A Theory of Topological Derivatives for Inverse Rendering of Geometry

We introduce a theoretical framework for differentiable surface evolution that allows discrete topology changes through the use of topological derivatives for variational optimization of image functionals. While prior methods for inverse rendering of geometry rely on silhouette gradients for topology changes, such signals are sparse. In contrast, our theory derives topological derivatives that relate the introduction of vanishing holes and phases to changes in image intensity. As a result, we enable differentiable shape perturbations in the form of hole or phase nucleation. We validate the proposed theory with optimization of closed curves in 2D and surfaces in 3D to lend insights into limitations of current methods and enable improved applications such as image vectorization, vector-graphics generation from text prompts, single-image reconstruction of shape ambigrams and multi-view 3D reconstruction.

A Level Set Theory for Neural Implicit Evolution under Explicit Flows

Coordinate-based neural networks parameterizing implicit surfaces have emerged as efficient representations of geometry. They effectively act as parametric level sets with the zero-level set defining the surface of interest. We present a framework that allows applying deformation operations defined for triangle meshes onto such implicit surfaces. Several of these operations can be viewed as energy-minimization problems that induce an instantaneous flow field on the explicit surface. Our method uses the flow field to deform parametric implicit surfaces by extending the classical theory of level sets. We also derive a consolidated view for existing methods on differentiable surface extraction and rendering, by formalizing connections to the level-set theory. We show that these methods drift from the theory and that our approach exhibits improvements for applications like surface smoothing, mean-curvature flow, inverse rendering and user-defined editing on implicit geometry.

Modulated Periodic Activations for Generalizable Local Functional Representations

Multi-Layer Perceptrons (MLPs) make powerful functional representations for sampling and reconstruction problems involving low-dimensional signals like images,shapes and light fields. Recent works have significantly improved their ability to represent high-frequency content by using periodic activations or positional encodings. This often came at the expense of generalization: modern methods are typically optimized for a single signal. We present a new representation that generalizes to multiple instances and achieves state-of-the-art fidelity. We use a dual-MLP architecture to encode the signals. A synthesis network creates a functional mapping from a low-dimensional input (e.g. pixel-position) to the output domain (e.g. RGB color). A modulation network maps a latent code corresponding to the target signal to parameters that modulate the periodic activations of the synthesis network. We also propose a local-functional representation which enables generalization. The signal's domain is partitioned into a regular grid,with each tile represented by a latent code. At test time, the signal is encoded with high-fidelity by inferring (or directly optimizing) the latent code-book. Our approach produces generalizable functional representations of images, videos and shapes, and achieves higher reconstruction quality than prior works that are optimized for a single signal.