Xifeng Gao

Hexahedral mesh re-parameterization from aligned base-complex

Recently, generating a high quality all-hex mesh of a given volume has gained much attention. However, little, if any, effort has been put into the optimization of the hex-mesh structure, which is equally important to the local element quality of a hex-mesh that may influence the performance and accuracy of subsequent computations. In this paper, we present a first and complete pipeline to optimize the global structure of a hex-mesh. Specifically, we first extract the base-complex of a hex-mesh and study the misalignments among its singularities by adapting the previously introduced hexahedral sheets to the base-complex. Second, we identify the valid removal base-complex sheets from the base-complex that contain misaligned singularities. We then propose an effective algorithm to remove these valid removal sheets in order. Finally, we present a structure-aware optimization strategy to improve the geometric quality of the resulting hex-mesh after fixing the misalignments. Our experimental results demonstrate that our pipeline can significantly reduce the number of components of a variety of hex-meshes generated by state-of-the-art methods, while maintaining high geometric quality.

A robust high-capacity affine-transformation-invariant scheme for watermarking 3D geometric models

In this article we propose a novel, robust, and high-capacity watermarking method for 3D meshes with arbitrary connectivities in the spatial domain based on affine invariants. Given a 3D mesh model, a watermark is embedded as affine-invariant length ratios of one diagonal segment to the residing diagonal intersected by the other one in a coplanar convex quadrilateral. In the extraction process, a watermark is recovered by combining all the watermark pieces embedded in length ratios through majority voting. Extensive experimental results demonstrate the robustness, high computational efficiency, high capacity, and affine-transformation-invariant characteristics of the proposed approach.

Robust hex-dominant mesh generation using field-guided polyhedral agglomeration

We propose a robust and efficient field-aligned volumetric meshing algorithm that produces hex-dominant meshes, i.e. meshes that are predominantly composed of hexahedral elements while containing a small number of irregular polyhedra. The latter are placed according to the singularities of two optimized guiding fields, which allow our method to generate meshes with an exceptionally high amount of isotropy. The field design phase of our method relies on a compact quaternionic representation of volumetric octa-fields and a corresponding optimization that explicitly models the discrete matchings between neighboring elements. This optimization naturally supports alignment constraints and scales to very large datasets. We also propose a novel extraction technique that uses field-guided mesh simplification to convert the optimized fields into a hexdominant output mesh. Each simplification operation maintains topological validity as an invariant, ensuring manifold output. These steps easily generalize to other dimensions or representations, and we show how they can be an asset in existing 2D surface meshing techniques. Our method can automatically and robustly convert any tetrahedral mesh into an isotropic hex-dominant mesh and (with minor modifications) can also convert any triangle mesh into a corresponding isotropic quad-dominant mesh, preserving its genus, number of holes, and manifoldness. We demonstrate the benefits of our algorithm on a large collection of shapes provided in the supplemental material along with all generated results.

Structured Volume Decomposition via Generalized Sweeping.

In this paper, we introduce a volumetric partitioning strategy based on a generalized sweeping framework to seamlessly partition the volume of an input triangle mesh into a collection of deformed cuboids. This is achieved by a user-designed volumetric harmonic function that guides the decomposition of the input volume into a sequence of two-manifold level sets. A skeletal structure whose corners correspond to corner vertices of a 2D parameterization is extracted for each level set. Corners are placed so that the skeletal structure aligns with features of the input object. Then, a skeletal surface is constructed by matching the skeletal structures of adjacent level sets. The surface sheets of this skeletal surface partition the input volume into the deformed cuboids. The collection of cuboids does not exhibit T-junctions, significantly simplifying the hexahedral mesh generation process, and in particular, it simplifies fitting trivariate B-splines to the deformed cuboids. Intersections of the surface sheets of the skeletal surface correspond to the singular edges of the generated hex-meshes. We apply our technique to a variety of 3D objects and demonstrate the benefit of the structure decomposition in data fitting.

Intraoperative registration of preoperative 4D cardiac anatomy with real-time MR images

Co-registering pre- and intra- operative MR data is an important yet challenging problem due to different acquisition parameters, resolutions, and plane orientations. Despite its importance, previous approaches are often computationally intensive and thus cannot be employed in real-time. In this paper, a novel three-step approach is proposed to dynamically register pre-operative 4D MR data with intra-operative 2D RT-MRI to guide intracardiac procedures. Specifically, a novel preparatory step, executed in the pre-operative phase, is introduced to generate bridging information that can be used to significantly speed up the on-the-fly registration in the intraoperative procedure. Our experimental results demonstrate an accuracy of 0.42 mm and a processing speed of 26 FPS of the proposed approach on an off-the-shelf PC. This approach, is in particularly developed for performing intra-cardiac procedures with real-time MR guidance.

Hexahedral mesh quality improvement via edge-angle optimization

Abstract We introduce a simple and practical technique to untangle and improve hexahedral (hex-) meshes. We achieve that by enabling the deformation of the boundary surfaces during the untangling process, which provides more space to reach a valid solution. To improve the element quality, an angle-based optimization strategy is proposed, which has much simpler formulation than the existing methods. The deformed volume after optimization is then pulled back to the original one using an inversion-free deformation. In contrast to the current methods, we perform the untangling and quality improvement within a few local regions surrounding elements with undesired quality, which can effectively improve the minimum scaled Jacobian (MSJ) quality of the mesh over the state-of-the-art method. We demonstrate the effectiveness of our methods by applying it to hex-meshes generated by a range of methods.

Hexahedral Meshing With Varying Element Sizes.

Hexahedral (or Hex‐) meshes are preferred in a number of scientific and engineering simulations and analyses due to their desired numerical properties. Recent state‐of‐the‐art techniques can generate high‐quality hex‐meshes. However, they typically produce hex‐meshes with uniform element sizes and thus may fail to preserve small‐scale features on the boundary surface. In this work, we present a new framework that enables users to generate hex‐meshes with varying element sizes so that small features will be filled with smaller and denser elements, while the transition from smaller elements to larger ones is smooth, compared to the octree‐based approach. This is achieved by first detecting regions of interest (ROIs) of small‐scale features. These ROIs are then magnified using the as‐rigid‐as‐possible deformation with either an automatically determined or a user‐specified scale factor. A hex‐mesh is then generated from the deformed mesh using existing approaches that produce hex‐meshes with uniform‐sized elements. This initial hex‐mesh is then mapped back to the original volume before magnification to adjust the element sizes in those ROIs. We have applied this framework to a variety of man‐made and natural models to demonstrate its effectiveness.

Evaluating Hex-mesh Quality Metrics via Correlation Analysis

Hexahedral (hex‐) meshes are important for solving partial differential equations (PDEs) in applications of scientific computing and mechanical engineering. Many methods have been proposed aiming to generate hex‐meshes with high scaled Jacobians. While it is well established that a hex‐mesh should be inversion‐free (i.e. have a positive Jacobian measured at every corner of its hexahedron), it is not well‐studied that whether the scaled Jacobian is the most effective indicator of the quality of simulations performed on inversion‐free hex‐meshes given the existing dozens of quality metrics for hex‐meshes. Due to the challenge of precisely defining the relations among metrics, studying the correlations among different quality metrics and their correlations with the stability and accuracy of the simulations is a first and effective approach to address the above question. In this work, we propose a correlation analysis framework to systematically study these correlations. Specifically, given a large hex‐mesh dataset, we classify the existing quality metrics into groups based on their correlations, which characterizes their similarity in measuring the quality of hex‐elements. In addition, we rank the individual metrics based on their correlations with the accuracy and stability metrics for simulations that solve a number of elliptic PDE problems. Our preliminary experiments suggest that metrics that assess the conditioning of the elements are more correlated to the quality of solving elliptic PDEs than the others. Furthermore, an inversion‐free hex‐mesh with higher average quality (measured by any quality metrics) usually leads to a more accurate and stable computation of elliptic PDEs. To support our correlation study and address the lack of a publicly available large hex‐mesh dataset with sufficiently varying quality metric values, we also propose a two‐level perturbation strategy to generate the desired dataset from a small number of meshes to exclude the influences of element numbers, vertex connectivity, and volume sizes to our study.

Robust structure simplification for hex re-meshing

We introduce a robust and automatic algorithm to simplify the structure and reduce the singularities of a hexahedral mesh. Our algorithm interleaves simplification operations to collapse sheets and chords of the base complex of the input mesh with a geometric optimization, which improves the elements quality. All our operations are guaranteed not to introduce elements with negative Jacobians, ensuring that our algorithm always produces valid hex-meshes, and not to increase the Hausdorff distance from the original shape more than a user-defined threshold, ensuring a faithful approximation of the input geometry. Our algorithm can improve meshes produced with any existing hexahedral meshing algorithm --- we demonstrate its effectiveness by processing a dataset of 194 hex-meshes created with octree-based, polycube-based, and field-aligned methods.