Kriging-Weighted Laplacian Kernels

<div>Laplacian kernels, which are widely used as sharpening filters in image processing, are isotropic and tend to over-highlight fine details with a sharp discontinuity in images. To address this issue, this paper introduces a method that integrates anisotropic averaging with the Laplacian kernels. </div><div>The proposed method can also be useful as a new type of image convolution for designing convolutional neural networks. </div>

Kriging-Weighted Laplacian Kernels

<div>Laplacian kernels, which are widely used as sharpening filters in image processing, are isotropic and tend to over-highlight fine details with a sharp discontinuity in images. To address this issue, this paper introduces a method that integrates anisotropic averaging with the Laplacian kernels. </div><div>The proposed method can also be useful as a new type of image convolution for designing convolutional neural networks. </div>

Towards a unified theory of fractional and nonlocal vector calculus

Nonlocal and fractional-order models capture effects that classical partial differential equations cannot describe; for this reason, they are suitable for a broad class of engineering and scientific applications that feature multiscale or anomalous behavior. This has driven a desire for a vector calculus that includes nonlocal and fractional gradient, divergence and Laplacian type operators, as well as tools such as Green’s identities, to model subsurface transport, turbulence, and conservation laws. In the literature, several independent definitions and theories of nonlocal and fractional vector calculus have been put forward. Some have been studied rigorously and in depth, while others have been introduced ad-hoc for specific applications. The goal of this work is to provide foundations for a unified vector calculus by (1) consolidating fractional vector calculus as a special case of nonlocal vector calculus, (2) relating unweighted and weighted Laplacian operators by introducing an equivalence kernel, and (3) proving a form of Green’s identity to unify the corresponding variational frameworks for the resulting nonlocal volume-constrained problems. The proposed framework goes beyond the analysis of nonlocal equations by supporting new model discovery, establishing theory and interpretation for a broad class of operators, and providing useful analogues of standard tools from the classical vector calculus.

Hyperspectral image spectral-spatial classification via weighted Laplacian smoothing constraint-based sparse representation

As a powerful tool in hyperspectral image (HSI) classification, sparse representation has gained much attention in recent years owing to its detailed representation of features. In particular, the results of the joint use of spatial and spectral information has been widely applied to HSI classification. However, dealing with the spatial relationship between pixels is a nontrivial task. This paper proposes a new spatial-spectral combined classification method that considers the boundaries of adjacent features in the HSI. Based on the proposed method, a smoothing-constraint Laplacian vector is constructed, which consists of the interest pixel and its four nearest neighbors through their weighting factor. Then, a novel large-block sparse dictionary is developed for simultaneous orthogonal matching pursuit. Our proposed method can obtain a better accuracy of HSI classification on three real HSI datasets than the existing spectral-spatial HSI classifiers. Finally, the experimental results are presented to verify the effectiveness and superiority of the proposed method.

Distance Laplacian spectra of joined union of graphs

The distance Laplacian matrix of a simple connected graph [Formula: see text] is defined as [Formula: see text], where [Formula: see text] is the distance matrix of [Formula: see text] and [Formula: see text] is the diagonal matrix whose main diagonal entries are the vertex transmissions in [Formula: see text] In this paper, we determine the distance Laplacian spectra of the graphs obtained by generalization of the join and lexicographic product of graphs (namely joined union). It is shown that the distance Laplacian spectra of these graphs not only depend on the distance Laplacian spectra of the participating graphs but also depend on the spectrum of another matrix of vertex-weighted Laplacian kind (analogous to the definition given by Chung and Langlands [A combinatorial Laplacian with vertex weights, J. Combin. Theory Ser. A 75 (1996) 316–327]).