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I'm looking for an algorithm that given a quad mesh with many a regular grid of small quads, would produce a similar looking quad mesh, but with as many of the small quads unified into larger quads, as much as possible.

For example, given a quad mesh that looks like this:

input mesh

The output would be something like this:

output mesh

I realise that producing a optimal solution is probably an NP-Hard problem, so the output would obviously need to be less than optimal.

What is this problem called? Are there established algorithms for it?

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Indeed the problem is NP-hard if you seek the minimum: "Tiling an orthogonal polygon with squares."

See this link for heuristic solutions:

PartSq
Key search phrase: partitioning an orthogonal polygon into squares.