MeshSDF: Differentiable Iso-Surface Extraction (2006.03997v2)

Abstract: Geometric Deep Learning has recently made striking progress with the advent of continuous Deep Implicit Fields. They allow for detailed modeling of watertight surfaces of arbitrary topology while not relying on a 3D Euclidean grid, resulting in a learnable parameterization that is not limited in resolution. Unfortunately, these methods are often not suitable for applications that require an explicit mesh-based surface representation because converting an implicit field to such a representation relies on the Marching Cubes algorithm, which cannot be differentiated with respect to the underlying implicit field. In this work, we remove this limitation and introduce a differentiable way to produce explicit surface mesh representations from Deep Signed Distance Functions. Our key insight is that by reasoning on how implicit field perturbations impact local surface geometry, one can ultimately differentiate the 3D location of surface samples with respect to the underlying deep implicit field. We exploit this to define MeshSDF, an end-to-end differentiable mesh representation which can vary its topology. We use two different applications to validate our theoretical insight: Single-View Reconstruction via Differentiable Rendering and Physically-Driven Shape Optimization. In both cases our differentiable parameterization gives us an edge over state-of-the-art algorithms.

• The paper introduces MeshSDF, a framework that computes derivatives of surface samples from deep implicit fields for gradient-based optimization.
• It derives a closed-form expression that transforms traditional non-differentiable iso-surface extraction into an end-to-end differentiable process.
• Application validations in single-view 3D reconstruction and aerodynamic shape optimization demonstrate improved accuracy and practical utility.

Differentiable Iso-Surface Extraction: A Technical Overview

The paper entitled "MeshSDF: Differentiable Iso-Surface Extraction" explores a significant development in the domain of Geometric Deep Learning, primarily addressing the challenge of converting implicit deep field representations into explicit surface mesh representations while maintaining differentiability. This approach, termed MeshSDF, is noteworthy for its capacity to bridge a gap in the realization of practical applications where implicit field models falter due to non-differentiable extraction mechanisms.

The core advancement presented in this paper lies in the ability to differentiate the 3D position of surface samples with respect to the underlying deep implicit field. This differentiation is achieved by understanding the local surface geometry impact due to perturbations in the implicit field, thus enabling the transformational use of Marching Cubes or other non-differentiable iso-surface extraction algorithms in a differentiable manner.

Contributions and Methodology

1. Theoretical Insight: The authors introduce a theoretical framework which demonstrates how differentiation can be successfully applied to implicit field-generated iso-surfaces. This is accomplished by deriving a closed-form expression for the derivative of surface samples concerning the implicit field. This aspect is crucial as it allows the use of implicit representations in applications where explicit surface representations are a necessity without compromising on differentiability.
2. MeshSDF Framework: The resultant of this theoretical insight is the MeshSDF, a differentiable mesh framework capable of adapting topology. The MeshSDF framework stands out as it allows for extraction and manipulation of surface meshes from implicit fields with end-to-end differentiability, a feat that was not previously attainable with existing methods that often suffered from limitations in resolution or topology.
3. Applications and Validation: The authors validate the broad applicability of MeshSDF through two distinct case studies – Single-View Reconstruction via Differentiable Rendering and Physically-Driven Shape Optimization for aerodynamic purposes. In the former, MeshSDF supports image-to-3D model refinement with enhanced accuracy, while in the latter, it facilitates complex shape optimization tasks for aerodynamic efficiency, achieving improvements over state-of-the-art techniques.

Implications and Future Directions

The implications of MeshSDF are manifold, both in theoretical and practical spheres. Theoretically, it enhances the utility of deep implicit fields, making them more suitable for a wider range of applications that demand explicit mesh modeling. Practically, it opens avenues in fields such as computational fluid dynamics, where explicit 3D mesh output is a necessity for simulations, and physically-based rendering, which benefits from MeshSDF's efficient topology manipulation abilities.

Future research could expand on this work by exploring more complex loss functions that can be directly formulated in terms of implicit field representations, thus increasing the scope of applications. Furthermore, enhancing the efficiency and scalability of the algorithms to handle higher-dimensional data or more intricate geometric shapes could broaden the adoption of these methodologies in industries such as automotive design, aerospace, and beyond.

In summary, MeshSDF represents a significant advancement in the field of geometric modeling and deep learning, offering robust solutions to previously challenging problems associated with differentiable modeling of complex 3D geometries. This research exemplifies the profound impact of theoretical insights on practical applications in computer science, paving the way for improved capabilities in digital design and simulation.