A Closed-Form Solution to Natural Image Matting

Interactive digital matting, the process of extracting a foreground object from an image based on limited user input, is an important task in image and video editing. From a computer vision perspective, this task is extremely challenging because it is massively ill-posed - at each pixel we must estimate the foreground and the background colors, as well as the foreground opacity ("alpha matte") from a single color measurement. Current approaches either restrict the estimation to a small part of the image, estimating foreground and background colors based on nearby pixels where they are known, or perform iterative nonlinear estimation by alternating foreground and background color estimation with alpha estimation. In this paper, we present a closed-form solution to natural image matting. We derive a cost function from local smoothness assumptions on foreground and background colors and show that in the resulting expression, it is possible to analytically eliminate the foreground and background colors to obtain a quadratic cost function in alpha. This allows us to find the globally optimal alpha matte by solving a sparse linear system of equations. Furthermore, the closed-form formula allows us to predict the properties of the solution by analyzing the eigenvectors of a sparse matrix, closely related to matrices used in spectral image segmentation algorithms. We show that high-quality mattes for natural images may be obtained from a small amount of user input.