From the second gradient operator and second class of integral theorems to Gaussian or spherical mapping invariants

By combining of the second gradient operator, the second class of integral theorems, the Gaussian-curvature-based integral theorems and the Gaussian （or spherical） mapping, a series of invariants or geometric conservation quantities under Gaussian （or spherical） mapping are revealed. From these mapping invariants important transformations between original curved surface and the spherical surface are derived. The potential applications of these invariants and transformations to geometry are discussed

Color Edge Detection Using Multidirectional Sobel Filter and Fuzzy Fusion

A new model is proposed in this paper on color edge detection that uses the second derivative operators and data fusion mechanism.The secondorder neighborhood shows the connection between the current pixel and the surroundings of this pixel.This connection is for each RGB component color of the input image.Once the image edges are detected for the three primary colors:red,green,and blue,these colors are merged using the combination rule.Then,the final decision is applied to obtain the segmentation.This process allows different data sources to be combined,which is essential to improve the image information quality and have an optimal image segmentation.Finally,the segmentation results of the proposed model are validated.Moreover,the classification accuracy of the tested data is assessed,and a comparison with other current models is conducted.The comparison results show that the proposed model outperforms the existing models in image segmentation.

On a Class of Second Order Partial Differential Operators of Non-principal Type

§1.IntroductionThis paper deals with linear partial differential operators with real principalsymbol.Let P(x,D)be such an operator of mth order with C~∞ coefficients definedin an open subset Ω of R^n and p_m(x,ξ)be its principal symbol.According to thedefinition given by Duistermaat and Hmander(see[1]),P(x,D)is called ofprincipal type at x^0 ∈Ω if for any ξ∈R^n\0 satisfying p_m(x^0,ξ)=0,x=x^0 is not theprojection in Ω of the bicharacteristic strip of P(x,D)through(x^0,ξ).Under thiscondition,they proved that there exists a neighborhood U of x^0,U,such that forany real number s,

Invariants for Parallel Mapping

This paper analyzes the geometric quantities that remain unchanged during parallel mapping （i.e., mapping from a reference curved surface to a parallel surface with identical normal direction）. The second gradient operator, the second class of integral theorems, the Gauss-curvature-based integral theorems, and the core property of parallel mapping are used to derive a series of parallel mapping invariants or geometrically conserved quantities. These include not only local mapping invariants but also global mapping invafiants found to exist both in a curved surface and along curves on the curved surface. The parallel mapping invariants are used to identify important transformations between the reference surface and parallel surfaces. These mapping invariants and transformations have potential applications in geometry, physics, biomechanics, and mechanics in which various dynamic processes occur along or between parallel surfaces.