Factorising the Subdivision Matrix

We therefore look first at what happens during a single step, with data at just a single spatial frequency. 

The subdivision matrix of a binary uniform scheme has each column a copy of the mask, shifted down two rows for each step to the right. Every row (stencil) sums to 1, but the mask entries sum to 2. 

We can reexpress this matrix as the product of two matrices. The first is a circulant matrix with every row29 and every column a copy of the mask divided by 2. The second is a diagonal matrix with the value 2 on the diagonal of slope -2. We call the latter the sampling matrix and the former the smoothing matrix. 

For example, the scheme whose mask is [1, 2, l]/2 can be written in matrix form as 

Essentially, alternate columns in the smoothing matrix are multiplied by the empty rows of the sampling matrix. 

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