Pub Date : 2021-01-31DOI: 10.1080/17415977.2021.1879805
Zheng Sun, Lisen Sun
Intravascular photoacoustic tomography (IVPAT) is a newly developed imaging modality for the diagnosis and intervention of coronary artery diseases. It is an ill-posed nonlinear least squares (NLS) problem to recover the absorbed optical energy density (AOED) and optical absorption coefficient (OAC) distribution in the vascular cross sections from pressure photoacoustically generated by tissues with variable speed of sound (SoS). The prior knowledge of the SoS is usually unavailable before IVPAT scanning. The ideal assumption of a constant SoS leads to degraded image quality. This paper focuses on improvement of image quality for IVPAT in tissues with variable SoS by simultaneously recovering the SoS, AOED and OAC from the measured time-dependent pressure series. The joint recovery is implemented by alternately minimizing the errors between the measured and theoretical pressure by forward simulation. The demonstration results indicate that the normalized mean square absolute distance (NMSAD) of the reconstructions produced by this method is decreased by about 15% in comparison to that of the reconstructions with a fixed SoS. Comparison results show that this method outperforms the delay compensation method in recovering the AOED and the two-step algorithm in estimating the OAC by about 20% and 25% in NMSAD respectively.
{"title":"Simultaneous reconstruction of optical absorption property and speed of sound in intravascular photoacoustic tomography","authors":"Zheng Sun, Lisen Sun","doi":"10.1080/17415977.2021.1879805","DOIUrl":"https://doi.org/10.1080/17415977.2021.1879805","url":null,"abstract":"Intravascular photoacoustic tomography (IVPAT) is a newly developed imaging modality for the diagnosis and intervention of coronary artery diseases. It is an ill-posed nonlinear least squares (NLS) problem to recover the absorbed optical energy density (AOED) and optical absorption coefficient (OAC) distribution in the vascular cross sections from pressure photoacoustically generated by tissues with variable speed of sound (SoS). The prior knowledge of the SoS is usually unavailable before IVPAT scanning. The ideal assumption of a constant SoS leads to degraded image quality. This paper focuses on improvement of image quality for IVPAT in tissues with variable SoS by simultaneously recovering the SoS, AOED and OAC from the measured time-dependent pressure series. The joint recovery is implemented by alternately minimizing the errors between the measured and theoretical pressure by forward simulation. The demonstration results indicate that the normalized mean square absolute distance (NMSAD) of the reconstructions produced by this method is decreased by about 15% in comparison to that of the reconstructions with a fixed SoS. Comparison results show that this method outperforms the delay compensation method in recovering the AOED and the two-step algorithm in estimating the OAC by about 20% and 25% in NMSAD respectively.","PeriodicalId":54926,"journal":{"name":"Inverse Problems in Science and Engineering","volume":"29 1","pages":"1764 - 1788"},"PeriodicalIF":1.3,"publicationDate":"2021-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/17415977.2021.1879805","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47200527","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-30DOI: 10.1080/17415977.2021.1999941
Yiqiu Dong, Chunlin Wu, Shi Yan
In this paper, we propose a new method to solve the minimization problem in a simultaneous reconstruction and segmentation (SRS) model for X-ray computed tomography (CT). The SRS model uses Bayes' rule and the maximum a posteriori (MAP) estimate on the hidden Markov measure field model (HMMFM). The original method [Romanov M, Dahl AB, Dong Y, Hansen PC. Simultaneous tomographic reconstruction and segmentation with class priors. Inverse Problems Sci Eng. 2016;24(8):1432–1453] includes a subproblem with logarithmic-summation (log-sum) term, which is non-separable to the classification index. This subproblem was solved by Frank–Wolfe algorithm, which is very time consuming especially when dealing with large-scale CT problems. The starting point of this paper is the commutativity of log-sum operations, where the log-sum problem could be transformed into a sum-log problem by introducing an auxiliary variable. The corresponding sum-log problem for the SRS model is separable. By applying the primal-dual algorithm, the sum-log problem turns into several easy-to-solve convex subproblems. In addition, we introduce an improved model by adding Tikhonov regularization on the SRS model, and give some convergence results for the proposed methods. Experimental results demonstrate that the proposed methods produce comparable results compared with the original SRS method with much less CPU time.
本文提出了一种新的方法来解决x射线计算机断层扫描(CT)的同步重建和分割(SRS)模型中的最小化问题。SRS模型在隐马尔可夫测量场模型(HMMFM)上使用贝叶斯规则和最大后验估计(MAP)。原方法[Romanov M, Dahl AB, Dong Y, Hansen PC.]同时层析重建和分割类先验。反问题科学与工程，2016;24(8):1432-1453]包含一个与分类指标不可分的对数求和项子问题。该子问题采用Frank-Wolfe算法求解，但该算法非常耗时，特别是在处理大规模CT问题时。本文的出发点是对数和运算的交换性，通过引入辅助变量，可以将对数和问题转化为和对数问题。SRS模型对应的和对数问题是可分的。通过应用原始对偶算法，将求和-对数问题转化为几个易于求解的凸子问题。此外，我们还在SRS模型上加入了Tikhonov正则化，给出了一个改进的模型，并给出了一些收敛结果。实验结果表明，该方法与原始的SRS方法相比具有可比性，且CPU时间大大减少。
{"title":"A fast method for simultaneous reconstruction and segmentation in X-ray CT application","authors":"Yiqiu Dong, Chunlin Wu, Shi Yan","doi":"10.1080/17415977.2021.1999941","DOIUrl":"https://doi.org/10.1080/17415977.2021.1999941","url":null,"abstract":"In this paper, we propose a new method to solve the minimization problem in a simultaneous reconstruction and segmentation (SRS) model for X-ray computed tomography (CT). The SRS model uses Bayes' rule and the maximum a posteriori (MAP) estimate on the hidden Markov measure field model (HMMFM). The original method [Romanov M, Dahl AB, Dong Y, Hansen PC. Simultaneous tomographic reconstruction and segmentation with class priors. Inverse Problems Sci Eng. 2016;24(8):1432–1453] includes a subproblem with logarithmic-summation (log-sum) term, which is non-separable to the classification index. This subproblem was solved by Frank–Wolfe algorithm, which is very time consuming especially when dealing with large-scale CT problems. The starting point of this paper is the commutativity of log-sum operations, where the log-sum problem could be transformed into a sum-log problem by introducing an auxiliary variable. The corresponding sum-log problem for the SRS model is separable. By applying the primal-dual algorithm, the sum-log problem turns into several easy-to-solve convex subproblems. In addition, we introduce an improved model by adding Tikhonov regularization on the SRS model, and give some convergence results for the proposed methods. Experimental results demonstrate that the proposed methods produce comparable results compared with the original SRS method with much less CPU time.","PeriodicalId":54926,"journal":{"name":"Inverse Problems in Science and Engineering","volume":"29 1","pages":"3342 - 3359"},"PeriodicalIF":1.3,"publicationDate":"2021-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45641653","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-17DOI: 10.1080/17415977.2021.1872563
C. C. Pacheco, C. R. de Lacerda
ABSTRACT This paper deals with the quantification of the different rates in epidemiological models from a function estimation framework, with the objective of identifying the desired unknowns without defining a priori basis functions for describing its behaviour. This approach is used to analyze data for the Covid-19 pandemic in Italy and Brazil. The forward problem is written in terms of the SIRD model, while the inverse problem is solved by combining the Levenberg–Marquardt method with Tikhonov regularization. A very good agreement was achieved between data and the calculated values. The resulting methodology is robust and very versatile, being easily applicable to other epidemiology models and data from other countries.
{"title":"Function estimation and regularization in the SIRD model applied to the COVID-19 pandemics","authors":"C. C. Pacheco, C. R. de Lacerda","doi":"10.1080/17415977.2021.1872563","DOIUrl":"https://doi.org/10.1080/17415977.2021.1872563","url":null,"abstract":"ABSTRACT This paper deals with the quantification of the different rates in epidemiological models from a function estimation framework, with the objective of identifying the desired unknowns without defining a priori basis functions for describing its behaviour. This approach is used to analyze data for the Covid-19 pandemic in Italy and Brazil. The forward problem is written in terms of the SIRD model, while the inverse problem is solved by combining the Levenberg–Marquardt method with Tikhonov regularization. A very good agreement was achieved between data and the calculated values. The resulting methodology is robust and very versatile, being easily applicable to other epidemiology models and data from other countries.","PeriodicalId":54926,"journal":{"name":"Inverse Problems in Science and Engineering","volume":"29 1","pages":"1613 - 1628"},"PeriodicalIF":1.3,"publicationDate":"2021-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/17415977.2021.1872563","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44702709","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-13DOI: 10.1080/17415977.2021.1872564
P. Broumand
Two efficient methods are presented to detect multiple cracks in 2D elastic bodies, based on the insights from Extended Finite Element. Adetection mesh is assigned to the cracked body and the responses are measured at the nodes. A finite element model with the same mesh is used to represent the uncracked state of the physical body. In the first method which is called Crack Detection based on Residual Error (CDRE), the residual error norm is calculated based on the uncracked body stiffness matrix and the cracked body responses. The contour of the error norm would show the crack pattern; the method is computationally efficient. In the second method that is coined as Crack Detection based on Stiffness Residual (CDSR), the crack locations are found based on the difference between the stiffness matrix of the cracked body and the uncracked body. The stiffness matrix of the cracked body is found by solving a dynamic inverse problem based on a modified Tikhonov regularization. The efficiency and accuracy of the identification method are enhanced by predicting the crack pattern by the CDRE method. Several examples are presented to show the accuracy and robustness of the methods in the presence of high noise levels.
{"title":"Inverse problem techniques for multiple crack detection in 2D elastic continua based on extended finite element concepts","authors":"P. Broumand","doi":"10.1080/17415977.2021.1872564","DOIUrl":"https://doi.org/10.1080/17415977.2021.1872564","url":null,"abstract":"Two efficient methods are presented to detect multiple cracks in 2D elastic bodies, based on the insights from Extended Finite Element. Adetection mesh is assigned to the cracked body and the responses are measured at the nodes. A finite element model with the same mesh is used to represent the uncracked state of the physical body. In the first method which is called Crack Detection based on Residual Error (CDRE), the residual error norm is calculated based on the uncracked body stiffness matrix and the cracked body responses. The contour of the error norm would show the crack pattern; the method is computationally efficient. In the second method that is coined as Crack Detection based on Stiffness Residual (CDSR), the crack locations are found based on the difference between the stiffness matrix of the cracked body and the uncracked body. The stiffness matrix of the cracked body is found by solving a dynamic inverse problem based on a modified Tikhonov regularization. The efficiency and accuracy of the identification method are enhanced by predicting the crack pattern by the CDRE method. Several examples are presented to show the accuracy and robustness of the methods in the presence of high noise levels.","PeriodicalId":54926,"journal":{"name":"Inverse Problems in Science and Engineering","volume":"29 1","pages":"1702 - 1728"},"PeriodicalIF":1.3,"publicationDate":"2021-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/17415977.2021.1872564","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47774884","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-13DOI: 10.1080/17415977.2020.1870971
M. Haskul, M. Kısa
This study presents the free vibration analysis of a double tapered beam having linearly varying both thickness and width, by using finite element and component mode synthesis methods. To determine the natural frequency and mode shape of the double tapered cracked beam, the stiffness and mass matrices of the beam have been obtained. The crack in the beam is modeled as a massless spring, and the beam is divided into two subcomponents from the crack section. The stiffness of spring has been derived from the linear elastic fracture mechanics theory as the inverse of the compliance matrix calculated using stress intensity factors and strain energy release rate expressions. It has been observed that natural frequencies and mode shapes vary depending on the location of the crack, the depth of the crack and the aspect ratios of the beam. The results of the present study and those in the literature are compared and a great deal of consistency has been found.
{"title":"Free vibration of the double tapered cracked beam","authors":"M. Haskul, M. Kısa","doi":"10.1080/17415977.2020.1870971","DOIUrl":"https://doi.org/10.1080/17415977.2020.1870971","url":null,"abstract":"This study presents the free vibration analysis of a double tapered beam having linearly varying both thickness and width, by using finite element and component mode synthesis methods. To determine the natural frequency and mode shape of the double tapered cracked beam, the stiffness and mass matrices of the beam have been obtained. The crack in the beam is modeled as a massless spring, and the beam is divided into two subcomponents from the crack section. The stiffness of spring has been derived from the linear elastic fracture mechanics theory as the inverse of the compliance matrix calculated using stress intensity factors and strain energy release rate expressions. It has been observed that natural frequencies and mode shapes vary depending on the location of the crack, the depth of the crack and the aspect ratios of the beam. The results of the present study and those in the literature are compared and a great deal of consistency has been found.","PeriodicalId":54926,"journal":{"name":"Inverse Problems in Science and Engineering","volume":"29 1","pages":"1537 - 1564"},"PeriodicalIF":1.3,"publicationDate":"2021-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/17415977.2020.1870971","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46688799","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-11DOI: 10.1080/17415977.2020.1870972
Hongbo Zhao, Bing-Rui Chen
Mechanical parameters of rock mass are essential in rock engineering for stability analysis, supporting design, and safety construction. The inverse analysis has been commonly used in rock engineering to determine the mechanical parameters of the rock mass. In this study, a novel inverse analysis approach was proposed through combing high dimensional model representation (HDMR), Excel solver, and numerical model. HDMR was employed to approximate the nonlinear function between the mechanical parameters of rock mass and the response of rock based on the numerical model. Excel Solver was adopted to search the mechanical parameters of rock mass based on the HDMR model for the inverse analysis. The proposed method was verified and illustrated the performance of the proposed method by two tunnels. The mechanical parameters of rock mass were determined based on the displacement of surrounding rock mass and HDMR model using the Excel solver for the tunnels. The displacement and stress of surrounding rock mass were computed based on the determined mechanical parameters of rock mass by the proposed method. There was an excellent agreement with the real value or contour that was computed based on the actual mechanical parameters of the rock mass. The results demonstrated that the proposed method was practical and accurate. It also made it convenient to be applied to determine mechanical parameters of rock mass based on monitored information.
{"title":"Inverse analysis for rock mechanics based on a high dimensional model representation","authors":"Hongbo Zhao, Bing-Rui Chen","doi":"10.1080/17415977.2020.1870972","DOIUrl":"https://doi.org/10.1080/17415977.2020.1870972","url":null,"abstract":"Mechanical parameters of rock mass are essential in rock engineering for stability analysis, supporting design, and safety construction. The inverse analysis has been commonly used in rock engineering to determine the mechanical parameters of the rock mass. In this study, a novel inverse analysis approach was proposed through combing high dimensional model representation (HDMR), Excel solver, and numerical model. HDMR was employed to approximate the nonlinear function between the mechanical parameters of rock mass and the response of rock based on the numerical model. Excel Solver was adopted to search the mechanical parameters of rock mass based on the HDMR model for the inverse analysis. The proposed method was verified and illustrated the performance of the proposed method by two tunnels. The mechanical parameters of rock mass were determined based on the displacement of surrounding rock mass and HDMR model using the Excel solver for the tunnels. The displacement and stress of surrounding rock mass were computed based on the determined mechanical parameters of rock mass by the proposed method. There was an excellent agreement with the real value or contour that was computed based on the actual mechanical parameters of the rock mass. The results demonstrated that the proposed method was practical and accurate. It also made it convenient to be applied to determine mechanical parameters of rock mass based on monitored information.","PeriodicalId":54926,"journal":{"name":"Inverse Problems in Science and Engineering","volume":"29 1","pages":"1565 - 1585"},"PeriodicalIF":1.3,"publicationDate":"2021-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/17415977.2020.1870972","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46484493","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-06DOI: 10.1080/17415977.2020.1862109
C. Fox, S. Dolgov, Malcolm Morrison, T. Molteno
Optimal continuous-discrete filtering for a nonlinear system requires evolving the forward Kolmogorov equation, that is a Fokker–Planck equation, in alternation with Bayes' conditional updating. We present two numerical grid-methods that represent density functions on a mesh, or grid. For low-dimensional, smooth systems the finite-volume method is an effective solver that gives estimates that converge to the optimal continuous-time values. We give numerical examples to show that this finite-volume filter is able to handle multi-modal filtering distributions that result from rank-deficient observations, and that Bayes-optimal parameter estimation may be performed within the filtering process. The naïve discretization of density functions used in the finite-volume filter leads to an exponential increase of computational cost and storage with increasing dimension, that makes the finite-volume filter unfeasible for higher-dimensional problems. We circumvent this ‘curse of dimensionality’ by using a tensor train representation (or approximation) of density functions and employ an efficient implicit PDE solver that operates on the tensor train representation. We present numerical examples of tracking n weakly coupled pendulums in continuous time to demonstrate filtering with complex density functions in up to 80 dimensions.
{"title":"Grid methods for Bayes-optimal continuous-discrete filtering and utilizing a functional tensor train representation","authors":"C. Fox, S. Dolgov, Malcolm Morrison, T. Molteno","doi":"10.1080/17415977.2020.1862109","DOIUrl":"https://doi.org/10.1080/17415977.2020.1862109","url":null,"abstract":"Optimal continuous-discrete filtering for a nonlinear system requires evolving the forward Kolmogorov equation, that is a Fokker–Planck equation, in alternation with Bayes' conditional updating. We present two numerical grid-methods that represent density functions on a mesh, or grid. For low-dimensional, smooth systems the finite-volume method is an effective solver that gives estimates that converge to the optimal continuous-time values. We give numerical examples to show that this finite-volume filter is able to handle multi-modal filtering distributions that result from rank-deficient observations, and that Bayes-optimal parameter estimation may be performed within the filtering process. The naïve discretization of density functions used in the finite-volume filter leads to an exponential increase of computational cost and storage with increasing dimension, that makes the finite-volume filter unfeasible for higher-dimensional problems. We circumvent this ‘curse of dimensionality’ by using a tensor train representation (or approximation) of density functions and employ an efficient implicit PDE solver that operates on the tensor train representation. We present numerical examples of tracking n weakly coupled pendulums in continuous time to demonstrate filtering with complex density functions in up to 80 dimensions.","PeriodicalId":54926,"journal":{"name":"Inverse Problems in Science and Engineering","volume":"29 1","pages":"1199 - 1217"},"PeriodicalIF":1.3,"publicationDate":"2021-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/17415977.2020.1862109","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47341426","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-02DOI: 10.1080/17415977.2021.2000604
A. Alekseev, A. Bondarev
In this paper, we consider the inverse problem for the estimation of a point-wise approximation error occurring at the discretization of the system of partial differential equations. We analyse the set of the solutions, obtained by the numerical algorithms of the dissimilar structures on the same grid. The differences between the numerical solutions are used as the input data for the inverse problem, which is posed in the variational statement with the zero-order Tikhonov regularization. The numerical tests, performed for the two-dimensional inviscid compressible flows corresponding to Edney-I and Edney-VI shock wave interference modes, are provided. The comparison of the estimated error and the exact error, obtained by subtraction of numerical and analytic solutions, is presented.
{"title":"The estimation of approximation error using inverse problem and a set of numerical solutions","authors":"A. Alekseev, A. Bondarev","doi":"10.1080/17415977.2021.2000604","DOIUrl":"https://doi.org/10.1080/17415977.2021.2000604","url":null,"abstract":"In this paper, we consider the inverse problem for the estimation of a point-wise approximation error occurring at the discretization of the system of partial differential equations. We analyse the set of the solutions, obtained by the numerical algorithms of the dissimilar structures on the same grid. The differences between the numerical solutions are used as the input data for the inverse problem, which is posed in the variational statement with the zero-order Tikhonov regularization. The numerical tests, performed for the two-dimensional inviscid compressible flows corresponding to Edney-I and Edney-VI shock wave interference modes, are provided. The comparison of the estimated error and the exact error, obtained by subtraction of numerical and analytic solutions, is presented.","PeriodicalId":54926,"journal":{"name":"Inverse Problems in Science and Engineering","volume":"29 1","pages":"3360 - 3376"},"PeriodicalIF":1.3,"publicationDate":"2021-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42178212","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-12-30DOI: 10.1080/17415977.2020.1867547
L. Afraites, A. Hadri, A. Laghrib, M. Nachaoui
In the present work, we investigate the inverse problem of identifying simultaneously the denoised image and the weighting parameter that controls the balance between two diffusion operators for an evolutionary partial differential equation (PDE). The problem is formulated as a non-smooth PDE-constrained optimization model. This PDE is constructed by second- and fourth-order diffusive tensors that combines the benefits from the diffusion model of Perona–Malik in the homogeneous regions, the Weickert model near sharp edges and the fourth-order term in reducing staircasing. The existence and uniqueness of solutions for the proposed PDE-constrained optimization system are provided in a suitable Sobolev space. Also, an optimization problem for the determination of the weighting parameter is introduced based on the Primal–Dual algorithm. Finally, simulation results show that the obtained parameter usually coincides with the better choice related to the best restoration quality of the image.
{"title":"A high order PDE-constrained optimization for the image denoising problem","authors":"L. Afraites, A. Hadri, A. Laghrib, M. Nachaoui","doi":"10.1080/17415977.2020.1867547","DOIUrl":"https://doi.org/10.1080/17415977.2020.1867547","url":null,"abstract":"In the present work, we investigate the inverse problem of identifying simultaneously the denoised image and the weighting parameter that controls the balance between two diffusion operators for an evolutionary partial differential equation (PDE). The problem is formulated as a non-smooth PDE-constrained optimization model. This PDE is constructed by second- and fourth-order diffusive tensors that combines the benefits from the diffusion model of Perona–Malik in the homogeneous regions, the Weickert model near sharp edges and the fourth-order term in reducing staircasing. The existence and uniqueness of solutions for the proposed PDE-constrained optimization system are provided in a suitable Sobolev space. Also, an optimization problem for the determination of the weighting parameter is introduced based on the Primal–Dual algorithm. Finally, simulation results show that the obtained parameter usually coincides with the better choice related to the best restoration quality of the image.","PeriodicalId":54926,"journal":{"name":"Inverse Problems in Science and Engineering","volume":"29 1","pages":"1821 - 1863"},"PeriodicalIF":1.3,"publicationDate":"2020-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/17415977.2020.1867547","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42627101","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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