Calculating Projections and Registration

Objectives of lecture:

Student contribution: matrix algebra, decompositions and mathemagian work (Zach)

Projections Part I
1 Standardized projections: UTM & State Plane, etc.

2 Numerical mechanics of projections
GCTP and other code for projections

3 Review of parameters required for metadata

Registration part II
1 Introduce basic geometric transformations: (rotation, translation, scale)
2 Application in orienting digitizers or material digitized (scanned/ sensed)
3 Least squares fit of "best" affine or projective
4 Influence of outliers ("blunders"); alternatives for fit

Projections

Sources of software

• Hunter College list. (some lack of maintenance...)
• Current location for GCTP at USGS. The README file tells you more about it.
• USGS has lots of others, like PROJ done at Woods Hole
  
• MICROCAM (Scott Loomer, now at Military Academy, West Point); now distributed by Illinois State Paul Andersons' site;

Standard Reference Systems

conventional choices of projection parameters; Overview Peter Dana Geographer's Craft site

• Universal Transverse mercator (factsheet from USGS, software at NOAA, (FORTRAN code); Originally a Defense Department thing)
• State Plane Coordinates (NOAA Geodesy & Charting branch has always been in charge of these; software for NAD 27; software for 83;

Zone of standardized system (UTM counts from Date Line eastward- we are in Zone 10)
State Plane zones have multiple numbering systems (alphabetic state or FIPS codes...)
state plane zones are aggregates of counties, eg. central Texas.

Most multi-zone states have a secret (usually unoffical) projection that covers the whole state. In Wisconsin, it is a TM zone positioned on the split between two zones in UTM (90 West), in Oregon, they have a special compromise Lambert rather openly adopted (including a set of error analysis, the area error image). Oregon law establishing coordinate systems (begin with 93.310)

Washington has the practice of extending South Zone to the North without any official sanction...

Official Coordinates for Washington State (RCW (the server) , Text (partially converted of Title 58 Chapter 20)

Other countries have similar choices, eg. Finland; UK; Ireland, (presumably M and N appear where expected...); France (3 Lambert zones);

Stephan Voser, great list of standard projections for Europe and elsewhere. (with ESRI parameters for most of them...)

Another summary with some specifics on European solutions

Bob Burth list of links on datums and coordinates (some links to specific country solutions)

the chatter it often takes to figure out that Switzerland is an oblique Mercator (1997 email, all links dead...)!

Mechanics of Projections:

Cylinders in normal aspect

longitude scaled onto the X axis; Latitude on Y, issue is scaling of Y axis.

Conics:

Centered over a Pole; R-Theta coordinate system from there; Parallels are concentric circles - the Radius of each circle, Angle scaled from longitude (at that parallel); then R-theta converted to X,Y.

Projection Parameters in Metadata:

Ellipsoid: Equat. radius, Polar radius, flattening (a,b,f)
eccentricity: e2 = 2f - f2 derived from flattening...
Horizontal and vertical datum(s?) [not data...] discussed lecture 08

Modification of "Normal Aspect":

Transverse (some meridian is new Equator);

Oblique; generally just recalculate lat/long based on new "pole" and "equator".

Cylindrics:

• central meridian,
• scale factor (permits cylinder inside, not tangent)

Conics:

• one parallel (tangent);
• 2 standard parallels (secant);
• central meridian
  
  

False origin

a location to be called (0,0) (given by latitude, longitude usually, and off in the far lower left of the study area...)

Registration:

(connecting source material to a spatial reference framework)

Registration involves transforming the coordinates of some collection of data into a desired coordinate system. This is usually based on having a key to the transformation by identifying some set of points ("control") on both sources.

Some systems (eg. ARC/INFO) build this into the process immediately (as suggested in Clarke page 69-70. I disagree, since I think the original space is needed to perform quality control, then transform later (with more ability to select how it happens)
How many points? ARC (see overheads) suggests 4, barely enough to cross-check an affine! 16 or 20?

Ground control:

What points provide the best connection?

Ultimately, geodetically surveyed control points (see Geodetic Control resources at National Geodetic Survey). Failing that, often special control points (panelled on ground for photographs) or simply points measured on maps (depends on which points you use...)

US National Geodetic Survey
Data sheets for control points

Monument data from WA DOT

GPS base station data (for differential corrections)

Numerical procedure to compute registration:

Affine: equation is simple, and general.
To translate, simply add (subtract) from X and Y; to scale multiply (either by the same value or by differential scales (removes a tilt from an airphoto, often, but not always...); rotation involves cross products.
Combined through use of matrix notation, 3 X 3 matrices of three components can be combined then applied to each coordinate.
alternatives to affine: projective (removes "barrel" distortion based on distance from a focal point (deals with lens distortions); higher order equations (usually impractical);

piecewise transformations (see Marvin White in his conflation days).

Solving for best fit

Least squares: apply regression analysis to problem. Compute transformation between the input X,Ys and the desired "control" values that minimizes the squared deviations. R squared (as in regression) is a measure of goodness of fit, and the distances (variances) of the residuals are interesting too.

Outliers: regression can be affected (sometimes heavily) by some single bad value. Common method: examine outliers, remove points, refit equation
"Robust" Alternative: Least-Median Squares (can be up to 50% contaminated)

versions of code shown...

Speaking Truth to Power presentation