The Prescribed $$L_p$$ Curvature Problem in Gaussian Probability Space

In this paper, we introduce a new approach in texture segmentation utilizing 2D wavelet transform. The HL and LH subbands coefficients as the features are mapped into the probability space by 2D Generalized Gaussian probability density function (GG-PDF) to achieve preliminary segmentation. The features in PDF are classified into homogenous regions via multilevel thresholding after wavelet de-noising. The edges can be extracted from the segmented images. To verify the accuracy of GG-PDF, 2D bootstrap algorithm is used. In addition, we test our algorithm in noisy environment to check its reliability. Finally the performance of the proposed algorithm is demonstrated on variety of Bordatz textures and some textual images.

Capacities on a numerical model — Tigtness of capacities — Quasicontinuous functions — Tchebycheff inequalities — Redefinitions — Positive generalized functionals, their representation by a Borel measure — Equilibrium potentials — Continuity of capacities — Capacitability of Borel stes — Equilibrium measures — Measures of finite energy — Charge and capacity — Slim sets are null sets for all measures of finite energy — Invariance of capacities under a change of numerical model, precise Gaussian probability space — Quasi-sure analysis on an abstract Wiener space.

We provide an overview of projection bodies in Gaussian probability space for sets of finite Gaussian perimeter and their corresponding applications in functions of Bounded variation space. On the one hand, we study the properties of Gaussian projection bodies for sets of finite Gaussian perimeter under Ehrhard symmetrization and establish a Gaussian projection-type inequality. This inequality concludes that Ehrhard symmetrization contracts the Minkowski symmetrization of the Gaussian projection body for set of finite Gaussian perimeter $E$. On the other hand, we investigate the functional “lifting” of Ehrhard symmetrization and establish the affine Gaussian Pólya–Szegö-type inequalities in terms of the functional Ehrhard symmetrization.

In this paper, it is proved that the weak convergence of the Lp Gaussian surface area measures implies the convergence of the corresponding convex bodies in the Hausdorff metric for p ≥ 1. Moreover, continuity of the solution to the Lp Gaussian Minkowski problem with respect to p is obtained.