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As-Rigid-As-Possible Regularization for Implicit Surfaces
Tobias Djuren, Markus Worchel, Ugo Finnendahl, Marc Alexa
TL;DR
Implicit-surface optimization needs regularization that preserves shape while supporting realistic deformation. The paper computes continuous ARAP energy from uniformly sampled surface points and differentiable implicit representations, reporting broad quality and efficiency benefits in neural shape processing. It also identifies constraint approximation and overlapping-region behavior as limitations.
Problem
Implicit-surface optimization needs regularization that encourages natural deformation and keeps the deformed surface close to its initial shape.
Method
The method directly evaluates continuous ARAP regularization by uniformly sampling the implicit surface and computing deformation differentials at each sample.
Results
Experiments report that direct point-sampled continuous spokes-and-rims ARAP outperforms competing methods in quality and efficiency across several applications.
Takeaways & Limitations
Differential quantities of the forward mapping enable direct energy computation without local patch meshing or additional per-sample resampling.
Takeaways & Limitations
Constraints are only approximated, and overlapping constraint regions may produce unexpected results.
Abstract
from arXiv · showhide
Implicit surface representations have regained popularity because of their use in machine learning. A common component in optimization is regularization, penalizing the deviation of the surface from its original shape. The popular as-rigid-aspossible (ARAP) energy strikes a good compromise between realistic deformation behavior and efficient computation, at least for piecewise linear meshes. We develop an approach for computing the ARAP energy of a deformation function based on point sampling of the surface. The implicit representation is exploited to provide differentials in each sample. The evaluation is efficient and exact in each sample (up to numerical precision). We demonstrate the general applicability of the method to neural shape processing in several applications and contrast its properties with alternatives from the literature.
1. Introduction
The paper transfers ARAP regularization from mesh deformation to differentiable implicit functions using continuous energy and point sampling. Experiments report improved quality and efficiency over competing methods, including quasi-interactive modeling.
- Regularization penalizes deviation from a given shape, supporting constrained interactive modeling and application-specific machine-learning optimization.
- The approach directly transfers ARAP principles into neural implicit functions.
- The method bases its formulation on continuous spokes-and-rims ARAP energy and uniformly samples the undeformed surface.Differentiability supplies per-sample differentials for energy and gradient computation.
- The method outperforms competing methods in both deformation quality and efficiency across several experiments.It is fast enough for quasi-interactive modeling with neural implicits and other ARAP-regularized applications.
2. Related Work
Related work spans mesh and space-based deformation, neural deformation fields, data-driven prediction, and matching methods. The paper distinguishes its approach by directly targeting continuous ARAP energy for implicit surfaces.
- Classical mesh deformation includes lattice-based freeform and cage-based methods that define deformation fields in space.These space deformers can make direct modification by selecting and dragging arbitrary surface points difficult.
- Neural deformation fields support geometry-processing tasks using differential quantities, but earlier work did not minimize the well-known ARAP energy.
- Rigidly constraining nearby points can penalize bending while unnecessarily restricting nearby surface parts that are not connected.
- Data-driven deformation can be very fast but is restricted to deformations observed in training data.Other shape-matching methods solve matching problems, whereas this work uses only a few correspondences or handles.
3. Method
The method derives a continuous ARAP regularizer for implicit surfaces and estimates its surface integral from uniformly sampled points on the undeformed surface. At each sample, deformation differentials quantify stretching and bending while stable rotation derivatives support efficient optimization.
- Deformation representation: The deformation field maps the initial implicit surface to its deformed counterpart, with f denoting the forward deformation function.The inverse f−1 expresses the deformed implicit surface as Φ(f−1(x)) = 0, although either function need not be invertible.
- Continuous energy: The continuous spokes-and-rims ARAP energy integrates stretching and bending over the undeformed surface.The stretching and bending contributions are balanced by r2, and the formulation also applies to curves embedded in R3.
- Stretching: Stretching compares the deformation Jacobian Jf with its closest rotation R, computed through an SVD that removes scaling and reflection.The determinant correction flips the smallest singular-value sign when necessary so R is a proper rotation.
- Bending: Bending penalizes changes in the closest rotation using its derivative tensor dR and the adjusted polar-decomposition factor Y.The bending weight is λbend = r2.
- Sampling and evaluation: The regularization estimates the continuous energy with Monte Carlo integration over uniformly sampled points on the initial implicit surface.Samples can be precomputed before optimization, allowing subsets to be reused instead of resampling the deformed surface each iteration.
- Derivatives: Rotation derivatives are computed through polar decomposition rather than automatic differentiation through iterative SVD, improving stability near repeated or small singular values.This is especially relevant for identity initialization, where U and V are not unique.
4. Experiments
The experiments evaluate implicit ARAP regularization with neural deformation and inverse networks, continuous-region constraints, and several implicit-surface applications. Results cover surface extraction, constrained modeling, comparisons with prior implicit ARAP, and Gauss stylization.
- Network setup: The experiments use signed distance networks trained from surface-near and uniform samples, with deformation networks parameterized by a forward function f.The deformation network can be initialized as the identity, while higher-order derivatives are computed through automatic-differentiation libraries.
- Surface extraction: Figure 4 shows that marching cubes on the initial signed distance function followed by f produces a deformed surface visually indistinguishable from marching cubes on the learned inverse-based field.The inverse-based field is defined using Φ(f^-1(x)).
- Network setup: An inverse network is trained with cycle consistency using deformed surface points and uniform samples in deformed space.Uniform deformed-space samples encourage invertibility and prevent off-surface positions from mapping accidentally onto the initial surface.
- Implicit surface modeling: Implicit ARAP modeling constrains surface points inside cuboids to follow specified transformations, while point-wise constraints remain available as a limiting case.The constraint loss uses precomputed initial-surface samples and their constrained positions.
- Implicit surface modeling: The method supports varied constraint sizes, counts, and movements, with lightweight networks and precomputed samples enabling interactive-rate optimization.The experiments also include strong deformations and supplemental-video trajectories, excluding surface-extraction time.
- Comparison and stylization: Compared with prior Implicit-ARAP, point-sample evaluation avoids local implicit-surface meshing and exposes a parameter that directly controls bending penalty.The comparison also discusses coupling between distant regions and the prior method's restriction to point-wise roto-translations.
5. Discussion
The method computes continuous ARAP regularization directly from differential quantities and point samples, avoiding local patch meshing while retaining numerical exactness. Its scope includes efficient implicit-surface processing, but constraint handling remains approximate and overlapping constraints can cause unexpected results.
- Direct differential computations avoid local patch meshing and additional resampling around each sample point.
- The method computes the exact ARAP energy at each sample up to numerical precision and can achieve almost interactive processing times.
- The approach is demonstrated for varied deformations and meshes, including optimization guidance for surface stylization.
- Constraints are only approximated, unlike mesh-based ARAP constraints, and higher interpolation-loss weights can only partially mitigate this difference.
- Overlapping constraint regions may produce unexpected results, while the method's focus on implicit surfaces is largely independent of the surface representation.