Definitions

I call ${\bf A}$ the convolution or combination operator with a bank of non-stationary filters. For the non-stationary convolution, the filters are in the column of ${\bf A_{conv}}$ (one filter corresponds to one point in the input space) whereas for the non-stationary combination, the filters lie in the rows of ${\bf A_{comb}}$ (one filter corresponds to one point in the output space). For the convolution matrix, I define a_i,j as the i^th coefficient of the filter for the j^th data point in the input space. For the combination matrix, I define a_i,j as the j^th coefficient of the filter for the i^th data point in the output space. Therefore, for the non-stationary convolution we have  

$$\bf{A}_{conv}=\left ( \begin{array} {cccc} 1 & 0 & 0 & \vdo... ... \\ \vdots & \vdots & \vdots & \vdots \end{array} \right )$$ (120)

and for the non-stationary combination we have  

$$\bf{A}_{comb}=\left ( \begin{array} {cccc} 1 & 0 & 0 & \vdo... ... \\ \vdots & \vdots & \vdots & \vdots \end{array} \right )$$ (121)

The size of both matrices is $(n\times m )$ where n is the size of an output vector (${\bf x}$) and m the size of an input vector (${\bf y}$) if  

$${\bf Ay=x}$$ (122)

The helical boundary conditions allow to generalize this 1-D convolution to higher dimensions. We can rewrite equation () for the convolution as follows:  

$$y_k = x_k + \sum_{i=1}^{\min(nf-1, k-1)} a_{i,(k-i)} \; x_{k-i}$$ (123)

where nf is the number of filter coefficients, and for the combination  

$$y_k = x_k + \sum_{i=1}^{\min(nf-1, k-1)} a_{i,k} \; x_{k-i}.$$ (124)

In the next section, I show how the non-stationary PEFs are estimated.