cs.GRAug 16, 2026

As-Rigid-As-Possible Regularization for Implicit Surfaces

Authors: Tobias Djuren, Markus Worchel, Ugo Finnendahl, Marc Alexa

Organizations: TU Berlin, Germany

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Abstract

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.

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Figure 1: Given a set of constrains (colored boxes) we compute the forward deformation function f\mathbf{f} . This allows (1) to train with samples from the initial space since the Arap energy is defined as integral over the initial surface and (2) all samples used for training can be computed before training without the need to resample points from the deformed space after each update of the deformation.
Figure 2: Regularization by the Arap energy penalizes stretching and bending of the surface/curve. The cost of bending is controlled by λbend\lambda_{bend} . Constrained regions are highlighted by colored boxes and should transform to the constrained target region (same color). Deformed implicit values are generated by learning f−1\mathbf{f}^{-1} using a cycle consistency loss.
Figure 3: We compute the deformation by optimizing the forward deformation f\mathbf{f} . To compute the deformed signed distance field, the inverse transformation is required Φ~(x)=Φ(f−1(x))\tilde{\Phi}(\mathbf{x})=\Phi(\mathbf{f}^{-1}(\mathbf{x})) . We suggest to learn f−1\mathbf{f}^{-1} using a circle consistency loss. The deformed surface (middle) can be computed by marching cubes on the initial signed distance function (left) and the forward transformation f\mathbf{f} . The result is typically indistinguishable from the marching cube mesh on Φ~\tilde{\Phi} (right) visually. Blue Whale by Bohdan Lvov under CC-BY Γ Δ .
Figure 4: Implicit Arap regularization allows to model implicit functions by defining constrained regions and minimizing the as-rigid-as possible energy. The top row shows the initial implicitly defined surfaces (extracted by marching cubes [ 23 ] ) and the deformed surfaces are in the bottom row. Whereas in original Arap for meshes [ 35 ] the constraints are usually defined as sets of vertices, for implicit Arap regularization we define the constraints as continuous regions (here as a set of cuboids). Each surface point in a cuboid of the initial surface space should undergo the same rigid transformation as the cuboid (same color). Frog by Bogdan Lapitsky , Tokay Gecko and Bottlenose Dolphin by DigitalLife3D , all under under CC-BY Γ Δ .
Figure 5: Implicit surface modeling with point constraints and controlled bending using different Arap -based approaches. Implicit- Arap [ 2 ] evaluates a discrete Arap energy by locally approximating level sets using patches (right column). Since the patches only approximate the level set surfaces, topologically distant regions on the surface may be coupled and are not deformed independently (all fingers move with the index finger). Increasing the patch size is the only way to penalize bending, which, however, amplifies the effect. Our regularization term is derived from the continuous energy and only requires point samples (middle column). The bending weight λbend\lambda_{\text{bend}} can be used to directly penalize bending without the above compromise.
Figure 6: Implicit surface modeling with point constraints using different Arap -based approaches. The Arap regularization term of Implicit- Arap [ 2 ] (right column) is restricted to deformation fields expressed as point-wise roto-translations, while our regularization term (middle column) does not impose such restrictions. When the same network architecture is used for the signed distance function, our regularization results in better deformations even with a more lightweight deformation network. By using a simple deformation network architecture (e.g., without strict invertibility) and precomputing surface samples, we achieve interactive optimization runtime.
Figure 7: In analogy to Gauss Stylization [ 18 ] and normal-driven spherical shape analogies [ 22 ] , we can use preference functions gg Eq. 5 on the Gauss map (first column) as well as implicit Arap regularization to create diverse, continuously defined stylizations for implicit surfaces. The implicitly defined surfaces (their marching cube meshes [ 23 ] ) that where used as input are in the first row. Dinosaur and Shiba Dog by zixisun02 , Frog by Bogdan Lapitsky , all under CC-BY Γ Δ .
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