This paper presents a method based on a kernel dictionary learning algorithm for segmenting brain tumor regions in magnetic resonance images (MRI). A set of first-order and second-order statistical feature vectors are extracted from patches of size 3 * 3 around pixels in the brain MRI scans. These feature vectors are utilized to train two kernel dictionaries separately for healthy and tumorous tissues. To enhance the efficiency of the dictionaries and reduce training time, a correlation-based sample selection technique is developed to identify the most informative and discriminative subset of feature vectors. This technique aims to improve the performance of the dictionaries by selecting a subset of feature vectors that provide valuable information for the segmentation task. Subsequently, a linear classifier is utilized to distinguish between healthy and unhealthy pixels based on the learned dictionaries. The results demonstrate that the proposed method outperforms other existing methods in terms of segmentation accuracy and significantly reduces both the time and memory required, resulting in a remarkably fast training process.
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We present a finite element approach for diffusion problems with thermal fluctuations based on a fluctuating hydrodynamics model. The governing transport equations are stochastic partial differential equations with a fluctuating forcing term. We propose a discrete formulation of the stochastic forcing term that has the correct covariance matrix up to a standard discretization error. Furthermore, to obtain a numerical solution with spatial correlations that converge to those of the continuum equation, we derive a linear mapping to transform the finite element solution into an equivalent discrete solution that is free from the artificial correlations introduced by the spatial discretization. The method is validated by applying it to two diffusion problems: a second-order diffusion equation and a fourth-order diffusion equation. The theoretical (continuum) solution to the first case presents spatially decorrelated fluctuations, while the second case presents fluctuations correlated over a finite length. In both cases, the numerical solution presents a structure factor that approximates well the continuum one.
In this paper, two novel classes of implicit exponential Runge-Kutta (ERK) methods are studied for solving highly oscillatory systems. First of all, we analyze the symplectic conditions of two kinds of exponential integrators, and present a first-order symplectic method. In order to solve highly oscillatory problems, the highly accurate implicit ERK integrators (up to order four) are formulated by comparing the Taylor expansions of numerical and exact solutions, it is shown that the order conditions of two new kinds of exponential methods are identical to the order conditions of classical Runge-Kutta (RK) methods. Moreover, we investigate the linear stability properties of these exponential methods. Finally, numerical results not only present the long time energy preservation of the first-order symplectic method, but also illustrate the accuracy and efficiency of these formulated methods in comparison with standard ERK methods.
We present HoVer-UNet, an approach to distill the knowledge of the multi-branch HoVerNet framework for nuclei instance segmentation and classification in histopathology. We propose a compact, streamlined single UNet network with a Mix Vision Transformer backbone, and equip it with a custom loss function to optimally encode the distilled knowledge of HoVerNet, reducing computational requirements without compromising performances. We show that our model achieved results comparable to HoVerNet on the public PanNuke and Consep datasets with a three-fold reduction in inference time. We make the code of our model publicly available at //github.com/DIAGNijmegen/HoVer-UNet.
We present an efficient matrix-free geometric multigrid method for the elastic Helmholtz equation, and a suitable discretization. Many discretization methods had been considered in the literature for the Helmholtz equations, as well as many solvers and preconditioners, some of which are adapted for the elastic version of the equation. However, there is very little work considering the reciprocity of discretization and a solver. In this work, we aim to bridge this gap. By choosing an appropriate stencil for re-discretization of the equation on the coarse grid, we develop a multigrid method that can be easily implemented as matrix-free, relying on stencils rather than sparse matrices. This is crucial for efficient implementation on modern hardware. Using two-grid local Fourier analysis, we validate the compatibility of our discretization with our solver, and tune a choice of weights for the stencil for which the convergence rate of the multigrid cycle is optimal. It results in a scalable multigrid preconditioner that can tackle large real-world 3D scenarios.
We propose a method for computing the Lyapunov exponents of renewal equations (delay equations of Volterra type) and of coupled systems of renewal and delay differential equations. The method consists in the reformulation of the delay equation as an abstract differential equation, the reduction of the latter to a system of ordinary differential equations via pseudospectral collocation, and the application of the standard discrete QR method. The effectiveness of the method is shown experimentally and a MATLAB implementation is provided.
The recovery of magnetic resonance (MR) images from undersampled measurements is a key problem that has seen extensive research in recent years. Unrolled approaches, which rely on end-to-end training of convolutional neural network (CNN) blocks within iterative reconstruction algorithms, offer state-of-the-art performance. These algorithms require a large amount of memory during training, making them difficult to employ in high-dimensional applications. Deep equilibrium (DEQ) models and the recent monotone operator learning (MOL) approach were introduced to eliminate the need for unrolling, thus reducing the memory demand during training. Both approaches require a Lipschitz constraint on the network to ensure that the forward and backpropagation iterations converge. Unfortunately, the constraint often results in reduced performance compared to unrolled methods. The main focus of this work is to relax the constraint on the CNN block in two different ways. Inspired by convex-non-convex regularization strategies, we now impose the monotone constraint on the sum of the gradient of the data term and the CNN block, rather than constrain the CNN itself to be a monotone operator. This approach enables the CNN to learn possibly non-monotone score functions, which can translate to improved performance. In addition, we only restrict the operator to be monotone in a local neighborhood around the image manifold. Our theoretical results show that the proposed algorithm is guaranteed to converge to the fixed point and that the solution is robust to input perturbations, provided that it is initialized close to the true solution. Our empirical results show that the relaxed constraints translate to improved performance and that the approach enjoys robustness to input perturbations similar to MOL.
We develop a novel discontinuous Galerkin method for solving the rotating thermal shallow water equations (TRSW) on a curvilinear mesh. Our method is provably entropy stable, conserves mass, buoyancy and vorticity, while also semi-discretely conserving energy. This is achieved by using novel numerical fluxes and splitting the pressure and convection operators. We implement our method on a cubed sphere mesh and numerically verify our theoretical results. Our experiments demonstrate the robustness of the method for a regime of well developed turbulence, where it can be run stably without any dissipation. The entropy stable fluxes are sufficient to control the grid scale noise generated by geostrophic turbulence, eliminating the need for artificial stabilization.
Parameter identification problems in partial differential equations (PDEs) consist in determining one or more unknown functional parameters in a PDE. Here, the Bayesian nonparametric approach to such problems is considered. Focusing on the representative example of inferring the diffusivity function in an elliptic PDE from noisy observations of the PDE solution, the performance of Bayesian procedures based on Gaussian process priors is investigated. Recent asymptotic theoretical guarantees establishing posterior consistency and convergence rates are reviewed and expanded upon. An implementation of the associated posterior-based inference is provided, and illustrated via a numerical simulation study where two different discretisation strategies are devised. The reproducible code is available at: //github.com/MattGiord.
In inverse scattering problems, a model that allows for the simultaneous recovery of both the domain shape and an impedance boundary condition covers a wide range of problems with impenetrable domains, including recovering the shape of sound-hard and sound-soft obstacles and obstacles with thin coatings. This work develops an optimization framework for recovering the shape and material parameters of a penetrable, dissipative obstacle in the multifrequency setting, using a constrained class of curvature-dependent impedance function models proposed by Antoine, Barucq, and Vernhet. We find that this constrained model improves the robustness of the recovery problem, compared to more general models, and provides meaningfully better obstacle recovery than simpler models. We explore the effectiveness of the model for varying levels of dissipation, for noise-corrupted data, and for limited aperture data in the numerical examples.
Deep learning is usually described as an experiment-driven field under continuous criticizes of lacking theoretical foundations. This problem has been partially fixed by a large volume of literature which has so far not been well organized. This paper reviews and organizes the recent advances in deep learning theory. The literature is categorized in six groups: (1) complexity and capacity-based approaches for analyzing the generalizability of deep learning; (2) stochastic differential equations and their dynamic systems for modelling stochastic gradient descent and its variants, which characterize the optimization and generalization of deep learning, partially inspired by Bayesian inference; (3) the geometrical structures of the loss landscape that drives the trajectories of the dynamic systems; (4) the roles of over-parameterization of deep neural networks from both positive and negative perspectives; (5) theoretical foundations of several special structures in network architectures; and (6) the increasingly intensive concerns in ethics and security and their relationships with generalizability.
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