Quasi-B-Splines

A slightly surprising result is that the limit curve of polygon P(z) under the binary scheme with mask 2ak(l + z2)/2z is identical to the limit curve of polygon aP(z) under the scheme whose mask is 2ak+i. It is a B-spline curve, but with a different control polygon. 

This can be traced back to the result that the two-slanted sampling matrix S encountered in the chapter on artifacts has the same product when multiplied on the right by the circulant matrix corresponding to (1 + z) as when multiplied on the left by that corresponding to (1 + z2). 

For example, we show here the first three refinements of cardinal data using the mask whose generating function is 2((1 + z)/2)2((l + z2)/2)2, and the argument above says that this has the same limit curve as applying the mask 2((1 + z)/2)4 to the polygon with vertices [l,2,l]/4. 

The polygons are shown as lines the dots are at points of the limit curve, using a unit row eigenvector obtained by convolving that of the normalised scheme [l,4,l]/6 with the prefix [l,2,l]/4. 

The limit curve does indeed consist of cubic spans, meeting C2. 

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