Flexible Isosurface Extraction for Gradient-Based Mesh Optimization (2308.05371v1)

Abstract: This work considers gradient-based mesh optimization, where we iteratively optimize for a 3D surface mesh by representing it as the isosurface of a scalar field, an increasingly common paradigm in applications including photogrammetry, generative modeling, and inverse physics. Existing implementations adapt classic isosurface extraction algorithms like Marching Cubes or Dual Contouring; these techniques were designed to extract meshes from fixed, known fields, and in the optimization setting they lack the degrees of freedom to represent high-quality feature-preserving meshes, or suffer from numerical instabilities. We introduce FlexiCubes, an isosurface representation specifically designed for optimizing an unknown mesh with respect to geometric, visual, or even physical objectives. Our main insight is to introduce additional carefully-chosen parameters into the representation, which allow local flexible adjustments to the extracted mesh geometry and connectivity. These parameters are updated along with the underlying scalar field via automatic differentiation when optimizing for a downstream task. We base our extraction scheme on Dual Marching Cubes for improved topological properties, and present extensions to optionally generate tetrahedral and hierarchically-adaptive meshes. Extensive experiments validate FlexiCubes on both synthetic benchmarks and real-world applications, showing that it offers significant improvements in mesh quality and geometric fidelity.

• The paper introduces FlexiCubes, a novel method for flexible isosurface extraction designed to improve gradient-based mesh optimization by increasing degrees of freedom in vertex positioning.
• FlexiCubes achieves flexibility through dynamic adjustments of vertex positions, quad splitting, and grid deformation using learned weights, enabling better capture of complex geometry.
• Experimental results demonstrate FlexiCubes outperforms existing methods in reconstruction accuracy and mesh quality across applications such as photogrammetry and 3D modeling.

Overview of FlexiCubes for Gradient-Based Mesh Optimization

The presented paper introduces FlexiCubes, a novel approach specifically designed to address challenges in gradient-based mesh optimization. Traditional isosurface extraction techniques such as Marching Cubes and Dual Contouring possess inherent limitations when applied in settings where the mesh needs to be iteratively refined and optimized, often leading to suboptimal mesh quality, restricted geometric fidelity, and issues with numerical stability. FlexiCubes aim to overcome these challenges by expanding the degrees of freedom available during mesh extraction without compromising the stability and quality of the resulting mesh.

Technical Contributions

The core technical contribution of FlexiCubes lies in its enhanced flexibility for vertex positioning within grid cells during the extraction process. This flexibility is achieved through several key innovations:

1. Flexible Vertex Positioning: Unlike classic methods that fix vertex positions based on predefined rules, FlexiCubes introduce interpolation weights ($\alpha$ and $\beta$) for grid corners and edges, respectively. These weights allow dynamic adjustment of vertex positions, increasing the capacity to capture sharp geometric features and improve mesh fit to complex surfaces.
2. Dynamic Quad Splitting: FlexiCubes provide a mechanism to optimize the splitting of quadrilateral faces into triangles through a splitting weight ($\gamma$). This enables adaptation to better align with surface curvatures and preserve geometric details.
3. Grid Deformation: By integrating displacement vectors ($\delta$) at each grid vertex, FlexiCubes facilitate alignment with features that require adaptive spacing, thereby enhancing mesh quality for nuanced structures.
4. Tetrahedral Mesh Extension: FlexiCubes extend their mesh extraction capabilities to include volumetric tetrahedralization, crucial for applications involving physical simulations.
5. Adaptive Meshing: The method supports hierarchical adaptive resolution by refining the grid structure based on task-specific criteria, thereby achieving significant gains in geometric fidelity where needed.

Experimental Validation

Extensive experiments conducted using FlexiCubes demonstrate substantial improvements in various application settings, including photogrammetry, generative 3D modeling, and inverse physics simulations. The research details how FlexiCubes surpasses existing methods in both reconstruction accuracy and intrinsic mesh quality. Quantitative metrics such as Chamfer Distance, F1-score, and edge metrics underscore the superior performance offered by FlexiCubes.

Implications and Future Directions

The enhancements provided by FlexiCubes underscore its potential impact on various research and practical domains. By enabling fine-grained control over mesh topology and geometry through flexible vertex positioning, FlexiCubes opens up avenues for sophisticated mesh-based regularizers that can be dynamically optimized to suit application-specific goals. This adaptability is particularly beneficial for physical simulation applications where interior mesh structures must adhere to stringent quality standards.

Future developments could focus on further integration of complex mesh-based loss terms for enhanced geometry processing, or exploring temporal extensions to address dynamic reconstruction in time-evolving scenarios. Integrating advanced neural operations that synergize with the grid-based FlexiCubes framework might offer promising avenues to push the boundaries of 3D shape synthesis and optimization.

FlexiCubes represent a significant step forward in enhancing the utility and efficacy of gradient-based mesh optimization methodologies, paving the way for more precise and reliable applications across diverse challenges in computer graphics and vision.