Spatial interpolation before or after data processing

[Oliver-BfS]

TL;DR

Shall spatial interpolation better be performed on original values or on processed values?

Let a set of values at several discrete points in 2D or 3D space be given. These values will be processed by an algorithm. At the end, processed values need not be known at the original locations but at grid points. Therefore, spatial interpolation needs to be applied.

Is there a general recommendation if spatial interpolation should better be performed on the original values before processing the values or on the processed values?

 

[pbuk]

Can I restate the question to check understanding?

You have a set of coordinates $x_i$ in a domain $X$ and corresponding values $f(x_i)$ in a codomain $F$. You have an operation $g$ that acts over $F$. You want to approximate the values $g(f(x))$ over a different set of coordinates $x_j$. You are considering 2 options:

1. approximate $f(x_j)$ with an interpolation function $q(f(x_i))$ and calculate $g(q(f(x_i)))$;
2. calculate $g(f(x_i))$ and approximate $g(f(x_j))$ with an interpolation function $q'(g(f(x_i)))$.

I don't think there is a general preference: if $f(x_i)$ are known exactly and can be accurately interpolated - in the extreme, let's say they are linear - then $f(x_j) = q(f(x_i))$ so you can't do better than option 1. If on the other hand $f(x_i)$ are only known approximately and $g$ applies some smoothing by acting over a subset of points then option 2 is probably better.

 

[Oliver-BfS]

Your understanding of the question is correct.

The values cannot be interpolated accurately because $f(x_i)$ is only known at the coordinates $x_i$. There is no knowledge of the underlying function $f$. $f(x_i)$ are measured values.