Quasiconformal Research Articles

A Unifying Framework for $n$-Dimensional Quasi-Conformal Mappings

With the advancement of computer technology, there is a surge of interest in effective mapping methods for objects in higher-dimensional spaces. To establish a one-to-one correspondence between objects, higher-dimensional quasi-conformal theory can be utilized for ensuring the bijectivity of the mappings. In addition, it is often desirable for the mappings to satisfy certain prescribed geometric constraints and possess low distortion in conformality or volume. In this work, we develop a unifying framework for computing $n$-dimensional quasi-conformal mappings. More specifically, we propose a variational model that integrates quasi-conformal distortion, volumetric distortion, landmark correspondence, intensity mismatch and volume prior information to handle a large variety of deformation problems. We further prove the existence of a minimizer for the proposed model and devise efficient numerical methods to solve the optimization problem. We demonstrate the effectiveness of the proposed framework using various experiments in two- and three-dimensions, with applications to medical image registration, adaptive remeshing and shape modeling.

Conformally invariant complete metrics

AbstractFor a domain G in the one-point compactification $\overline{\mathbb{R}}^n = {\mathbb{R}}^n \cup \{ \infty\}$ of ${\mathbb{R}}^n, n \geqslant 2$ , we characterise the completeness of the modulus metric $\mu_G$ in terms of a potential-theoretic thickness condition of $\partial G\,,$ Martio’s M-condition [35]. Next, we prove that $\partial G$ is uniformly perfect if and only if $\mu_G$ admits a minorant in terms of a Möbius invariant metric. Several applications to quasiconformal maps are given.

Hölder continuity of quasiconformal harmonic mappings from the unit ball to a spatial domain with [formula omitted] boundary

We prove that every quasiconformal mapping from the harmonic β-Bloch space between the unit ball and a spatial domain with C1 boundary is globally α-Hölder continuous for α<1−β, with the Hölder coefficient that does not depend neither on the mapping nor on β. An analogous result also holds for Lipschitz continuous, quasiconformal harmonic mappings for α<1. This is an approach towards the extension of some results from the complex plane obtained by Warschawski (1951) for conformal mappings and Kalaj (2022) for quasiconformal harmonic mappings.

On the energy of quasiconformal mappings and pseudoholomorphic curves in complex projective spaces

We prove that the energy density of uniformly continuous, quasiconformal mappings, omitting two points in $\mathbb{C} \mathbb{P}^1$, is equal to zero. We also prove the sharpness of this result, constructing a family of uniformly continuous, quasiconformal mappings, whose areas grow asymptotically quadratically. Finally, we prove that the energy density of pseudoholomorphic Brody curves, omitting three “complex lines” in general position in $\mathbb{C} \mathbb{P}^2$, is equal to zero.

Quasiconformal flows on non-conformally flat spheres

We study integral curvature conditions for a Riemannian metric g on S4 that quantify the best bilipschitz constant between (S4,g) and the standard metric on S4. Our results show that the best bilipschitz constant is controlled by the L2-norm of the Weyl tensor and the L1-norm of the Q-curvature, under the conditions that those quantities are sufficiently small, g has a positive Yamabe constant and the Q-curvature is mean-positive. The proof of the result is achieved in two steps. Firstly, we construct a quasiconformal map between two conformally related metrics in a positive Yamabe class. Secondly, we apply the Ricci flow to establish the bilipschitz equivalence from such a conformal class to the standard conformal class on S4.

Triangular Ratio Metric Under Quasiconformal Mappings in Sector Domains

The hyperbolic metric and different hyperbolic type metrics are studied in open sector domains of the complex plane. Several sharp inequalities are proven for them. Our main result describes the behavior of the triangular ratio metric under quasiconformal maps from one sector onto another one.

Triangular ratio metric in the unit disk

The triangular ratio metric is studied in a domain , . Several sharp bounds are proven for this metric, especially in the case where the domain is the unit disk of the complex plane. The results are applied to study the Hölder continuity of quasiconformal mappings.

Introducing a New Intrinsic Metric

A new intrinsic metric called the t-metric is introduced. Several sharp inequalities between this metric and the most common hyperbolic type metrics are proven for various domains Gsubsetneq mathbb {R}^n. The behaviour of the new metric is also studied under a few examples of conformal and quasiconformal mappings, and the differences between the balls drawn with all the metrics considered are compared by both computational and analytical means.

Erratum to “Harmonic quasiconformal mappings between $\mathcal{C}^1$ smooth Jordan domains”

Erratum to “Harmonic quasiconformal mappings between $\mathcal{C}^1$ smooth Jordan domains”

Digital image correlation method based on quasi-conformal mapping for large deformation measurement

Digital image correlation (DIC) has the advantages of non-contact full-field measurement and high measurement accuracy in optical mechanics, but still faces challenges in large deformation measurement. This work developed an initial estimation algorithm, using geometric distortion as an indicator. And the more accurate deformation is adjusted by the modified inverse compositional Gauss-Newton (IC-GN) algorithm that introduces the inverse mapping of the initial estimation. The simulation experiment proves that the proposed method has good measurement ability for large deformation, and the accuracy is about 0.02 pixels. The uniaxial tensile test of vulcanized rubber and the three-point bending test of the polycarbonate sheet are used to verify the effectiveness of the method in this paper.

Beltrami Equations on Rossi Spheres

Beltrami equations L¯t(g)=μ(·,t)Lt(g) on S3 (where Lt, |t|&lt;1, are the Rossi operators i.e., Lt spans the globally nonembeddable CR structure H(t) on S3 discovered by H. Rossi) are derived such that to describe quasiconformal mappings f:S3→N⊂C2 from the Rossi sphere S3,H(t). Using the Greiner–Kohn–Stein solution to the Lewy equation and the Bargmann representations of the Heisenberg group, we solve the Beltrami equations for Sobolev-type solutions gt such that gt−v∈WF1,2S3,θ with v∈CR∞S3,H(0).

Weighted Hardy Spaces of Quasiconformal Mappings

We study the integral characterizations of weighted Hardy spaces of quasiconformal mappings on the n-dimensional unit ball using the weight \((1-r)^{n-2 + \alpha }\). We extend the known results for univalent functions on the unit disk. Some of our results are new even in the unweighted setting for quasiconformal mappings.

Quantitative characterization of the human retinotopic map based on quasiconformal mapping.

The retinotopic map depicts the cortical neurons' response to visual stimuli on the retina and has contributed significantly to our understanding of human visual system. Although recent advances in high field functional magnetic resonance imaging (fMRI) have made it possible to generate the in vivo retinotopic map with great detail, quantifying the map remains challenging. Existing quantification methods do not preserve surface topology and often introduce large geometric distortions to the map. In this study, we developed a new framework based on computational conformal geometry and quasiconformal Teichmüller theory to quantify the retinotopic map. Specifically, we introduced a general pipeline, consisting of cortical surface conformal parameterization, surface-spline-based cortical activation signal smoothing, and vertex-wise Beltrami coefficient-based map description. After correcting most of the violations of the topological conditions, the result was a "Beltrami coefficient map" (BCM) that rigorously and completely characterizes the retinotopic map by quantifying the local quasiconformal mapping distortion at each visual field location. The BCM provided topological and fully reconstructable retinotopic maps. We successfully applied the new framework to analyze the V1 retinotopic maps from the Human Connectome Project (n=181), the largest state of the art retinotopy dataset currently available. With unprecedented precision, we found that the V1 retinotopic map was quasiconformal and the local mapping distortions were similar across observers. The new framework can be applied to other visual areas and retinotopic maps of individuals with and without eye diseases, and improve our understanding of visual cortical organization in normal and clinical populations.

The Quasisymmetric Mappings under the Quasihyperbolic Metric in Real Banach Spaces

Suppose that E, E′ are Q‐regular Banach spaces, that G⊂E and G′⊂E′ are domains, and that f : G⟶G′ is a homeomorphism. The aim of this study is to prove that the quasisymmetric mapping under the quasihyperbolic metric is equivalent to the freely quasiconformal mapping under the condition E, E′ being Q‐regular Banach spaces. This result is a partial solution to an open problem raised by Väisälä.

Stretching and Rotation of Planar Quasiconformal Mappings on a Line

In this article, we examine stretching and rotation of planar quasiconformal mappings on a line. We show that for almost every point on the line, the set of complex stretching exponents (describing stretching and rotation jointly) is contained in the disk overline {B}(1/(1-k^{4}),k^{2}/(1-k^{4})). This yields a quadratic improvement over the known optimal estimate for general sets of Hausdorff dimension 1. Our proof is based on holomorphic motions and estimates for dimensions of quasicircles. We also give a lower bound for the dimension of the image of a 1-dimensional subset of a line under a quasiconformal mapping.

Identifying the Structure of Soil Massifs by Numerical Quasiconformal Mapping Methods

Identifying the Structure of Soil Massifs by Numerical Quasiconformal Mapping Methods

Quasi-symmetric mappings and their generalizations on Riemannian manifolds

A relation between 𝜂-quasi-symmetric homomorphisms and K-quasiconformal mappings on n-dimensional smooth connected Riemannian manifolds has been studied. The main results of the research are presented in Theorems 2.6 and 2.7. Several conditions for the boundary behavior of 𝜂-quasi-symmetric homomorphisms between two arbitrary domains with weakly flat boundaries and compact closures, QED and uniform domains on the Riemannian manifolds, which satisfy the obtained results, were also formulated. In addition, quasiballs, c-locally connected domains, and the corresponding results were also considered.

Diffeomorphic Registration of Retinotopic Maps with Quasiconformal Mapping

Diffeomorphic Registration of Retinotopic Maps with Quasiconformal Mapping

Quantitative Characterization of the Human Retinotopic Map Based on Quasiconformal Mapping

Abstract The retinotopic map depicts the cortical neurons’ response to visual stimuli on the retina and has contributed significantly to our understanding of human visual system. Although recent advances in high field functional magnetic resonance imaging (fMRI) have made it possible to generate the in vivo retinotopic map with great detail, quantifying the map remains challenging. Existing quantification methods do not preserve surface topology and often introduce large geometric distortions to the map. In this study, we developed a new framework based on computational conformal geometry and quasiconformal Teichmuller theory to quantify the retinotopic map. Specifically, we introduced a general pipeline, consisting of cortical surface conformal parameterization, surface-spline-based cortical activation signal smoothing, and vertex-wise Beltrami coefficient-based map description. After correcting most of the violations of the topological conditions, the result was a “Beltrami coefficient map” (BCM) that rigorously and completely characterizes the retinotopic map by quantifying the local quasiconformal mapping distortion at each visual field location. The BCM provided topological and fully reconstructable retinotopic maps. We successfully applied the new framework to analyze the V1 retinotopic maps from the Human Connectome Project (n=181), the largest state of the art retinotopy dataset currently available. With unprecedented precision, we found that the V1 retinotopic map was quasiconformal and the local mapping distortions were similar across observers. The new framework can be applied to other visual areas and retinotopic maps of individuals with and without eye diseases, and improve our understanding of visual cortical organization in normal and clinical populations.

Quasiconformal extension for harmonic mappings on finitely connected domains

We prove that a harmonic quasiconformal mapping defined on a finitely connected domain in the plane, all of whose boundary components are either points or quasicircles, admits a quasiconformal extension to the whole plane if its Schwarzian derivative is small. We also make the observation that a univalence criterion for harmonic mappings holds on uniform domains.