Having chosen the radial direction, we can think of some different ways of implementing our radially-smoothed filters. An obvious one is putting a PEF at every point on the data grid, and devising a derivative filter which adjusts its direction to point at the origin. An alternative is to overlay a radial grid on the data grid, and arrange PEFs on the radial grid. Here we compare the two smoothing schemes.
Our goal is to assume stationarity in a small enough region that we can interpolate well where the data do not fit a plane-wave model. In the method of independent patches, individual patches are treated as separate problems. A patch can not be arbitrarily small, because it must provide enough fitting equations that the filter coefficients are well overdetermined. In 1-D, filtering with a PEF looks like this:
In moving to the method of gradually-varying PEFs, we replace the notion of extracting a subset of the data with that of dereferencing the data coordinates to find the appropriate filter, as in
We have lots of freedom in dereferencing ai. In the limiting cases, all the data may share one PEF, or we can choose ai to be a different set of coefficients for each data point. In the case where we have a PEF at every data point, we call pixel-wise smoothing.
Choosing a separate PEF for every input sample is a possibility, but not necessary. Our motivation for moving away from independent patches was to use one PEF in a region small enough that we do not have trouble with nonstationarity. Some amount of patching may still make sense, provided the patches may be small. It is easy to implement small patches as a generalization of the case above where each data sample has its own PEF. A particular ai can be the same for any number of values of i without complication. Because they may be small, we refer to the new patches as ``micropatches'' to distinguish between them and independent patches.
To subdivide a CMP gather into micropatches, we choose a web-like grid made up of radial lines and circular lines. Radial lines are a natural choice because we want to smooth in the radial direction. Circles are somewhat arbitrary; we could choose flat lines or reflection-like hyperbolas to cross the radial lines. Circles have the attractive property that they make equal-area micropatches at a given radius.