Piecewise linear approximation of vector-valued images and curves via second-order variational model

Variational models are known to work well for addressing image restoration/regularization problems. However, most of the methods proposed in literature are defined for scalar inputs and are used on multiband images (such as RGB or multispectral imagery) by the composition of a simple band-wise processing. This involves suboptimal results and may introduce artifacts. Only in a few cases variational models are extended to the case of vector-valued inputs. However, the known implementations are restricted to 1st-order models, while 2nd-order models are never considered. Thus, typical problems of 1st-order models such as the staircasing effect cannot be overtaken. This paper considers a 2nd-order functional model to function approximation with free discontinuities given by Blake-Zisserman (BZ) and proposes an efficient minimization algorithm in the case of vector-valued inputs. In the BZ model, the Hessian of the solution is penalized outside a set of finite length, therefore the solution is forced to be piecewise linear. Moreover, the model allows the formation of free discontinuities and free gradient discontinuities. The proposed algorithm is applied to difficult color image restoration/regularization problems and to piecewise linear approximation of curves in space.