I can answer the DBE question for sure, since I designed and wrote that tool. DBE uses two-dimensional smoothing surface splines, also known as
thin plates. Second-order splines are used by default (that is, the generated interpolation device guarantees continuity of second derivatives by default). Our implementation of thin plates on the PixInsight/PCL platform is based on the solution given in the 1996 book
Numerical Algorithms with C by Giesela Engeln-Müllges and Frank Uhlig. I have used different adaptations of the same algorithms in other PixInsight tools such as StarAlignment and DynamicAlignment.
The AutomaticBackgroundExtractor (ABE) tool was designed and initially implemented by Carlos Milovic, so you better ask him for details. AFAIK, ABE uses a global 2-D polynomial interpolation in rectangular coordinates. I think it does not have derivative continuity constraints, but I may be wrong. This interpolation scheme, as all global (that is, non-piecewise) polynomial-based interpolations, tends to generate ripple artifacts using high orders. However, polynomials are much more rigid than splines, which is potentially more robust in some cases, especially where adaptation to local variations is not a desirable property of the solution. Each scheme has its own strong and weak points.
