BACKGROUND

A PEF is traditionally estimated by minimizing the output of the known data d, convolved (D) with a filter f which is unknown except for the first coefficient, which is constrained to 1. This is expressed below, with K representing a mask which is 1 when all filter coefficients lie on known data, and when coefficients lie on missing data.  

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Fitting goal works well for estimating the PEF if there are sufficiently contiguous data. However, if the data are irregularly sampled so that there are an inadequate number of fitting equations, a different method is used (), where more fitting equations are generated by regridding the data (D) with an operator S_i (normalized linear interpolation followed by its adjoint), and then simultaneously estimating a single filter (f) on all of the versions of the scaled data (S_i d). The mask for the known data (K_i) must also be regridded accordingly for each scale.  

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Several user-defined parameters must be set during this procedure, specifically the choice of scales to be used in the estimation as well as the size of the PEF. The only constraint on these parameters is that the aspect ratio of the data remain constant from scale to scale, meaning that the ratio of the number of bins in each dimension remain constant. For example, a 50*40 data set should not be regridded to 26*21, since the aspect ratio is changed by the round-off from 20.8 to 21. A better choice would be to use 25*20 as a scale. The PEF size is only constrained by the size of the coarsest scale of data.

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.