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IPI
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.
IPI
We study the inverse problem of deducing the dynamical characteristics
(such as the potential field) of large systems from kinematic observations.
We show that, for a class of steady-state systems,
the solution is unique even with fragmentary data, dark matter,
or selection (bias) functions. Using spherically symmetric
models for simulations, we investigate solution convergence and
the roles of data noise and
regularization in the inverse problem. We also present a method,
analogous to tomography,
for comparing the observed data with a model probability
distribution such that the latter can be determined.
IPI
We discuss shape reconstruction methods for data presented in various image spaces. We demonstrate the usefulness of the Fourier transform in transferring image data and shape model projections to a domain more suitable for shape inversion. Using boundary contours in images to represent minimal information, we present uniqueness results for shapes recoverable from interferometric and range-Doppler data. We present applications of our methods to adaptive optics, interferometry, and range-Doppler images.
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