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Geodesic-informed Generative Diffusion Model For Topology-preserved Image Video Generation
Nian Wu, Nivetha Jayakumar, Jiarui Xing, Miaomiao Zhang
TL;DR
Existing diffusion models underrepresent object geometry and explicit deformation dynamics, limiting topology preservation and interpretability in structure-sensitive applications. IGG integrates geodesic-informed registration with latent geometric diffusion to generate template-conditioned deformations, reporting improved fidelity, diversity, topology preservation, and low-data segmentation augmentation. Its effectiveness depends on accurate registration and on anatomical changes remaining approximately diffeomorphic.
Problem
Existing diffusion models focus mainly on intensity and texture, without guaranteeing geometric consistency or explicitly representing shape changes during generation.
Method
IGG combines geodesic-informed registration with latent geometric diffusion to generate text-guided image deformations from template images using learned diffeomorphic transformations.
Results
IGG improves sample fidelity and diversity, preserves topology, and yields substantial downstream segmentation gains when generated samples augment low-data training sets.
Takeaways & Limitations
Geodesic deformation spaces provide a framework for synthesizing anatomically consistent samples while quantifying topological consistency and shape-changing processes.
Takeaways & Limitations
IGG depends on accurate registration and assumes anatomically meaningful correspondences with diffeomorphic structural variation; noise, pathology, or non-diffeomorphic changes may violate this assumption.
Abstract
from arXiv · showhide
Generative diffusion models have emerged as a class of powerful techniques for various imaging applications, including but not limited to synthesis, reconstruction, and segmentation. Despite their success, current generative models pose two key limitations. First, they primarily rely on image intensity and texture information, with limited attention to underlying object geometry. As a result, they do not guarantee geometric or topological consistency during the generation process, which is a crucial requirement for high-stakes domains such as computational anatomy, biology, and robotics, where preserving object structure is critical. Second, existing models fail to explicitly learn or represent shape changes in the generative process. Such deformation dynamics remain occluded within network parameters; hence leaving the transformation process uninterpretable and physically uninformed. To address these challenges, we introduce IGG (Image Generation informed by Geodesic dynamics), a novel framework that integrates topology-preserving geodesic principles into the diffusion-based generative process. In contrast to conventional methods that operate in image intensity space, IGG learns and synthesizes diverse samples within geodesic deformation spaces, where geometric object changes are learned as smooth and invertible smooth mappings from a given template/source image. Our code is publicly available at https://github.com/nellie689/IGG.
1. Introduction
Existing diffusion models often prioritize intensity and texture over object geometry, leaving topological consistency and deformation dynamics insufficiently represented. IGG addresses these gaps by generating template-conditioned images along geodesic, topology-preserving deformations and evaluates fidelity, topology, and downstream segmentation utility.
- Motivation: Existing diffusion models may produce biologically implausible samples because they explicitly model image intensity and texture while giving little attention to object geometry.This limitation is especially relevant to medical imaging, robotics, and autonomous systems, where geometric preservation matters.
- Motivation: Common visual-quality metrics such as FID, IS, and SSIM do not measure preservation of object geometry and topology.This raises questions about their validity for structure-sensitive applications.
- Motivation: Existing diffusion models also fail to explicitly represent shape changes during generation, leaving deformation dynamics physically uninterpretable.Geodesic-constrained deformation can instead provide interpretable trajectories and regularization against folding, tearing, and crossing.
- IGG framework: IGG generates images by deforming a template along random geodesics using learned diffeomorphic transformations and text guidance.Its two components are a geodesic-informed registration representation and a latent geometric diffusion model.
- Evaluation: IGG evaluates fidelity and topological consistency, and uses synthesized image-label pairs for downstream segmentation experiments.The study includes Komatsuna plant growth data and brain MRIs, with results indicating preserved topology and potential utility for segmentation augmentation.
2. Background: Geodesics In Deformation Spaces
Geodesics provide a principled framework for modeling smooth image deformations while preserving structural integrity. They are obtained by minimizing transformation energy subject to diffeomorphic dynamics and image-matching constraints.
- Geodesics in deformation spaces: A geodesic is a shortest path with locally minimized transformation energy between images in a deformation space.Deforming a template along such paths provides descriptors for objects represented as deformed variants of that template.
- Diffeomorphic transformations: Smooth vector fields form the tangent space of diffeomorphisms, while time-varying velocity fields generate transformations between images.The resulting diffeomorphisms evolve from the identity map to a target transformation.
- Geodesic optimization: The operator L maps velocity fields to momentum vectors, with the Laplacian choice L = (−α∆ + e)^3 controlling transformation-field smoothness.Its dual pairing supplies the transformation-energy term used by the geodesic formulation.
- Geodesic optimization: Geodesic paths can be computed by forward shooting an Euler–Poincaré differential equation from an initial velocity.The minimum of the geodesic functional is determined by solving the EPDiff equation with a given initial condition.
- Geodesic optimization: Given source and target images, the initial velocity is optimized under EPDiff constraints to derive the geodesic transformation.This formulation links image matching to physically structured deformation dynamics.
- Geodesic optimization: The deformation energy combines a velocity-field regularization term with an image dissimilarity term weighted by λ.The image distance uses sum-of-squared intensity differences in this paper.
3. Method: IGG
IGG learns geodesic deformation representations with GIR and generates text-conditioned transformation sequences with LGD. Its diffusion process operates in latent deformation space, then applies sampled diffeomorphic transformations to a template image.
- Framework overview: IGG comprises GIR, which encodes geodesic deformation paths, and LGD, which generates time-sequences of image transformations conditioned on text.GIR learns from paired template/source and target images, while LGD models latent geodesic flows.
- Geodesic-informed image registration: GIR encodes velocity-field sequences between template and target images into a compact latent representation.Its encoder maps input data to latent velocity representations, and its decoder projects them back to full-dimensional velocity fields.
- Latent geodesic evolution: The NeurEPDiff operator propagates a latent initial velocity along a geodesic path through successive latent states.A neural operator approximates the EPDiff mapping during training using local transformations, global convolutions, and a smoothed GeLU activation.
- Latent geodesic evolution: Latent geodesic evolution updates each state by composing J neural-operator layers, producing a sequence from z_v^t to z_v^(t+1).The decoder converts the latent sequence into velocity fields, which generate corresponding deformations through the geodesic flow.
- Training GIR: The GIR loss combines unsupervised registration, geodesic consistency guided by numerically integrated EPDiff solutions, and network regularization.λ and η balance image matching and geodesic terms, respectively.
- Latent geodesic diffusion: LGD diffuses concatenated time-dependent latent geodesic representations rather than image intensities, then samples topology-preserving diffeomorphic transformations for template deformation.The forward process progressively adds Gaussian noise over T steps, while reverse diffusion reconstructs clean latent geodesic flows.
- Reverse diffusion with geometric conditioning: Reverse diffusion uses image and geodesic conditioning together with text embeddings integrated through cross-attention.The denoising network predicts removed noise, and iterative sampling yields a clean geodesic deformation embedding.
- Training and inference: IGG first trains geodesic representation learning and then optimizes the latent geodesic diffusion model; testing samples flows conditioned on a template and text.The training and sampling procedures are summarized in Algorithms 1 and 2.
4. Experimental Evaluation
IGG is evaluated on geodesic learning, topology-preserving image generation, confidence patterns, and downstream brain-MRI segmentation augmentation. Across these experiments, it is compared with numerical EPDiff solutions and established generative or segmentation baselines.
- Evaluation setup: The evaluation uses plant-growth and longitudinal brain-MRI datasets, with text conditions describing image content and temporal edits.The plant data contain 300 RGB-D label-map frames from five plants, while the MRI experiment uses 2,618 longitudinal T1-weighted scans.
- Geodesic learning: GIR learns geodesic mapping functions comparable to numerical EPDiff solutions, based on visual agreement and per-pixel velocity errors over time.The comparison evaluates transformations, velocity fields, and deformed images against Euler-integrated numerical solutions.
- Generated samples: IGG preserves object topology and geometric progression more effectively than baselines that diffuse directly in image-intensity space.Baseline plant images can show unnaturally merged or overlapping leaves, while brain samples can contain ventricular regions inconsistent with the source structure.
- Generated samples: IGG achieves lower FVD, FID, and KID and higher SSIM than baselines, while achieving similar IS and approximately 10 times better scores overall.The reported metrics assess sample quality, distributional similarity, diversity, and pixel-level fidelity.
- Downstream segmentation: Increasing IGG-generated augmentation from 2× to 4× consistently improves Dice scores across segmentation models, including evaluations on OASIS-3 and OASIS-1.Representative segmentation predictions more closely match ground truth with augmented data, especially under limited-data conditions.
5. Conclusion & Discussion
IGG performs diffusion in geodesic deformation spaces, using learned diffeomorphic transformations to generate topology-consistent image sequences. Experiments report improved fidelity, diversity, and low-data segmentation performance, while the method depends on reliable registration and diffeomorphic anatomical variation.
- Contribution: IGG learns geodesic-informed deformations and synthesizes diverse samples through smooth, interpretable image transformations.Its components include a registration network for time-dependent diffeomorphic transformations and a latent geometric diffusion model conditioned on text and template images.
- Results: IGG-generated samples improve segmentation performance across dataset configurations, models, and anatomical structures.The experiments vary augmentation size and report average Dice scores as well as structure-specific results.
- Results: IGG improves sample fidelity and diversity compared with state-of-the-art methods and yields gains when generated samples augment segmentation training.The reported downstream benefit is particularly pronounced in low-data regimes.
- Limitations: The fidelity of synthesized data depends on registration accuracy and reliably established anatomically meaningful correspondences.Large registration errors can limit synthesized-result quality.
- Limitations: Strict topology preservation may not capture anatomical variability when imaging noise, pathology, or other non-diffeomorphic changes violate the framework’s assumptions.The authors identify combining deformation fields with image-texture modeling and extending the method to 4D data as future directions.
Ethical Standards
The paper states that its research and manuscript preparation followed applicable ethical standards, laws, and regulations for animal or human subjects.
- Ethical standards: The work follows applicable ethical standards, laws, and regulations for research involving animals or human subjects.The statement covers both research conduct and manuscript writing.