The approach of Li and Zhou(2014)is adopted to find the Laplace transform of occupation time over interval(0,a)and joint occupation times over semi-infinite intervals(-∞,a)and(b,∞)for a time-homogeneous diffusion pr...The approach of Li and Zhou(2014)is adopted to find the Laplace transform of occupation time over interval(0,a)and joint occupation times over semi-infinite intervals(-∞,a)and(b,∞)for a time-homogeneous diffusion process up to an independent exponential time e_(q)for 0<a<b.The results are expressed in terms of solutions to the differential equations associated with the diffusion generator.Applying these results,we obtain explicit expressions on the Laplace transform of occupation time and joint occupation time for Brownian motion with drift.展开更多
近年来,Transformer在众多监督式计算机视觉任务中取得了显著进展,然而由于高质量医学标注图像的缺乏,其在半监督图像分割领域的性能仍有待提高。为此,提出了一种基于多尺度和多视图Transformer的半监督医学图像分割框架:MSMVT(multi-sc...近年来,Transformer在众多监督式计算机视觉任务中取得了显著进展,然而由于高质量医学标注图像的缺乏,其在半监督图像分割领域的性能仍有待提高。为此,提出了一种基于多尺度和多视图Transformer的半监督医学图像分割框架:MSMVT(multi-scale and multi-view transformer)。鉴于对比学习在Transformer的预训练中取得的良好效果,设计了一个基于伪标签引导的多尺度原型对比学习模块。该模块利用图像金字塔数据增强技术,为无标签图像生成富有语义信息的多尺度原型表示;通过对比学习,强化了不同尺度原型之间的一致性,从而有效缓解了由标签稀缺性导致的Transformer训练不足的问题。此外,为了增强Transformer模型训练的稳定性,提出了多视图一致性学习策略。通过弱扰动视图,以校正多个强扰动视图。通过最小化不同视图之间的输出差异性,使得模型能够对不同扰动保持多层次的一致性。实验结果表明,当仅采用10%的标注比例时,提出的MSMVT框架在ACDC、LIDC和ISIC三个公共数据集上的DSC图像分割性能指标分别达到了88.93%、84.75%和85.38%,优于现有的半监督医学图像分割方法。展开更多
Hessian matrices are square matrices consisting of all possible combinations of second partial derivatives of a scalar-valued initial function. As such, Hessian matrices may be treated as elementary matrix systems of ...Hessian matrices are square matrices consisting of all possible combinations of second partial derivatives of a scalar-valued initial function. As such, Hessian matrices may be treated as elementary matrix systems of linear second-order partial differential equations. This paper discusses the Hessian and its applications in optimization, and then proceeds to introduce and derive the notion of the Jaffa Transform, a new linear operator that directly maps a Hessian square matrix space to the initial corresponding scalar field in nth dimensional Euclidean space. The Jaffa Transform is examined, including the properties of the operator, the transform of notable matrices, and the existence of an inverse Jaffa Transform, which is, by definition, the Hessian matrix operator. The Laplace equation is then noted and investigated, particularly, the relation of the Laplace equation to Poisson’s equation, and the theoretical applications and correlations of harmonic functions to Hessian matrices. The paper concludes by introducing and explicating the Jaffa Theorem, a principle that declares the existence of harmonic Jaffa Transforms, which are, essentially, Jaffa Transform solutions to the Laplace partial differential equation.展开更多
基金Supported by the National Natural Science Foundation of China(12271062,11731012)by the Hunan Provincial National Natural Science Foundation of China(2019JJ50405)。
文摘The approach of Li and Zhou(2014)is adopted to find the Laplace transform of occupation time over interval(0,a)and joint occupation times over semi-infinite intervals(-∞,a)and(b,∞)for a time-homogeneous diffusion process up to an independent exponential time e_(q)for 0<a<b.The results are expressed in terms of solutions to the differential equations associated with the diffusion generator.Applying these results,we obtain explicit expressions on the Laplace transform of occupation time and joint occupation time for Brownian motion with drift.
文摘近年来,Transformer在众多监督式计算机视觉任务中取得了显著进展,然而由于高质量医学标注图像的缺乏,其在半监督图像分割领域的性能仍有待提高。为此,提出了一种基于多尺度和多视图Transformer的半监督医学图像分割框架:MSMVT(multi-scale and multi-view transformer)。鉴于对比学习在Transformer的预训练中取得的良好效果,设计了一个基于伪标签引导的多尺度原型对比学习模块。该模块利用图像金字塔数据增强技术,为无标签图像生成富有语义信息的多尺度原型表示;通过对比学习,强化了不同尺度原型之间的一致性,从而有效缓解了由标签稀缺性导致的Transformer训练不足的问题。此外,为了增强Transformer模型训练的稳定性,提出了多视图一致性学习策略。通过弱扰动视图,以校正多个强扰动视图。通过最小化不同视图之间的输出差异性,使得模型能够对不同扰动保持多层次的一致性。实验结果表明,当仅采用10%的标注比例时,提出的MSMVT框架在ACDC、LIDC和ISIC三个公共数据集上的DSC图像分割性能指标分别达到了88.93%、84.75%和85.38%,优于现有的半监督医学图像分割方法。
文摘Hessian matrices are square matrices consisting of all possible combinations of second partial derivatives of a scalar-valued initial function. As such, Hessian matrices may be treated as elementary matrix systems of linear second-order partial differential equations. This paper discusses the Hessian and its applications in optimization, and then proceeds to introduce and derive the notion of the Jaffa Transform, a new linear operator that directly maps a Hessian square matrix space to the initial corresponding scalar field in nth dimensional Euclidean space. The Jaffa Transform is examined, including the properties of the operator, the transform of notable matrices, and the existence of an inverse Jaffa Transform, which is, by definition, the Hessian matrix operator. The Laplace equation is then noted and investigated, particularly, the relation of the Laplace equation to Poisson’s equation, and the theoretical applications and correlations of harmonic functions to Hessian matrices. The paper concludes by introducing and explicating the Jaffa Theorem, a principle that declares the existence of harmonic Jaffa Transforms, which are, essentially, Jaffa Transform solutions to the Laplace partial differential equation.
