The classical TV (Total Variation) model has been applied to gray texture image denoising and inpainting previously based on the non local operators, but such model can not be directly used to color texture image inpa...The classical TV (Total Variation) model has been applied to gray texture image denoising and inpainting previously based on the non local operators, but such model can not be directly used to color texture image inpainting due to coupling of different image layers in color images. In order to solve the inpainting problem for color texture images effectively, we propose a non local CTV (Color Total Variation) model. Technically, the proposed model is an extension of local TV model for gray images but we take account of the coupling of different layers in color images and make use of concepts of the non-local operators. As the coupling of different layers for color images in the proposed model will in-crease computational complexity, we also design a fast Split Bregman algorithm. Finally, some numerical experiments are conducted to validate the performance of the proposed model and its algorithm.展开更多
Deep neural network is a powerful tool for many tasks.Understanding why it is so successful and providing a mathematical explanation is an important problem and has been one popular research direction in past years.In...Deep neural network is a powerful tool for many tasks.Understanding why it is so successful and providing a mathematical explanation is an important problem and has been one popular research direction in past years.In the literature of mathematical analysis of deep neural networks,a lot of works is dedicated to establishing representation theories.How to make connections between deep neural networks and mathematical algorithms is still under development.In this paper,we give an algorithmic explanation for deep neural networks,especially in their connections with operator splitting.We show that with certain splitting strategies,operator-splitting methods have the same structure as networks.Utilizing this connection and the Potts model for image segmentation,two networks inspired by operator-splitting methods are proposed.The two networks are essentially two operator-splitting algorithms solving the Potts model.Numerical experiments are presented to demonstrate the effectiveness of the proposed networks.展开更多
In this work, we try to use the so-called Piecewise Constant Level Set Method （PCLSM） for the Mumford-Shah segmentation model. For image segmentation, the Mumford-Shah model needs to find the regions and the constan...In this work, we try to use the so-called Piecewise Constant Level Set Method （PCLSM） for the Mumford-Shah segmentation model. For image segmentation, the Mumford-Shah model needs to find the regions and the constant values inside the regions for the segmen- tation. In order to use PCLSM for this purpose, we need to solve a minimization problem using the level set function and the constant values as minimization variables. In this work, we test on a model such that we only need to minimize with respect to the level set function, i.e., we do not need to minimize with respect to the constant values. Gradient descent method and Newton method are used to solve the Euler-Lagrange equation for the minimization problem. Numerical experiments are given to show the efficiency and advantages of the new model and algorithms.展开更多
In this paper,we present a surface reconstruction via 2D strokes and a vector field on the strokes based on a two-step method.In the first step,from sparse strokes drawn by artists and a given vector field on the stro...In this paper,we present a surface reconstruction via 2D strokes and a vector field on the strokes based on a two-step method.In the first step,from sparse strokes drawn by artists and a given vector field on the strokes,we propose a nonlinear vector interpolation combining total variation(TV)and H1 regularization with a curl-free constraint for obtaining a dense vector field.In the second step,a height map is obtained by integrating the dense vector field in the first step.Jump discontinuities in surface and discontinuities of surface gradients can be well reconstructed without any surface distortion.We also provide a fast and efficient algorithm for solving the proposed functionals.Since vectors on the strokes are interpreted as a projection of surface gradients onto the plane,different types of strokes are easily devised to generate geometrically crucial structures such as ridge,valley,jump,bump,and dip on the surface.The stroke types help users to create a surface which they intuitively imagine from 2D strokes.We compare our results with conventional methods via many examples.展开更多
In this paper,we propose a generalized penalization technique and a convex constraint minimization approach for the p-harmonic flow problem following the ideas in[Kang&March,IEEE T.Image Process.,16(2007),2251–22...In this paper,we propose a generalized penalization technique and a convex constraint minimization approach for the p-harmonic flow problem following the ideas in[Kang&March,IEEE T.Image Process.,16(2007),2251–2261].We use fast algorithms to solve the subproblems,such as the dual projection methods,primal-dual methods and augmented Lagrangian methods.With a special penalization term,some special algorithms are presented.Numerical experiments are given to demonstrate the performance of the proposed methods.We successfully show that our algorithms are effective and efficient due to two reasons:the solver for subproblem is fast in essence and there is no need to solve the subproblem accurately(even 2 inner iterations of the subproblem are enough).It is also observed that better PSNR values are produced using the new algorithms.展开更多
Image fusion is an imaging technique to visualize information from multiple imaging sources in one single image,which is widely used in remote sensing,medical imaging etc.In this work,we study two variational approach...Image fusion is an imaging technique to visualize information from multiple imaging sources in one single image,which is widely used in remote sensing,medical imaging etc.In this work,we study two variational approaches to image fusion which are closely related to the standard TV-L_(2) and TV-L_(1) image approximation methods.We investigate their convex optimization formulations,under the perspective of primal and dual,and propose their associated new image decomposition models.In addition,we consider the TV-L_(1) based image fusion approach and study the specified problem of fusing two discrete-constrained images f_(1)(x)∈L_(1) and f_(2)(x)∈L_(2),where L_(1) and L_(2) are the sets of linearly-ordered discrete values.We prove that the TV-L_(1) based image fusion actually gives rise to the exact convex relaxation to the corresponding nonconvex image fusion constrained by the discretevalued set u(x)∈L_(1)∪L_(2).This extends the results for the global optimization of the discrete-constrained TV-L_(1) image approximation[8,36]to the case of image fusion.As a big numerical advantage of the two proposed dual models,we show both of them directly lead to new fast and reliable algorithms,based on modern convex optimization techniques.Experiments with medical images,remote sensing images and multi-focus images visibly show the qualitative differences between the two studied variational models of image fusion.We also apply the new variational approaches to fusing 3D medical images.展开更多
In this work,we study gradient-based regularization methods for neural networks.We mainly focus on two regularization methods:the total variation and the Tikhonov regularization.Adding the regularization term to the t...In this work,we study gradient-based regularization methods for neural networks.We mainly focus on two regularization methods:the total variation and the Tikhonov regularization.Adding the regularization term to the training loss is equivalent to using neural networks to solve some variational problems,mostly in high dimensions in practical applications.We introduce a general framework to analyze the error between neural network solutions and true solutions to variational problems.The error consists of three parts:the approximation errors of neural networks,the quadrature errors of numerical integration,and the optimization error.We also apply the proposed framework to two-layer networks to derive a priori error estimate when the true solution belongs to the so-called Barron space.Moreover,we conduct some numerical experiments to show that neural networks can solve corresponding variational problems sufficiently well.The networks with gradient-based regularization are much more robust in image applications.展开更多
In this work,we present a new method for convex shape representation,which is regardless of the dimension of the concerned objects,using level-set approaches.To the best of our knowledge,the proposed prior is the firs...In this work,we present a new method for convex shape representation,which is regardless of the dimension of the concerned objects,using level-set approaches.To the best of our knowledge,the proposed prior is the first one which can work for high dimensional objects.Convexity prior is very useful for object completion in computer vision.It is a very challenging task to represent high dimensional convex objects.In this paper,we first prove that the convexity of the considered object is equivalent to the convexity of the associated signed distance function.Then,the second order condition of convex functions is used to characterize the shape convexity equivalently.We apply this new method to two applications:object segmentation with convexity prior and convex hull problem(especially with outliers).For both applications,the involved problems can be written as a general optimization problem with three constraints.An algorithm based on the alternating direction method of multipliers is presented for the optimization problem.Numerical experiments are conducted to verify the effectiveness of the proposed representation method and algorithm.展开更多
In this paper,we propose an algorithm based on augmented Lagrangian method and give a performance comparison for two segmentation models that use the L^(1)-and L^(2)-Euler’s elastica energy respectively as the regula...In this paper,we propose an algorithm based on augmented Lagrangian method and give a performance comparison for two segmentation models that use the L^(1)-and L^(2)-Euler’s elastica energy respectively as the regularization for image seg-mentation.To capture contour curvature more reliably,we develop novel augmented Lagrangian functionals that ensure the segmentation level set function to be signed dis-tance functions,which avoids the reinitialization of segmentation function during the iterative process.With the proposed algorithm and with the same initial contours,we compare the performance of these two high-order segmentation models and numerically verify the different properties of the two models.展开更多
This paper presents an approach to model the solvent-excluded surface(SES)of 3D protein molecular structures using the geometric PDE-based level-set method.The level-set method embeds the shape of 3D molecular objects...This paper presents an approach to model the solvent-excluded surface(SES)of 3D protein molecular structures using the geometric PDE-based level-set method.The level-set method embeds the shape of 3D molecular objects as an isosurface or level set corresponding to some isovalue of a scattered dense scalar field,which is saved as a discretely-sampled,rectilinear grid,i.e.,a volumetric grid.Our level-set model is described as a class of tri-cubic tensor product B-spline implicit surface with control point values that are the signed distance function.The geometric PDE is evolved in the discrete volume.The geometric PDE we use is the mean curvature specified flow,which coincides with the definition of the SES and is geometrically intrinsic.The technique of speeding up is achieved by use of the narrow band strategy incorporated with a good initial approximate construction for the SES.We get a very desirable approximate surface for the SES.展开更多
文摘The classical TV (Total Variation) model has been applied to gray texture image denoising and inpainting previously based on the non local operators, but such model can not be directly used to color texture image inpainting due to coupling of different image layers in color images. In order to solve the inpainting problem for color texture images effectively, we propose a non local CTV (Color Total Variation) model. Technically, the proposed model is an extension of local TV model for gray images but we take account of the coupling of different layers in color images and make use of concepts of the non-local operators. As the coupling of different layers for color images in the proposed model will in-crease computational complexity, we also design a fast Split Bregman algorithm. Finally, some numerical experiments are conducted to validate the performance of the proposed model and its algorithm.
基金supported by HKBU 179356,NSFC 12201530 and HKRGC ECS 22302123supported by NSFC/RGC grant N-HKBU214-19 and NORCE Kompetanseoppbygging programsupported by HKRGC GRF grants CityU1101120,CityU11309922,CRF grant C1013-21GF,and HKRGC-NSFC Grant N CityU214/19.
文摘Deep neural network is a powerful tool for many tasks.Understanding why it is so successful and providing a mathematical explanation is an important problem and has been one popular research direction in past years.In the literature of mathematical analysis of deep neural networks,a lot of works is dedicated to establishing representation theories.How to make connections between deep neural networks and mathematical algorithms is still under development.In this paper,we give an algorithmic explanation for deep neural networks,especially in their connections with operator splitting.We show that with certain splitting strategies,operator-splitting methods have the same structure as networks.Utilizing this connection and the Potts model for image segmentation,two networks inspired by operator-splitting methods are proposed.The two networks are essentially two operator-splitting algorithms solving the Potts model.Numerical experiments are presented to demonstrate the effectiveness of the proposed networks.
文摘In this work, we try to use the so-called Piecewise Constant Level Set Method （PCLSM） for the Mumford-Shah segmentation model. For image segmentation, the Mumford-Shah model needs to find the regions and the constant values inside the regions for the segmen- tation. In order to use PCLSM for this purpose, we need to solve a minimization problem using the level set function and the constant values as minimization variables. In this work, we test on a model such that we only need to minimize with respect to the level set function, i.e., we do not need to minimize with respect to the constant values. Gradient descent method and Newton method are used to solve the Euler-Lagrange equation for the minimization problem. Numerical experiments are given to show the efficiency and advantages of the new model and algorithms.
基金The research is supported by MOE(Ministry of Education)Tier II project T207N2202and National Research Foundation grant,which is administered by the Media Development Authority Interactive Digital Media Programme Office,MDA(IDMPO).
文摘In this paper,we present a surface reconstruction via 2D strokes and a vector field on the strokes based on a two-step method.In the first step,from sparse strokes drawn by artists and a given vector field on the strokes,we propose a nonlinear vector interpolation combining total variation(TV)and H1 regularization with a curl-free constraint for obtaining a dense vector field.In the second step,a height map is obtained by integrating the dense vector field in the first step.Jump discontinuities in surface and discontinuities of surface gradients can be well reconstructed without any surface distortion.We also provide a fast and efficient algorithm for solving the proposed functionals.Since vectors on the strokes are interpreted as a projection of surface gradients onto the plane,different types of strokes are easily devised to generate geometrically crucial structures such as ridge,valley,jump,bump,and dip on the surface.The stroke types help users to create a surface which they intuitively imagine from 2D strokes.We compare our results with conventional methods via many examples.
基金The authors’research was supported by MOE IDM project NRF2007IDM-IDM002-010,SingaporeThe first author was partially supported by PHD Program Scholarship Fund of ECNU with Grant No.2010026Overseas Research Fund of East China Normal University,China.Discussions with Dr.Zhifeng Pang,Dr.Haixia Liang and Dr.Yuping Duan are helpful.
文摘In this paper,we propose a generalized penalization technique and a convex constraint minimization approach for the p-harmonic flow problem following the ideas in[Kang&March,IEEE T.Image Process.,16(2007),2251–2261].We use fast algorithms to solve the subproblems,such as the dual projection methods,primal-dual methods and augmented Lagrangian methods.With a special penalization term,some special algorithms are presented.Numerical experiments are given to demonstrate the performance of the proposed methods.We successfully show that our algorithms are effective and efficient due to two reasons:the solver for subproblem is fast in essence and there is no need to solve the subproblem accurately(even 2 inner iterations of the subproblem are enough).It is also observed that better PSNR values are produced using the new algorithms.
基金J.Yuan and A.Fenster gratefully acknowledge funding from the Canadian Institutes of Health Research,and the Ontario Institute of Cancer ResearchB.Miles gratefully acknowledges funding from the Graduate Program in BioMedical Engineering at the University of Western Ontario and the Computer Assisted Medical Intervention Training Program,which is funded by the Natural Sciences and Engineer-ing Research Council of Canada.A.Fenster holds a Canada Research Chair in Biomedi-cal Engineering,and acknowledges the support of the Canada Research Chair Program.
文摘Image fusion is an imaging technique to visualize information from multiple imaging sources in one single image,which is widely used in remote sensing,medical imaging etc.In this work,we study two variational approaches to image fusion which are closely related to the standard TV-L_(2) and TV-L_(1) image approximation methods.We investigate their convex optimization formulations,under the perspective of primal and dual,and propose their associated new image decomposition models.In addition,we consider the TV-L_(1) based image fusion approach and study the specified problem of fusing two discrete-constrained images f_(1)(x)∈L_(1) and f_(2)(x)∈L_(2),where L_(1) and L_(2) are the sets of linearly-ordered discrete values.We prove that the TV-L_(1) based image fusion actually gives rise to the exact convex relaxation to the corresponding nonconvex image fusion constrained by the discretevalued set u(x)∈L_(1)∪L_(2).This extends the results for the global optimization of the discrete-constrained TV-L_(1) image approximation[8,36]to the case of image fusion.As a big numerical advantage of the two proposed dual models,we show both of them directly lead to new fast and reliable algorithms,based on modern convex optimization techniques.Experiments with medical images,remote sensing images and multi-focus images visibly show the qualitative differences between the two studied variational models of image fusion.We also apply the new variational approaches to fusing 3D medical images.
基金partially supported by the National Science Foundation of China and Hong Kong RGC Joint Research Scheme(NSFC/RGC 11961160718)the fund of the Guangdong Provincial Key Laboratory of Computational Science and Material Design(No.2019B030301001)+1 种基金supported by the National Science Foundation of China(NSFC-11871264)the Shenzhen Natural Science Fund(RCJC20210609103819018).
文摘In this work,we study gradient-based regularization methods for neural networks.We mainly focus on two regularization methods:the total variation and the Tikhonov regularization.Adding the regularization term to the training loss is equivalent to using neural networks to solve some variational problems,mostly in high dimensions in practical applications.We introduce a general framework to analyze the error between neural network solutions and true solutions to variational problems.The error consists of three parts:the approximation errors of neural networks,the quadrature errors of numerical integration,and the optimization error.We also apply the proposed framework to two-layer networks to derive a priori error estimate when the true solution belongs to the so-called Barron space.Moreover,we conduct some numerical experiments to show that neural networks can solve corresponding variational problems sufficiently well.The networks with gradient-based regularization are much more robust in image applications.
基金supported by RG(R)-RC/17-18/02-MATHHKBU 12300819+2 种基金NSF/RGC grant N-HKBU214-19RC-FNRA-IG/19-20/SCI/01supported by Programs for Science and Technology Development of Henan Province(192102310181)。
文摘In this work,we present a new method for convex shape representation,which is regardless of the dimension of the concerned objects,using level-set approaches.To the best of our knowledge,the proposed prior is the first one which can work for high dimensional objects.Convexity prior is very useful for object completion in computer vision.It is a very challenging task to represent high dimensional convex objects.In this paper,we first prove that the convexity of the considered object is equivalent to the convexity of the associated signed distance function.Then,the second order condition of convex functions is used to characterize the shape convexity equivalently.We apply this new method to two applications:object segmentation with convexity prior and convex hull problem(especially with outliers).For both applications,the involved problems can be written as a general optimization problem with three constraints.An algorithm based on the alternating direction method of multipliers is presented for the optimization problem.Numerical experiments are conducted to verify the effectiveness of the proposed representation method and algorithm.
基金X.C.Tai was supported by the startup grant at Hong Kong Baptist University,grant RG(R)-RC/17-18/02-MATH and FRG2/17-18/033.
文摘In this paper,we propose an algorithm based on augmented Lagrangian method and give a performance comparison for two segmentation models that use the L^(1)-and L^(2)-Euler’s elastica energy respectively as the regularization for image seg-mentation.To capture contour curvature more reliably,we develop novel augmented Lagrangian functionals that ensure the segmentation level set function to be signed dis-tance functions,which avoids the reinitialization of segmentation function during the iterative process.With the proposed algorithm and with the same initial contours,we compare the performance of these two high-order segmentation models and numerically verify the different properties of the two models.
基金We acknowledge the support from:NSFC grant 10701071Key Laboratory of Computational&Stochastic Mathematics and Their Applications at Universities of Hunan Province,MOE(Ministry of Education)Tier II project T207N2202IDM project NRF2007IDM-IDM002-010 and NTU project SUG 20/07.
文摘This paper presents an approach to model the solvent-excluded surface(SES)of 3D protein molecular structures using the geometric PDE-based level-set method.The level-set method embeds the shape of 3D molecular objects as an isosurface or level set corresponding to some isovalue of a scattered dense scalar field,which is saved as a discretely-sampled,rectilinear grid,i.e.,a volumetric grid.Our level-set model is described as a class of tri-cubic tensor product B-spline implicit surface with control point values that are the signed distance function.The geometric PDE is evolved in the discrete volume.The geometric PDE we use is the mean curvature specified flow,which coincides with the definition of the SES and is geometrically intrinsic.The technique of speeding up is achieved by use of the narrow band strategy incorporated with a good initial approximate construction for the SES.We get a very desirable approximate surface for the SES.
