1,720,995 research outputs found
Note on Calderón's inverse problem for measurable conductivities
No full textThe unique determination of a measurable conductivity from the Dirichlet-to-Neumann map of the equation div(sigma del u) = 0 is the subject of this note. A new strategy, based on Clifford algebras and a higher dimensional analogue of the Beltrami equation, is here proposed. This represents a possible first step for a proof of uniqueness for the Calderon problem in three and higher dimensions in the L-infinity case
On an inverse Robin spectral problem
No full textWe consider the problem of the recovery of a Robin coefficient on a part γ ⊂ ∂Ω of the boundary of a bounded domain Ω from the principal eigenvalue and the boundary values of the normal derivative of the principal eigenfunction of the Laplace operator with Dirichlet boundary condition on ∂Ωγ. We prove the uniqueness, as well as local Lipschitz stability of the inverse problem. Moreover, we present an iterative reconstruction algorithm with numerical computations in two dimensions showing the accuracy of the method
Infinite dimensional compressed sensing from anisotropic measurements and applications to inverse problems in PDE
No full textWe consider a compressed sensing problem in which both the measurement and the sparsifying systems are assumed to be frames (not necessarily tight) of the underlying Hilbert space of signals, which may be finite or infinite dimensional. The main result gives explicit bounds on the number of measurements in order to achieve stable recovery, which depends on the mutual coherence of the two systems. As a simple corollary, we prove the efficiency of nonuniform sampling strategies in cases when the two systems are not incoherent, but only asymptotically incoherent, as with the recovery of wavelet coefficients from Fourier samples. This general framework finds applications to inverse problems in partial differential equations, where the standard assumptions of compressed sensing are often not satisfied. Several examples are discussed, with a special focus on electrical impedance tomography
Calderón's inverse problem with a finite number of measurements II: independent data
No full textWe prove a local Lipschitz stability estimate for Gel'fand-Calderón's inverse problem for the Schrödinger equation. The main novelty is that only a finite number of boundary input data is available, and those are independent of the unknown potential, provided it belongs to a known finite-dimensional subspace of L∞. A similar result for Calderón's problem is obtained as a corollary. This improves upon two previous results of the authors on several aspects, namely the number of measurements and the stability with respect to mismodeling errors. A new iterative reconstruction scheme based on the stability result is also presented, for which we prove exponential convergence in the number of iterations and stability with respect to noise in the data and to mismodeling errors
Gel'fand-Calderón's inverse problem for anisotropic conductivities on bordered surfaces in R 3
No full textLet X be a smooth bordered surface in R 3 with a smooth boundary and σ̂ a smooth anisotropic conductivity on X. If the genus of X is given, then starting from the Dirichlet-to-Neumann operator Λ σ̂ on ∂X, we give an explicit procedure to find a unique Riemann surface Y (up to a biholomorphism), an isotropic conductivity σ on Y and a quasiconformal diffeomorphism F:X→Y which transforms σ̂ into σ.As a corollary, we obtain the following uniqueness result: if σ 1 and σ 2 are two smooth anisotropic conductivities on X with Λ σ1= Λ σ2, then there exists a smooth diffeomorphism Φ:X̄ → X̄ such that Φ|∂X=Id and Φ*σ 1=σ 2. © The Author(s) 2011
Continuous Generative Neural Networks: A Wavelet-Based Architecture in Function Spaces
No full textIn this work, we present and study Continuous Generative Neural Networks (CGNNs), namely, generative models in the continuous setting: the output of a CGNN belongs to an infinite-dimensional function space. The architecture is inspired by DCGAN, with one fully connected layer, several convolutional layers and nonlinear activation functions. In the continuous L2 setting, the dimensions of the spaces of each layer are replaced by the scales of a multiresolution analysis of a compactly supported wavelet. We present conditions on the convolutional filters and on the nonlinearity that guarantee that a CGNN is injective. This theory finds applications to inverse problems, and allows for deriving Lipschitz stability estimates for (possibly nonlinear) infinite-dimensional inverse problems with unknowns belonging to the manifold generated by a CGNN. Several numerical simulations, including signal deblurring, illustrate and validate this approach
Assessing the Use of Diffusion Models for Motion Artifact Correction in Brain MRI
No full textMagnetic Resonance Imaging generally requires long exposure times, while being sensitive to patient motion, resulting in artifacts in the acquired images, which may hinder their diagnostic relevance. Despite research efforts to decrease the acquisition time, and designing efficient acquisition sequences, motion artifacts are still a persistent problem, pushing toward the need for the development of automatic motion artifact correction techniques. Recently, diffusion models have been proposed as a solution for the task at hand. While diffusion models can produce high-quality reconstructions, they are also susceptible to hallucination, which poses risks in diagnostic applications. In this study, we critically evaluate the use of diffusion models for correcting motion artifacts in 2D brain MRI scans. Using a popular benchmark dataset, we compare a diffusion model-based approach with state-of-the-art methods consisting of Unets trained in a supervised fashion on motion-affected images to reconstruct ground truth motion-free images. Our findings reveal mixed results: diffusion models can produce accurate predictions or generate harmful hallucinations in this context, depending on data heterogeneity and the acquisition planes considered as input
Compressed Sensing Photoacoustic Tomography Reduces to Compressed Sensing for Undersampled Fourier Measurements
No full textPhotoacoustic tomography (PAT) is an emerging imaging modality that aims at measuring the high-contrast optical properties of tissues by means of high-resolution ultrasonic measurements. The interaction between these two types of waves is based on the thermoacoustic effect. In recent years, many works have investigated the applicability of compressed sensing to PAT in order to reduce measuring times while maintaining a high reconstruction quality. However, in most cases, theoretical guarantees are missing. In this work, we show that in many measurement setups of practical interest, compressed sensing PAT reduces to compressed sensing for undersampled Fourier measurements. This is achieved by applying known reconstruction formulae in the case of the free-space model for wave propagation, and by applying the theories of Riesz bases and nonuniform Fourier series in the case of the bounded domain model. Extensive numerical simulations illustrate and validate the approach
On an inverse problem for anisotropic conductivity in the plane
No full textLet Ω̂ ⊂ R2 be a bounded domain with a smooth boundary and σ̂ a smooth anisotropic conductivity on Ω̂. Starting from the Dirichlet-to-Neumann operator A σ̂ on ∂Ω̂, we give an explicit procedure to find a unique (up to a biholomorphism) domain Ω, an isotropic conductivity σ on Ω and the boundary values of a quasiconformal diffeomorphism F : Ω̂ Ω which transforms σ̂ into σ. © 2010 IOP Publishing Ltd
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