We can see that the threshold map distributes perturbations more optimally than purely random noise, resulting in a clearer and more detailed final image. The algorithm itself is extremely simple and trivially parallelisable, requiring only a few operations per pixel.
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Notice how the highlighted region shrinks at each step. The algorithm never examines points outside the narrowing window. In a balanced tree with nnn points, this takes about log4(n)\log_4(n)log4(n) steps. For a million points, that's roughly 10 steps instead of a million comparisons.