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Register a new userLame Parameter Estimation from Static Displacement Field Measurements in the Framework of Nonlinear Inverse Problems
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We consider a problem of quantitative static elastography, the estimation of the Lame parameters from internal displacement field data. This problem is formulated as a nonlinear operator equation. To solve this equation, we investigate the Landweber iteration both analytically and numerically. The main result of this paper is the verification of a nonlinearity condition in an infinite dimensional Hilbert space context. This condition guarantees convergence of iterative regularization methods. Furthermore, numerical examples for recovery of the Lame parameters from displacement data simulating a static elastography experiment are presented.
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We propose a fully discrete variational scheme for nonlinear evolution equations with gradient flow structure on the space of finite Radon measures on an interval with respect to a generalized version of the Wasserstein distance with nonlinear mobility. Our scheme relies on a spatially discrete approximation of the semi-discrete (in time) minimizing movement scheme for gradient flows. Performing a finite-volume discretization of the continuity equation appearing in the definition of the distance, we obtain a finite-dimensional convex minimization problem usable as an iterative scheme. We prove that solutions to the spatially discrete minimization problem converge to solutions of the spatially continuous original minimizing movement scheme using the theory of $Gamma$-convergence, and hence obtain convergence to a weak solution of the evolution equation in the continuous-time limit if the minimizing movement scheme converges. We illustrate our result with numerical simulations for several second- and fourth-order equations.
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In this paper, we consider the problem of estimating the internal displacement field of an object which is being subjected to a deformation, from Optical Coherence Tomography (OCT) images before and after compression. For the estimation of the internal displacement field we propose a novel algorithm, which utilizes particular speckle information to enhance the quality of the motion estimation. We present numerical results based on both simulated and experimental data in order to demonstrate the usefulness of our approach, in particular when applied for quantitative elastography, when the material parameters are estimated in a second step based on the internal displacement field.
Data assisted reconstruction algorithms, incorporating trained neural networks, are a novel paradigm for solving inverse problems. One approach is to first apply a classical reconstruction method and then apply a neural network to improve its solution. Empirical evidence shows that such two-step methods provide high-quality reconstructions, but they lack a convergence analysis. In this paper we formalize the use of such two-step approaches with classical regularization theory. We propose data-consistent neural networks that we combine with classical regularization methods. This yields a data-driven regularization method for which we provide a full convergence analysis with respect to noise. Numerical simulations show that compared to standard two-step deep learning methods, our approach provides better stability with respect to structural changes in the test set, while performing similarly on test data similar to the training set. Our method provides a stable solution of inverse problems that exploits both the known nonlinear forward model as well as the desired solution manifold from data.
Partial differential equations are central to describing many physical phenomena. In many applications these phenomena are observed through a sensor network, with the aim of inferring their underlying properties. Leveraging from certain results in sampling and approximation theory, we present a new framework for solving a class of inverse source problems for physical fields governed by linear partial differential equations. Specifically, we demonstrate that the unknown field sources can be recovered from a sequence of, so called, generalised measurements by using multidimensional frequency estimation techniques. Next we show that---for physics-driven fields---this sequence of generalised measurements can be estimated by computing a linear weighted-sum of the sensor measurements; whereby the exact weights (of the sums) correspond to those that reproduce multidimensional exponentials, when used to linearly combine translates of a particular prototype function related to the Greens function of our underlying field. Explicit formulae are then derived for the sequence of weights, that map sensor samples to the exact sequence of generalised measurements when the Greens function satisfies the generalised Strang-Fix condition. Otherwise, the same mapping yields a close approximation of the generalised measurements. Based on this new framework we develop practical, noise robust, sensor network strategies for solving the inverse source problem, and then present numerical simulation results to verify their performance.
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