Inverse Problems and Imaging最新文献

{"title":"Reproducible kernel Hilbert space based global and local image segmentation","authors":"Liam Burrows, Weihong Guo, Ke-long Chen, F. Torella","doi":"10.3934/ipi.2020048","DOIUrl":"https://doi.org/10.3934/ipi.2020048","url":null,"abstract":"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.","PeriodicalId":50274,"journal":{"name":"Inverse Problems and Imaging","volume":"1 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83400205","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A fast explicit diffusion algorithm of fractional order anisotropic diffusion for image denoising","authors":"Zhiguang Zhang, Qiang Liu, Tianling Gao","doi":"10.3934/IPI.2021018","DOIUrl":"https://doi.org/10.3934/IPI.2021018","url":null,"abstract":"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.","PeriodicalId":50274,"journal":{"name":"Inverse Problems and Imaging","volume":"66 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85063136","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"IPI special issue on 'mathematical/statistical approaches in data science' in the Inverse Problem and Imaging","authors":"Weihong Guo, Y. Lou, Jing Qin, Ming Yan","doi":"10.3934/ipi.2021007","DOIUrl":"https://doi.org/10.3934/ipi.2021007","url":null,"abstract":"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.","PeriodicalId":50274,"journal":{"name":"Inverse Problems and Imaging","volume":"21 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73000965","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Quantum tomography and the quantum Radon transform","authors":"A. Ibort, A. López-Yela","doi":"10.3934/IPI.2021021","DOIUrl":"https://doi.org/10.3934/IPI.2021021","url":null,"abstract":"<p style='text-indent:20px;'>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 <inline-formula><tex-math id=\"M1\">begin{document}$ C^* $end{document}</tex-math></inline-formula>-algebras is presented. Given a <inline-formula><tex-math id=\"M2\">begin{document}$ C^* $end{document}</tex-math></inline-formula>-algebra, the main ingredients for a tomographical description of its states are identified: A generalized sampling theory and a positive transform. A generalization of the notion of dual tomographic pair provides the background for a sampling theory on <inline-formula><tex-math id=\"M3\">begin{document}$ C^* $end{document}</tex-math></inline-formula>-algebras and, an extension of Bochner's theorem for functions of positive type, the positive transform.</p><p style='text-indent:20px;'>The abstract theory is realized by using dynamical systems, that is, groups represented on <inline-formula><tex-math id=\"M4\">begin{document}$ C^* $end{document}</tex-math></inline-formula>-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.</p>","PeriodicalId":50274,"journal":{"name":"Inverse Problems and Imaging","volume":"31 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81044020","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"An efficient multi-grid method for TV minimization problems","authors":"Zhe Zhang, Xue Li, Y. Duan, K. Yin, X. Tai","doi":"10.3934/IPI.2021034","DOIUrl":"https://doi.org/10.3934/IPI.2021034","url":null,"abstract":"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.","PeriodicalId":50274,"journal":{"name":"Inverse Problems and Imaging","volume":"23 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74479887","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Phase retrieval from Fourier measurements with masks","authors":"Huiping Li, Song Li","doi":"10.3934/IPI.2021028","DOIUrl":"https://doi.org/10.3934/IPI.2021028","url":null,"abstract":"<p style='text-indent:20px;'>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 <inline-formula><tex-math id=\"M1\">begin{document}$ mathit{boldsymbol{x}}_0in mathbb{R}^d $end{document}</tex-math></inline-formula> in noiseless case when <inline-formula><tex-math id=\"M2\">begin{document}$ d $end{document}</tex-math></inline-formula> is even. It is demonstrated that <inline-formula><tex-math id=\"M3\">begin{document}$ O(log^2d) $end{document}</tex-math></inline-formula> real random masks are able to ensure accurate recovery of <inline-formula><tex-math id=\"M4\">begin{document}$ mathit{boldsymbol{x}}_0 $end{document}</tex-math></inline-formula>. Then we find that such real masks are not adaptable to reconstruct complex signals of even dimension. Subsequently, we prove that <inline-formula><tex-math id=\"M5\">begin{document}$ O(log^4d) $end{document}</tex-math></inline-formula> complex masks are enough to stably estimate a general signal <inline-formula><tex-math id=\"M6\">begin{document}$ mathit{boldsymbol{x}}_0in mathbb{C}^d $end{document}</tex-math></inline-formula> under bounded noise interference, which extends E. Candès et al.'s work. Meanwhile, we establish tighter error estimations for real signals of even dimensions or complex signals of odd dimensions by using <inline-formula><tex-math id=\"M7\">begin{document}$ O(log^2d) $end{document}</tex-math></inline-formula> real masks. Finally, we intend to tackle with the noisy phase problem about an <inline-formula><tex-math id=\"M8\">begin{document}$ s $end{document}</tex-math></inline-formula>-sparse signal by a robust and efficient approach, namely, two-stage algorithm. Based on the stable guarantees for general signals, we show that the <inline-formula><tex-math id=\"M9\">begin{document}$ s $end{document}</tex-math></inline-formula>-sparse signal <inline-formula><tex-math id=\"M10\">begin{document}$ mathit{boldsymbol{x}}_0 $end{document}</tex-math></inline-formula> can be stably recovered from composite measurements under near-optimal sample complexity up to a <inline-formula><tex-math id=\"M11\">begin{document}$ log $end{document}</tex-math></inline-formula> factor, namely, <inline-formula><tex-math id=\"M12\">begin{document}$ O(slog(frac{ed}{s})log^4(slog(frac{ed}{s}))) $end{document}</tex-math></inline-formula></p>","PeriodicalId":50274,"journal":{"name":"Inverse Problems and Imaging","volume":"23 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84670105","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Weighted area constraints-based breast lesion segmentation in ultrasound image analysis","authors":"Qianting MA, T. Zeng, D. Kong, Jianwei Zhang","doi":"10.3934/ipi.2021057","DOIUrl":"https://doi.org/10.3934/ipi.2021057","url":null,"abstract":"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.","PeriodicalId":50274,"journal":{"name":"Inverse Problems and Imaging","volume":"85 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76703829","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Velocity modeling based on Rayleigh wave dispersion curve and sparse optimization inversion","authors":"Yan Cui, Yanfei Wang","doi":"10.3934/IPI.2021031","DOIUrl":"https://doi.org/10.3934/IPI.2021031","url":null,"abstract":"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.","PeriodicalId":50274,"journal":{"name":"Inverse Problems and Imaging","volume":"23 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86524200","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Nonconvex regularization for blurred images with Cauchy noise","authors":"Xiao Ai, Guoxi Ni, T. Zeng","doi":"10.3934/ipi.2021065","DOIUrl":"https://doi.org/10.3934/ipi.2021065","url":null,"abstract":"<p style='text-indent:20px;'>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 <inline-formula><tex-math id=\"M1\">begin{document}$ l_1 $end{document}</tex-math></inline-formula>-norm and <inline-formula><tex-math id=\"M2\">begin{document}$ l_2 $end{document}</tex-math></inline-formula>-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.</p>","PeriodicalId":50274,"journal":{"name":"Inverse Problems and Imaging","volume":"126 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89883354","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Large region inpainting by re-weighted regularized methods","authors":"Yiting Chen, Jia Li, Qingyun Yu","doi":"10.3934/IPI.2021015","DOIUrl":"https://doi.org/10.3934/IPI.2021015","url":null,"abstract":"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.","PeriodicalId":50274,"journal":{"name":"Inverse Problems and Imaging","volume":"1 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78176441","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}