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FUSE: A Flow-based Mapping Between Shapes

Conference: ECCV 2026
Paper: ECCV
Code: https://github.com/LorenzoOlearo/FUSE-Flow-based-Mapping-Between-Shapes
Area: 3D Vision
Keywords: shape matching, flow matching, neural representation, geodesic feature embedding, invertible flow

TL;DR

Each shape is represented as an invertible continuous flow from a shared unit-Gaussian anchor (trained with flow matching); composing the source shape's inverse flow with the target shape's forward flow yields a pointwise map between them β€” no paired training and no test-time optimization, plus cross-representation matching across meshes, point clouds, SDFs and tetrahedral meshes, reaching the lowest or near-lowest correspondence error on landmark-supervised benchmarks such as FAUST and SMAL.

Background & Motivation

Shape matching has long been recognized as a "key problem" in geometry processing: once accurate correspondences are available, downstream tasks such as pose transfer and shape interpolation follow almost for free. The mainstream line of work has been built on triangle meshes, where the functional maps framework represents a correspondence as a linear operator between two functional spaces. It is smooth and insensitive to deformation, but for exactly that reason it relies on spectral bases β€” which are only natural on meshes β€” and such methods struggle to guarantee bijectivity. Another line is optimal transport (OT), which writes the correspondence as a transport plan moving a feature distribution from one shape to the other; however, under sampling imbalance or feature ambiguity it exhibits mass splitting, where the probability mass of a single point is spread over several targets, degrading the sharpness and stability of the recovered discrete map. Registration-based approaches instead optimize 3D deformation fields directly, achieving good accuracy at the cost of solving heavy optimization problems and depending on rigid initialization or learned priors. Meanwhile, neural representations (NeRF, DeepSDF, Geometry Distributions, …) have rapidly diversified how 3D geometry is encoded, yet they do not come with the suite of analysis and manipulation tools that meshes enjoy.

The real tension is this: geometric representations keep multiplying, while almost every matching pipeline hard-wires its prior into one specific representation. Diff3F must first render meshes before distilling semantic features from a large vision model; ULRSSM needs large-scale training data to learn spectral features; functional-map methods need spectral bases. Establishing a correspondence between two different representations therefore either requires retraining a model or falling back to 3D coordinates β€” an embedding available everywhere but encoding neither semantics nor intrinsic geometry. What this paper wants is a map representation that is both general and accurate: it asks only two things of a shape β€” that points can be sampled from its surface, and that a pointwise embedding can be computed β€” regardless of whether that shape is a mesh, a point cloud, an SDF or a volumetric mesh, and it requires no pairwise optimization at all.

Core idea: let each shape independently learn an invertible flow from a shared Gaussian anchor to its own embedding distribution, and connect the two shapes' flows through that same Gaussian; composing the source's inverse flow with the target's forward flow gives the map β€” the only shared information is a handful of landmarks, so no paired training and no test-time optimization are needed.

Method

Overall Architecture

The inputs are two shapes \(S_1\), \(S_2\) of a common semantic class (in any representation, as long as surface points can be sampled) plus a few landmarks. The pipeline has four steps: sample points on both surfaces and compute a pointwise embedding for each point (by default, geodesic distance features to the landmarks); then train one flow model per shape independently, transporting a fixed shared Gaussian anchor \(p_0 = \mathcal{N}(0,1)^d\) to that shape's embedding distribution \(p_\ell\); then define the map from source to target as the composition of the two flows β€” push the embeddings of \(S_1\) back to the Gaussian with the source's inverse flow, then forward to \(S_2\) with the target's forward flow, yielding a continuous map whose images already lie in the target embedding space; finally, run a single nearest-neighbor search in embedding space to discretize the continuous map into a pointwise correspondence. There is no paired training and no test-time fine-tuning anywhere, and the only information the two shapes share is the set of landmarks.

%%{init: {'flowchart': {'rankSpacing': 24, 'nodeSpacing': 28, 'padding': 6, 'wrappingWidth': 400}}}%%
flowchart TD
    A["Shapes S1 / S2<br/>meshΒ·point cloudΒ·SDFΒ·volume"] --> B["Geodesic feature embedding & landmarks<br/>surface sampling β†’ per-point features"]
    B --> C["Per-shape invertible flow & shared Gaussian anchor<br/>embedding distribution ↔ N(0, I)"]
    C --> D["Flow composition is the map<br/>Φ = ψ2 ∘ (ψ1)⁻¹"]
    D --> E["Nearest neighbor in embedding space<br/>pointwise map T12"]
    E --> F["Downstream tasks<br/>UV mappingΒ·scan fittingΒ·volume matching"]

Key Designs

1. Geodesic feature embedding with landmark priors: a minimal prior that buys a representation-agnostic embedding

Every shape matching pipeline needs some prior to resolve fundamental ambiguities, the most notable being intrinsic symmetry β€” pure geometry cannot tell "left hand" from "right hand." Existing methods place that prior in different spots: functional-map methods rely on spectral signatures, Diff3F on rendered views plus a pretrained vision model, ULRSSM on a feature extractor trained over a large collection of shapes. Each choice introduces a representation-specific dependency (spectral basis, rendering pipeline, training data) that limits where it applies. Plain 3D coordinates, by contrast, are universally computable but encode neither intrinsic geometry nor semantics: as an embedding they leave both KNN and FUSE with high error (0.1572 and 0.1488 on FAUST) and cannot capture non-rigid deformation. This paper chooses the geodesic feature embedding (GFE): each surface point is represented by the vector of its geodesic distances to a small set of landmarks \(L\), so the dimensionality equals the number of landmarks, the encoding is intrinsic, and it is naturally cross-representation β€” Dijkstra or the heat method on meshes, heat geodesics on point clouds, voxelization plus Dijkstra on SDFs. Landmarks are a "minimal yet sufficient" prior: they provably resolve intrinsic symmetries and anchor the semantic orientation of a shape, exactly as axiomatic functional-map pipelines do. This work uses 5 landmarks for humans and 6 for animals, picked manually on one shape and propagated to the rest of the dataset through ground-truth correspondences.

Landmarks alone do not solve the problem at the distribution level: a non-rigid deformation substantially changes geodesic distances between points, so the GFE distributions of two shapes are not naturally aligned β€” which is precisely where flow composition takes over (see design 3). More importantly, what decides whether FUSE works is not the embedding's dimensionality but its smoothness. Table 2 gives clean evidence: GFE has extremely low Dirichlet energy (0.0001 on FAUST), and the flow immediately drives KNN's 0.0454 down to 0.0289; WKS and the learned ULRSSM embeddings carry energy an order of magnitude higher (0.0056 / 0.1847), and FUSE barely improves on WKS (0.1007 β†’ 0.1015) or even degrades on ULRSSM (0.0723 β†’ 0.0829). The authors call this the curse of frequency: high-frequency embeddings induce complex distributions that the flow finds hard to align, mirroring long-standing difficulties in functional-map optimization. The reverse also holds β€” projecting an embedding onto a small Laplace–Beltrami eigenbasis to smooth it (WKS* and WKS-L* in Table 2) makes FUSE consistently better than KNN again (0.1125 β†’ 0.0901, 0.0522 β†’ 0.0416), suggesting that "smooth learned features + flow-based alignment" is a viable direction. FUSE is also not afraid of higher dimensionality: growing the landmark count from 3 to 30 decreases the error monotonically while staying ahead of KNN, and the advantage persists when landmark positions are perturbed by a random fraction of the shape diameter.

2. Per-shape invertible flow with a shared Gaussian anchor: representing a shape as a continuous flow out of a Gaussian

This design answers how a shape itself is represented. Given the embedding distribution \(p_\ell\) of shape \(\ell\) (the distribution of the rows of its embedding matrix), the method trains an independent flow \(\psi^\ell: [0,1] \times \mathbb{R}^d \to \mathbb{R}^d\) that transports the fixed shared Gaussian \(p_0 = \mathcal{N}(0,1)^d\) to \(p_\ell\): sample a point from the Gaussian, integrate forward to \(t = 1\), and the endpoint follows that shape's embedding distribution. The flow is given by a neural-network-parameterized velocity field satisfying the ODE \(\frac{d}{dt}\psi_t(x) = u_t(\psi_t(x))\) with the identity as initial condition, \(\psi_0(x) = x\); the flow is therefore invertible and diffeomorphic by construction, and backward integration takes the shape back to the Gaussian. Since the true velocity field is intractable, training uses Conditional Flow Matching: regress a tractable conditional velocity field along the simplest linear-interpolation path (\(x_t=(1-t)x_0+tx_1\), \(u_t=x_1-x_0\)), giving the objective

\[\mathcal{L}_{\text{CFM}}(\theta) = \mathbb{E}_{t,\, x_0 \sim p_0,\, x_1 \sim p_1}\left\lVert v^\theta_t(x_t) - u_t(x_t \mid x_1) \right\rVert^2\]

(⚠️ Equation (1) of the original paper is corrupted by OCR in this cache; the standard form from the original Flow Matching formulation is given here.) Training is simulation-free β€” only mapping/generation requires integrating the ODE β€” and each flow is trained solely on its own shape's embeddings, needing nothing but the ability to sample embeddings from the surface. That is why the framework is fully agnostic to the geometric representation, and why the very same machinery consumes SDFs and tetrahedral meshes without modification.

There is also an easily overlooked design choice that matters for the whole paper: flow matching rather than diffusion. The predecessor Geometry Distributions represents shapes with DDIM as a Gaussian-to-surface diffusion process, but DDIM only approaches the Gaussian asymptotically through a long noise schedule, so inverting back to the anchor leaves a deviation and "all shapes share one latent space" holds only approximately. Flow matching establishes an explicit relationship between the Gaussian and the data distribution, so the inverse flow deterministically maps a shape back to the exact Gaussian. Table 3 quantifies this on FAUST: FUSE's anchor-space Euclidean / geodesic errors are 0.042 / 0.039 versus 0.099 / 0.095 for FUSE DDIM, and the mean KL divergence is 0.0375 versus 0.3519 with mean JS 0.0829 versus 0.2128. The divergence gap carries straight through to final matching accuracy (0.0289 for FUSE versus 0.0402 for FUSE DDIM in Table 4), confirming that accurate anchoring is what makes accurate matching possible.

3. Flow composition is the map: no paired training and no test-time optimization

Once two flows exist, the map between the shapes is just one composition: \(\Phi^{12} = \psi^2 \circ (\psi^1)^{-1}\). Its semantics are transparent β€” first take \(S_1\)'s embeddings back to the shared Gaussian with the source's inverse flow, then push the same points forward into \(S_2\)'s embedding space with the target's forward flow. The result is therefore not "another similarity matrix" but a map that actually deposits points in the target embedding space. To land on discrete points, one nearest-neighbor search suffices: \(T_{12} = \mathrm{NearestSearch}(\Phi^{12}(E(S_1)), E(S_2))\).

This construction holds because of a property of flow-matching solutions, not by luck. FM training uses random couplings between Gaussian and data samples, but that randomness only enters the conditional velocity fields and does not affect the learned flow itself: Theorem 2 of [25], cited by the authors, states that marginalizing the conditional fields over \(x_1 \sim p_1\) recovers the unique marginal velocity field generating the probability path, and that field is determined by the pair \((p_1, p_0)\) alone β€” hence so is the induced flow. Consequently, when \(E(S_1) = E(S_2)\) the two flows coincide exactly; when the two distributions are similar, the composition \(\psi^2 \circ (\psi^1)^{-1}\) aligns \(E(S_1)\) to \(E(S_2)\), and points with similar embeddings on different shapes land in nearby regions of the anchor space. Three practical benefits follow: the map automatically inherits the invariances of the embedding (if the embedding is invariant to some transformation, so is the map); any two shapes whose flows are already trained can be paired immediately, with no retraining; and building the bridge only requires sampling a surface, so cross-representation behavior is intrinsic rather than patched on.

The same argument also delineates the method's limits: when \(S_1\) and \(S_2\) differ by a transformation to which the embedding is not invariant, the two embedding distributions become dissimilar and flow composition has nothing to align; conversely, the less sensitive the embedding to deformation and the closer the two distributions, the more reliable the composition β€” which is exactly the mechanism behind the empirical finding that FUSE's advantage grows as isometry weakens. One more point must be stated plainly: the invertibility of the flows does not by itself induce a bijection between the shapes. The final KNN step assigns each source point its nearest target independently, so the matching is not strictly bijective, consistent with standard practice in point-based correspondence. The authors note that since both the GFE and the flow are defined continuously over the surface, extending the construction to a continuous, strictly bijective map is a natural direction, but this paper does not do it.

Loss & Training

Each shape gets its own flow model, trained independently on its own embeddings with the CFM objective above (linear-interpolation path, simulation-free, needing only sampled embeddings). At test time the method only performs backward/forward ODE integration plus one KNN search β€” no pairwise optimization, no fine-tuning, and no module trained with ground-truth correspondences. Optionally, FUSE's output can initialize functional-map refinement methods: obtain correspondences with FUSE first, then attach ZoomOut (FUSE ZO) or Neural Adjoint Maps (FUSE NAM) for pairwise optimization, which buys smoother maps and lower error. As for landmarks, 5 are used for humans and 6 for animals, manually picked on the first shape and propagated via ground-truth correspondences. ⚠️ The cached text does not report network architecture, ODE solver steps, or training scale; refer to the original paper and its supplementary material for those details.

Key Experimental Results

Main Results

Evaluation covers four non-rigid shape matching benchmarks: FAUST (near-isometric human meshes), SMAL (non-isometric quadruped meshes), SHREC20 (strongly non-isometric animal meshes) and KINECT (human point clouds). Four metrics are used: Euclidean error, geodesic error and Dirichlet energy (lower is better), plus coverage (higher is better). Baselines span several paradigms: KNN with the same geodesic features, Sinkhorn optimal transport, functional maps with geodesic descriptors and their refined variants (FMaps ZO, FMaps NAM), classic functional maps with WKS, the Neural Deformation Pyramid (NDP), vision-foundation features (Diff3F), and the data-driven ULRSSM. The table below summarizes Euclidean error and coverage from Table 4.

Method FAUST Eucl.↓ SMAL Eucl.↓ SHREC20 Eucl.↓ Kinect Eucl.↓ FAUST Cov.↑ SMAL Cov.↑ SHREC20 Cov.↑ Kinect Cov.↑
KNN 0.0454 0.0879 0.1083 0.1119 0.2452 0.1364 0.1446 0.0995
OT 0.0926 0.0779 0.1252 0.1327 0.0890 0.1129 0.1001 0.0748
FMaps 0.0401 0.0756 0.1381 0.0810 0.3236 0.2162 0.2037 0.1683
FMaps WKS 0.0331 0.0647 0.1197 0.0723 0.3759 0.2585 0.2380 0.2211
FMaps ZO 0.0207 0.0625 0.1454 0.0706 0.6969 0.4706 0.3495 0.4043
FMaps NAM 0.0245 0.0617 0.1041 0.0689 0.6323 0.4694 0.2860 0.3494
NDP 0.0702 0.0555 0.0698 0.0457 0.4893 0.4818 0.5016 0.4190
Diff3F 0.0697 0.1594 0.0876 0.1394 0.3075 0.2146 0.1579 0.1448
ULRSSM 0.0092 0.0370 0.3286 N/A 0.7302 0.4936 0.1337 N/A
FUSE 0.0289 0.0595 0.0975 0.0969 0.5320 0.5156 0.5278 0.4931
FUSE ZO 0.0200 0.0437 0.1275 0.0690 0.7127 0.5361 0.3763 0.4173
FUSE NAM 0.0179 0.0422 0.0658 0.0770 0.7027 0.5520 0.5342 0.3904

Geodesic error follows the same trend: on FAUST, FUSE cuts KNN's 0.0434 down to 0.0274, reaching 0.0168 with refinement (FUSE NAM); on SHREC20, FUSE NAM's 0.0603 is the lowest of all methods (NDP: 0.0649).

Ablation Study

Embedding type (Table 2, Euclidean error on FAUST / SMAL and Dirichlet energy on FAUST). This table validates both why GFE was chosen and the curse of frequency.

Embedding Dim KNN (FAUST) FUSE (FAUST) KNN (SMAL) FUSE (SMAL) Dirichlet (FAUST)
(x, y, z) 3 0.1572 0.1488 0.2903 0.3226 0.0002
GFE 5 0.0454 0.0289 0.0879 0.0595 0.0001
WKS 20 0.1007 0.1015 0.2736 0.3039 0.0056
WKS* (smoothed) 20 0.1125 0.0901 β€” β€” 0.0008
WKS-L 20 0.0428 0.0556 0.0697 0.0668 0.0022
WKS-L* (smoothed) 20 0.0522 0.0416 β€” β€” 0.0004
ULRSSM 256 0.0723 0.0829 0.1061 0.1117 0.1847

Anchoring scheme (Table 3, FAUST, GFE). Comparing nearest neighbors on raw embeddings (Raw), DDIM inversion (FUSE DDIM) and flow matching (FUSE): anchor-space accuracy and distribution divergence point the same way.

Metric Raw FUSE DDIM FUSE
Euclidean error (anchor) ↓ 0.045 0.099 0.042
Geodesic error (anchor) ↓ 0.043 0.095 0.039
Mean KL divergence ↓ 0.2528 0.3519 0.0375
Mean JS divergence ↓ 0.1928 0.2128 0.0829

Prior type and cross-representation (Tables 1 and 5). Table 1 shows that with only 5–6 landmarks as prior, FUSE reaches 0.028 / 0.059 on FAUST / SMAL, below ULRSSM which uses a training-data prior (0.043 / 0.063) and Diff3F which uses a vision-model prior (0.089 / 0.189). Table 5 is the SDF matching benchmark introduced by this paper (derived from FAUST; at evaluation the original mesh vertices are projected onto the corresponding SDF surface so that error is always measured on the original mesh geometry; for FUSE, about \(10^5\) points are uniformly sampled from the zero level set and each surface is voxelized to compute geodesic distances via Dijkstra):

Method Eucl.↓ Geod.↓ Dirichlet↓ Cov.↑
KNN 0.0562 0.0569 0.0023 0.1744
NDP 0.0525 0.0643 0.0016 0.5139
OT 0.0921 0.0881 0.0018 0.0936
FUSE ANCHOR 0.0472 0.0469 0.0175 0.2784
FUSE 0.0375 0.0385 0.0069 0.4653

Key Findings

  • The most direct gain from flow composition is coverage. FUSE lifts coverage from KNN's 0.2452 / 0.1446 to 0.5320 / 0.5278 on FAUST / SHREC20, approaching the heavily refined methods that rely on pairwise optimization. The reason is mechanistic: flow composition is an almost everywhere single-valued operation that genuinely pushes points into the target embedding space, whereas KNN merely lets each point pick its own nearest neighbor.
  • The more non-isometric the pair, the larger FUSE's advantage. On near-isometric FAUST, FUSE (0.0289) trails ULRSSM (0.0092) and FMaps ZO (0.0207); on strongly non-isometric SHREC20, ULRSSM collapses to 0.3286 while FUSE reaches 0.0975 and FUSE NAM 0.0658 β€” the best of all. This matches the explanation in design 3: the less the embedding reacts to deformation, the closer the two distributions and the more reliable the composition.
  • FUSE makes a better initialization. Initializing ZoomOut / NAM with FUSE beats the standard FMaps initialization, especially on SHREC20 (FUSE NAM 0.0658 / 0.0603 versus FMaps NAM 0.1041 / 0.0962), indicating that the map it provides already starts from a more sensible region.
  • FUSE's maps are not smooth, and the cause is localized to flow inversion. FUSE ANCHOR (no flow composition, just nearest neighbors in anchor space) is marginally more accurate in Euclidean error on non-isometric SMAL and SHREC20 (0.0579 / 0.0912 versus 0.0595 / 0.0975), but at considerably higher Dirichlet energy (0.0185 / 0.0538 versus 0.0075 / 0.0339) and much lower coverage (0.3708 / 0.3619 versus 0.5156 / 0.5278). The authors conclude that the lack of smoothness in FUSE's maps comes from the inversion step from the surface back to the anchor distribution, which is itself non-smooth.
  • Point clouds remain the weak spot. Point clouds guarantee neither reliable geodesic distances nor sufficient surface coverage (surface information off the vertices is unknown), so FUSE loses to the extrinsic NDP (0.0969 versus 0.0457). Notably, on the KINECT column FMaps' Euclidean / geodesic errors (0.0810 / 0.0779) are also lower than FUSE's (0.0969 / 0.0914), with FUSE's advantage resting mainly on coverage (0.4931 versus 0.1683); the paper's broader claim that "FUSE still outperforms non-refined geodesic methods" should be read with care, ⚠️ refer to the original paper and supplementary material for the complete metrics.
  • Cross-representation matching largely holds up. On the constructed FAUST-like collection (vertices taken from the original meshes as point clouds, plus SDFs), as long as at least one of the two shapes is a mesh or an SDF, FUSE matches mesh-to-mesh performance; it degrades noticeably only when both shapes are point clouds.
  • Volume matching is better on average but not everywhere. Against Volumetric FMaps on tetrahedral meshes (Table 6, lower Accuracy is better), the method is slightly behind on nearly isometric Octopus (0.0215 versus 0.0151) and Cat (0.0422 versus 0.0237), clearly ahead on strongly non-isometric Dino_skel (0.0426 versus 0.0652) and Airplane (0.0942 versus 0.1273), averaging 0.0501 versus 0.0578. The authors read this as consistent with the surface scenario: functional approaches struggle under strong non-isometry.

Highlights & Insights

  • Splitting "the map between two shapes" into "two independent flows plus one composition" turns a pairwise optimization problem into two single-shape distribution-fitting problems. The benefits compound: no retraining across shapes, no framework change across representations, and a new shape only needs its own flow trained. This "fit each side to a common prior, then compose" paradigm transfers to any correspondence problem between two domains β€” cross-lingual embedding alignment, cross-modal retrieval β€” provided there is a per-sample embedding and a shared prior.
  • Treating the shared anchor distribution as a pluggable connector is the cleverest step in the paper: instead of pulling the two shapes' features together directly, both sides align to a fixed \(\mathcal{N}(0,1)\), so the difficulty becomes "how far is each side from the Gaussian," which can be quantified directly with KL/JS divergence (Table 3).
  • The curse of frequency is a reusable diagnostic. The authors identify the spectral complexity / smoothness of the embedding as the decisive variable for flow-based alignment and provide an immediately effective remedy (smoothing the embedding with a small Laplace–Beltrami eigenbasis, which in Table 2 flips WKS* / WKS-L* to beating KNN). Any work that aligns feature distributions with a generative model can self-check with Dirichlet energy first.
  • The motivation for replacing DDIM is refreshingly concrete: DDIM's inverse process only approaches the Gaussian asymptotically, so after inversion the shapes are not actually in the same latent space β€” and "all shapes share one anchor" is precisely what the entire FUSE construction rests on. This style of argument, where a small choice at the representation level decides whether a method stands at all, is worth borrowing.

Limitations & Future Work

  • The authors acknowledge three limitations: symmetry disambiguation depends entirely on the embedding, which GFE with landmarks handles well but embeddings without symmetry-breaking information (e.g. WKS) do not; the reliance on consistent geodesic distances limits effectiveness on point clouds and makes the method unsuitable for partial matching in its current form; and the final KNN retrieval step makes the matching not strictly bijective, although both the GFE and the flow are continuous, so a continuous solution is already within reach.
  • Limitations I would add: landmarks are few but still require manual selection on a first shape plus propagation through ground-truth correspondences β€” on a new category without correspondence data this prior is not free, and the paper only analyzes random perturbation of landmark positions, not what happens when a landmark itself is misplaced. All evaluations rest on small benchmarks with ground-truth correspondences, so evidence on large-scale real scans with noise, holes and topological variation is thin. Moreover, the argument that the flow is determined by the distributions alone β€” and hence that composition is reliable β€” presupposes the two embedding distributions are similar enough; whether that holds for cross-category or topologically different pairs is never given a quantitative criterion. Finally, the cached text lacks the reproducibility-critical details of network architecture, ODE solver steps and training scale.
  • Concrete improvements: combine smooth learned features with flow-based alignment, as the authors suggest; pin down points in the anchor space to support partial matching; replace KNN with a consistent continuous map to recover strict bijectivity; and provide an a-priori measure of whether two embedding distributions are alignable, to predict when FUSE will fail.
  • vs Functional Maps (incl. ZoomOut / NAM): FMap represents correspondences as linear operators between functional spaces β€” smooth, with low Dirichlet energy β€” but relies on spectral bases (natural only on meshes) and struggles to guarantee bijectivity. FUSE represents them as the composition of two invertible flows, naturally representation-agnostic with much higher coverage, at the price of a non-smooth map. The two are complementary: FUSE can directly initialize ZO / NAM, beating the standard FMaps initialization (on SHREC20, FUSE NAM 0.0658 / 0.0603 versus FMaps NAM 0.1041 / 0.0962).
  • vs OT (Sinkhorn): OT exhibits mass splitting under sampling imbalance or feature ambiguity, so the recovered discrete correspondences are not sharp; empirically OT trails clearly on every dataset (FAUST 0.0926, Kinect 0.1327, with coverage of only 0.0890 / 0.0748).
  • vs NDP: NDP optimizes 3D deformation fields directly and bypasses geodesics, making it the best on point clouds (Kinect 0.0457), but it fails under strong non-rigid deformation (FAUST 0.0702); FUSE is exactly complementary to it.
  • vs ULRSSM: data-driven unsupervised spectral feature learning, best on FAUST / SMAL (0.0092 / 0.0370), but it collapses to 0.3286 under strong non-isometry and is restricted to meshes with large-scale training data; FUSE works across representations with 5–6 landmarks.
  • vs Diff3F: distills semantic features from rendered views with a vision foundation model, requiring a rendering pipeline and a representation dependency; FUSE needs only surface sampling and a pointwise embedding.
  • vs Geometry Distributions: shares the probabilistic "Gaussian β†’ shape" representation, but that work uses DDIM to represent a single shape, whereas FUSE switches to flow matching, builds the flow in intrinsic feature space, and changes the target to representing the map between shapes.

Rating

  • Novelty: ⭐⭐⭐⭐ First to use flow matching for a map representation between shapes, with a shared Gaussian anchor enabling paired-training-free, cross-representation composed maps.
  • Experimental Thoroughness: ⭐⭐⭐⭐ Four mesh/point-cloud benchmarks plus self-built SDF and volume benchmarks and multiple embedding/anchoring ablations, but network and training hyper-parameters are missing and the point-cloud column deserves more scrutiny.
  • Writing Quality: ⭐⭐⭐⭐ The mechanism is argued clearly (why composition works, how FM differs from DDIM), with restrained formulas and well-matched figures and tables.
  • Value: ⭐⭐⭐⭐ Offers a new paradigm for representing maps between arbitrary geometric representations; its coverage and cross-representation ability are directly useful for UV mapping, scan fitting and partial matching.