In this paper, we present a method for constructing multivariate tight framelet packets associated with an arbitrary dilation matrix using unitary extension principles. We also prove how to construct various tight fra...In this paper, we present a method for constructing multivariate tight framelet packets associated with an arbitrary dilation matrix using unitary extension principles. We also prove how to construct various tight frames for L2(Rd) by replacing some mother framelets.展开更多
Generalizing wavelets by adding desired redundancy and flexibility,framelets(i.e.,wavelet frames)are of interest and importance in many applications such as image processing and numerical algorithms.Several key proper...Generalizing wavelets by adding desired redundancy and flexibility,framelets(i.e.,wavelet frames)are of interest and importance in many applications such as image processing and numerical algorithms.Several key properties of framelets are high vanishing moments for sparse multiscale representation,fast framelet transforms for numerical efficiency,and redundancy for robustness.However,it is a challenging problem to study and construct multivariate nonseparable framelets,mainly due to their intrinsic connections to factorization and syzygy modules of multivariate polynomial matrices.Moreover,all the known multivariate tight framelets derived from spline refinable scalar functions have only one vanishing moment,and framelets derived from refinable vector functions are barely studied yet in the literature.In this paper,we circumvent the above difficulties through the approach of quasi-tight framelets,which behave almost identically to tight framelets.Employing the popular oblique extension principle(OEP),from an arbitrary compactly supported M-refinable vector functionφwith multiplicity greater than one,we prove that we can always derive fromφa compactly supported multivariate quasi-tight framelet such that:(i)all the framelet generators have the highest possible order of vanishing moments;(ii)its associated fast framelet transform has the highest balancing order and is compact.For a refinable scalar functionφ(i.e.,its multiplicity is one),the above item(ii)often cannot be achieved intrinsically but we show that we can always construct a compactly supported OEP-based multivariate quasi-tight framelet derived fromφsatisfying item(i).We point out that constructing OEP-based quasi-tight framelets is closely related to the generalized spectral factorization of Hermitian trigonometric polynomial matrices.Our proof is critically built on a newly developed result on the normal form of a matrix-valued filter,which is of interest and importance in itself for greatly facilitating the study of refinable vector functions and multiwavelets/multiframelets.This paper provides a comprehensive investigation on OEP-based multivariate quasi-tight multiframelets and their associated framelet transforms with high balancing orders.This deepens our theoretical understanding of multivariate quasi-tight multiframelets and their associated fast multiframelet transforms.展开更多
The Sobolev space H^(■)(R^(d)),where■>d/2,is an important function space that has many applications in various areas of research.Attributed to the inertia of a measurement instrument,it is desirable in sampling t...The Sobolev space H^(■)(R^(d)),where■>d/2,is an important function space that has many applications in various areas of research.Attributed to the inertia of a measurement instrument,it is desirable in sampling theory to recover a function by its nonuniform sampling.In the present paper,based on dual framelet systems for the Sobolev space pair(H^(s)(R^(d)),H^(-s)(R^(d))),where d/2<s<■,we investigate the problem of constructing the approximations to all the functions in H^(■)(R^(d))by nonuniform sampling.We first establish the convergence rate of the framelet series in(H^(s)(R^(d)),H^(-s)(R^(d))),and then construct the framelet approximation operator that acts on the entire space H^(■)(R^(d)).We examine the stability property for the framelet approximation operator with respect to the perturbations of shift parameters,and obtain an estimate bound for the perturbation error.Our result shows that under the condition d/2<s<■,the approximation operator is robust to shift perturbations.Motivated by Hamm(2015)’s work on nonuniform sampling and approximation in the Sobolev space,we do not require the perturbation sequence to be in■^(α)(Z^(d)).Our results allow us to establish the approximation for every function in H^(■)(R^(d))by nonuniform sampling.In particular,the approximation error is robust to the jittering of the samples.展开更多
In this paper,two framelet based deconvolution algorithms are proposed.The basicidea of framelet based approach is to convert the deconvolution problem to the problemof inpainting in a frame domain by constructing a f...In this paper,two framelet based deconvolution algorithms are proposed.The basicidea of framelet based approach is to convert the deconvolution problem to the problemof inpainting in a frame domain by constructing a framelet system with one of the masksbeing the given (discrete) convolution kernel via the unitary extension principle of [26],as introduced in [6-9].The first algorithm unifies our previous works in high resolutionimage reconstruction and infra-red chopped and nodded image restoration,and the secondone is a combination of our previous frame-based deconvolution algorithm and the iterativethresholding algorithm given by [14,16].The strong convergence of the algorithmsin infinite dimensional settings is given by employing proximal forward-backward splitting(PFBS) method.Consequently,it unifies iterative algorithms of infinite and finitedimensional setting and simplifies the proof of the convergence of the algorithms of [6].展开更多
Homogeneous wavelets and framelets have been extensively investigated in the classical theory of wavelets and they are often constructed from refinable functions via the multiresolution analysis. On the other hand, no...Homogeneous wavelets and framelets have been extensively investigated in the classical theory of wavelets and they are often constructed from refinable functions via the multiresolution analysis. On the other hand, nonhomogeneous wavelets and framelets enjoy many desirable theoretical properties and are often intrinsically linked to the refinable structure and multiresolution analysis. In this paper, we provide a comprehensive study on connecting homogeneous wavelets and framelets to nonhomogeneous ones with the refinable structure. This allows us to understand better the structure of homogeneous wavelets and framelets as well as their connections to the refinable structure and multiresolution analysis.展开更多
Frameworks are time-tested highly reusable architectural skeleton structures. They are designed ‘abstract’ and ‘inco- mplete’ and are designed with predefined points of variability, known as hot spots, to be custo...Frameworks are time-tested highly reusable architectural skeleton structures. They are designed ‘abstract’ and ‘inco- mplete’ and are designed with predefined points of variability, known as hot spots, to be customized later at the time of framework reuse. Frameworks are reusable entities thus demand stricter and rigorous testing in comparison to one- time use application. The overall cost of framework development may be reduced by designing frameworks with high testability. This paper aims at discussing a few metric models for testability analysis of object-oriented frameworks in an attempt to having quantitative data on testability to be used to plan and monitor framework testing activities so that the framework testing effort and hence the overall framework development effort may be brought down.展开更多
文摘In this paper, we present a method for constructing multivariate tight framelet packets associated with an arbitrary dilation matrix using unitary extension principles. We also prove how to construct various tight frames for L2(Rd) by replacing some mother framelets.
基金supported by the Natural Sciences and Engineering Research Council of Canada(NSERC)(Grant No.RGPIN-2019-04276)。
文摘Generalizing wavelets by adding desired redundancy and flexibility,framelets(i.e.,wavelet frames)are of interest and importance in many applications such as image processing and numerical algorithms.Several key properties of framelets are high vanishing moments for sparse multiscale representation,fast framelet transforms for numerical efficiency,and redundancy for robustness.However,it is a challenging problem to study and construct multivariate nonseparable framelets,mainly due to their intrinsic connections to factorization and syzygy modules of multivariate polynomial matrices.Moreover,all the known multivariate tight framelets derived from spline refinable scalar functions have only one vanishing moment,and framelets derived from refinable vector functions are barely studied yet in the literature.In this paper,we circumvent the above difficulties through the approach of quasi-tight framelets,which behave almost identically to tight framelets.Employing the popular oblique extension principle(OEP),from an arbitrary compactly supported M-refinable vector functionφwith multiplicity greater than one,we prove that we can always derive fromφa compactly supported multivariate quasi-tight framelet such that:(i)all the framelet generators have the highest possible order of vanishing moments;(ii)its associated fast framelet transform has the highest balancing order and is compact.For a refinable scalar functionφ(i.e.,its multiplicity is one),the above item(ii)often cannot be achieved intrinsically but we show that we can always construct a compactly supported OEP-based multivariate quasi-tight framelet derived fromφsatisfying item(i).We point out that constructing OEP-based quasi-tight framelets is closely related to the generalized spectral factorization of Hermitian trigonometric polynomial matrices.Our proof is critically built on a newly developed result on the normal form of a matrix-valued filter,which is of interest and importance in itself for greatly facilitating the study of refinable vector functions and multiwavelets/multiframelets.This paper provides a comprehensive investigation on OEP-based multivariate quasi-tight multiframelets and their associated framelet transforms with high balancing orders.This deepens our theoretical understanding of multivariate quasi-tight multiframelets and their associated fast multiframelet transforms.
基金supported by National Natural Science Foundation of China(Grant Nos.61961003,61561006 and 11501132)Natural Science Foundation of Guangxi(Grant Nos.2018JJA110110 and 2016GXNSFAA380049)+1 种基金the talent project of Education Department of Guangxi Government for Young-Middle-Aged Backbone Teacherssupported by National Science Foundation of USA(Grant No.DMS-1712602)。
文摘The Sobolev space H^(■)(R^(d)),where■>d/2,is an important function space that has many applications in various areas of research.Attributed to the inertia of a measurement instrument,it is desirable in sampling theory to recover a function by its nonuniform sampling.In the present paper,based on dual framelet systems for the Sobolev space pair(H^(s)(R^(d)),H^(-s)(R^(d))),where d/2<s<■,we investigate the problem of constructing the approximations to all the functions in H^(■)(R^(d))by nonuniform sampling.We first establish the convergence rate of the framelet series in(H^(s)(R^(d)),H^(-s)(R^(d))),and then construct the framelet approximation operator that acts on the entire space H^(■)(R^(d)).We examine the stability property for the framelet approximation operator with respect to the perturbations of shift parameters,and obtain an estimate bound for the perturbation error.Our result shows that under the condition d/2<s<■,the approximation operator is robust to shift perturbations.Motivated by Hamm(2015)’s work on nonuniform sampling and approximation in the Sobolev space,we do not require the perturbation sequence to be in■^(α)(Z^(d)).Our results allow us to establish the approximation for every function in H^(■)(R^(d))by nonuniform sampling.In particular,the approximation error is robust to the jittering of the samples.
文摘In this paper,two framelet based deconvolution algorithms are proposed.The basicidea of framelet based approach is to convert the deconvolution problem to the problemof inpainting in a frame domain by constructing a framelet system with one of the masksbeing the given (discrete) convolution kernel via the unitary extension principle of [26],as introduced in [6-9].The first algorithm unifies our previous works in high resolutionimage reconstruction and infra-red chopped and nodded image restoration,and the secondone is a combination of our previous frame-based deconvolution algorithm and the iterativethresholding algorithm given by [14,16].The strong convergence of the algorithmsin infinite dimensional settings is given by employing proximal forward-backward splitting(PFBS) method.Consequently,it unifies iterative algorithms of infinite and finitedimensional setting and simplifies the proof of the convergence of the algorithms of [6].
基金supported by the Natural Sciences and Engineering Research Council of Canada (NSERC Canada) (Grant No. RGP 228051)
文摘Homogeneous wavelets and framelets have been extensively investigated in the classical theory of wavelets and they are often constructed from refinable functions via the multiresolution analysis. On the other hand, nonhomogeneous wavelets and framelets enjoy many desirable theoretical properties and are often intrinsically linked to the refinable structure and multiresolution analysis. In this paper, we provide a comprehensive study on connecting homogeneous wavelets and framelets to nonhomogeneous ones with the refinable structure. This allows us to understand better the structure of homogeneous wavelets and framelets as well as their connections to the refinable structure and multiresolution analysis.
文摘Frameworks are time-tested highly reusable architectural skeleton structures. They are designed ‘abstract’ and ‘inco- mplete’ and are designed with predefined points of variability, known as hot spots, to be customized later at the time of framework reuse. Frameworks are reusable entities thus demand stricter and rigorous testing in comparison to one- time use application. The overall cost of framework development may be reduced by designing frameworks with high testability. This paper aims at discussing a few metric models for testability analysis of object-oriented frameworks in an attempt to having quantitative data on testability to be used to plan and monitor framework testing activities so that the framework testing effort and hence the overall framework development effort may be brought down.
