# Signal and image approximation with level-set constraints

- First Online:

- Received:
- Accepted:

DOI: 10.1007/s00607-007-0246-y

- Cite this article as:
- Schnörr, C. Computing (2007) 81: 137. doi:10.1007/s00607-007-0246-y

## Summary

We present a novel variational approach to signal and image approximation using filter statistics (histograms) as constraints. Given a set of linear filters, we study the problem to determine the closest point to given data while constraining the level-sets of the filter outputs. This criterion and the constraints are formulated as a bilevel optimization problem. We develop an algorithm by representing the lower-level problem through complementarity constraints and by applying an interior-penalty relaxation method. Based on a decomposition of the penalty term into the difference of two convex functions, the resulting algorithm approximates the data by solving a sequence of convex programs. Our approach allows to model and to study the generation of image structure through the interaction of two convex processes for spatial approximation and for preserving filter statistics, respectively.