Example of bad Tiger data in Stanislaus County: Red lines are the original road network, green lines are the corrected road network.

 
The Problem
The US Census does a nice job of collecting all sorts of geographic and demographic information every 10 years. This data is available free of charge in the rather complex and soon to be replaced TIGER/LINE format. While this data covers the entire US down to the local road level, there are numerous errors and even extreme cases of coordinate-shift. Here is an example from Stanislaus County, California. The original TIGER data (red lines) are offset several hundred meters from the imagery. While it is not clear what may have caused the problem, it can be fixed without much effort using an affine transformation. We do not have the transformation matrix, however it can be 'fit' to a set of control points by several methods. The general form of the affine transform can be conveniently represented in homogeneous coordinates as:

Affine Matrix Form in homogeneous coordinates:New coordinates on the left-hand side, old coordinates on the right-hand side. The transformation matrix is the 3x3 matrix in the center.

 
The Solution

• transform using GRASS and a set of control points with v.transform. This is the natural choice if the data is already in GRASS.
• Using the same control points, compute the transformation matrix in R, then apply in PostGIS with the Affine() function. This is the ideal approach for our scenario. Note that the format of the 'fitted' transformation matrix returned by R is in a slightly different format:

  
  

  Transformation Matrix from R

We first need a set of control points, good and bad coordinates. This can be accomplished in several ways, we used the d.where command in GRASS:

Computing the transformation matrix can be done with a simple regression between 'good' and 'bad' coordinates in R. Note that this approach was suggested by Prof. Brian Ripley on the R-help mailing list.

 
Compute the Affine Transformation Matrix in R

Establishing the transformation based on control points: Red points represent where the coordinates should be. Black points are the original and incorect coordinates.

 
Check Affine Transform Results in PostGIS

 
Perform Affine Transformation in PostGIS

 
Regression Diagnostics

Response nx :

Call:
lm(formula = nx ~ x + y, data = d)

Residuals:
     Min       1Q   Median       3Q      Max 
-207.088  -23.856    8.614   21.245  161.610 

Coefficients:
             Estimate Std. Error  t value Pr(>|t|)    
(Intercept) 5.017e+03  1.369e+03    3.666  0.00079 ***
x           1.002e+00  6.654e-04 1506.386  < 2e-16 ***
y           9.190e-03  9.419e-04    9.756 1.20e-11 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 

Residual standard error: 52.25 on 36 degrees of freedom
Multiple R-Squared:     1,      Adjusted R-squared:     1 
F-statistic: 1.322e+06 on 2 and 36 DF,  p-value: < 2.2e-16 


Response ny :

Call:
lm(formula = ny ~ x + y, data = d)

Residuals:
    Min      1Q  Median      3Q     Max 
-39.835 -18.459  -4.556  15.311  94.226 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept) -2.814e+04  7.409e+02  -37.98   <2e-16 ***
x           -1.352e-02  3.602e-04  -37.54   <2e-16 ***
y            9.974e-01  5.099e-04 1956.22   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 

Residual standard error: 28.28 on 36 degrees of freedom
Multiple R-Squared:     1,      Adjusted R-squared:     1 
F-statistic: 2.187e+06 on 2 and 36 DF,  p-value: < 2.2e-16