How to reconstruct a 2D field from its integral?

Question

General question

I work on the plane where I have a two-dimensional shape $V$ that is cut in a collection of parts $\{V_i\}$ that do not overlap

$ V_i ~~\text{s.t.}~~ \bigcup_i \overline{V}_i = \overline{V} ~~\text{and}~~ \bigcap_i V_i = \{\emptyset\} $

I know the value of the integral of a scalar field $q(\pmb{x})$ over each part $V_i$

$ \int_{V_i} q(\pmb{x}) \, \pmb{dx} ~~ \text{known for all \(i\)} $

and I would like to get an estimation of the field $q(\pmb{x})$ for any point $\pmb{x}$ of space.

What methods can do that?

My particular case

Here is an example of the shape $V$ cut into non-overlapping (triangular) subshapes $\{V_i\}$:

where the colors in the plot corresponds to the value of a packing factor $P(V_i)$ calculated from a characteristic function $\chi$ as $ P(V_i) := \frac{\int_{V_i} \chi(\pmb{x})}{\int_{V_i} 1} $

I would like to find a function $p$ such that $ P(V_i) = \int_{V_i} p(\pmb{x}) ~~ \forall i $
Of course, being in a general 2D case, there is no such thing as a primitive function… How can I estimate function $p$?

Answer 1

There are infinite functions that have the same integral over a given domain, so you would need to make assumptions on the type of functions that you want to allow.

The easiest approach that comes to my mind is assuming a constant function in each cell, then

$$p_i \equiv p(x) = \frac{P(V_i)}{V_i} \quad \forall x \in V_i\, .$$

Then you can use this information to obtain the values for $p(x)$ in other regions of interest for you. For example:

1. You can average the values from neighboring cells to the nodes to have a piece-wise linear approximation of $p(x)$.

2. You could compute the Voronoi dual of your triangulation and use an interpolation where $p_i$ is assigned to each centroid.

3. Do a non-local interpolation with the values.

You could also use another assumption for the behavior of $p(x)$ and enforce some properties using an optimization approach, but I will first try the approach suggested above.

Answer 2

I think there's a couple of questions here:

1. How can we estimate $q$ for any point inside the support and

2. how can we estimate $q$ for any point outside the support.
   For 1, I'd try taking some inspiration from the Radon transformation, in particular, something like Algebraic reconstruction

Discretize the support into a whole bunch of points $x_i$; we'll try to recover $q(x_i)$. Use the values of the integrals to get linear constraints for $\sum_j a_{ij} q(x_j) = \int_{V_i} q$. This will be under-determined, so add in a smoothness penalty to bound $\frac{q(x_i)-q(x_j)}{x_i-x_j}$ for neighboring $x_i$, $x_j$, and whatever other bounds you thing might be fun. Plug it all into your favorite solver and see what comes out.

For 2., that's going to be a lot harder, but maybe once you have a solution for 1. you can try kriging.