A Closed Form Solution to Natural Image Matting

by Anat Levin, Dani Lischinski and Yair Weiss

Matting is an interesting problem, which is very useful in digital image processing. The digital photos could be interpreted with a layer of foreground (including the subject) and another of background. The matting problem is that we have to decide a foreground image and a background image and the probability of each pixel belonging to foreground (something like an alpha channel). It is different from image segmentation (something like a soft-classifier) though. But it can also be solved with spectral clustering.

The idea is quite simple. The probability ci at pixel i is a linear combination of the intensity of each channel (RGB). Therefore c_i = a_i^r I_i^r + a_i^g I_i^g + a_i^b I_i^b + b. So we have to minimize their sum of squared residue with a regularizer of e a². This could be formulated with a quadratic form of a graph's Laplacian matrix. Therefore, given some constraints (e.g. users' input), the matting problem could be solve as an eigen value problem.