Estimating piecewise linear models using combinatorial optimization techniques

A wide range of image and signal processing problems have been formulated as ill-posed linear inverse problems. Due to the importance of discontinuities and non-stationarity, piecewise linear models are a natural step towards more realistic results. Although there have been some attempts to extend classical approaches to deal with discontinuities, finding at the same time the piecewise decomposition and the corresponding model parameters remains a major challenge. A new approach based on partitioning inconsistent linear systems into a minimum number of consistent subsystems (MIN PCS) is proposed for solving ill-posed problems whose formulation as linear inverse problems with discrete data fails to take into account discontinuities. In spite of the NP-hardness of MIN PCS, satisfactory approximate solutions can be obtained using simple but effective variants of an algorithm which has been extensively studied in the artificial neural network literature. Our approach presents various advantages compared to classical alternatives, including a wider range of applicability and a lower computational complexity.