Search Results (10)

In freeform optical metrology, wavefront fitting over non-circular apertures is hindered by the loss of Zernike polynomial orthogonality and severe sampling grid distortion inherent in standard conformal mappings. To address the resulting numerical instability and fitting bias, we propose a unified framework curve-shortening flow (CSF)-guided progressive quasi-conformal mapping (CSF-QCM), which integrates geometric boundary evolution with topology-aware parameterization. CSF-QCM first smooths complex boundaries via curve-shortening flow, then solves a sparse Laplacian system for harmonic interior coordinates, thereby establishing a stable diffeomorphism between physical and canonical domains. For doubly connected apertures, it preserves topology by computing the conformal modulus via Dirichlet energy minimization and simultaneously mapping both boundaries. Benchmarked against state-of-the-art methods (e.g., Fornberg, Schwarz–Christoffel, and Ricci flow) on representative irregular apertures, CSF-QCM suppresses area distortion and restores discrete orthogonality of the Zernike basis, reducing the Gram matrix condition number from >900 to <8. This enables high-precision reconstruction with RMS residuals as low as $3\times {10}^{-3}\lambda$ and up to 92% lower fitting errors than baselines. The framework provides a unified, computationally efficient, and numerically stable solution for wavefront reconstruction in complex off-axis and freeform optical systems. Full article

In this paper, we study the behavior of the m-th$(m\ge 0)$ derivatives of general algebraic polynomials in weighted Bergman spaces defined in regions of the complex plane G bounded by piecewise smooth curves $L=\partial G$ with $\lambda \pi$$(0<\lambda \le 2)$ exterior angles relative to G. Upper bounds are found for the growth of the m-th derivatives of the polynomials not only inside the unbounded region but also on the closures of this region with both exterior non-zero angles $\lambda \pi$$(0<\lambda <2)$ and interior zero angles (i.e., exterior angles $2\pi$). The influence of the boundary angles $\lambda \pi$$(0<\lambda \le 2)$ of the region G and the “growth rate” of the weight function on the behavior of the moduli of polynomials and their derivatives in regions of the complex plane that are “symmetric” with respect to L (bounded and unbounded) is found. Full article

In this paper, we study asymptotic bounds on the m-th derivatives of general algebraic polynomials in weighted Bergman spaces. We consider regions in the complex plane defined by bounded, piecewise, asymptotically conformal curves with strictly positive interior angles. We first establish asymptotic bounds on the growth in the exterior of a given unbounded region. We then extend our analysis to the closures of the region and derive the corresponding growth bounds. Combining these bounds with those for the corresponding exterior, we obtain comprehensive bounds on the growth of the m-th derivatives of arbitrary algebraic polynomials in the whole complex plane. Full article

This paper concerns the old problem of the connection between the dilatations of a given quasisymmetric homeomorphism h of a circle and the associated polygonal quasiconformal maps with a fixed finite number of boundary points, namely whether $k\left(h\right)=sup{k}_n$, where the supremum is taken over all possible n-gons formed by the disk with n distinguished boundary points. A still open question is whether such equality is valid under the additional assumption that the naturally related univalent functions with quasiconformal extensions have equal Grunsky and Teichmüller norms. We solved this problem in the negative for $n\ge 4$. Full article

Compared to traditional technology, image classification technology possesses a superior capability for quantitative analysis of the target and background, and holds significant applications in the domains of ground target reconnaissance, marine environment monitoring, and emergency response to sudden natural disasters, among others. Currently, the enhancement of spatial spectral resolution heightens the difficulty and reduces the efficiency of classification, posing a substantial challenge to the aforementioned applications. Hence, the classification algorithm is required to take both computing power and classification accuracy into account. Research indicates that the deep kernel mapping network can accommodate both computing power and classification accuracy. By employing the kernel mapping function as the network node function of deep learning, it effectively enhances the classification accuracy under the condition of limited computing power. Therefore, to address the issue of network structure optimization of deep mapping networks and the insufficient application of line feature learning and expression in existing network structures, considering the adaptive optimization of network structures, deep quasiconformal kernel network learning (DQKNet) is proposed for image classification. Firstly, the structural parameters and learning parameters of the deep kernel mapping network are optimized. This approach can adaptively adjust the network structure based on the distribution characteristics of the data and enhance the performance of image classification. Secondly, the computational network node optimization method of quasiconformal kernel learning is applied to this network, further elevating the performance of the deep kernel learning mapping network in image classification. The experimental results demonstrate that the improvement in the deep kernel mapping network from the perspectives of accounting children, mapping network nodes, and network structure can effectively enhance the feature extraction and classification performance of the data. On the five public datasets, the average AA, OA, and KC values of our algorithm are 91.99, 91.25, and 85.99, respectively, outperforming the currently most-advanced algorithms. Full article

Univalent functions with asymptotically conformal extension to the boundary form a subclass of functions with quasiconformal extension with rather special features. Such functions arise in various questions of geometric function theory and Teichmüller space theory and have important applications involving conformal and quasiconformal maps. The paper provides an approximative characterization of local conformality and its connection with univalent polynomials. Also, some other quantitative applications of this connection are given. Full article

The classical Belinskii theorem implies that any sufficiently regular function $\mu \left(z\right)$ on the extended complex plane $\widehat{\mathbb{C}}$ with a small $C^{1+\alpha }$ norm generates via the two-dimensional Cauchy integral a quasiconformal automorphism w of $\widehat{\mathbb{C}}$ with the Beltrami coefficient $\tilde{\mu }=\mu +{O(\parallel \mu \parallel }^2)$. We consider $\mu$ supported in arbitrary bounded quasiconformal disks and show that under appropriate assumptions of $\mu$, this automorphism explicitly provides the basic curvelinear quasi-invariants associated with conformal and quasiconformal maps, advancing an old problem of quasiconformal analysis. Full article

Beltrami equations ${\overline{L}}_t\left(g\right)=\mu (·,t)L_t\left(g\right)$ on $S^3$ (where $L_t$, $\left|t\right|<1$, are the Rossi operators i.e., $L_t$ spans the globally nonembeddable CR structure $\mathcal{H}\left(t\right)$ on $S^3$ discovered by H. Rossi) are derived such that to describe quasiconformal mappings $f:S^3\to N\subset {\mathbb{C}}^2$ from the Rossi sphere $\left(S^3,\mathcal{H}\left(t\right)\right)$. 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 $g_t$ such that $g_t-v\in W_F^{1,2}\left(S^3,\theta \right)$ with $v\in {\mathrm{CR}}^{\infty }\left(S^3,\mathcal{H}\left(0\right)\right)$. Full article

Conventional splines offer powerful means for modeling surfaces and volumes in three-dimensional Euclidean space. A one-dimensional quaternion spline has been applied for animation purpose, where the splines are defined to model a one-dimensional submanifold in the three-dimensional Lie group. Given two surfaces, all of the diffeomorphisms between them form an infinite dimensional manifold, the so-called diffeomorphism space. In this work, we propose a novel scheme to model finite dimensional submanifolds in the diffeomorphism space by generalizing conventional splines. According to quasiconformal geometry theorem, each diffeomorphism determines a Beltrami differential on the source surface. Inversely, the diffeomorphism is determined by its Beltrami differential with normalization conditions. Therefore, the diffeomorphism space has one-to-one correspondence to the space of a special differential form. The convex combination of Beltrami differentials is still a Beltrami differential. Therefore, the conventional spline scheme can be generalized to the Beltrami differential space and, consequently, to the diffeomorphism space. Our experiments demonstrate the efficiency and efficacy of diffeomorphism splines. The diffeomorphism spline has many potential applications, such as surface registration, tracking and animation. Full article

We investigate the interplay between the existence of fat triangulations, P L approximations of Lipschitz–Killing curvatures and the existence of quasiconformal mappings. In particular we prove that if there exists a quasiconformal mapping between two P L or smooth n-manifolds, then their Lipschitz–Killing curvatures are bilipschitz equivalent. An extension to the case of almost Riemannian manifolds, of a previous existence result of quasimeromorphic mappings on manifolds due to the first author is also given. Full article