633#633 | (258) |
634#634 | (259) |
A non-stationary filter varies with position, so instead of only having indices corresponding to the lag of the filter, there are also indices corresponding to the position of the filter. The filter would go from looking like a(ia) to a(ia,id), where ia is the lag of the filter, and id is the position of the filter. An illustration of this concept is shown in figure . Only two indices are used for lag and position, thanks to the helical coordinate system ().
nstat
Figure 1 An illustration of non-stationary convolution. The shaded boxes represent the data, and the hollow boxes represent the filter at various positions. The two indices on each filter point correspond to the data position (id) and the filter lag (ia), respectively. At each point in the convolution, the filter is different. |
Since we have moved from estimating a single filter with na filter coefficients to estimating a non-stationary filter with na*nd filter coefficients, PEF estimation becomes an under-determined problem instead of an over-determined problem. As a result, we need to incorporate some type of regularization into the estimation in order to get enough equations. Laplacian or radial rougheners of common filter coefficients (constant ia) across the spatial axes (id) are both used to ensure a filter bank that varies smoothly spatially ().
Another method used to constrain the filter coefficients is called micro-patching (). Instead of the filter varying at every data point, micro-patching uses the same filter within a small region within the data, reducing the number of filter coefficients that need to be estimated. This has two benefits: the PEF estimation problem becomes less under-determined, and the amount of memory required for the filter, which was na times the size of the data nd, is now the number of micro-patches, np times na.