GeodesicNVS: Probability Density Geodesic Flow Matching for Novel View Synthesis

Xuqin Wang; Tao Wu; Yanfeng Zhang; Lu Liu; Mingwei Sun; Yongliang Wang; Niclas Zeller; Daniel Cremers

cs.CVMar 1, 2026

GeodesicNVS: Probability Density Geodesic Flow Matching for Novel View Synthesis

Authors: Xuqin Wang, Tao Wu, Yanfeng Zhang, Lu Liu, Mingwei Sun, Yongliang Wang, Niclas Zeller, Daniel Cremers

Organizations: Huawei Hilbert Research Center (Dresden) · Technical University of Munich · Karlsruhe University of Applied Sciences

Abstract

Recent advances in generative modeling have substantially enhanced novel view synthesis, yet maintaining consistency across viewpoints remains challenging. Diffusion-based models rely on stochastic noise-to-data transitions, which obscure deterministic structures and yield inconsistent view predictions. We advocate a Data-to-Data Flow Matching framework that learns deterministic transformations between paired views, enhancing view-consistent synthesis through explicit data coupling. Building on this, we propose Probability Density Geodesic Flow Matching (PDG-FM), which aligns interpolation trajectories with density-based geodesics of a data manifold. To enable tractable geodesic estimation, we employ a teacher-student framework that distills density-based geodesic interpolants into an efficient ambient-space predictor. Empirically, our method surpasses diffusion-based baselines on Objaverse and GSO30 datasets, demonstrating improved structural coherence and smoother transitions across views. These results highlight the advantages of incorporating data-dependent geometric regularization into deterministic flow matching for consistent novel view generation.

Figures & tables

Panel of Figure 1 — Figure 1: From Conditional Diffusion Model to Probability Density Geodesic Flow Matching. Conventional diffusion models learn stochastic noise-to-data transitions, often losing deterministic structure. We instead train a Data-to-Data Flow Matching network to learn continuous deterministic transformations between paired data samples $(x_{0},x_{1})$ . To ensure geometric consistency, we propose to deploy a probability-density-based geodesic to align flow matching interpolants with high-density regions of the data manifold. This unified design couples accurate data coupling with manifold-aware regularization, yielding realistic, view-consistent transformations.

Figure 2: Overview of the Probability Density Geodesic Flow Matching (PDG-FM) framework. (a) Data-to-Data Flow Matching framework learns deterministic mappings between paired samples $(x_{0},x_{1})$ by encoding source and target views into latent space and predicting intermediate states $x_{t}$ through either linear interpolants (Eq. 6 ) or geodesic interpolants (Eq. 17 ) predicted by the GeodesicNet $\phi_{\eta}$ . The velocity field $v_{\theta}(x_{t},t,c)$ is conditioned on source view features and Plücker ray embeddings, and its decoded outputs yield novel views consistent with the target pose. (b) Variational Distillation of Geodesics trains GeodesicNet $\phi_{\eta}$ to produce manifold-aligned interpolants by minimizing path energy defined under the probability density geodesic metric $\|x\|_{\mathcal{G}(x)}=\sqrt{x^{\top}x/p(x)^{2}}$ , where the density $p(x)$ is estimated using a pretrained diffusion score function (Eq. 12 ). (c) The unified PDG-FM integrates both components, resulting in geometry-aware and manifold-consistent transformations for view-consistent novel view synthesis.

	Objaverse					GSO30
Method	FID $\downarrow$	CLIP-S $\uparrow$	SSIM $\uparrow$	PSNR $\uparrow$	LPIPS $\downarrow$	FID $\downarrow$	CLIP-S $\uparrow$	SSIM $\uparrow$	PSNR $\uparrow$	LPIPS $\downarrow$
Zero-1-to-3	5.9959	88.6625	0.8446	19.5850	0.0962	12.5796	90.9683	0.8536	19.3038	0.0638
Zero-1-to-3-XL	7.2006	87.1176	0.7916	17.1722	0.1505	15.7340	89.0756	0.8280	18.6022	0.0910
EscherNet	7.9058	87.9020	0.8363	18.7560	0.1061	14.7044	90.1891	0.8595	20.1754	0.0689
Free3D	5.5434	88.9575	0.8537	20.3222	0.0873	12.0620	91.1424	0.8581	19.5610	0.0666
Naive FM	5.5093	88.8786	0.8622	20.8246	0.0809	15.2763	89.8226	0.8631	19.7273	0.0641

Table 1: Novel view synthesis performance on Objaverse and GSO datasets. Comparison of Linear-D2D-FM against diffusion and flow baselines on Objaverse and GSO30.

Panel of Figure 3 — Figure 3: Qualitative comparisons on Objaverse. Comparison of Linear-D2D-FM, the Noise-to-Data FM (Naive FM) baseline and the diffusion-based Free3D model.

Panel of Figure 4 — Figure 4: Qualitative comparisons on GSO30. Comparison of Free3D and our method (“Ours”) given the same input view.

	Objaverse
Method	FID $\downarrow$	CLIP-S $\uparrow$	SSIM $\uparrow$	LPIPS $\downarrow$
Free3D	22.4533	77.9569	0.4470	0.4372
Free3D	8.9367	84.4197	0.7894	0.1493
Naive FM	5.7768	88.9998	0.8679	0.0779
Linear-D2D-FM	5.8223	88.9749	0.8688	0.0782

Table 2: Novel view synthesis performance with 10 NFE. Linear-D2D-FM preserves superior performance with 10 NFE.

Panel of Figure 5 — Figure 5: Qualitative comparisons between PDG-FM (geodesic) and Linear-D2D-FM (linear) on Objaverse. Visual results showing that PDG-FM generates more geometrically faithful novel views. The improvement reflects the effect of energy-guided optimization along data-dependent geodesics.

	Objaverse
Method	FID $\downarrow$	CLIP-S $\uparrow$	SSIM $\uparrow$	LPIPS $\downarrow$
Linear-D2D-FM	11.8124	94.3476	0.8736	0.0809
Metric FM	11.9609	92.0370	0.8663	0.0900
PDG-FM	10.4010	92.3368	0.8768	0.0804

Table 3: Effect of probability density geodesic alignment (PDG-FM) compared to linear interpolants (Linear-D2D-FM) and Metric FM under a reduced training setup.

Panel of Figure 6 — Figure 6: Geodesic Interpolants vs Linear Interpolants. Geodesic interpolants traverse semantically meaningful regions of the data manifold, producing perceptually consistent interpolants. In contrast, linear interpolants merely blend the two endpoints, resulting in limited structural continuity and unrealistic transitions.

Method	Setting	PPL $\downarrow$	AOFM $\uparrow$
Ground-Truth	6 objs	0.2575	14.4716
Linear	6 objs	0.2130	1.0411
Initial	6 objs	0.5710	6.4773
Geodesic	6 objs	0.5023	13.6968

Table 4: Geodesic Path Study Results. We report PPL and AOFM for linear, DDIM-initialized, and geodesic interpolants, with ground-truth camera trajectories as the reference.

Figure 7: Comparison of Geodesic Gradient Norm across Time for geodesic interpolants, their pre-optimization counterparts, and linear interpolants on training dataset. The optimized geodesic paths have lower residuals thus indicating better satisfaction of the Euler-Lagrange condition, aligning with improved perceptual and geometric consistency.

	Objaverse
Method	FID $\downarrow$	CLIP-S $\uparrow$	SSIM $\uparrow$	PSNR $\uparrow$
Linear-D2D-FM	5.4324	88.9855	0.8634	20.8447
Linear-D2D-FM(eps=50)	5.8050	88.3582	0.8532	20.0291
Linear-D2D-FM	5.8223	88.9749	0.8688	21.3045
Linear-D2D-FM, $U_{10}$	5.5146	88.9185	0.8706	21.5952
Linear-D2D-FM, $U_{4}$	7.4414	88.1954	0.8784	22.2092

Table 5: Ablation study of noise augmentation (top) and timestep sampling (bottom) in Linear-D2D-FM.

Panel of Figure 8 — Figure 8: Extended Qualitative Results of PDG-FM (geodesic) and Linear-D2D-FM on Objaverse.

Panel of Figure 9 — Figure 9: Extended Qualitative Results on Objaverse.

Panel of Figure 10 — Figure 10: Extended Qualitative Results on GSO30.

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Novel view synthesis from sparse images must reconcile faithful reconstruction of observed regions with plausible completion of unseen content, while maintaining world consistency across viewpoints. Existing geometry-based methods preserve observed scene structure but often struggle to complete unseen regions, whereas video generative models offer rich appearance priors but accumulate inconsistencies during sequential view generation. We propose GeoVerse, a framework that synthesizes world-consistent novel views by performing generation within the geometric latent space of a pretrained 3D foundation model and injecting appearance priors from a video generative model. Specifically, GeoVerse extracts multilevel features from Wan2.2 VACE and injects them into the geometric latent diffusion model via a ControlNet-style adapter, incorporating video-learned appearance priors to enhance structural completion. To enforce cross-view coherence, a global spatial memory continuously aggregates observed and synthesized content, reprojecting target-aligned guidance to anchor subsequent predictions to a shared scene representation. Extensive experiments across diverse datasets demonstrate improved visual quality and geometric consistency, with a 2.23 dB higher PSNR on DL3DV and 32.4% lower ATE on Mip-NeRF360 compared to GLD.

Abstract

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