  Many works on inverse problems in the imaging sciences consider
regularization via one or more penalty functions or constraint sets. When the
models/images are not easily described using one or a few penalty
functions/constraints, additive model descriptions for regularization lead to
better imaging results. These include cartoon-texture decomposition,
morphological component analysis, and robust principal component analysis;
methods that typically rely on penalty functions. We propose a regularization
framework, based on the Minkowski set, that merges the strengths of additive
models and constrained formulations. We generalize the Minkowski set, such that
the model parameters are the sum of two components, each of which is
constrained to an intersection of sets. Furthermore, the sum of the components
is also an element of another intersection of sets. These generalizations allow
us to include multiple pieces of prior knowledge on each of the components, as
well as on the sum of components, which is necessary to ensure physical
feasibility of partial-differential-equation based parameters estimation
problems. We derive the projection operation onto the generalized Minkowski
sets and construct an algorithm based on the alternating direction method of
multipliers. We illustrate how we benefit from using more prior knowledge in
the form of the generalized Minkowski set using seismic waveform inversion and
video background-anomaly separation.
