Lecture 15Transformations - 2

Learning Objectives

15.1 What are the four schemes for surface transformations and how do they differ?

15.2 How are scheme entries applied as part of surface transformations?

15.3 How are space and attribute considerations used in surface transformations?

15.4 How do we discover relationships from geometry?

15.5 Once relationships are discovered from geometry what do we do with the attributes?

In Table 9-1 of surface transformations (previous lecture), the table entries described the approach for transformation.

Table 9-1: Surface Oriented Transformations
Input \ Output / Points (w.Z) / Isoline / DEM / TIN
Points (w.Z) / Interpolation / Interp. & Trace / Interpolation / Triangulation
Isoline / Interpolation / Interp. & Trace / Interpolation / Triangulation*
DEM / Interpolation / Interp. & Trace / Resampling / Triangulation*
TIN / Extraction / Tracing / Tracing / Refinement

*denotes an operation that might produce overly dense triangles without filtering

15.1 What are four schemes for surface transformations and how do they differ?

A scheme outlines the steps in an approach to transformation, see Table 9-2, herein each called a ‘case’ that connects input data structure and output data structure. Case codes 0, 1, 2 refer to the relative amount of computation required to perform scheme and use of Neighborhood (N) Relationships andAttribute (A) Assumptions information.

Table 9-2: Scheme for Transformations
Attribute Assumptions
Neighborhood Relationships / Implicit / External
Implicit – stored within / Case 0 / Case 1A
Discovered – computed / Case 1N / Case 2
Case 0: transformation by extraction:

Source already contains information required to compute solution, e.g. topological structure with geometry already contains 'spaghetti'(coordinate list) geometry – no need to derive it, just extract it. But the inverse is more complex, i.e., generating topology from the geometry.

Case 1A: attribute assumptions required

Geometry stays same, attributes are changed. Simplest case: Groupings; add information; rules do not have to remain local. Remotely sensed imagery: multiple continuous measures=> clusters (also called classes). Two ways to obtain: 1) information from ‘ground truth’ of pixels using training set (verify data with world); and 2) cluster from data valueswithout knowing exactly what they are on the ground, but could use other data as insight, e.g. land cover.

Case 1N: geometric processing only

Construct neighborhood in some manner: e.g. TIN operations locate point in triangle.

Case 2: complete processing

Requires geometry and attribute assumptions be put to use. Good example is a surface interpolation from scatter of points, and then the attribute rule generates new attribute data value.

15.2 How are scheme entries applied as part of surface transformations?

Table 9-3 presents conversions between various measurement frameworks reclassified into case groups using case codes 0, 1, 2 and N: Neighborhood Relationships or A:Attribute Assumptions' per Table 9-2.

Table 9-3: Surface Oriented Transformations Reclassified (Revised version)
Input \ Output / Points (w.Z) / Isoline / DEM / TIN
Points (w.Z) / 2: N Found / A Imported / 2: N Found / A Imported / 2: N Found / A Imported / 1N: Found / Flat triangle
Isoline / 2: N Found / A Imported / 2N: N Found / A Imported / 2: N Found / A Imported
[Not implemented directly in 3D Analyst] / 1N: Found / Flat triangle
[very dense triangles]
DEM / 1A: N- implicit /
A Imported / 1A: N- implicit /
A Imported / 1A: N- implicit /
A Imported / 0: N- implicit /
A- Flat triangle
TIN / 0: N- Point in Triangle
/ A- Flat triangle / 1A: tracing rule [if linear, 0, otherwise smoothness imported] / 0: N- Point in Triangle
/ A- Flat triangle / 2: N Simplified / A Refined

Transformations from minimal information stored within a data structure (e.g.,points w.Z as isolated objects) require the full transformation of space and attribute (2: found / imported). The points and isolated objects carry little information, so software uses spatial exploration and discovery to create the information.

For Points with Z and Isoline to TIN, neighborhoods are found and attributes are drawn from flat triangles.

For DEM to DEM,geometry staysthe same and attributes are imported into structure.

For TIN to others and DEM to TIN, geometry is implicit and the attributes are retrieved from the flat triangle.

When transforming from minimal information input to maximal output, considerable effort is needed to develop the relationship between the data structures. This means substantial data processing using more sophisticated software procedures, requiring more processing effort.

15.3 How are space and attribute considerations used in surface transformations?

Case 1A: Classification of Remotely Sensed Imagery

Forest Mapping for the United States

Case is based on the.pdf version of the Southern Forest Experiment Station report SO-280 for maps. Information from two different resolution remote sensing images were used to estimate a percentage of forest cover. A selected set of more detailed Thematic Mapper images (30 m pixels) are used to calibrate the relationship between forests and the 1 km pixels of Advanced Very High Resolution Radiometer images. The regression analysis is performed on each of the 15 physiographic regions in Hammond's Atlas of the 48 states (See pdf p. 6 Fig 2). The result is a percent forest cover per 1 km pixel, based on the spectral values of that pixel and the regression analysis for the physiographic region (See pdf p 9 Fig 5). From Zhu and Evans, 1994: US Forest types and Predicted percent forest cover from AVHRR data, Photogrammetric Engineering and Remote Sensing, 60(5), 525-531.

Case 1N: Geometric Measures Converted to Attributes

Wastelands versus Wetlands in Westport, Wisconsin

Wisconsin statutes set up two regimes to recognize wetlands.

  1. For taxation: 'wastelands' assessed at lower rate. (See Figure 9-7Exploring GIS)
  2. For conservation: 'wetlands' protected by statewide zoning. (See Figure 9-8)

The first regime works through the assessment of parcels, while the second regime works through an inventory of wetlands.To examine the effectiveness of these two programs to serve as reciprocal 'carrot and stick', the two views of the landscape must be combined. A transformation of wetland acreage onto the parcels demonstrates that there are some large discrepancies between the two processes. (See Table 9-4)

Case 2: Areal Interpolation

Dasymetric Mapping of Population DensityJohn K. Wright 1936 Cape CodWright wanted to produce a map of population density that reflected the places where people actually lived. He had population by “townships”. So, he constructed a land use map, and tried to generate a reasonable pop density to assign to each land use class. He then took the total township population and reallocated it to the various land use class zones in the township, that is, using a finer resolution of information. This method he termed dasymetric, which means “mapping density using a proxy to upgrade information” across the surface.Cartographers continue to cite Wright's 1936 study. We need to be careful about our assumptions. The land use map was used because Wright assumed that the same land use class (say residential, two story) had similar densities in adjacent towns, thus permitting him to estimate the density from the sources he had. His study was a form of areal interpolation, using of sparse data to solve a set of simultaneous linear equations. (See Figure 9-10 in Exploring GIS)

Discovering Information from Neighborhoods and Attributes for Transformations

15.4 How do we discover relationships from neighborhoods?

Generally speaking, there are two groups of processing procedures:

Processing coordinates - neighborhood based on coordinate containership

Processing relationships - neighborhood based on adjacency and connectedness

Note: coordinates can be used to discover relationships; relationships can be used to discover coordinates.

Neighborhood based on containership using coordinate site (as in overlay)

A form of spatial JOIN where the "key" is sharing a containership site location (not situation). Simple form of container: Point in Polygon; this is coordinate coincidence
Neighborhood based on situation relationships(as in buffer)

Reaching out from the coordinate site to broader distances, we use various techniques to process situation:

-Buffers

-Cellular neighborhoods in raster can be distance based since topology is implicit

-Explicit topological relationships using adjacency and connectedness in vector network

15.5 Once relationships are discovered from neighborhood what do we do with the attributes?

Example: point in polygon

Aggregation

  • Total the number of points inside some collection zone, thereby convertingan occurrence (i.e., a nominal event) into a kind of ratio (or count) measure for the zone
  • Summarize, e.g. total, average, maximum, minimum..., an attribute of the points in the zone

Disaggregation

  • Assign "formal" properties of the zone to the points, which works for administrative districts, like building inspections for hazardous waste containers across an area to form work packages for inspectors.From building counts to inspection counts.

Example: polygon in polygon

Aggregation

  • attribute data values for more general (higher level)geographic objects can be computed from sums or averages of the more primitive (finer grain) polygons; census blocks aggregate to block groups, block groups to census tracts

Disaggregation

  • If the attributes are not “categorical properties as observed” there may be a need to interpolate (estimate) information, and then assign to finer grain polygons. For example, as in Wright’s dasymetric mapping of population density from density and land use.