Abstract

Inverse problems are of significant interest across a broad range of science and engineering applications. The primary objective in an inverse problem is to extract the unknown composition and structure of a medium based on a set of indirect observations which are related to the unknown via a physical model. In many cases, one seeks only the identification and characterization of ``regions of interest’’ (ROIs), such as cancerous tumors from Xray CT or diffuse optical data, subsurface contaminants from hydrological data or buried objects from electromagnetic data. These problems are often solved by first forming an image and then post processing to identify the ROIs.  Although this can be effective, it is computationally expensive. Moreover, for data limited problems, the initial image formation stage will require potentially complex regularization methods and user tuning to overcome the ill-posed nature of these problems. We propose an alternative approach in the context of inverse image reconstruction/restoration of piecewise constant objects whereby we parameterize the image model itself. Our PaLEnTIR model is a significantly enhanced parametric level set image model.  Instead of solving for pixel/voxel values, we optimize for a very small set of parameters and the regularization is built into the model itself.   Given upper and lower bounds on the contrast, our approach can recover objects with any distribution of contrasts and eliminates the need to know either the number of contrasts in a given scene or their values.  Our inversion algorithm therefore optimizes for the model parameters while iteratively estimating the necessary space-varying contrast limits.  We discuss the properties of our PaLEnTIR model and the inversion algorithm and demonstrate the performance on several 2D and 3D linear and non-linear inverse problems.