HotSpot: Signed Distance Function Optimization with an Asymptotically Sufficient Condition

TL;DR

The paper addresses the limitation that eikonal-based regularization is only a necessary condition and can fail to yield a true signed distance function, often causing unstable optimization. It introduces HotSpot, a neural SDF optimization framework grounded in the screened Poisson equation, deriving a heat loss via $h(oldsymbol{x}) = e^{-\lambda |u(oldsymbol{x})|}$ and combining boundary, eikonal, and heat terms to achieve an asymptotically sufficient distance function as $\lambda \to \infty$. The authors prove spatial and temporal stability of the approach and show that the heat loss naturally penalizes surface area, avoiding distortions seen with prior area-based regularization. Empirically, HotSpot yields superior surface reconstructions and distance approximations on 2D and 3D datasets, including high-genus geometries, and enhances sphere-tracing efficiency due to more accurate near-surface fields.

Abstract

We propose a method, HotSpot, for optimizing neural signed distance functions. Existing losses, such as the eikonal loss, act as necessary but insufficient constraints and cannot guarantee that the recovered implicit function represents a true distance function, even if the output minimizes these losses almost everywhere. Furthermore, the eikonal loss suffers from stability issues in optimization. Finally, in conventional methods, regularization losses that penalize surface area distort the reconstructed signed distance function. We address these challenges by designing a loss function using the solution of a screened Poisson equation. Our loss, when minimized, provides an asymptotically sufficient condition to ensure the output converges to a true distance function. Our loss also leads to stable optimization and naturally penalizes large surface areas. We present theoretical analysis and experiments on both challenging 2D and 3D datasets and show that our method provides better surface reconstruction and a more accurate distance approximation.

Figures (24)

• Figure 1: We propose HotSpot, a neural signed distance function optimization method that establishes an asymptotic sufficient condition to guarantee convergence to a true distance function, enabling precise surface reconstruction and level set representation for complex shapes. Here we show a reconstruction from a point cloud sampled from the reference bunny (taken from Mehta et al. Mehta:2022:LST) on the right. In the inset, we visualize the recovered signed distance function on a horizontal slice, using warm colors for positive values and cool for negative (zoom in for details). Our reconstruction is significantly more accurate than prior works (SAL atzmon2020sal, DiGS ben2022digs, and StEik yang2023steik).
• Figure 2: Existing constraints are only necessary but insufficient conditions. That is, there exist solutions, like \ref{['fig:1d']}, that satisfy them almost everywhere yet fail to be distance functions. In contrast, we use a heat loss, modeled by a screened Poisson equation with parameter $\lambda$. As the parameter $\lambda$ goes to infinity, the minimizer converges to the distance function and excludes other solutions. Thus, our method is asymptotically both necessary and sufficient.
• Figure 3: We illustrate a 1D neural signed distance function optimization using the classical eikonal loss gropp2020implicit and our model. The $x$ axis is the domain and the $y$ axis shows the output of the implicit function. The middle and bottom rows show the intermediate and final states of the optimization respectively. The eikonal loss, as a necessary but insufficient condition, is incapable of converging to the actual signed distance function (dashed line), even when the function satisfies the eikonal equation almost everywhere.
• Figure 4: We show an illustration of the relation between screened Poisson equation and a distance field. The top shows a 1D example, where on the left we show a solution to \ref{['eq:heat']} with different absorption $\lambda$, and on the right we show the reconstructed distance $-\frac{1}{\lambda} \ln(h)$ (\ref{['eq:heat_to_distance']}). Boundary loss for real boundary points acts as an isothermal heat source, while heat loss diffuses the heat. When $\lambda$ is increased, the heat decays faster, and the error of the reconstructed function approaches 0. The bottom shows a 2D example.
• Figure 5: We visualize the direction of the changes of the implicit function during one iteration of optimization. Left shows the result with only the eikonal loss and right shows the result after adding our loss. Our loss helps to escape the local minimum in \ref{['fig:1d']}. The heat diffusion encourages the reconstructed absolute distance to increase monotonically away from the boundary.

Theorems & Definitions (2)

• Proposition 1
• Proposition 2