Auditing Training-Free 3D Shape Retrieval with Diffused Geodesic Moments
Authors: Zhicheng Du, Changyue Liu, Wenji Xi, Zhaotian Xie, Zhuo Deng, Ziheng Zhang, Yang Liu, Lan Ma
Organizations: Tsinghua Shenzhen International Graduate School, Tsinghua University · Guangzhou International Economics College · School of Electrical and Electronic Engineering, The University of Sheffield
Abstract
Reported retrieval scores for training-free shape descriptors conflate local signal design, normalization, aggregation, codebook fitting, and metric choices, making isolated component evaluation difficult. This paper reframes descriptor evaluation as a {\em protocol audit}. We introduce Diffused Geodesic Moments (DGM), a seed-conditioned descriptor that computes sparse implicit heat responses, converts them to distance-like fields, and summarizes each vertex by low-order moments across seeds and scales. DGM is used both as a practical non-spectral baseline and as an instrument for isolating protocol effects. On the registered FAUST benchmark split (FAUST-Reg) and the TOSCA shape collection, aggregation-matched experiments show that an independent Geometric Moment Shape Descriptor baseline built on Heat Kernel Signature features (GMSD-HKS) obtains the highest scores in this implementation (0.621/0.820 and 0.865/0.963 mean average precision (mAP)/top-1), Wave Kernel Signature (WKS) remains a strong classical signal, and DGM is useful mainly when sparse solves, non-spectral deployment, or symmetry-informative seed frames are priorities. The broader finding is methodological: the input field and aggregation protocol can dominate the moment formula. The paper contributes a reproducible protocol-cascade analysis, a cross-shape alignment diagnostic for functional-map compatibility, and concrete recommendations for designing and reporting training-free shape descriptors.
Classical spectral descriptors such as the Heat Kernel Signature and Wave Kernel Signature are widely used for non-rigid 3D shape retrieval, yet their failure modes remain poorly understood. We present a frequency-scale saliency framework that audits these descriptors by quantifying the retrieval-level contribution of each descriptor scale interval through ablation. We introduce class spectral fingerprints to characterize category-level scale dependence, and show that descriptor similarity between class pairs is substantially correlated with retrieval failure, with a Spearman correlation of 0.479. Experiments on SHREC'11 demonstrate that short scales dominate retrieval performance while long scales are harmful, that HKS and WKS exhibit distinct scale dependence patterns, and that saliency-weighted retrieval improves mAP on hard categories by 0.156, with cross-fold and random-weight controls confirming that the gain is stable and not due to arbitrary reweighting.
PCA can be used for rotation invariant features, describing a shape with its pab=E[(xi−E[xa])(xb−E[xb])] covariance matrix approximating shape by ellipsoid, allowing for rotation invariants like its traces of powers. However, real shapes are usually much more complicated, hence there is proposed its extension to e.g. pabc=E[(xa−E[xa])(xb−E[xb])(xc−E[xc])] order-3 or higher tensors describing central moments, or polynomial times Gaussian allowing decodable shape descriptors of arbitrarily high accuracy, and their analogous rotation invariants. Its practical applications could be rotation-invariant features to include shape modulo rotation e.g. for molecular shape descriptors, or for up to rotation object recognition in 2D images/3D scans maybe also for 3D scene understanding, or shape similarity metric allowing inexpensive comparison of objects modulo rotation avoiding costly optimization over rotations.
Large pretrained vision models have substantially improved appearance-based 3D shape retrieval, but they still confuse shapes that look similar while differing in geometry. Although geometry-aware features can reduce these errors, naive fusion of geometry and appearance may hurt retrieval when the two modalities are already well aligned. We propose GATE-3D, a lightweight query-adaptive reranking method that incorporates geometry without retraining the retrieval backbone. For each query, GATE-3D predicts how much a geometry-aware score should adjust the appearance-based ranking using features that capture disagreement between the two modalities. This selective design lets geometry contribute where it helps and stay silent where it would hurt. Experiments on three open-set 3D retrieval benchmarks show that GATE-3D improves over appearance-only retrieval and is more robust than always-on fusion. On the primary benchmark, it improves mAP@10 by 2.00 points over appearance-only retrieval (p=0.041); it also improves leave-one-category-out generalization and reduces geometric false positives by 10.8%. GATE-3D achieves competitive zero-shot results against DAC-based baselines. We further find that simple linear routing is more effective than a small MLP in the low-data regime, suggesting that cross-modal disagreement features matter more than model capacity for adaptive routing.