parametric curve representation

This work addresses the challenge of automatically converting vascular segmentation results into editable parametric models. We propose the first end-to-end differentiable vascular shape modeling framework. Methodologically, it jointly parameterizes the centerline and radius using cubic B-splines and introduces a differentiable voxelization layer to enable unsupervised mapping from segmentation masks to parametric models, unifying voxel, mesh, and parametric representations. The framework directly generates high-fidelity, smooth, and backpropagation-enabled meshes from segmentation inputs. Its key innovation lies in eliminating reliance on ground-truth parametric annotations—enabling shape-level gradient backpropagation and post-hoc geometric editing. Evaluated on aortic, aneurysmal, and cerebral vascular datasets, our method significantly reduces volumetric segmentation matching error and achieves state-of-the-art performance in geometric reconstruction accuracy and mesh quality.

To address error accumulation in two-stage methods for reconstructing 3D parametric curves from multi-view edge maps—caused by decoupled optimization—this paper proposes CurveGaussian, an end-to-end differentiable framework. Our key contributions are: (1) a novel curve-Gaussian bidirectional coupling representation that jointly models parametric curves and edge-guided Gaussian splatting; (2) a differentiable dynamic topology optimization framework enabling linearization, merging, splitting, and pruning of curve structures; and (3) integration of differentiable rendering, multi-view geometric constraints, and adaptive topology updates. Evaluated on the ABC dataset and real-world scenes, CurveGaussian achieves significant improvements in reconstruction accuracy and robustness. It reduces model parameters by 37%, accelerates training by 2.1×, and yields cleaner outputs with stronger geometric consistency compared to prior approaches.

This paper addresses the unsupervised extraction of curves and one-dimensional structures in images with known endpoints. The proposed method integrates geometric modeling and variational principles: curves are lifted to a position-orientation space, where sub-Riemannian or Finsler metrics naturally encode curvature-dependent energy; a differentiable energy functional is constructed based on Smirnov’s vector field decomposition theorem, and a bilevel optimization framework is designed to compute globally optimal paths under endpoint constraints. By jointly discretizing the energy and enforcing geometric regularization, the approach significantly improves connectivity robustness in weak-boundary and low signal-to-noise ratio scenarios. Experiments demonstrate superior accuracy and generalizability compared to state-of-the-art variational and learning-based methods—without requiring manual parameter tuning or ground-truth annotations—making it particularly effective for challenging 1D structure extraction tasks such as biological image analysis and vascular segmentation.

Computer vision and medical image classification often suffer from symmetry-induced ambiguities—such as rotation, scaling, and reparameterization—whose invariance is typically enforced via data augmentation, lacking geometric interpretability and sustainability. This paper proposes a geometry-aware learning framework that eliminates the need for data augmentation. Grounded in the theory of principal fiber bundles, we construct a two-parameter family of canonical parametrizations for planar curves—including constant-speed parametrization as a special case—and learn discriminative representations by optimizing bundle sections to maximize inter-class separation. Coupled with an orbit-space metric design, our approach achieves strict invariance to translation, rotation, scaling, and reparameterization. Empirical evaluation on object contour datasets demonstrates that the learned shape representations are symmetry-invariant and yield significant classification improvements. The implementation, including code and tutorials, is publicly available to support reproducible research and practical deployment.

Extracting slender geometric structures from Gaussian splatting scenes for physical simulation is highly challenging due to the absence of explicit topological connectivity and significant noise inherent in Gaussian primitives. This work proposes a user-sketch-guided, screen-space shortest path approach that leverages dynamic programming to efficiently construct polyline meshes, enabling robust reconstruction of slender object geometries. To the best of our knowledge, this is the first method to achieve interactive extraction of coherent slender structures directly from Gaussian splatting representations, effectively overcoming the difficulties posed by missing topology and noisy input. Extensive experiments across multiple real-world scenes demonstrate the robustness and practical utility of the proposed technique.

In high-dimensional parametric shape design, linear dimensionality reduction struggles to capture nonlinear geometric variations effectively, limiting optimization and exploration efficiency. This work proposes a nonlinear latent variable representation that generalizes linear subspaces to an interpretable nonlinear latent space while preserving an explicit inverse mapping back to the original design parameters. The approach integrates nonlinear latent modeling, a parameter decoder, and a CAD-driven forward geometry generation pipeline, achieving both strong compression performance and engineering interpretability. Demonstrated on an underwater glider case study, the method reconstructs shapes with only 5% error using just five latent variables—outperforming linear PME, which requires eight—and achieves 1% reconstruction error with nine latent variables compared to fifteen for linear methods, significantly surpassing both conventional linear techniques and black-box deep autoencoders.

Existing interpolation methods for generative models lack a universal, principle-driven definition—often relying on strong assumptions or requiring architectural modifications. Method: We propose a general interpolation framework that models interpolation paths as Riemannian geodesics constrained by the data distribution: leveraging gradient information of the probability density, it computes high-density, quasi-geodesic paths directly in the pre-trained model’s latent space, without additional training or fine-tuning. Contribution/Results: Theoretically, the path satisfies the geodesic equation locally under a Riemannian metric induced by the data density. Algorithmically, the method is agnostic to the choice of distance metric and data distribution. Empirically, it significantly improves interpolation smoothness and semantic plausibility across diverse generative models—including VAEs, GANs, and diffusion models—and on both image and text datasets, outperforming state-of-the-art baselines.

Existing aerial image road extraction methods suffer from geometric distortion inherent in polyline representations or rely heavily on scarce curve-level ground-truth annotations. To address these limitations, we propose the Bézier Graph—a fully differentiable, parametric vector representation—formulating road network generation as an alternating geometric and topological optimization problem. Geometric refinement of Bézier control points is performed via differentiable rendering (DiffAlign), while discrete topological operations (TopoAdapt) enforce structural connectivity. The framework jointly trains with segmentation priors, eliminating the need for curve annotations and enabling direct learning of high-fidelity vector maps from binary masks. Evaluated on SpaceNet and CityScale benchmarks, our method achieves state-of-the-art performance, significantly improving both geometric accuracy and topological completeness.

This work addresses the challenge of achieving safe, interpretable, and real-time trajectory tracking for domestic service robots while preserving the geometric structure of variables such as SE(3) poses and SPD(n) stiffness/damping matrices—a balance that existing methods struggle to maintain between stability and accuracy. To this end, we propose the Curve-Induced Dynamical System on Manifolds (CDSM), which, for the first time, integrates a curve-induced mechanism into dynamical system modeling on Riemannian manifolds and Lie groups. By decomposing motion into tangential progression and normal attraction components, CDSM unifies stable convergence, online adaptability, and high-precision trajectory generation. Experiments demonstrate that CDSM significantly improves trajectory accuracy, reduces path deviation, and accelerates query speed on the S2 benchmark, with successful real-time adaptive control of both SE(3) and SPD(n) variables validated on robotic arms and mobile platforms.

This work addresses the structural fragmentation and physical unrealizability inherent in traditional 3D sketching methods, which rely on numerous discrete curve segments. To overcome these limitations, the authors propose representing complex 3D geometry through a single, continuous, differentiable 4D B-spline that jointly encodes spatial coordinates and variable stroke width, thereby reformulating modeling as a global routing problem. This approach achieves, for the first time, topologically consistent sparse 3D abstractions and incorporates strong inductive biases to enhance structural coherence and aesthetic quality. Differentiable rendering is employed to rasterize width-varying curves, with optimization guided by projection error constraints and semantic signals from CLIP and Score Distillation Sampling (SDS). The method demonstrates significant improvements in semantic fidelity and structural integrity across tasks including image-to-3D abstraction, multi-view line drawing generation, and stylized surface filling.