Area Distortion Research Articles

Subdivision connectivity remeshing via Teichmüller extremal map

Curvilinear surfaces in 3D Euclidean spaces are commonly represented by triangular meshes. The structure of the triangulation is important, since it affects the accuracy and efficiency of the numerical computation on the mesh. Remeshing refers to the process of transforming an unstructured mesh to one with desirable structures, such as the subdivision connectivity. This is commonly achieved by parameterizing the surface onto a simple parameter domain, on which a structured mesh is built. The 2D structured mesh is then projected onto the surface via the parameterization. Two major tasks are involved. Firstly, an effective algorithm for parameterizing, usually conformally, surface meshes is necessary. However, for a highly irregular mesh with skinny triangles, computing a folding-free conformal parameterization is difficult. The second task is to build a structured mesh on the parameter domain that is adaptive to the area distortion of the parameterization while maintaining good shapes of triangles. This paper presents an algorithm to remesh a highly irregular mesh to a structured one with subdivision connectivity and good triangle quality. We propose an effective algorithm to obtain a conformal parameterization of a highly irregular mesh, using quasi-conformal Teichmuller theories. Conformality distortion of an initial parameterization is adjusted by a quasi-conformal map, resulting in a folding-free conformal parameterization. Next, we propose an algorithm to obtain a regular mesh with subdivision connectivity and good triangle quality on the conformal parameter domain, which is adaptive to the area distortion, through the landmark-matching Teichmuller map. A remeshed surface can then be obtained through the parameterization. Experiments have been carried out to remesh surface meshes representing real 3D geometric objects using the proposed algorithm. Results show the efficacy of the algorithm to optimize the regularity of an irregular triangulation.

Optimized quasiconformal parameterization with user-defined area distortions

Optimized quasiconformal parameterization with user-defined area distortions

Iterative Closest Conformal Maps between Planar Domains

AbstractConformal maps between planar domains are an important tool in geometry processing, used for shape deformation and image warping. The Riemann mapping theorem guarantees that there exists a conformal map between any two simply connected planar domains, yet computing this map efficiently remains challenging. In practice, one of the main algorithmic questions is the correspondence between the boundaries of the domains. On the one hand, there exist a number of conformal maps between any two domains, thus many potential boundary correspondences, yet on the other, given full boundary prescription a conformal map might not exist. Furthermore, an approximate boundary fitting can be enough for many applications. We therefore propose an alternating minimization algorithm for finding a boundary‐approximating conformal map given only an initial global alignment of the two input domains. We utilize the Cauchy‐Green complex barycentric coordinates to parameterize the space of conformal maps from the source domain, and thus compute a continuous map without requiring the discretization of the domain, and without mapping to intermediate domains. This yields a very efficient method which allows to interactively modify additional user‐provided constraints, such as point‐to‐point and stroke‐to‐stroke correspondences. Furthermore, we show how to easily generalize this setup to quasi‐conformal maps, thus enriching the space of mappings and reducing the area distortion. We compare our algorithm to state‐of‐the‐art methods for mapping between planar domains, and demonstrate that we achieve less distorted maps on the same inputs. Finally, we show applications of our approach to stroke based deformation and constrained texture mapping.

Lightning Mapping Array flash detection performance with variable receiver thresholds.

This study characterizes Lightning Mapping Array performance for networks that participated in the Deep Convective Clouds and Chemistry field program using new Monte Carlo and curvature matrix model simulations. These open‐source simulation tools are readily adapted to real‐time operations or detailed studies of performance. Each simulation accounted for receiver threshold and location, as well as a reference distribution of source powers and flash sizes based on thunderstorm observations and the mechanics of station triggering. Source and flash detection efficiency were combined with solution bias and variability to predict flash area distortion at long ranges. Location errors and detection efficiency were highly dependent on the station configuration and thresholds, especially at longer ranges, such that performance varied more than expected across different networks and with azimuth within networks. Error characteristics matched prior studies, which led to an increase in flash distortion with range. Predicted flash detection efficiency exceeded 95% within 100 km of all networks.

Area-preserving mesh parameterization for poly-annulus surfaces based on optimal mass transportation

This work proposes a novel method for computing area-preserving parameterization for genus zero surfaces with multiple boundaries (poly-annuli), which is based on discrete optimal mass transportation and surface Ricci Flow. We first begin with a conformal mapping (which may greatly distort area) by Ricci Flow and then correct the area distortion using the mass transport procedure via a convex optimization. The method is intrinsic and stable, and the resulting parameterization preserves area element and minimizes angle distortion. Comparing with existing algorithms, our method is more general and flexible. It can handle surfaces with more complicated topology, and gives users full control of the target measure, such as the areas of the holes. We have tested the method for applications in various fields. Our experimental results demonstrate the efficiency and efficacy of the proposed method.

Optimizing the balance between area and orientation distortions for variable-scale maps

Applying a variable scale transformation to maps leads to area distortions of the maps, i.e. enlarging interesting areas on maps to larger scales and shrinking other areas. Such area distortions improve the clarity of the interesting areas. However, it may result in the over-distortion of line orientations so as to reduce map recognition. This article developed an optimization method to correct orientation distortions for variable-scale maps generated by any existing variable scale transformations, while preserving the map clarity as much as possible. The proposed method is tested with the variable-scale resultant maps of two real-life datasets and evaluated by statistical analysis and perceptual tests. The experimental results indicate that the proposed method is able to effectively reduce the orientation distortions so as to improve map recognition, while map clarity has been sufficiently achieved.

Spherical Parameterization Balancing Angle and Area Distortions.

This work presents a novel framework for spherical mesh parameterization. An efficient angle-preserving spherical parameterization algorithm is introduced, which is based on dynamic Yamabe flow and the conformal welding method with solid theoretic foundation. An area-preserving spherical parameterization is also discussed, which is based on discrete optimal mass transport theory. Furthermore, a spherical parameterization algorithm, which is based on the polar decomposition method, balancing angle distortion and area distortion is presented. The algorithms are tested on 3D geometric data and the experiments demonstrate the efficiency and efficacy of the proposed methods.

Explicit Bounds and Sharp Results for the Composition Operators Preserving the Exponential Class

Letf:Ω⊂Rn→Rnbe aquasiconformal mappingwhose Jacobian is denoted byJfand letEXP(Ω)be the space of exponentially integrable functions onΩ. We give an explicit bound for the norm of the composition operatorTf:u∈EXP(Ω)↦u∘f-1∈EXP(f(Ω))and, as a related question, we study the behaviour of the norm oflog⁡Jfin the exponential class. TheA∞property ofJfis the counterpart in higher dimensions of the area distortion formula due to Astala in the plane and it is the key tool to prove the sharpness of our results.

Global three‐dimensional‐mesh indexing based on structural analysis and geometrical signatures

This study presents a new local feature matching approach that relies upon Reeb graph (RG)-based representation as well as a simple and accurate similarity estimation. The central contribution of this work is to reinforce the topological consistency conditions of the graph-based description. Formally, the RGs are enriched with geometry signatures based on parameterisation approaches. After RG construction, the shape is segmented into Reeb charts of controlled topology mapped to its canonical planar domain. Then, two stretching signatures, corresponding to the area and angle distortion, are determined and taken as three-dimensional-shape descriptor. The similarity estimation is performed in two steps. The first one consists in forming the pairs of similar Reeb charts, according to the minimal distance between their corresponding signatures. The second step is to measure the global similarity which quantifies the similitude degree between all the matched Reeb charts. Retrieval experiments conducted on four publicly available databases have shown that the proposed matching scheme yields satisfactory results. Among observations, it can be noticed that despite its rapidity, the method provides an overall retrieval efficiency gain compared to very recent state-of-the-art methods.

Dark solitonic interaction and conservation laws for a higher-order [formula omitted]-dimensional nonlinear Schrödinger-type equation in a Heisenberg ferromagnetic spin chain with bilinear and biquadratic interaction

In this paper, a higher-order (2+1)-dimensional nonlinear Schrödinger-type equation is investigated, which describes the nonlinear spin dynamics of the (2+1)-dimensional Heisenberg ferromagnetic spin chain with bilinear and biquadratic interaction. Lax pair and infinite-number conservation laws are constructed, which could prove the existence of the multi-soliton solutions. Via the auxiliary function, bilinear forms and dark-soliton solutions are derived. Properties and interaction patterns for the dark solitons are investigated: (i) Effects on the dark solitons arising from the bilinear interaction, biquadratic interaction and lattice parameter are discussed. (ii) Through the asymptotic analysis, elastic and inelastic interaction between the two solitons is discussed analytically and graphically, respectively. Due to the elastic interaction, amplitudes and velocities of the two dark solitons remain unchanged with the distortion of the interaction area, except for certain phase shifts. However, in virtue of the inelastic interaction, amplitudes of the dark solitons reduce to zero, without the distortion. (iii) Elastic interaction among the three dark solitons is investigated, which implies that the properties of the elastic interaction among the three are similar to that between the two, except for the more complicated distortion. Inelastic–elastic interaction is also investigated, which implies that the interaction between the inelastic region and the dark soliton is elastic. (iv) Linear stability analysis is proposed, which is used to analyze the properties of modulation instability and proves that the dark solitons are modulationally stable.

Optimal mass transport for shape matching and comparison.

Surface based 3D shape analysis plays a fundamental role in computer vision and medical imaging. This work proposes to use optimal mass transport map for shape matching and comparison, focusing on two important applications including surface registration and shape space. The computation of the optimal mass transport map is based on Monge-Brenier theory, in comparison to the conventional method based on Monge-Kantorovich theory, this method significantly improves the efficiency by reducing computational complexity from O(n(2)) to O(n) . For surface registration problem, one commonly used approach is to use conformal map to convert the shapes into some canonical space. Although conformal mappings have small angle distortions, they may introduce large area distortions which are likely to cause numerical instability thus resulting failures of shape analysis. This work proposes to compose the conformal map with the optimal mass transport map to get the unique area-preserving map, which is intrinsic to the Riemannian metric, unique, and diffeomorphic. For shape space study, this work introduces a novel Riemannian framework, Conformal Wasserstein Shape Space, by combing conformal geometry and optimal mass transport theory. In our work, all metric surfaces with the disk topology are mapped to the unit planar disk by a conformal mapping, which pushes the area element on the surface to a probability measure on the disk. The optimal mass transport provides a map from the shape space of all topological disks with metrics to the Wasserstein space of the disk and the pullback Wasserstein metric equips the shape space with a Riemannian metric. We validate our work by numerous experiments and comparisons with prior approaches and the experimental results demonstrate the efficiency and efficacy of our proposed approach.

Analysis of transverse RMS emittance growth of a beam induced by spherical and chromatic aberration in a solenoidal field

In a medium energy beam transport line transverse rms emittance growth associated with spherical aberration is analysed. An analytical expression is derived for beam optics in a solenoid field considering terms up to the third order in the radial displacement. Two important phenomena: effect of spherical aberrations in axial-symmetric focusing lens and influence of nonlinear space charge forces on beam emittance growth are discussed for different beam distributions. In the second part nonlinear effect associated with chromatic aberration that describes the growth of emittance and distortion of phase space area is discussed.

Fusion of infrared intensity and polarization images using embedded multi-scale transform

Multi-scale transforms are considered to be the effective methods in image fusion. However, infrared intensity and polarization images fusion based on these methods frequently arise the issues such as limited contrast ratio improvement and marginal area distortion. The embedded multi-scale decomposition has been proposed to integrate the advantages of different methods in this paper, where the low-frequency image continues to be decomposed by another multi-scale transform. The detailed procedure is shown as the following. Firstly, the low-frequency components and support value image sequences of infrared intensity and polarization images are obtained respectively with support value transform (SVT). Secondly, multi-scale bright and dim information are extracted from the last layer of the low-frequency images using the multi-scale top-hat decomposition, respectively. Thirdly, the resulted images are respectively fused and enhanced, and then fused with the two low-frequency images to get the fused low-frequency image. After that, the image is reversely transformed with the support value image sequences fused by extracting maximum gray to get the final image. Compared with the simple SVT and the multi-scale top-hat decomposition, the method suggested successfully improves the contrast ratio, the distortion of marginal area and the local coarseness. The validity of the method proposed is proven.

On Astala's theorem for martingales and Fourier multipliers

We exhibit a large class of symbols m on Rd, d≥2, for which the corresponding Fourier multipliers Tm satisfy the following inequality. If D, E are measurable subsets of Rd with E⊆D and |D|<∞, then∫D∖E|TmχE(x)|dx≤{|E|+|E|ln⁡(|D|2|E|),if |E|<|D|/2,|D∖E|+12|D∖E|ln⁡(|E||D∖E|),if |E|≥|D|/2. Here |⋅| denotes the Lebesgue measure on IRd. When d=2, these multipliers include the real and imaginary parts of the Beurling–Ahlfors operator B and hence the inequality is also valid for B with the right-hand side multiplied by 2. The inequality is sharp for the real and imaginary parts of B. This work is motivated by K. Astala's celebrated results on the Gehring–Reich conjecture concerning the distortion of area by quasiconformal maps. The proof rests on probabilistic methods and exploits a family of appropriate novel sharp inequalities for differentially subordinate martingales. These martingale bounds are of interest on their own right.

Stretch-Minimizing Volumetric Parameterization

Not many methods for parameterization guarantee bijectivity or local injectivity, which is essential for foldoverfree mappings. Stretch-minimizing parameterization which is widely used for surface parameterization, provides foldover-free mappings and is capable of trading off between angle and area distortions. We extend its usage to volumetric parameterization in this paper by deriving a 3D version of stretch-distortion energy and incorporating fixed boundary conditions. Our energy definition includes a natural barrier term which effectively prevents elements from collapsing and folding over. It saves the effort in other methods of formulating additional energy or constrains to ensure the local injectivity. We propose to minimize the overall energy integrated over the whole mesh with a relaxation-enhanced solver and optimize the energy globally. This is different from the conventional approach of surface parameterization where mesh nodes are optimized individually. Compared with other volumetric parameterizations, our approach bears the advantages of stretch-minimizing method, being foldover-free and offering a good trade-off between angle and volume distortions.

Enlarging Image by Constrained Least Square Approach with Shape Preserving

From a visual point of view, the shape of an image is mainly determined by the edges. Conventional polynomial interpolation of image enlarging methods would produce blurred edges, while edge-directed interpolation based methods would cause distortion in the non-edge areas. A new method for image enlarging is presented. The image is enlarged in two steps. In the first step, a fitting surface is constructed to interpolate the image data. To remove the zigzagging artifact, for each pixel, a fitting patch is constructed using edge information as constraints. The combination of all the patches forms the fitting surface which has the shape suggested by image data. Each point on the fitting surface can be regarded as a sampling point taken from a unit square domain, which means that when the fitting surface is used to enlarge the image, each sampling domain of the enlarged pixels is also a unit square, causing the enlarged image to lose some details. To make the enlarged image keep the details as many as possible, the sampling domain of the enlarged pixels should be less than a unit square. Then, in the second step, using the points taken from the fitting surface, new pixels are computed using constrained optimization technique to form the enlarged image, and the size of the sampling domain of the enlarged pixels is inversely proportional to the size of the enlarged image. The image enlarged by the new method has a quadratic polynomial precision. Comparison results show that the new method produces resized image with better quality.

Lengths, areas and Lipschitz-type spaces of planar harmonic mappings

In this paper, we give bounds for length and area distortion for harmonic K-quasiconformal mappings, and investigate certain Lipschitz-type spaces on harmonic mappings.

IMPROVEMENT THE SPATIAL RESOLUTION OF MULTICHANNEL AEROSPACE HIGH SPATIAL RESOLUTION IMAGES ON THE BASE OF HYPERSPHERICAL TRANSFORM

In this paper we solve an actual problem of development of information technology to improve the visual quality of multi-channel space images of high spatial resolution. The object of the research is the process of fusing panchromatic and multispectral photogrammetric images obtained coordinate-sensitive sensors in the visible and infrared regions of the electromagnetic radiation. The subject of research are methods make preliminary and synergistic multi-channel data processing to improve the quality of the resulting image and reduce the color distortion. The purpose of the work is to increase the spatial resolution of the automated primary multi-channel images and compared with existing methods of eliminating color distortion in the local areas. In addition, the proposed technology will effectively carry out further recognition and real-time monitoring infrastructure. In the paper we propose a new information pansharpening technology based on HSV-converting and hyperspherical color conversion, which allows not only to improve the spatial resolution of the primary digital image, but also avoid the spectral distortion. This is achieved, in particular, by image pre-equalization, data processing localized spectral bases, optimized performance information, and the information contained in the infrared image. The software implementing proposed method is developed. The experiments to study the properties of the proposed algorithm are conducted. Experimental evaluation performed on eight-channel images obtained WorldView-2 satellite. Test results confirmed that the proposed approach can achieve high spectral and spatial quality multichannel images and outperforms existing methods.

Equitable mirrors.

Mirror surfaces used in catadioptric sensors are often designed so as to minimize one particular kind of image distortion. In this article we explore some finer properties of equi-areal mirrors, those that feature no area distortion, and we propose novel ways to measure compound forms of distortion. Specifically, we develop new mirror surfaces with large fields of view that simultaneously minimize angular and areal distortion with respect to different cost functions.

Effect of Heat Treatment Parameters in Plasma Arc Surface Hardening of AISI 4340 Steel

Flame hardening has been traditionally used for selective surface hardening of steel. This technique frequently resulting in imprecise harden area and part distortion when overheated. Focused heating source such as plasma arc can be an alternative to overcome this problem. In this work, plasma arc is scanned at the 4340 steel surface to improve the hardness. The variable parameters investigated were at scanning speed and operating current. Four types of surfaces were observed after they are scanned with plasma arc. They are fully-melted, partially-melted, non-melted continuous and non-melted intermittent where each type of surface has different roughness value. This work found that scanning speed and operating current has significantly influence the type of surface and roughness values. Analysis on non-melted surface gives the maximum depth of hardened layer of about 187 μm as well as hardness values of about 990 HV50. It is also observed that the depth of hardened layer and hardness value is significantly decreasing with increase in scanning speed or the decrease in operating current. Microstructure examination on hardened layer revealed that the increase of hardness is due to formation of fine grain martensitie structure.