds use UTM (Universal Transverse Mercator), local 'State Plane' is either Lambert Conformal Conic (as in Wa) or some other Transverse Mercator; Washington State Plane is a particular Lambert Conformal

The 'grid north' on these projections is NOT 'true north' except along the central meridian. [An example from Texas - capitol] The scale is not true except along particular lines and at sea level. GIS often assumes that these corrections are not important... There really should be corrections built into calculation of area and perimeter for "grid to ground" (distance from projection surface to ground elevation) as well as known distortions (such as using conformal projections that are not equal area...).

Why are these maps tilted?

USGS response [not online today...]

Projections matter; grid north is NOT true North; see 465 course

Generalization

all models simplify.
Geographic models choose which component to control and which to measure.
Representations also impose further modifications.

Beard presents a system of generalization:

1. Operations to reduce the number of objects (eg. selection)
2. Operations to simplify the spatial information (measurements)
3. Operations to reduce the attribute information (eg. reclassify)
4. All combinations of 1,2,3

MacMaster and Shea:

Specific examples

Verification and Quality Control:

either a lot of visual inspection, or use software to verify relationships (integrity constraints) expected to occur - the topological model for geometry; attributes verified by other relationships (completeness from list of all objects) or brute force inspection.

Building larger coverages:

stitch map sheets together into sheetless coverage.

'Rubber sheet' forces certain points to match, interpolates between;
'Conflation' matches attributes.
Zipper interpolates across sheet edges. [slides]