GenAnalysis: Joint Shape Analysis by Learning Man-Made Shape Generators with Deformation Regularizations

TL;DR

GenAnalysis introduces an implicit shape generator governed by an as-affine-as-possible (AAAP) deformation prior to learn a manifold of man-made shapes. The method enables joint shape analysis through tangent-space vector fields for single-shape segmentation and via interpolations on the manifold for shape matching, while achieving consistent segmentation by aggregating cues across inter-shape correspondences. It combines a data loss, a KL prior, and the AAAP regularization with a lightweight test-time optimization, and demonstrates state-of-the-art performance on ShapeNetPart for both shape matching and co-segmentation. The results validate the effectiveness of AAAP priors and tangent-space analysis in handling large, structured variations in man-made shapes, with practical implications for robust 3D segmentation and correspondence tasks.

Abstract

We present GenAnalysis, an implicit shape generation framework that allows joint analysis of man-made shapes, including shape matching and joint shape segmentation. The key idea is to enforce an as-affine-as-possible (AAAP) deformation between synthetic shapes of the implicit generator that are close to each other in the latent space, which we achieve by designing a regularization loss. It allows us to understand the shape variation of each shape in the context of neighboring shapes and also offers structure-preserving interpolations between the input shapes. We show how to extract these shape variations by recovering piecewise affine vector fields in the tangent space of each shape. These vector fields provide single-shape segmentation cues. We then derive shape correspondences by iteratively propagating AAAP deformations across a sequence of intermediate shapes. These correspondences are then used to aggregate single-shape segmentation cues into consistent segmentations. We conduct experiments on the ShapeNet dataset to show superior performance in shape matching and joint shape segmentation over previous methods.

Figures (24)

• Figure 1: GenAnalysis learns a shape manifold from a collection of man-made shapes with a novel as-affine-as-possible deformation regularization loss. The learned shape manifold supports shape analysis by analyzing the tangent space of each shape, shape matching through intermediate shapes on this manifold, and consistent segmentation by aggregating single-shape analysis results using inter-shape correspondences.
• Figure 2: Piece-wise affine assumption. Each shape part from the source shape, approximated by its bounding box, undergoes an affine transformation $A_i$ to the corresponding part in the target shape.
• Figure 3: The pipeline of GenAnalysis pipeline, which consists of four stages. (a) The first stage learns an implicit shape generator to fit the input shapes by combing a data loss and an as-affine-as possible deformation loss. (b) The second stage extracts piece-wise affine structures in vector fields of each shape derived from the tangent space of each shape. (c) The third stage computes pairwise shape correspondences obtained by composing correspondences along intermediate shapes defined by the generator. Points with similar color are in correspondence. (d) The last stage performs consistent segmentation using the correspondences obtained in stage three to integrate single-shape segmentation cues derived from stage two.
• Figure 4: As-affine-as-possible (AAAP) regularization. (a) We study infinitesimal perturbation $\mathbf{v}$ in the tangent space at each shape with latent code $\mathbf{z}$. (b) Due to constraint shown in Eq. (\ref{['Eq:Implicit:Cons']}), we can not determine the correspondence $\mathbf{d}_i^{\mathbf{v}}(\mathbf{z})$ that lies on $g^{\theta} (\mathbf{x},\mathbf{z}+\epsilon\mathbf{v}) = 0$ directly. (c) We instead jointly compute all $\mathbf{d}_i^{\mathbf{v}}(\mathbf{z})$ by solving an constrained optimization problem using the objective function in Eq. (\ref{['Eq:E:Obj']}). (d) We show resulting 3D correspondences between source shape colored in white, and a neighboring perturbed shape colored in transparent blue. (e) After derivation in section \ref{['Subsubsec:AAAP:Deformation']}, we arrived at closed form solution shown in Eq. (\ref{['Eq:d:Explicit:Expression']}) where each perturbation $v_i$ corresponds to variation at $\mathbf{d}^{\mathbf{v_i}}(\mathbf{z})$. We integrate over all directions to obtain the regularization term in Eq. (\ref{['Eq:Geo:Regu']}).
• Figure 5: Effectiveness of the AAAP formulation for identifying correspondences between two implicit shapes that have two parts, each of which undergoes an affine transformation. Correspondence errors are color-coded. (Left) The source shape is colored in black. The target shape is colored in red. (Middle) Deform the source shape to align with the target shape under the AAAP model, resulting in accurate correspondences. (Right) Correspondences drift under the ACAP model, i.e., $\mathbf{a}_i = 0$. Errors are color coded.