Higher Arity Schemes

For higher arity schemes we have the two complications noted in the previous chapter, that the equivalent of the (l + z)/2 found in the binary case becomes (1 + z +. .. + za )/a. 

Then we find that the eigenanalysis can be performed at a — 1 mark points spread along a span of the polygon. 

---

It then turns out that the discontinuities found at such points may be totally different from those at the dyadic points. We can always make schemes of higher arity by considering two or more refinements as a single step. We call this squaring or taking a higher power of the scheme. 

Quaternary and higher arities are significantly more complex. Quaternary schemes can, of course be created by squaring a binary scheme, but there are others which are not so created. Note that the square of a weighted mean of two binary schemes is not the same as the weighted mean of the squares of those schemes. 

The artifact analysis above is limited, essentially to binary and ternary schemes. Higher arities can bring in artifacts at higher frequencies which might spoil or improve the shapes of the limit curves, and it is not obvious how these can most sensibly be handled. 

Both of the above examples approximately double the number of vertices in the polygon with each step of refinement. They are binary schemes. It is also possible to have schemes in which the number of vertices trebles or quadruples or is multiplied by a still higher factor. As mentioned above, we call that factor the arity, so that binary schemes have an arity of 2, ternary of 3, quaternary of 4 etc. Some of the mathematics applies to all arities, and in such cases we will denote the arity by the letter a. 

The slope of the subdivision matrix is equal to the arity, and so when we have a scheme with an arity higher than two, not all diagonals are equal. We find a — 1 different possible matrices to analyse. For a = 3 we get analyses at limit points corresponding to the control points and to the middles of the spans. In each case the choice of a diagonal identifies a labelling, which can then be viewed as defining a centre of symmetry. 

---