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1. Difference of anisotropic and isotropic TV for segmentation under blur and Poisson noise

   https://doi.org/10.3389/fcomp.2023.1131317  

   Bui, Kevin; Lou, Yifei; Park, Fredrick; Xin, Jack (June 2023, Frontiers in Computer Science)

   In this paper, we aim to segment an image degraded by blur and Poisson noise. We adopt a smoothing-and-thresholding (SaT) segmentation framework that finds a piecewise-smooth solution, followed by k -means clustering to segment the image. Specifically for the image smoothing step, we replace the least-squares fidelity for Gaussian noise in the Mumford-Shah model with a maximum posterior (MAP) term to deal with Poisson noise and we incorporate the weighted difference of anisotropic and isotropic total variation (AITV) as a regularization to promote the sparsity of image gradients. For such a nonconvex model, we develop a specific splitting scheme and utilize a proximal operator to apply the alternating direction method of multipliers (ADMM). Convergence analysis is provided to validate the efficacy of the ADMM scheme. Numerical experiments on various segmentation scenarios (grayscale/color and multiphase) showcase that our proposed method outperforms a number of segmentation methods, including the original SaT. 

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2. An a priori error analysis of adjoint-based super-convergent Galerkin approximations of linear functionals

   https://doi.org/10.1093/imanum/draa102  

   Cockburn, Bernardo; Xia, Shiqiang (January 2021, IMA Journal of Numerical Analysis)

   Abstract We present the first a priori error analysis of a new method proposed in Cockburn & Wang (2017, Adjoint-based, superconvergent Galerkin approximations of linear functionals. J. Comput. Sci., 73, 644–666), for computing adjoint-based, super-convergent Galerkin approximations of linear functionals. If $J(u)$ is a smooth linear functional, where $$u$$ is the solution of a steady-state diffusion problem, the standard approximation $$J(u_h)$$ converges with order $$h^{2k+1}$$, where $$u_h$$ is the Hybridizable Discontinuous Galerkin approximation to $$u$$ with polynomials of degree $k>0$. In contrast, numerical experiments show that the new method provides an approximation that converges with order $$h^{4k}$$, and can be computed by only using twice the computational effort needed to compute $$J(u_h)$$. Here, we put these experimental results in firm mathematical ground. We also display numerical experiments devised to explore the convergence properties of the method in cases not covered by the theory, in particular, when the solution $$u$$ or the functional $$J(\cdot )$$ are not very smooth. We end by indicating how to extend these results to the case of general Galerkin methods. 

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3. An Integral Equation Model for PET Imaging

   Tang, Xinhuang; Schmidtlein, Charles Ross; Li, Si; Xu, Yuesheng (January 2021, International journal of numerical analysis and modeling)

   Positron emission tomography (PET) is traditionally modeled as discrete systems. Such models may be viewed as piecewise constant approximations of the underlying continuous model for the physical processes and geometry of the PET imaging. Due to the low accuracy of piecewise constant approximations, discrete models introduce an irreducible modeling error which fundamentally limits the quality of reconstructed images. To address this bottleneck, we propose an integral equation model for the PET imaging based on the physical and geometrical considerations, which describes accurately the true coincidences. We show that the proposed integral equation model is equivalent to the existing idealized model in terms of line integrals which is accurate but not suitable for numerical approximation. The proposed model allows us to discretize it using higher accuracy approximation methods. In particular, we discretize the integral equation by using the collocation principle with piecewise linear polynomials. The discretization leads to new ill-conditioned discrete systems for the PET reconstruction, which are further regularized by a novel wavelet-based regularizer. The resulting non-smooth optimization problem is then solved by a preconditioned proximity fixed-point algorithm. Convergence of the algorithm is established for a range of parameters involved in the algorithm. The proposed integral equation model combined with the discretization, regularization, and optimization algorithm provides a new PET image reconstruction method. Numerical results reveal that the proposed model substantially outperforms the conventional discrete model in terms of the consistency to simulated projection data and reconstructed image quality. This indicates that the proposed integral equation model with appropriate discretization and regularizer can significantly reduce modeling errors and suppress noise, which leads to improved image quality and projection data estimation. 

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4. Deep Variational Instance Segmentation

   Yuan, Jialin; Chen, Chao; Li, Fuxin (October 2020, Advances in neural information processing systems)

   null (Ed.)

   Instance segmentation, which seeks to obtain both class and instance labels for each pixel in the input image, is a challenging task in computer vision. State-ofthe-art algorithms often employ a search-based strategy, which first divides the output image with a regular grid and generate proposals at each grid cell, then the proposals are classified and boundaries refined. In this paper, we propose a novel algorithm that directly utilizes a fully convolutional network (FCN) to predict instance labels. Specifically, we propose a variational relaxation of instance segmentation as minimizing an optimization functional for a piecewise-constant segmentation problem, which can be used to train an FCN end-to-end. It extends the classical Mumford-Shah variational segmentation algorithm to be able to handle the permutation-invariant ground truth in instance segmentation. Experiments on PASCAL VOC 2012 and the MSCOCO 2017 dataset show that the proposed approach efficiently tackles the instance segmentation task. The source code and trained models are released at https://github.com/jia2lin3yuan1/2020-instanceSeg. 

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5. Finite element approximation of the Einstein tensor

   https://doi.org/10.1093/imanum/draf004  

   Gawlik, Evan_S; Neunteufel, Michael (March 2025, IMA Journal of Numerical Analysis)

   Abstract We construct and analyse finite element approximations of the Einstein tensor in dimension $$N \ge 3$$. We focus on the setting where a smooth Riemannian metric tensor $$g$$ on a polyhedral domain $$\varOmega \subset \mathbb{R}^{N}$$ has been approximated by a piecewise polynomial metric $$g_{h}$$ on a simplicial triangulation $$\mathcal{T}$$ of $$\varOmega $$ having maximum element diameter $$h$$. We assume that $$g_{h}$$ possesses single-valued tangential–tangential components on every codimension-$$1$$ simplex in $$\mathcal{T}$$. Such a metric is not classically differentiable in general, but it turns out that one can still attribute meaning to its Einstein curvature in a distributional sense. We study the convergence of the distributional Einstein curvature of $$g_{h}$$ to the Einstein curvature of $$g$$ under refinement of the triangulation. We show that in the $$H^{-2}(\varOmega )$$-norm this convergence takes place at a rate of $$O(h^{r+1})$$ when $$g_{h}$$ is an optimal-order interpolant of $$g$$ that is piecewise polynomial of degree $$r \ge 1$$. We provide numerical evidence to support this claim. In the process of proving our convergence results we derive a few formulas for the evolution of certain geometric quantities under deformations of the metric. 

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