Multimodal inverse problems: Maximum compatibility estimate and shape reconstruction
Mikko Kaasalainen
Inverse Problems & Imaging 2011, 5(1): 37-57 doi: 10.3934/ipi.2011.5.37
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.
keywords: computational geometry machine vision computational methods in astronomy. Inverse problems three-dimensional polytopes
Dynamical tomography of gravitationally bound systems
Mikko Kaasalainen
Inverse Problems & Imaging 2008, 2(4): 527-546 doi: 10.3934/ipi.2008.2.527
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.
keywords: $n$-body problems Quasi-periodic motions and invariant tori Mathematical physics Inverse problems Dynamical systems Hamiltonian systems Galactic and stellar dynamics
Shape reconstruction from images: Pixel fields and Fourier transform
Matti Viikinkoski Mikko Kaasalainen
Inverse Problems & Imaging 2014, 8(3): 885-900 doi: 10.3934/ipi.2014.8.885
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.
keywords: interferometry generalized projections image analysis three-dimensional polytopes radar. adaptive optics Inverse problems

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