Applications of multiresolution analysis in Besov-Q type spaces and Triebel-Lizorkin-Q type spaces

In this survey,we give a neat summary of the applications of the multi-resolution analysis to the studies of Besov-Q type spaces B_(p,q)^(γ1,γ2) (R^(n))and Triebel-Lizorkin-Q type spaces B_(p,q)^(γ1,γ2) (R^(n)).We will state briefly the recent progress on the wavelet characterizations,the boundedness of Calderon-Zygmund operators,the boundary value problem of B_(p,q)^(γ1,γ2) (R^(n)) and F_(p,q)^(γ1,γ2) (R^(n)).We also present the recent developments on the well-posedness of fluid equations with small data in B_(p,q)^(γ1,γ2) (R^(n))and F_(p,q)^(γ1,γ2) (R^(n)).

Multimodality Medical Image Fusion Based on Pixel Significance with Edge-Preserving Processing for Clinical Applications

Multimodal medical image fusion has attained immense popularity in recent years due to its robust technology for clinical diagnosis.It fuses multiple images into a single image to improve the quality of images by retaining significant information and aiding diagnostic practitioners in diagnosing and treating many diseases.However,recent image fusion techniques have encountered several challenges,including fusion artifacts,algorithm complexity,and high computing costs.To solve these problems,this study presents a novel medical image fusion strategy by combining the benefits of pixel significance with edge-preserving processing to achieve the best fusion performance.First,the method employs a cross-bilateral filter(CBF)that utilizes one image to determine the kernel and the other for filtering,and vice versa,by considering both geometric closeness and the gray-level similarities of neighboring pixels of the images without smoothing edges.The outputs of CBF are then subtracted from the original images to obtain detailed images.It further proposes to use edge-preserving processing that combines linear lowpass filtering with a non-linear technique that enables the selection of relevant regions in detailed images while maintaining structural properties.These regions are selected using morphologically processed linear filter residuals to identify the significant regions with high-amplitude edges and adequate size.The outputs of low-pass filtering are fused with meaningfully restored regions to reconstruct the original shape of the edges.In addition,weight computations are performed using these reconstructed images,and these weights are then fused with the original input images to produce a final fusion result by estimating the strength of horizontal and vertical details.Numerous standard quality evaluation metrics with complementary properties are used for comparison with existing,well-known algorithms objectively to validate the fusion results.Experimental results from the proposed research article exhibit superior performance compared to other competing techniques in the case of both qualitative and quantitative evaluation.In addition,the proposed method advocates less computational complexity and execution time while improving diagnostic computing accuracy.Nevertheless,due to the lower complexity of the fusion algorithm,the efficiency of fusion methods is high in practical applications.The results reveal that the proposed method exceeds the latest state-of-the-art methods in terms of providing detailed information,edge contour,and overall contrast.

Multiresolution Finite Element Method Based on a New Locking-Free Rectangular Mindlin Plate Element

A locking-free rectangular Mindlin plate element with a new multi-resolution analysis (MRA) is proposed and a new finite element method is hence presented. The MRA framework is formulated out of a mutually nesting displacement subspace sequence whose basis functions are constructed of scaling and shifting on the element domain of basic full node shape function. The basic full node shape function is constructed by extending the split node shape function of a traditional Mindlin plate element to other three quadrants around the coordinate zero point. As a result, a new rational MRA concept together with the resolution level (RL) is constituted for the element. The traditional 4-node rectangular Mindlin plate element and method is a mono-resolution one and also a special case of the proposed element and method. The meshing for the monoresolution plate element model is based on the empiricism while the RL adjusting for the multiresolution is laid on the rigorous mathematical basis. The analysis clarity of a plate structure is actually determined by the RL, not by the mesh. Thus, the accuracy of a plate structural analysis is replaced by the clarity, the irrational MRA by the rational and the mesh model by the RL that is the discretized model by the integrated.

A Wavelet-Free Approach for Multiresolution-Based Grid Adaptation for Conservation Laws

In recent years the concept of multiresolution-based adaptive discontinuous Galerkin(DG)schemes for hyperbolic conservation laws has been developed.The key idea is to perform a multiresolution analysis of the DG solution using multiwavelets defined on a hierarchy of nested grids.Typically this concept is applied to dyadic grid hierarchies where the explicit construction of the multiwavelets has to be performed only for one reference element.For non-uniform grid hierarchies multiwavelets have to be constructed for each element and,thus,becomes extremely expensive.To overcome this problem a multiresolution analysis is developed that avoids the explicit construction of multiwavelets.

The Biorthogonality of Multiple Vector-valued Bivariate Wavelet Packets

The notion of a sort of biorthogonal multiple vector-valued bivariate wavelet packets,which are associated with a quantity dilation matrix,is introduced.The biorthogonality property of the multiple vector-valued wavelet packets in higher dimensions is studied by means of Fourier transform and integral transform biorthogonality formulas concerning these wavelet packets are obtained.