Formulation and test of a model of positional distortion fields

Chris Funk, Kevin Curtin, Michael Goodchild, Dan Montello, UCSB; Val Noronha, Digital Geographic

The model

• Vector field • Measured positions x+x, y+y

• Vector field continuous and smooth– Kiiveri: function of coordinates– but function loses generality

• If is known everywhere then distortion can be removed– variation in magnitude of could be visualized

The data

• Street centerline files– multiple vendors, sources

• many different

• Ambiguous messages about location if origin, destination have different databases– which street is (x,y) on?

• Applications in transportation, generalizes to other domains

Figure 1: Plot of a section of the two databases, superimposed on an interpolated field showing the magnitude of the distortion vector.

Obtaining a sample of

• Match points between databases– easiest at nodes

• Provides a sample set of observations– poor in rural areas

Determining a complete

• Interpolate a continuous field

• But what model to use for the surface?– Kiiveri - function of coordinates– spatial interpolation (e.g. Kriging)

• maximally smooth

– piecewise with linear breaks• mosaic of patches

Figure 2: Effect of a cliff on a linear feature (left); editing with a smooth line (right)

Why a mosaic of patches?

• Constant or linear or affine within each patch

• Breaks where there are no features– causes no cartographic offense

• Fits production methods– photogrammetric mosaic, edgematching of

different sources

Clustering the error field

• Variogram of angular differences

• Ratio of areal dependence– compares variation within lag with predictions

from variogram

• Cluster using RAD

0.15

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0 300 600 900 1200 1500 1800 2100

Distance

Gam

ma

Figure 3: Semivariogram of angular distortion values.

Figure 4: A plot of the RAD values associated with angular distortions.

Figure 5: Initial clustering based on RAD values.

Figure 6: Clustering using only high RAD values.

Conclusions

• Piecewise approach to modeling • Observable at points

• Identification of patches– piecewise constant

• Transportation application generalizes

• Errors highest where field is least observable