Imaging & inverse problems (IMAGINE) OneWorld seminars

Title: I can see clearer now: the optimal regularization parameter

Abstract: Selecting the regularization parameter in the image restoration variational framework is of crucial importance, since it can highly influence the quality of the final restoration. In this talk, after a short review of traditional methods, we discuss a fully automatic approach for selecting the regularization parameter when the blur is space-invariant and known and the noise is additive white Gaussian with unknown standard deviation. The proposed method, which is based on the so-called residual whiteness principle, can be applied to a wide class of variational models, such as those including in their formulation regularizers of Tikhonov and Total Variation type. For non-quadratic regularizers, the residual whiteness principle is nested in an iterative optimization scheme based on the alternating direction method of multipliers. The strategy can also be successfully used for the more challenging image super-resolution inverse problem.

This is joint work with L. Calatroni (CNRS), M. Pragliola and A. Lanza (University of Bologna).

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Title: Shape optimization approach for sharp-interface reconstructions in inverse problems.

Abstract: Working within the class of piecewise constant or piecewise smooth models, inverse problems can be recast as shape optimization problems where the discontinuity interface is the unknown. The sensitivity analysis of the cost functional relies on shape optimization techniques and in particular on the concept of shape derivatives. I will show several recent developments including a shape-Lagrangian approach for point measurements, and distributed shape derivatives for geometries with low regularity. Numerical results using methods based on the distributed shape derivative will be presented for the inverse problems of electrical impedance tomography and full waveform inversion.

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Title: Wave enhancement through spectral optimization

Abstract: We present an efficient approach for optimizing the transmission signal between two points in a cavity at a given frequency, by changing boundary conditions from Dirichlet to Neumann. The optimization makes use of recent results on the monotonicity of the eigenvalues of the mixed boundary value problem and on the sensitivity of Green's function to small changes in the boundary conditions. Highly accurate calculations of the mixed Dirichlet-Neumann eigenvalues are performed via a layer potential approach.

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Title: Convergence and Non-Convergence of Algebraic Iterative Reconstruction Methods

Abstract: Algebraic iterative reconstruction methods – such as ART (Kaczmarz), SART, and SIRT – are suited to handle limited-angle and limited-data X-ray CT problems. These methods utilize forward and back projection in each iteration. We represent the forward projection by a matrix A while, in principle, we represent the back projection by B = A^T (the transpose of A).

When we implement the iterative methods with a focus on computational efficiency, we use different discretization schemes for the forward and back projections. Hence, there is a mismatch between the back projection matrix B and the transposed A^T of the forward projection matrix. The use of such an unmatched pair has two consequences: the accuracy (compared to when using a matched pair) may deteriorate, and the iteration may fail to converge.

In this survey talk, I illustrate these issues with recent theoretical and computational results, and I present a two novel approaches to "fixing" the non-convergence – either by modifying the iterative method or by using the AB- and BA-GMRES methods.

This is joint work with Yiqiu Dong, Tommy Elfving, Ken Hayami, Michiel Hochstenbach, Keiichi Morikuni, and Nicolai Riis.

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Title: Orthogonality sampling methods for solving electromagnetic inverse scattering problems

Abstract: Broadly speaking, inverse scattering problems are the problems of determining information about an object (scatterer) from measurements of the field scattered from that object. Solving these inverse problems is challenging since they are in general highly nonlinear and severely ill-posed problems. In this talk, we will discuss our recent results on orthogonality sampling methods for solving electromagnetic inverse scattering problems. Compared with classical sampling methods (e.g. linear sampling method, factorization method) the orthogonality sampling methods are simpler to implement, do not require regularizations, and are more robust with respect to noise in the data. Numerical studies for 2D and 3D experimental data will also be presented.

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Title: Methods for $\ell_p$-$\ell_q$ minimization with applications to image restoration and regression with nonconvex loss and penalty.

Abstract: Minimization problems whose objective function consists of the sum of a fidelity term and a regularization term, that are determined by a $p$-norm and a $q$-norm, respectively, with $0<p,q\leq 2$, find applications in image restoration as well as in regression. When $p$ or $q$ are strictly smaller than unity, the minimization problem is not convex. The choice of the parameters $p$ and $q$ is informed by properties of the error in the data and by the sparsity of the desired solution. A regularization parameter balances the influence of the fidelity and regularization terms. Methods for the solution of the minimization problem and for determinaing the regularization parameter are discussed. Several applications are presented.

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Title: Infinite-dimensional inverse problems with finite measurements

Abstract: In this talk, I will discuss uniqueness, stability and reconstruction for infinite-dimensional nonlinear inverse problems with finite measurements, under the a priori assumption that the unknown lies in, or is well-approximated by, a finite-dimensional subspace or submanifold. The methods are based on the interplay of applied harmonic analysis, in particular sampling theory and compressed sensing, machine learning and the theory of inverse problems for partial differential equations. Several examples, including the Calderón problem and scattering, will be discussed.

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