Scalable Matting: A Sub-linear Approach
  Natural image matting, which separates foreground from background, is a very important intermediate step in recent computer vision algorithms. However, it is severely underconstrained and difficult to solve. State-of-the-art approaches include matting by graph Laplacian, which significantly improves the underconstrained nature by reducing the solution space. However, matting by graph Laplacian is still very difficult to solve and gets much harder as the image size grows: current iterative methods slow down as $\mathcal{O}\left(n^2 \right)$ in the resolution $n$. This creates uncomfortable practical limits on the resolution of images that we can matte. Current literature mitigates the problem, but they all remain super-linear in complexity. We expose properties of the problem that remain heretofore unexploited, demonstrating that an optimization technique originally intended to solve PDEs can be adapted to take advantage of this knowledge to solve the matting problem, not heuristically, but exactly and with sub-linear complexity. This makes ours the most efficient matting solver currently known by a very wide margin and allows matting finally to be practical and scalable in the future as consumer photos exceed many dozens of megapixels, and also relieves matting from being a bottleneck for vision algorithms that depend on it. 
Philip G. Lee and Ying Wu
cs.CV