61 research outputs found

    Real-time Ultrasound Signals Processing: Denoising and Super-resolution

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    Ultrasound acquisition is widespread in the biomedical field, due to its properties of low cost, portability, and non-invasiveness for the patient. The processing and analysis of US signals, such as images, 2D videos, and volumetric images, allows the physician to monitor the evolution of the patient's disease, and support diagnosis, and treatments (e.g., surgery). US images are affected by speckle noise, generated by the overlap of US waves. Furthermore, low-resolution images are acquired when a high acquisition frequency is applied to accurately characterise the behaviour of anatomical features that quickly change over time. Denoising and super-resolution of US signals are relevant to improve the visual evaluation of the physician and the performance and accuracy of processing methods, such as segmentation and classification. The main requirements for the processing and analysis of US signals are real-time execution, preservation of anatomical features, and reduction of artefacts. In this context, we present a novel framework for the real-time denoising of US 2D images based on deep learning and high-performance computing, which reduces noise while preserving anatomical features in real-time execution. We extend our framework to the denoise of arbitrary US signals, such as 2D videos and 3D images, and we apply denoising algorithms that account for spatio-temporal signal properties into an image-to-image deep learning model. As a building block of this framework, we propose a novel denoising method belonging to the class of low-rank approximations, which learns and predicts the optimal thresholds of the Singular Value Decomposition. While previous denoise work compromises the computational cost and effectiveness of the method, the proposed framework achieves the results of the best denoising algorithms in terms of noise removal, anatomical feature preservation, and geometric and texture properties conservation, in a real-time execution that respects industrial constraints. The framework reduces the artefacts (e.g., blurring) and preserves the spatio-temporal consistency among frames/slices; also, it is general to the denoising algorithm, anatomical district, and noise intensity. Then, we introduce a novel framework for the real-time reconstruction of the non-acquired scan lines through an interpolating method; a deep learning model improves the results of the interpolation to match the target image (i.e., the high-resolution image). We improve the accuracy of the prediction of the reconstructed lines through the design of the network architecture and the loss function. %The design of the deep learning architecture and the loss function allow the network to improve the accuracy of the prediction of the reconstructed lines. In the context of signal approximation, we introduce our kernel-based sampling method for the reconstruction of 2D and 3D signals defined on regular and irregular grids, with an application to US 2D and 3D images. Our method improves previous work in terms of sampling quality, approximation accuracy, and geometry reconstruction with a slightly higher computational cost. For both denoising and super-resolution, we evaluate the compliance with the real-time requirement of US applications in the medical domain and provide a quantitative evaluation of denoising and super-resolution methods on US and synthetic images. Finally, we discuss the role of denoising and super-resolution as pre-processing steps for segmentation and predictive analysis of breast pathologies

    A note on the spectral analysis of matrix sequences via GLT momentary symbols: from all-at-once solution of parabolic problems to distributed fractional order matrices

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    The first focus of this paper is the characterization of the spectrum and the singular values of the coefficient matrix stemming from the discretization of a parabolic diffusion problem using a space-time grid and secondly from the approximation of distributed-order fractional equations. For this purpose we use the classical GLT theory and the new concept of GLT momentary symbols. The first permits us to describe the singular value or eigenvalue asymptotic distribution of the sequence of the coefficient matrices. The latter permits us to derive a function that describes the singular value or eigenvalue distribution of the matrix of the sequence, even for small matrix sizes, but under given assumptions. The paper is concluded with a list of open problems, including the use of our machinery in the study of iteration matrices, especially those concerning multigrid-type techniques

    Preconditioned iterative methods for optimal control problems with time-dependent PDEs as constraints

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    In this work, we study fast and robust solvers for optimal control problems with Partial Differential Equations (PDEs) as constraints. Speci cally, we devise preconditioned iterative methods for time-dependent PDE-constrained optimization problems, usually when a higher-order discretization method in time is employed as opposed to most previous solvers. We also consider the control of stationary problems arising in uid dynamics, as well as that of unsteady Fractional Differential Equations (FDEs). The preconditioners we derive are employed within an appropriate Krylov subspace method. The fi rst key contribution of this thesis involves the study of fast and robust preconditioned iterative solution strategies for the all-at-once solution of optimal control problems with time-dependent PDEs as constraints, when a higher-order discretization method in time is employed. In fact, as opposed to most work in preconditioning this class of problems, where a ( first-order accurate) backward Euler method is used for the discretization of the time derivative, we employ a (second-order accurate) Crank-Nicolson method in time. By applying a carefully tailored invertible transformation, we symmetrize the system obtained, and then derive a preconditioner for the resulting matrix. We prove optimality of the preconditioner through bounds on the eigenvalues, and test our solver against a widely-used preconditioner for the linear system arising from a backward Euler discretization. These theoretical and numerical results demonstrate the effectiveness and robustness of our solver with respect to mesh-sizes and regularization parameter. Then, the optimal preconditioner so derived is generalized from the heat control problem to time-dependent convection{diffusion control with Crank- Nicolson discretization in time. Again, we prove optimality of the approximations of the main blocks of the preconditioner through bounds on the eigenvalues, and, through a range of numerical experiments, show the effectiveness and robustness of our approach with respect to all the parameters involved in the problem. For the next substantial contribution of this work, we focus our attention on the control of problems arising in fluid dynamics, speci fically, the Stokes and the Navier-Stokes equations. We fi rstly derive fast and effective preconditioned iterative methods for the stationary and time-dependent Stokes control problems, then generalize those methods to the case of the corresponding Navier-Stokes control problems when employing an Oseen approximation to the non-linear term. The key ingredients of the solvers are a saddle-point type approximation for the linear systems, an inner iteration for the (1,1)-block accelerated by a preconditioner for convection-diffusion control problems, and an approximation to the Schur complement based on a potent commutator argument applied to an appropriate block matrix. Through a range of numerical experiments, we show the effectiveness of our approximations, and observe their considerable parameter-robustness. The fi nal chapter of this work is devoted to the derivation of efficient and robust solvers for convex quadratic FDE-constrained optimization problems, with box constraints on the state and/or control variables. By employing an Alternating Direction Method of Multipliers for solving the non-linear problem, one can separate the equality from the inequality constraints, solving the equality constraints and then updating the current approximation of the solutions. In order to solve the equality constraints, a preconditioner based on multilevel circulant matrices is derived, and then employed within an appropriate preconditioned Krylov subspace method. Numerical results show the e ciency and scalability of the strategy, with the cost of the overall process being proportional to N log N, where N is the dimension of the problem under examination. Moreover, the strategy presented allows the storage of a highly dense system, due to the memory required being proportional to N

    A note on the spectral analysis of matrix sequences via GLT momentary symbols: from all-at-once solution of parabolic problems to distributed fractional order matrices

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    The first focus of this paper is the characterization of the spectrum and the singular values of the coefficient matrix stemming from the discretization with space-time grid for a parabolic diffusion problem and from the approximation of distributed order fractional equations. For this purpose we will use the classical GLT theory and the new concept of GLT momentary symbols. The first permits to describe the singular value or eigenvalue asymptotic distribution of the sequence of the coefficient matrices, the latter permits to derive a function, which describes the singular value or eigenvalue distribution of the matrix of the sequence, even for small matrix-sizes but under given assumptions. The note is concluded with a list of open problems, including the use of our machinery in the study of iteration matrices, especially those concerning multigrid-type techniques

    Regularization by inexact Krylov methods with applications to blind deblurring

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