The algorithm maintains a running "best distance" that starts at infinity. As it walks the tree, it checks each visited point and updates the best distance if it finds something closer. Before recursing into a child node, it checks whether the closest possible point in that child's bounding box is farther than the current best. If so, the entire subtree gets pruned.
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Uniform regions (solid backgrounds, smooth gradients) get stored as large blocks. Detailed regions (edges, textures) get subdivided into smaller blocks. You end up with a compressed representation that preserves detail where it exists and simplifies where it can.
Transforms can be stateless or stateful. A stateless transform is just a function that takes chunks and returns transformed chunks: