**Introduction**

This paper introduces a ARAPReg(As-Rigid-As Possible Regularization Loss) for learning parametric mesh generator from a collection of deformable shapes with significant geometric variations but shared topology.

It proposes an unsupervised loss functional to model the local rigidity constraint for generative modeling. This loss can be combined with standard mesh generators like variational auto-encoders (VAEs) and auto-decoders (ADs). An important feature of ARAPReg's loss functional is its consistency with other training losses, offering advantages such as insensitivity to tradeoff parameters and faster convergence.

The main key components of ARAPReg include utilizing the Hessian of the ARAP deformation model to derive a regularizer for the Jacobian of the shape generator and employing a robust norm on the Hessian to capture pose and shape variations.

**ARAPReg Loss**

Note: The paper specifically focuses on 3D generative models using the mesh representation.

Here,the unsupervised loss, $L_{\text{reg}}(\theta)$, is defined as:

$L_{\text{reg}}(\theta) = E_{z-Nk}(E_{{\delta}z-sNk}\|g^{\theta}(z + \delta z) - 2g^{\theta}(z) + g^{\theta}(z - \delta z)\|^2 + \lambda_R \cdot r_R(g^{\theta}(z), \frac{\partial g^{\theta}(z)}{\partial z}))$

**Decoupling Smoothness and Jacobian regularization**

ARAPReg decouples the enforcement of local rigidity into two terms.

The first term in the loss, involving the generator $g_{\theta}$, promotes smoothness by minimizing the difference between shape reconstructions. $\delta z$ represents an infinitesimal displacement in the parameter space.

The second term, $r_R(g_{\theta}(z), \frac{\partial g_{\theta}(z)}{\partial z})$, formulates the regularization loss. It enforces the preservation of local rigidity by considering the generated mesh $g_{\theta}(z)$ and infinitesimal perturbations specified by the Jacobian $\frac{\partial g_{\theta}(z)}{\partial z}$.$λ_R$ is another hyper-parameter of ARAPReg.

The local rigidity term, denoted as $r_R$, which regularizes the Jacobian of the generator in shape generation. The formulation draws inspiration from the as-rigid-as-possible (ARAP) potential function. The ARAP deformation between two meshes, represented by vertex position vectors (g) and (g + x), is minimized by considering the latent rotations $O_i$ associated with each vertex. The ARAP potential depends on the edge set ${\epsilon}$ of the mesh generator.

The ARAP potential function is defined as:

$f_R(g, x) := min_{O_i{∈} SO(3)}{\sum}_{(i,j){∈}{\epsilon}} ∥r_{ij}(O_i, g, x)∥^2$

To formulate $r_R$ based solely on the Jacobian, the Taylor expansion of the ARAP potential energy is considered.

By enforcing the preservation of local rigidity in the tangent space specified by the Jacobian, ARAPReg achieves an accurate first-order approximation of the shape space. This formulation allows for efficient network training, as it only requires the computation of first-order derivatives of the generator, while still promoting the rigidity constraint in the shape space's local neighborhood.

To introduce a formulation that only depends on the Jacobian of the generator, the Taylor expansion of the as-rigid-as possible potential energy is considered.

**Proposition 1** establishes the zero and first-order derivatives of the ARAP potential and defines the Hessian matrix. Approximating the ARAP potential for infinitesimal perturbations $\epsilon Jy$, where J represents the Jacobian, leads to a rigidity potential formulation.

The Hessian matrix $H_R(g)$ is given by:

${\partial}^2f_R(g, 0)/{\partial x}^2 = H_R(g)$,

$H_R(g) = L ⊗ I_3 - A(g)ᵀD(g)^{-1} A(g)$

where $L ∈ R^{n×n}$ is the graph Laplacian associated to ${\epsilon}$; A(g) is a sparse n × n block matrix; D(g) is a diagonal block matrix.

To integrate the rigidity potential over all possible directions $y$, the local rigidity term $r_L$ is introduced by integrating $y^T{\overline{H}_R(g,J)} y$ over the unit sphere $S_k$ in $\mathbb{R}^k$.

$r^{L^2}_R (g,J):= \frac{k}{Vol(S^k)}{\int_{y∈S^k}} y^T{\overline{H}}_R(g,J)ydy$

**Proposition 2** shows that $r_L^2(g, J)$ is equal to the trace of the Hessian matrix $H_R(g, J)$, which is the sum of the eigenvalues of $H(g, J)$.

$r^{L^2}_R(g,J) = Tr({\overline{H}_R} (g,J)) = {\sum}^{k}_{i=1} λ_i ({\overline{H}_R} (g,J))$

where $λ_i({\overline{H}_R} (g,J))$ is the i-th eigenvalue of ${\overline{H}_R (g,J)}$.

**Pose and Shape Variation Modeling**

A simple formulation is also presented to decouple pose and shape variations in the context of enforcing the local rigidity constraint. The eigenvalues $λ_i({\overline{H}_R (g,J)}) = u_i^{T}({\overline{H}_R (g,J)})u_i$, where $u_i$ is the corresponding eigenvector of $λ_i{\overline{H}_R (g,J)}$,, capture the deformations in different directions of the tangent space. It is observed that eigenvectors corresponding to small eigenvalues primarily represent pose variations, while eigenvectors corresponding to large eigenvalues represent shape variations.

To address the limitation of penalizing all directions equally, ARAPReg employs a robust norm to model the local rigidity loss.

$r_R(g, J) = {\sum_{i=1}^k}λ_i^{\alpha}({\overline{H}_R} (g,J))$

where ${\alpha} = 1/2$ .This formulation assigns smaller weights to the subspace spanned by eigenvectors associated with large eigenvalues, which correspond to shape variations. By minimizing this term, the small eigenvalues, related to pose variations, are automatically minimized as well.

It is noted that while prior works aimed to decouple pose and shape in the latent space, the goal here is to model the regularization term by considering both pose and shape variations.

**Final Loss Term**

$L_{\text{reg}}(\theta) = E_{z-Nk}(E_{{\delta}z-sNk}\|g^{\theta}(z + \delta z) - 2g^{\theta}(z) + g^{\theta}(z - \delta z)\|^2 +$
$\lambda_R \cdot {\sum_{i=1}^k}λ_i^{\alpha}({\overline{H}_R} (g^{\theta}(z), \frac{\partial g^{\theta}(z)}{\partial z}))$

In this paper, the hyperparameters s and $λ_R$ are set to 0.05 and 1, respectively, for all experiments.$

This formulation of $L_{reg}(θ)$ combines the smoothness of the generator, the gradient of the generator, and the regularization term based on the eigenvalues of the Hessian matrix. It provides a comprehensive loss function for training the mesh generator, taking into account both local rigidity and smoothness constraints.

The main challenge in using this formulation for training was computing the gradient of the Jacobian regularization term. To address this, a gradient computation approach is introduced that only requires computing the derivatives of $g_θ$.It is in the material shared.

**Application in Learning Mesh Generators**

The network architecture used in the experiments is introduced, followed by the integration of the unsupervised loss into two formulations of training mesh generators: variational auto-encoders (VAEs) and auto-decoders.

The network architecture focuses on the decoder network $g^θ$, while VAEs also utilize an encoder network $h^φ$. The decoder network has six layers, with the first layer concatenating latent features associated with each vertex of the coarse mesh as input.

For VAEs, the optimization problem for training the auto-encoder combining $g^θ$ and $h^φ$ is solved.The loss consists of the standard VAE loss terms, where $λ_{KL}$ and $λ_{reg}$ are hyperparameters.

For auto-decoders, the encoder is replaced with latent variables $z_i$ associated with the training meshes. The optimization problem (equation 11) is solved using alternating minimization. The latent parameters zi are initialized as an empirical distribution.

In both formulations, the unsupervised loss $L_{reg}(θ)$ is incorporated, ensuring the preservation of local rigidity. The hyperparameters and optimization methods are specified for each formulation.

So, these formulations demonstrate how the unsupervised loss can be effectively integrated into variational auto-encoders and auto-decoders, facilitating the training of mesh generators while considering local rigidity constraints.

**Shape Interpolation**

Given two shapes, g1 and g2, their corresponding latent parameters, z1 and z2, are obtained. For the VAE model, z1 and z2 come from the encoder, while for the AD model, they are obtained by optimizing the reconstruction error. Interpolation is performed by linearly interpolating z1 and z2.

Notably, the approach better preserves prominent shape features, such as fingers, and introduces less distortion among joint regions.

**Shape Extrapolation**

In the evaluation of shape extrapolation, the focus is on extending the capabilities of the shape generator. A center shape, g, is chosen, and its corresponding latent parameters, z, are obtained.Extrapolation is performed by randomly sampling new latent parameters from a distribution centered around z.

The comparison reveals that this approach generates smoother and more realistic shapes, particularly in areas such as animal tails and human hands and arms.

The proposed method outperforms state-of-the-art techniques, providing substantial advancements in preserving geometric features and generating realistic deformable shapes.

**Conclusion**

In Conclusion, ARAPReg, the introduced unsupervised loss functional, enhances the performance of shape generators. It enables the generation of novel shapes that preserve their characteristics at multiple scales.

**Limitation**

It currently requires datasets with predefined correspondences and has not been explored for unorganized shape collections. Future research could focus on addressing these limitations and extending the formulation to handle synthetic shapes of various forms and functions.