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4 . Teichmüller parameterization of a lion point cloud. Left: The input point cloud. Middle: The parameterization result. The landmark constraints are represented by the green and
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In recent decades, the use of 3D point clouds has been widespread in computer
industry. The development of techniques in analyzing point clouds is
increasingly important. In particular, mapping of point clouds has been a
challenging problem. In this paper, we develop a discrete analogue of the
Teichm\"{u}ller extremal mappings, which guarantee unif...
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... Experimental Results. In this section, we demonstrate the effectiveness of our proposed TEMPO method by various examples. Our proposed algorithms are implemented in MATLAB. The point clouds are adapted from the AIM@SHAPE shape repository [51], the TF3DM repository [54], the Laboratory for Computational Longitudinal Neuroimaging (LCLN) shape database [53], the human face dataset [52] and the Spacetime Face Data [55]. Some additional facial point clouds are sampled using Kinect for the shape analysis experiment. The sparse linear systems are solved using the built-in backslash operator ( \ ) in MATLAB. All experiments are performed on a PC with an Intel(R) Core(TM) i7-4770 CPU @3.40 GHz processor and 16.00 GB RAM. 7.1. The performance of our proposed approximation schemes. In this subsection, we evaluate the performance of our proposed approximation schemes for computing quasi-conformal (including conformal) mappings on point clouds with disk topology. We first compare our proposed scheme with the local mesh method [24] and the moving least square method with special weight [27, 28] for computing conformal mappings. In each experiment, we first generate a random point cloud on a planar rectangular domain. Then, we transform the point cloud using a conformal mapping with an explicit formula. An example is given in Figure 7.1. On the transformed point cloud, we approximate the Laplace-Beltrami operator using the aforementioned schemes. Using the approximated Laplace-Beltrami operator, we solve the Laplace equation to map the point cloud back onto the rectangular domain. The resulting position errors show the accuracies of the approximation schemes. Table 7.1 lists the statistics of several experiments. It can be observed that our proposed combined scheme provides better approximations for the conformal mappings on point clouds with disk topology. Then, we compare the mentioned approximation schemes for computing quasiconformal mappings with prescribed Beltrami coefficients. This time, in each experiment, we transform a randomly generated point cloud using a quasi-conformal mapping with an explicit formula. An example is given in Figure 7.2. On the transformed point cloud, we approximate the generalized Laplacian using different schemes and solve the generalized Laplace equation to map the point cloud back onto the rectangular domain. Table 7.2 lists the statistics of several experiments. Again, our combined approximation scheme produces results with higher accuracy. 7.2. Landmark constrained Teichmüller Parameterizations. After demon- strating the advantage of our proposed approximation scheme for quasi-conformal mappings, we illustrate the effectiveness of our landmark constrained Teichmüller parameterization algorithm using numerous experiments. In all experiments, the stop- ping criterion is set to be = 10 − 6 and the initial balancing parameter γ in the hybrid equation (6.8) is set to be γ = 1 . 2 Figure 7.3 shows the landmark-constrained Teichmüller parameterization result of a twisted point cloud. The green points and the blue points represent the prescribed landmark constraints on the planar domain. Even under the large landmark deformations, our parameterization result ensures a uniform conformality distortion. This indicates the Teichmüller property of our proposed algorithm. More examples are shown in Figure 7.4 and Figure 7.5. It can be easily observed that in all of our experiments, the norms of the Beltrami coefficients always accumulate at a specific value. This indicates that our parameterization results of point clouds closely resemble the continuous Teichmüller parameterizations of simply-connected open surfaces. Extensive experiments of our proposed landmark constrained Teichmüller parameterization algorithm have been carried out using different point clouds with disk topology. The statistics of the experiments are listed in Table 7.3. Our proposed method is highly efficient. The computations of the landmark-matching Teichmüller parameterizations complete within 1 minute on average for point clouds with ...
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... Early studies investigate various specialized parameterization strategies for disk-type [11,1,41], genus-0 [45,17], genus-1 [38], well-sampled [14] or low-quality [22] point clouds. Later approaches achieve fixed-boundary spherical/rectangular parameterization through approximating the Laplace-Beltrami operators [5] or Teichmüller extremal mappings [23]. More recently, FBCP-PC [4] further pursues free-boundary conformal parameterization, in which a new point cloud Laplacian approximation scheme is proposed for handling boundary non-convexity. ...
Surface parameterization plays an essential role in numerous computer graphics and geometry processing applications. Traditional parameterization approaches are designed for high-quality meshes laboriously created by specialized 3D modelers, thus unable to meet the processing demand for the current explosion of ordinary 3D data. Moreover, their working mechanisms are typically restricted to certain simple topologies, thus relying on cumbersome manual efforts (e.g., surface cutting, part segmentation) for pre-processing. In this paper, we introduce the Flatten Anything Model (FAM), an unsupervised neural architecture to achieve global free-boundary surface parameterization via learning point-wise mappings between 3D points on the target geometric surface and adaptively-deformed UV coordinates within the 2D parameter domain. To mimic the actual physical procedures, we ingeniously construct geometrically-interpretable sub-networks with specific functionalities of surface cutting, UV deforming, unwrapping, and wrapping, which are assembled into a bi-directional cycle mapping framework. Compared with previous methods, our FAM directly operates on discrete surface points without utilizing connectivity information, thus significantly reducing the strict requirements for mesh quality and even applicable to unstructured point cloud data. More importantly, our FAM is fully-automated without the need for pre-cutting and can deal with highly-complex topologies, since its learning process adaptively finds reasonable cutting seams and UV boundaries. Extensive experiments demonstrate the universality, superiority, and inspiring potential of our proposed neural surface parameterization paradigm. The code will be publicly available.
... As a normal extension of conformal mapping, quasi-conformal mapping with bounded conformality distortion has been extensively studied in the field of graphics (e.g. Meng et al. 2016;Ho and Lui 2016;Xu et al. 2018;Zhu et al. 2022;Pan and Chen 2022), where textures generated with quasi-conformal mapping enjoy sufficient design freedom, and the structural bijectivity is ensured. Quasi-conformal mapping converts an infinitesimally small circle into an ellipse with the bounds for its major-to-minor axis ratio kept under control (Gu et al. 2010). ...
For spatially varying multiscale configurations (SVMSCs) that increasingly gain engineering perspective, a design scheme enabling the direct tuning of their microstructural layouts is still desired. Challenges lie behind the fact that an arbitrary setting on the (field) distributions about the microstructural geometric features is likely to violate the underlying conditions of total differential, and the article is aimed to resolve the issue systematically. This is done by representing (two-dimensional) SVMSCs from the viewpoint of quasi-conformal mapping, where the microstructural profile can be summarised by a (complete) family consisting of four geometric feature fields. By modifying the so-called Beltrami equations that automatically satisfy the total differential condition, the relationships among these four geometric feature fields can be explicitly derived, and their degrees of freedom are reduced to two. Different choices for which two field quantities as the free design variables result in various geometric-feature-based modes, which almost cover all scenarios of SVMSC design by means of directly tuning their microstructural profiles. Furthermore, we can now directly manipulate the stretch ratio, one of the four field variables, to avoid ill microstructural distortion, which has been a challenging issue for SVMSCs representation. Fueled by an asymptotic homogenisation scheme (Zhu et al. in J Mech Phys Solids 124:612–633, 2019, https://doi.org/10.1016/j.jmps.2018.11.008), the present method is then employed for geometric-feature-based compliance design of SVMSC.
... where (s, t) ∈ R 2 is a parametric location mapped, e.g., using conformal mapping [36] (or simply normalized x and y coordinates), from a Cartesian measured point (x, y, z) to a parametric space; P is an n-row CP matrix, each row of which represents a CP p i ∈ R 3 ; b is a T-spline blending [17] vector with the i-th blending function entry b i (s, t) in the form of: ...
... where ( , ) ∈ ℝ is a parametric location mapped, e.g., using conformal mapping [36] (or simply normalized and coordinates), from a Cartesian measured point ( , , ) to a parametric space; is an n-row CP matrix, each row of which represents a CP ∈ ℝ ; is a T-spline blending [17] vector with the i-th blending function entry ( , ) in the form of: ...
... Providing an input 3D Cartesian point set to properly map to a parametric space [36], e.g., = 0,1 ⊗ 0,1 , this step divides the cloud into K minor data patches in the parametric space simply according to sample sizes for subsequent fitting within each patch. Each local patch then comprises an independent T-spline with local-support CPs. ...
Parametric splines are popular tools for precision optical metrology of complex freeform surfaces. However, as a promising topologically unconstrained solution, existing T-spline fitting techniques, such as improved global fitting, local fitting, and split-connect algorithms, still suffer the problems of low computational efficiency, especially in the case of large data scales and high accuracy requirements. This paper proposes a speed-improved algorithm for fast, large-scale freeform point cloud fitting by stitching locally fitted T-splines through three steps of localized operations. Experiments show that the proposed algorithm produces a three-to-eightfold efficiency improvement from the global and local fitting algorithms, and a two-to-fourfold improvement from the latest split-connect algorithm, in high-accuracy and large-scale fitting scenarios. A classical Lena image study showed that the algorithm is at least twice as fast as the split-connect algorithm using fewer than 80% control points of the latter.
... Since the geometries of objects are commonly described by the mesh models, conformal mapping receives much more attention owing to its property of preserving mesh quality. In the present work, the computational method for solving conformal mappings developed in [57][58][59] is adopted to achieve an MMC-friendly mapping. The core procedures are summarized as follows. ...
... The conformality and bijectivity (means no folding or overlaps of meshes exist in the parameterization results) of the acquired mapping are ensured, which are essential for alleviating nonlinearity and avoiding singularity in the problem formulation of structural optimization. For theoretical fundamentals and technical details of the CCM technique, please refer to [57][58][59] and references therein. ...
... Step 3: Conformality Correction. To address potential angular distortion and mesh overlapping, we employ the Linear Beltrami Solver (LBS) scheme [57][58][59] for rectifying conformal distortion and non-bijectivity, as depicted in Fig. A.1(d). Steps 3 and 2 collectively constitute the procedure for computing a conformal mapping from the pertinent surface to a planar rectangular parametric domain. ...
In this article, a novel explicit approach for designing complex thin-walled structures based on the Moving Morphable Component (MMC) method is proposed, which provides a unified framework to systematically address various design issues, including topology optimization, reinforced-rib layout optimization, and sandwich structure design problems. The complexity of thin-walled structures mainly comes from flexible geometries and the variation of thickness. On the one hand, the geometric complexity of thin-walled structures leads to the difficulty in automatically describing material distribution (e.g., reinforced ribs). On the other hand, thin-walled structures with different thicknesses require various hypotheses (e.g., Kirchhoff-Love shell theory and Reissner-Mindlin shell theory) to ensure the precision of structural responses. Whereas for cases that do not fit the shell hypothesis, the precision loss of response solutions is non-negligible in the optimization process since the accumulation of errors will cause entirely different designs. Hence, the current article proposes a novel embedded solid component to tackle these challenges. The geometric constraints that make the components fit to the curved thin-walled structure are whereby satisfied. Compared with traditional strategies, the proposed method is free from the limit of shell assumptions of structural analysis and can achieve optimized designs with clear load transmission paths at the cost of few design variables and degrees of freedom for finite element analysis (FEA). Finally, we apply the proposed method to several representative examples to demonstrate its effectiveness, efficiency, versatility, and potential to handle complex industrial structures.
... Partial differential equations on manifolds have been applied in many areas including material science [9,16], fluid flow [19,21], biology physics [6,17,29], machine learning [7,11,24,27,33] and image processing [10,20,22,23,30,32,40]. Among all the manifold PDEs in the literature, the Poisson model has been studied most frequently as it is mathematically interesting and usually reveals much information of the manifold. ...
In this paper, we analyze a class of nonlocal Poisson model on manifold under Dirichlet boundary condition. To get second order convergence, the geometric information of the manifold has to be incorporated in the nonlocal model. So the manifold is assumed to be given explicitly such that all geometric information can be calculated explicitly. Under this assumption, the well-posedness of nonlocal model is studied. Moreover, the second-order convergence rate of model is attained by combining the stability argument and the truncation error analysis. Such rate is currently optimal among all nonlocal models. Numerical tests are conducted to verify the convergence.
... Additionally, LBS has been utilized in various segmentation algorithms for different applications [2,27,33,35]. Besides, it has also been applied to shape analysis [1,3,6,7,24,25]. However, due to the limitation of the quasiconformal theory the LBS is based on, these useful techniques have not been generalized to process 3D data. ...
The analysis of mapping relationships and distortions in multidimensional data poses a significant challenge in contemporary research. While Beltrami coefficients offer a precise description of distortions in two-dimensional mappings, current tools lack this capability in the context of three-dimensional space. This paper presents a novel approach: a 3D quasiconformal representation that captures the local dilation of 3D mappings, along with an algorithm that establishes a connection between this representation and the corresponding mapping. Experimental results showcase the algorithm's effectiveness in eliminating foldings in 3D mappings, as well as in mapping reconstruction and generation. These features bear a resemblance to the 2D Linear Beltrami Solver technique. The work presented in this paper offers a promising solution for the precise analysis and adjustment of distortions in 3D data and mappings.
... The computational conformal mapping technique (Meng, Choi and Lui, 2016;Huo, Liu, Du, Jiang, Liu and Guo, 2022) has been implemented using our in-house code to model the 3D sponge-like lattice in ABAQUS. Periodic boundary conditions are applied to the paired nodes of the top and bottom of the lattice model according to the following equations: ...
Euplectella aspergillum, as a textbook example presented by nature, is famous for its superior buckling resistance and stiffness subject to compression of arbitrary directions and its double-diagonal reinforced square microstructure has attracted considerable interest. In the present work, a parametric model is developed for a class of sponge-like lattices, and a neural network model is trained to realize accurate and efficient mapping from the geometry parameters to the corresponding effective stiffness and buckling resistance under uniaxial compression along all directions. Then an optimization formulation is proposed for designing a sponge-like lattice with on-demand requirements. Using the same amount of material, in the present design framework, the average buckling resistance of the original sponge-lattice model in literature is increased by about 40% and the design efficiency is improved by about 3-4 orders as compared to the traditional optimization process with exact mechanical modeling. Combining the intelligence of nature, humans and computers, the present AI-enhanced bioinspiration paradigm provides a novel pathway for designing innovative materials and structural systems.
... For example, parametric coordinates describe the sphere, the cylinder, Mobius strip and helicoids amongst others 92 . In computer graphics this is realised in the common practice of mapping between surfaces through simpler intermediaries: for example, the texture mapping of arbitrary surfaces by optimal surface cutting and mapping of the cuts into individual 2D shapes 57 and ( , ) surface parameterization by cutting and gluing individually mapped 2D planar patches 93 or mapping to canonical shapes such as the triangle 94 , plane 39 or polyhedra 95 to minimize distortion. u-Unwrap3D draws inspiration from this thinking. ...
... To convert a unit disk parameterization to an x pixel image, we 'square' the disk using the elliptical grid mapping formula 113 , multiply the resulting vertex coordinates by /2 and interpolate the coordinates and associated vertex quantities onto a x pixel integer grid. This gives similar results to but is significantly faster than solving the Beltrami equation 94 . ...
Signal transduction and cell function are governed by the spatiotemporal organization of membrane-associated molecules. Despite significant advances in visualizing molecular distributions by 3D light microscopy, cell biologists still have limited quantitative understanding of the processes implicated in the regulation of molecular signals at the whole cell scale. In particular, complex and transient cell surface morphologies challenge the complete sampling of cell geometry, membrane-associated molecular concentration and activity and the computing of meaningful parameters such as the cofluctuation between morphology and signals. Here, we introduce u-Unwrap3D, a framework to remap arbitrarily complex 3D cell surfaces and membrane-associated signals into equivalent lower dimensional representations. The mappings are bidirectional, allowing the application of image processing operations in the data representation best suited for the task and to subsequently present the results in any of the other representations, including the original 3D cell surface. Leveraging this surface-guided computing paradigm, we track segmented surface motifs in 2D to quantify the recruitment of Septin polymers by blebbing events; we quantify actin enrichment in peripheral ruffles; and we measure the speed of ruffle movement along topographically complex cell surfaces. Thus, u-Unwrap3D provides access to spatiotemporal analyses of cell biological parameters on unconstrained 3D surface geometries and signals.
... To achieve the geometric fit of the infill microstructures to the swept volume, the isoparametric surface of a swept volume must first be parameterized. In this work, the computational conformal mapping technology [65,66] is used to minimize the angular distortion, which conformally transforms a complex 3D surface onto a regular 2D Euclidean plane (and vice versa) to facilitate the geometric processing of the graded infill design. ...
Swept volumes are widely existed in various significant engineering applications. However, an effective design method for the lightweight swept volume with porous infills is lacking. This paper develops a novel topology optimization method for the design of curved volumes filled with spatially-varying microstructures. The infill optimization formulation for the curved swept volume is established to concurrently optimize both the microstructural geometry and the macroscopic distribution of the graded infills, aiming at minimizing the structural compliance subject to a global volume constraint. A conformal sweeping approach is developed to map the curved volume into a regular parametrization cube along the sweeping trajectory. In this fashion, the graded infills defined in a regular unit cell can be first filled into this cubic parameterization space, and then reversely mapped into the original volume to conformally match the curved geometry. A numerical homogenization method for parallelepiped microstructures is introduced to evaluate several sampling microstructures. The local level sets approach is used to create a family of connectable microstructures, while a polynomial interpolation is employed to rapidly interpolate the graded property of the generated microstructures based on the properties of sampling microstructures. Two different sweeping volume types are investigated. Several examples with respect to the infill optimization design of curved swept volumes are provided to illustrate the characteristics of the proposed method.
... Meng-Lui [23] developed the theory of computational quasiconformal geometry on point clouds. Using approximations of the differential operators on point clouds, Liang et al. [24,25] and Choi et al. [26] computed the spherical conformal parameterizations of genus-0 closed surfaces, and Meng et al. [27] computed quasiconformal maps on topological disks. Li-Shi-Sun [28] computed quasiconformal maps from surfaces to planar domain, using the so-called point integral method for discretizing integral equations for point clouds. ...
... The idea of Eq. (5.5) has been used in computing harmonic maps in various previous works, such as [27] and [90]. These two equations in (5.5) can be solved efficiently by the preconditioned conjugate gradient method. ...
... We compare our cubic lattice method with Meng-Choi-Lui's [27] method, which is also a meshless method for computing conformal maps from a topological disk to a rectangle. Our method has a comparable accuracy with Meng-Choi-Lui's method and is a bit slower. ...
