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Reproducible kernel Hilbert space based global and local image segmentation 基于全局和局部图像分割的可复制核希尔伯特空间
IF 1.3 4区 数学 Q2 MATHEMATICS, APPLIED Pub Date : 2021-01-01 DOI: 10.3934/ipi.2020048
Liam Burrows, Weihong Guo, Ke-long Chen, F. Torella
Image segmentation is the task of partitioning an image into individual objects, and has many important applications in a wide range of fields. The majority of segmentation methods rely on image intensity gradient to define edges between objects. However, intensity gradient fails to identify edges when the contrast between two objects is low. In this paper we aim to introduce methods to make such weak edges more prominent in order to improve segmentation results of objects of low contrast. This is done for two kinds of segmentation models: global and local. We use a combination of a reproducing kernel Hilbert space and approximated Heaviside functions to decompose an image and then show how this decomposition can be applied to a segmentation model. We show some results and robustness to noise, as well as demonstrating that we can combine the reconstruction and segmentation model together, allowing us to obtain both the decomposition and segmentation simultaneously.
图像分割是将图像分割成单个对象的任务,在广泛的领域中有许多重要的应用。大多数分割方法依赖于图像强度梯度来定义物体之间的边缘。然而,当两个物体之间的对比度较低时,强度梯度无法识别边缘。在本文中,我们的目标是引入一些方法,使这种弱边缘更加突出,以提高低对比度目标的分割效果。这适用于两种分割模型:全局和局部。我们使用再现核希尔伯特空间和近似Heaviside函数的组合来分解图像,然后展示如何将这种分解应用于分割模型。我们展示了一些结果和对噪声的鲁棒性,并证明了我们可以将重建和分割模型结合在一起,使我们可以同时获得分解和分割。
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引用次数: 8
A fast explicit diffusion algorithm of fractional order anisotropic diffusion for image denoising 一种用于图像去噪的分数阶各向异性快速显式扩散算法
IF 1.3 4区 数学 Q2 MATHEMATICS, APPLIED Pub Date : 2021-01-01 DOI: 10.3934/IPI.2021018
Zhiguang Zhang, Qiang Liu, Tianling Gao
In this paper, we mainly show a novel fast fractional order anisotropic diffusion algorithm for noise removal based on the recent numerical scheme called the Fast Explicit Diffusion. To balance the efficiency and accuracy of the algorithm, the truncated matrix method is used to deal with the iterative matrix in the model and its error is also estimated. In particular, we obtain the stability condition of the iteration by the spectrum analysis method. Through implementing the fast explicit format iteration algorithm with periodic change of time step size, the efficiency of the algorithm is greatly improved. At last, we show some numerical results on denoising tasks. Many experimental results confirm that the algorithm can more quickly achieve satisfactory denoising results.
本文主要在快速显式扩散的基础上,提出了一种新的快速分数阶各向异性扩散去噪算法。为了平衡算法的效率和准确性,采用截断矩阵法处理模型中的迭代矩阵,并对其误差进行了估计。特别地,我们用谱分析法得到了迭代的稳定性条件。通过实现时间步长周期性变化的快速显式格式迭代算法,大大提高了算法的效率。最后给出了去噪任务的一些数值结果。大量实验结果表明,该算法能够更快地获得满意的去噪效果。
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引用次数: 1
IPI special issue on 'mathematical/statistical approaches in data science' in the Inverse Problem and Imaging 关于“数据科学中的数学/统计方法”的IPI特刊在逆问题和成像中
IF 1.3 4区 数学 Q2 MATHEMATICS, APPLIED Pub Date : 2021-01-01 DOI: 10.3934/ipi.2021007
Weihong Guo, Y. Lou, Jing Qin, Ming Yan
Data science is an interdisciplinary field about extracting knowledge or insights from data. It involves computational and applied mathematics, statistics, computer science, engineering, and domain sciences. In an effort to bring together researchers from different disciplines to report on cutting-edge methodologies in data science, Dr. Yifei Lou at the University of Texas at Dallas (UTD), together with Drs. Weihong Guo (Case Western Reserve University), Jing Qin (University of Kentucky), and Ming Yan (Michigan State University), organized a workshop, entitled “Recent Developments on Mathematical/Statistical Approaches in Data Science,” held at the UTD’s campus, on June 1-June 2 2019. To better disseminate the results, this special issue in the journal of Inverse Problems and Imaging (IPI) assembles peer reviewed articles from some of the invited speakers. The scope of the special issue is centered at data science, aiming to collect state-of-the-art computational algorithms and novel applications in data processing. The topics range from compressive sensing, machine learning, image processing, variational and PDE-based models, large-scale optimization, and data-driven applications.
数据科学是一个从数据中提取知识或见解的跨学科领域。它涉及计算和应用数学、统计学、计算机科学、工程学和领域科学。为了将不同学科的研究人员聚集在一起,报告数据科学的前沿方法,德克萨斯大学达拉斯分校(University of Texas at Dallas, UTD)的楼亦菲博士(Yifei Lou)和dr。郭卫红(凯斯西储大学),秦靖(肯塔基大学)和闫明(密歇根州立大学),组织了一个研讨会,题为“在数据科学数学/统计方法的最新发展,”在UTD的校园举行,于2019年6月1日至6月2日。为了更好地传播这些结果,《逆问题与成像》(IPI)杂志的这一期特刊汇集了一些受邀演讲者的同行评议文章。本期特刊的范围以数据科学为中心,旨在收集最新的计算算法和数据处理中的新应用。主题包括压缩感知、机器学习、图像处理、基于变分和pde的模型、大规模优化和数据驱动的应用。
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引用次数: 0
Quantum tomography and the quantum Radon transform 量子层析成像和量子氡变换
IF 1.3 4区 数学 Q2 MATHEMATICS, APPLIED Pub Date : 2021-01-01 DOI: 10.3934/IPI.2021021
A. Ibort, A. López-Yela

A general framework for the tomographical description of states, that includes, among other tomographical schemes, the classical Radon transform, quantum state tomography and group quantum tomography, in the setting of begin{document}$ C^* $end{document}-algebras is presented. Given a begin{document}$ C^* $end{document}-algebra, the main ingredients for a tomographical description of its states are identified: A generalized sampling theory and a positive transform. A generalization of the notion of dual tomographic pair provides the background for a sampling theory on begin{document}$ C^* $end{document}-algebras and, an extension of Bochner's theorem for functions of positive type, the positive transform.

The abstract theory is realized by using dynamical systems, that is, groups represented on begin{document}$ C^* $end{document}-algebra. Using a fiducial state and the corresponding GNS construction, explicit expressions for tomograms associated with states defined by density operators on the corresponding Hilbert spade are obtained. In particular a general quantum version of the classical definition of the Radon transform is presented. The theory is completed by proving that if the representation of the group is square integrable, the representation itself defines a dual tomographic map and explicit reconstruction formulas are obtained by making a judiciously use of the theory of frames. A few significant examples are discussed that illustrates the use and scope of the theory.

A general framework for the tomographical description of states, that includes, among other tomographical schemes, the classical Radon transform, quantum state tomography and group quantum tomography, in the setting of begin{document}$ C^* $end{document}-algebras is presented. Given a begin{document}$ C^* $end{document}-algebra, the main ingredients for a tomographical description of its states are identified: A generalized sampling theory and a positive transform. A generalization of the notion of dual tomographic pair provides the background for a sampling theory on begin{document}$ C^* $end{document}-algebras and, an extension of Bochner's theorem for functions of positive type, the positive transform.The abstract theory is realized by using dynamical systems, that is, groups represented on begin{document}$ C^* $end{document}-algebra. Using a fiducial state and the corresponding GNS construction, explicit expressions for tomograms associated with states defined by density operators on the corresponding Hilbert spade are obtained. In particular a general quantum version of the classical definition of the Radon transform is presented. The theory is completed by proving that if the representation of the group is square integrable, the representation itself defines a dual tomographic map and explicit reconstruction formulas are obtained by making a judiciously use of the theory of frames. A few significant examples are discussed that illustrates the use and scope of the theory.
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引用次数: 4
An efficient multi-grid method for TV minimization problems 一种求解电视最小化问题的高效多网格方法
IF 1.3 4区 数学 Q2 MATHEMATICS, APPLIED Pub Date : 2021-01-01 DOI: 10.3934/IPI.2021034
Zhe Zhang, Xue Li, Y. Duan, K. Yin, X. Tai
We propose an efficient multi-grid domain decomposition method for solving the total variation (TV) minimization problems. Our multi-grid scheme is developed based on the piecewise constant function spanned subspace correction rather than the piecewise linear one in [17], which ensures the calculation of the TV term only occurs on the boundaries of the support sets. Besides, the domain decomposition method is implemented on each layer to enable parallel computation. Comprehensive comparison results are presented to demonstrate the improvement in CPU time and image quality of the proposed method on medium and large-scale image denoising and reconstruction problems.
提出了一种求解总变分最小化问题的多网格域分解方法。我们的多网格方案是基于分段常数函数跨子空间修正而不是[17]中的分段线性修正而开发的,这确保了TV项的计算仅发生在支持集的边界上。并在各层上实现了域分解方法,实现了并行计算。综合对比结果表明,该方法在处理中、大规模图像去噪和重建问题时,在CPU时间和图像质量上都有显著改善。
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引用次数: 3
Phase retrieval from Fourier measurements with masks 相位检索从傅立叶测量与掩模
IF 1.3 4区 数学 Q2 MATHEMATICS, APPLIED Pub Date : 2021-01-01 DOI: 10.3934/IPI.2021028
Huiping Li, Song Li

This paper concerns the problem of phase retrieval from Fourier measurements with random masks. Here we focus on researching two kinds of random masks. Firstly, we utilize the Fourier measurements with real masks to estimate a general signal begin{document}$ mathit{boldsymbol{x}}_0in mathbb{R}^d $end{document} in noiseless case when begin{document}$ d $end{document} is even. It is demonstrated that begin{document}$ O(log^2d) $end{document} real random masks are able to ensure accurate recovery of begin{document}$ mathit{boldsymbol{x}}_0 $end{document}. Then we find that such real masks are not adaptable to reconstruct complex signals of even dimension. Subsequently, we prove that begin{document}$ O(log^4d) $end{document} complex masks are enough to stably estimate a general signal begin{document}$ mathit{boldsymbol{x}}_0in mathbb{C}^d $end{document} under bounded noise interference, which extends E. Candès et al.'s work. Meanwhile, we establish tighter error estimations for real signals of even dimensions or complex signals of odd dimensions by using begin{document}$ O(log^2d) $end{document} real masks. Finally, we intend to tackle with the noisy phase problem about an begin{document}$ s $end{document}-sparse signal by a robust and efficient approach, namely, two-stage algorithm. Based on the stable guarantees for general signals, we show that the begin{document}$ s $end{document}-sparse signal begin{document}$ mathit{boldsymbol{x}}_0 $end{document} can be stably recovered from composite measurements under near-optimal sample complexity up to a begin{document}$ log $end{document} factor, namely, begin{document}$ O(slog(frac{ed}{s})log^4(slog(frac{ed}{s}))) $end{document}

This paper concerns the problem of phase retrieval from Fourier measurements with random masks. Here we focus on researching two kinds of random masks. Firstly, we utilize the Fourier measurements with real masks to estimate a general signal begin{document}$ mathit{boldsymbol{x}}_0in mathbb{R}^d $end{document} in noiseless case when begin{document}$ d $end{document} is even. It is demonstrated that begin{document}$ O(log^2d) $end{document} real random masks are able to ensure accurate recovery of begin{document}$ mathit{boldsymbol{x}}_0 $end{document}. Then we find that such real masks are not adaptable to reconstruct complex signals of even dimension. Subsequently, we prove that begin{document}$ O(log^4d) $end{document} complex masks are enough to stably estimate a general signal begin{document}$ mathit{boldsymbol{x}}_0in mathbb{C}^d $end{document} under bounded noise interference, which extends E. Candès et al.'s work. Meanwhile, we establish tighter error estimations for real signals of even dimensions or complex signals of odd dimensions by using begin{document}$ O(log^2d) $end{document} real masks. Finally, we intend to tackle with the noisy phase problem about an begin{document}$ s $end{document}-sparse signal by a robust and efficient approach, namely, two-stage algorithm. Based on the stable guarantees for general signals, we show that the begin{document}$ s $end{document}-sparse signal begin{document}$ mathit{boldsymbol{x}}_0 $end{document} can be stably recovered from composite measurements under near-optimal sample complexity up to a begin{document}$ log $end{document} factor, namely, begin{document}$ O(slog(frac{ed}{s})log^4(slog(frac{ed}{s}))) $end{document}
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引用次数: 5
Weighted area constraints-based breast lesion segmentation in ultrasound image analysis 超声图像分析中基于加权面积约束的乳腺病变分割
IF 1.3 4区 数学 Q2 MATHEMATICS, APPLIED Pub Date : 2021-01-01 DOI: 10.3934/ipi.2021057
Qianting MA, T. Zeng, D. Kong, Jianwei Zhang
Breast ultrasound segmentation is a challenging task in practice due to speckle noise, low contrast and blurry boundaries. Although numerous methods have been developed to solve this problem, most of them can not produce a satisfying result due to uncertainty of the segmented region without specialized domain knowledge. In this paper, we propose a novel breast ultrasound image segmentation method that incorporates weighted area constraints using level set representations. Specifically, we first use speckle reducing anisotropic diffusion filter to suppress speckle noise, and apply the Grabcut on them to provide an initial segmentation result. In order to refine the resulting image mask, we propose a weighted area constraints-based level set formulation (WACLSF) to extract a more accurate tumor boundary. The major contribution of this paper is the introduction of a simple nonlinear constraint for the regularization of probability scores from a classifier, which can speed up the motion of zero level set to move to a desired boundary. Comparisons with other state-of-the-art methods, such as FCN-AlexNet and U-Net, show the advantages of our proposed WACLSF-based strategy in terms of visual view and accuracy.
由于斑点噪声、低对比度和模糊的边界,乳房超声分割在实践中是一项具有挑战性的任务。虽然目前已经有许多方法来解决这一问题,但由于分割区域的不确定性,大多数方法在没有专业领域知识的情况下无法得到满意的结果。在本文中,我们提出了一种新的乳房超声图像分割方法,该方法结合了加权面积约束,使用水平集表示。具体而言,我们首先使用散斑减小各向异性扩散滤波器来抑制散斑噪声,并对其应用Grabcut来提供初始分割结果。为了改进生成的图像掩模,我们提出了一种基于加权面积约束的水平集公式(WACLSF)来提取更准确的肿瘤边界。本文的主要贡献是为分类器的概率分数的正则化引入了一个简单的非线性约束,它可以加速零水平集移动到期望边界的运动。与其他最先进的方法(如FCN-AlexNet和U-Net)相比,我们提出的基于waclsf的策略在视觉视图和精度方面具有优势。
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引用次数: 2
Velocity modeling based on Rayleigh wave dispersion curve and sparse optimization inversion 基于瑞利波频散曲线和稀疏优化反演的速度建模
IF 1.3 4区 数学 Q2 MATHEMATICS, APPLIED Pub Date : 2021-01-01 DOI: 10.3934/IPI.2021031
Yan Cui, Yanfei Wang
This paper studies the S wave velocity modeling based on the Rayleigh wave dispersion curve inversion. We first discuss the forward simulation, and present a fast root-finding method with cubic-order of convergence speed to obtain the Rayleigh wave dispersion curve. With the Rayleigh wave dispersion curve as the observation data, and considering the prior geological anomalies structural information, we establish a sparse constraint regularization model, and propose an iterative solution method to solve for the S wave velocity. Experimental tests are performed both on the theoretical models and on the field data. It indicates from the experimental results that our new inversion scheme possesses the characteristics of easy calculation, high computational efficiency and high precision for model characterization.
本文研究了基于瑞利波频散曲线反演的横波速度模拟方法。我们首先讨论了正演模拟,并提出了一种三阶收敛速度的快速求根方法来获得瑞利波频散曲线。以瑞利波频散曲线为观测资料,考虑先验地质异常构造信息,建立了稀疏约束正则化模型,提出了求解S波速度的迭代求解方法。对理论模型和现场数据进行了实验验证。实验结果表明,该方法具有计算简便、计算效率高、模型表征精度高等特点。
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引用次数: 1
Nonconvex regularization for blurred images with Cauchy noise 柯西噪声模糊图像的非凸正则化
IF 1.3 4区 数学 Q2 MATHEMATICS, APPLIED Pub Date : 2021-01-01 DOI: 10.3934/ipi.2021065
Xiao Ai, Guoxi Ni, T. Zeng

In this paper, we propose a nonconvex regularization model for images damaged by Cauchy noise and blur. This model is based on the method of the total variational proposed by Federica, Dong and Zeng [SIAM J. Imaging Sci.(2015)], where a variational approach for restoring blurred images with Cauchy noise is used. Here we consider the nonconvex regularization, namely a weighted difference of begin{document}$ l_1 $end{document}-norm and begin{document}$ l_2 $end{document}-norm coupled with wavelet frame, the alternating direction method of multiplier is carried out for this minimization problem, we describe the details of the algorithm and prove its convergence. Numerical experiments are tested by adding different levels of noise and blur, results show that our method can denoise and deblur the image better.

In this paper, we propose a nonconvex regularization model for images damaged by Cauchy noise and blur. This model is based on the method of the total variational proposed by Federica, Dong and Zeng [SIAM J. Imaging Sci.(2015)], where a variational approach for restoring blurred images with Cauchy noise is used. Here we consider the nonconvex regularization, namely a weighted difference of begin{document}$ l_1 $end{document}-norm and begin{document}$ l_2 $end{document}-norm coupled with wavelet frame, the alternating direction method of multiplier is carried out for this minimization problem, we describe the details of the algorithm and prove its convergence. Numerical experiments are tested by adding different levels of noise and blur, results show that our method can denoise and deblur the image better.
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引用次数: 4
Large region inpainting by re-weighted regularized methods 采用重加权正则化方法绘制大面积区域
IF 1.3 4区 数学 Q2 MATHEMATICS, APPLIED Pub Date : 2021-01-01 DOI: 10.3934/IPI.2021015
Yiting Chen, Jia Li, Qingyun Yu
In the development of imaging science and image processing request in our daily life, inpainting large regions always plays an important role. However, the existing local regularized models and some patch manifold based non-local models are often not effective in restoring the features and patterns in the large missing regions. In this paper, we will apply a strategy of inpainting from outside to inside and propose a re-weighted matching algorithm by closest patch (RWCP), contributing to further enhancing the features in the missing large regions. Additionally, we propose another re-weighted matching algorithm by distance-based weighted average (RWWA), leading to a result with higher PSNR value in some cases. Numerical simulations will demonstrate that for large region inpainting, the proposed method is more applicable than most canonical methods. Moreover, combined with image denoising methods, the proposed model is also applicable for noisy image restoration with large missing regions.
在成像科学的发展和我们日常生活中对图像处理的要求中,大面积的图像绘制一直扮演着重要的角色。然而,现有的局部正则化模型和一些基于补丁流形的非局部模型往往不能有效地恢复大面积缺失区域的特征和模式。在本文中,我们将采用从外到内的补图策略,并提出一种基于最接近补丁的重加权匹配算法(RWCP),有助于进一步增强缺失大区域的特征。此外,我们提出了另一种基于距离加权平均(RWWA)的重新加权匹配算法,在某些情况下得到了更高的PSNR值。数值模拟结果表明,对于大面积的喷漆,本文提出的方法比大多数标准方法更适用。此外,结合图像去噪方法,该模型也适用于缺失区域较大的噪声图像恢复。
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引用次数: 2
期刊
Inverse Problems and Imaging
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