# American Institute of Mathematical Sciences

February  2011, 5(1): 37-57. doi: 10.3934/ipi.2011.5.37

## Multimodal inverse problems: Maximum compatibility estimate and shape reconstruction

 1 Department of Mathematics, Tampere University of Technology, P.O. Box 553, 33101 Tampere, Finland

Received  April 2009 Revised  September 2010 Published  February 2011

We present an optimal strategy for the relative weighting of multiple data modalities in inverse problems, and derive the maximum compatibility estimate (MCE) that corresponds to the maximum likelihood or maximum a posteriori estimates in the case of a single data mode. MCE is not explicitly dependent on the noise levels, scale factors or numbers of data points of the complementary data modes, and can be determined without the mode weight parameters. We also discuss discontinuities in the solution estimates in multimodal inverse problems, and derive a corresponding self-consistency criterion. As a case study, we consider the problem of reconstructing the shape and the spin state of a body in $\R^3$ from the boundary curves (profiles) and volumes (brightness values) of its generalized projections in $\R^2$. We also show that the generalized profiles uniquely determine a large class of shapes. We present a solution method well suitable for adaptive optics images in particular, and discuss various choices of regularization functions.
Citation: Mikko Kaasalainen. Multimodal inverse problems: Maximum compatibility estimate and shape reconstruction. Inverse Problems & Imaging, 2011, 5 (1) : 37-57. doi: 10.3934/ipi.2011.5.37
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