A quadratic programming approach to image labelling

Image labelling tasks are usually formulated within the framework of discrete Markov random fields where the optimal labels are recovered by extremising a discrete energy function. The authors present an alternative continuous relaxation approach to image labelling, which makes use of a quadratic cost function over the class labels. The cost function to be minimised is convex and its discrete version is equivalent up to a constant additive factor to the target function used in discrete MRF approaches. Moreover, its corresponding Hessian matrix is given by the graph Laplacian of the adjacency matrix. Therefore the optimisation of the cost function is governed by the pairwise interactions between pixels in the local neighbourhood. This leads to a sparse Hessian matrix for which the global minimum of the continuous relaxation problem can be efficiently found by solving a system of linear equations using the Cholesky factorisation. The authors elaborate on the links between the method and other techniques elsewhere in the literature and provide results on synthetic and real-world imagery. The authors also provide a comparison with competing approaches.