Polyakov Action for Efficient Color Image Processing∗

The Laplace-Beltrami operator is an extension of the Laplacian from flat domains to curved manifolds. It was proven to be useful for color image processing as it models a meaningful coupling between the color channels. This coupling is naturally expressed in the Beltrami framework in which a color image is regarded as a two dimensional manifold embedded in a hybrid, five dimensional, spatial-chromatic (x, y, R, G, B) space. The Beltrami filter defined by this framework minimizes the Polyakov action, adopted from high-energy physics, which measures the area of the image manifold. Minimization is usually obtained through a geometric heat equation defined by the Laplace-Beltrami operator. Though efficient simplifications such as the bilateral filter have been proposed for the single channel case, so far, the coupling between the color channel posed a nontrivial obstacle when designing fast Beltrami filters. Here, we propose to use an augmented Lagrangian approach to design an efficient and accurate regularization framework for color image processing by minimizing the Polyakov action. We extend the augmented Lagrangian framework for total variation (TV) image denoising to the more general Polyakov action case for color images, and apply the proposed framework to denoise and deblur color images.