Multimodal inverse problems: Maximum compatibility estimateand shape reconstruction

We present an optimal strategy for the relative weighting of multiple datamodalities in inverse problems, and derive the maximum compatibilityestimate (MCE) that corresponds to the maximum likelihood or maximuma posteriori estimates in the case of a single data mode. MCE is notexplicitly dependent on the noise levels, scale factors or numbers of datapoints of the complementary data modes, and can be determined without themode weight parameters. We also discuss discontinuities in thesolution estimatesin multimodal inverse problems, and derive acorresponding self-consistency criterion.As a case study, we consider the problem ofreconstructing the shape and the spin stateof a body in ³ from the boundary curves(profiles) and volumes (brightness values) of its generalized projectionsin ².We also show that the generalized profilesuniquely determine a large class of shapes.We present a solution method well suitablefor adaptive optics images in particular, and discuss various choices ofregularization functions.