5.10 Multi Level B-Spline Approximation

In this section a set of irregularily distributed data points is approximated via Multi Level B-Spline approximation.

The MBA algorithm has the advantage that it can be used to create very smooth transition functions from both very small and very big data sets. It is often used for image or surface warping (see Wolberg 4.3).

Note that the Mba algorithm basically creates a B-Spline surface of the form z = f(x,y). I only bring it to a nurbs surface of the general form (x,y,z) = S(u,v) to be able to display it (i.e. x and y are only linear mappings of u and v).

In Example 26 the point set is approximated with 5 iterations, the result is shown in Figure 5.25. The point set contains the height changes of 11335 points in a certain area between the years 1994 and 1996. The maximum height change was 50cm, and the heights were measured with a precision of +/-20cm.

Example 26: (taken from: data/big/nubmbas3.dat (7.1))

surf1 = NUBMBAS( 
  5,       /* number of iterations */
  1,       /* minimize area of base rectangle in XY-plane */
  3,       /* degree in U-direction the surface shall have */
  3,       /* degree in V-direction the surface shall have */
  /* list of data points: */
  (
    (1330.76, 1650.00, 220.00),
    (2198.50, 2033.20, 20.00),
    ...
    (4334.40, 5134.66, 170.00),
    (4977.55, 5458.02, -120.00)
  )
);

Figure 5.25: MBA of 11335 data points with 5 iterations

Figure 5.26, Figure 5.27 and Figure 5.28 were created by setting the number of iterations in Example 26 to 4, 6 and 7.

Figure 5.26: MBA of 11335 data points with 4 iterations

Figure 5.27: MBA of 11335 data points with 6 iterations

Figure 5.28: MBA of 11335 data points with 7 iterations