Computational Quantification of Map Projection Distortion by Fractal Dimension of Coastlines

Maps, essential tools for portraying the Earth’s surface, inherently introduce distortions to geographical features. While various quantification methods exist for assessing these distortions, they often fall short when evaluating actual geographic features. In our study, we took a novel approach by analyzing map projection distortion from a geometric perspective. We computed the fractal dimensions of different stretches of coastline before and after projection using the divide-and-conquer algorithm and image processing. Our findings revealed that map projections, even when preserving basic shapes, inevitably stretch and compress coastlines in diverse directions. This analysis method provides a more realistic and practical way to measure map-induced distortions, with significant implications for cartography, geographic information systems (GIS), and geomorphology. By bridging the gap between theoretical analysis and real-world features, this method greatly enhances accuracy and practicality when evaluating map projections.

Binary Image Steganalysis Based on Distortion Level Co-Occurrence Matrix

In recent years,binary image steganography has developed so rapidly that the research of binary image steganalysis becomes more important for information security.In most state-of-the-art binary image steganographic schemes,they always find out the flippable pixels to minimize the embedding distortions.For this reason,the stego images generated by the previous schemes maintain visual quality and it is hard for steganalyzer to capture the embedding trace in spacial domain.However,the distortion maps can be calculated for cover and stego images and the difference between them is significant.In this paper,a novel binary image steganalytic scheme is proposed,which is based on distortion level co-occurrence matrix.The proposed scheme first generates the corresponding distortion maps for cover and stego images.Then the co-occurrence matrix is constructed on the distortion level maps to represent the features of cover and stego images.Finally,support vector machine,based on the gaussian kernel,is used to classify the features.Compared with the prior steganalytic methods,experimental results demonstrate that the proposed scheme can effectively detect stego images.

A unified approach to the weighted Grtzsch and Nitsche problems for mappings of finite distortion

This note deals with the existence and uniqueness of a minimiser of the following Grtzsch-type problem inf f ∈F∫∫_(Q_1)φ(K(z,f))λ(x)dxdyunder some mild conditions,where F denotes the set of all homeomorphims f with finite linear distortion K(z,f)between two rectangles Q_1 and Q_2 taking vertices into vertices,φ is a positive,increasing and convex function,and λ is a positive weight function.A similar problem of Nitsche-type,which concerns the minimiser of some weighted functional for mappings between two annuli,is also discussed.As by-products,our discussion gives a unified approach to some known results in the literature concerning the weighted Grtzsch and Nitsche problems.

Distortion Theorem for Bloch Mappings on the Unit Ball ■~n

In this paper, we obtain a version of subordination lemma for hyperbolic disk relative to hyperbolic geometry on the unit disk D. This subordination lemma yields the distortion theorem for Bloch mappings f ∈ H（B^n） satisfying ｜｜f｜｜0 = 1 and det f＇（0） = α ∈ （0, 1], where｜｜f｜｜0 = sup{（1 - ｜z｜^2 ）n＋1/2n det（f＇（z））[1/n ： z ∈ B^n}. Here we establish the distortion theorem from a unified perspective and generalize some known results. This distortion theorem enables us to obtain a lower bound for the radius of the largest univalent ball in the image of f centered at f（0）. When a = 1, the lower bound reduces to that of Bloch constant found by Liu. When n = 1, our distortion theorem coincides with that of Bonk, Minda and Yanagihara.

Regularity of the Inverse of a Homeomorphism with Finite Inner Distortion

Let f ： Ω→ f（Ω） belong to R^n be a W^1,1-homeomorphism with L^1-inegrable inner We show that finiteness of min{lipf（x）, kf（x））, for every x∈ Ω／E, implies that f^-1 ∈ W^1,n and has finite distortion, provided that the exceptional set E has σ-finite H^1-measure.Moreover, f has finite distortion, differentiable a.e. and the Jacobian Jf 〉 0 a.e.