Deep functional map frameworks (DFM) for shape correspondence are powerful, yet fundamentally limited by their reliance on end-to-end differentiability. This constraint prevents the integration of highly accurate, non-differentiable refinement techniques, capping their overall performance, especially on challenging non-isometric shapes. To overcome this, we introduce MDND, a novel DFM paradigm built on the principle of merging differentiable and non-differentiable components. Our framework facilitates unsupervised learning guided by an internal, non-differentiable refinement. Specifically, MDND employs a dual-branch architecture: a non-differentiable refinement branch leverages a novel, multiscale iterative solver to produce highly robust correspondences, acting as a refined target. Concurrently, a fully differentiable branch learns to predict correspondences from features. The entire system is trained end-to-end without supervision by enforcing a consistency loss that compels the differentiable branch to learn from the superior, refined results of the non-differentiable branch. Extensive experiments show that MDND sets a new state-of-the-art, demonstrating remarkable robustness on shapes with non-isometric deformations and topological noise.

In recent years, deep functional maps (DFM) have emerged as a leading learning-based framework for non-rigid shape-matching problems, offering diverse network architectures for this domain. This richness also makes exploring better and novel design beliefs for existing powerful DFM components to promote performance meaningful and engaging. This paper delves into this problem and successfully produces the SEDFMNet, a simple yet highly efficient DFM pipeline. To achieve this, we systematically deconstruct the core modules of the general DFM framework and analyze key design choices in existing approaches to identify the most critical components through extensive experiments. By reassembling these crucial components, we culminate in developing our SEDFMNet, which features a simpler structure than conventional DFM pipelines while delivering superior performance. Our approach is rigorously validated through comprehensive experiments on diverse datasets, where the SEDFMNet consistently achieves state-of-the-art results, even in challenging scenarios such as non-isometric shape matching and shape matching with topological noise. Our work offers fresh insights into DFM research and opens new avenues for advancing this field.

A large number of 3D spectral descriptors have been proposed in the literature, which act as an essential component for 3D deformable shape matching and related applications. An outstanding descriptor should have desirable natures including high-level descriptive capacity, cheap storage, and robustness to a set of nuisances. It is, however, unclear which descriptors are more suitable for a particular application. This paper fills the gap by comprehensively evaluating nine state-of-the-art spectral descriptors on ten popular deformable shape datasets as well as perturbations such as mesh discretization, geometric noise, scale transformation, non-isometric setting, partiality, and topological noise. Our evaluated terms for a spectral descriptor cover four major concerns, i.e., distinctiveness, robustness, compactness, and computational efficiency. In the end, we present a summary of the overall performance and several interesting findings that can serve as guidance for the following researchers to construct a new spectral descriptor and choose an appropriate spectral feature in a particular application.

Traditional deep functional map frameworks are widely used for 3D shape matching; however, many methods fail to adaptively capture the relevant frequency information required for functional map estimation in complex scenarios, leading to poor performance, especially under significant deformations. To address these challenges, we propose a novel unsupervised learning-based framework, Deep Frequency Awareness Functional Maps (DFAFM), specifically designed to tackle diverse shape-matching problems. Our approach introduces the Spectral Filter Operator Preservation constraint, which ensures the preservation of critical frequency information. These constraints promote frequency awareness by learning a set of spectral filters and incorporating them as a loss function to jointly supervise the functional maps, pointwise maps, and spectral filters. The spectral filters are constructed using orthonormal Jacobi polynomials with learnable coefficients, enabling adaptive and efficient frequency representation. Furthermore, we propose a refinement strategy that leverages the learned spectral filters and constraints to enhance the accuracy of the final pointwise map. Extensive experiments conducted on multiple benchmark datasets demonstrate that our method outperforms state-of-the-art approaches, particularly in challenging scenarios involving non-isometric deformations and inconsistent topology.

Abstract In deep functional maps, the regularizer computing the functional map is especially crucial for ensuring the global consistency of the computed pointwise map. As the regularizers integrated into deep learning should be differentiable, it is not trivial to incorporate informative axiomatic structural constraints into the deep functional map, such as the orientation‐preserving term. Although commonly used regularizers include the Laplacian‐commutativity term and the resolvent Laplacian commutativity term, these are limited to single‐scale analysis for capturing geometric information. To this end, we propose a novel and theoretically well‐justified regularizer commuting the functional map with the multiscale spectral manifold wavelet operator. This regularizer enhances the isometric constraints of the functional map and is conducive to providing it with better structural properties with multiscale analysis. Furthermore, we design an unsupervised deep functional map with the regularizer in a fully differentiable way. The quantitative and qualitative comparisons with several existing techniques on the (near‐)isometric and non‐isometric datasets show our method's superior accuracy and generalization capabilities. Additionally, we illustrate that our regularizer can be easily inserted into other functional map methods and improve their accuracy.