We present TrackGS, a novel method to integrate global feature tracks with 3D Gaussian Splatting (3DGS) for COLMAP-free novel view synthesis. While 3DGS delivers impressive rendering quality, its reliance on accurate precomputed camera parameters remains a significant limitation. Existing COLMAP-free approaches depend on local constraints that fail in complex scenarios. Our key innovation lies in leveraging feature tracks to establish global geometric constraints, enabling simultaneous optimization of camera parameters and 3D Gaussians. Specifically, we: (1) introduce track-constrained Gaussians that serve as geometric anchors, (2) propose novel 2D and 3D track losses to enforce multi-view consistency, and (3) derive differentiable formulations for camera intrinsics optimization. Extensive experiments on challenging real-world and synthetic datasets demonstrate state-of-the-art performance, with much lower pose error than previous methods while maintaining superior rendering quality. Our approach eliminates the need for COLMAP preprocessing, making 3DGS more accessible for practical applications.

Computing the Hausdorff distance between triangle meshes with guaranteed accuracy is a computationally intensive task. Conventional Branch-and-Bound (B&B) approaches are fundamentally ill-suited for massive parallelism. Their reliance on a global priority queue (PQ) for both best-first scheduling and global termination checks creates a serial bottleneck that prevents scalable performance. We introduce PQ-Free HD , a parallel B&B framework that eliminates this dependency by decoupling the algorithm's termination logic from its scheduling order. This is achieved by relaxing the culling criterion, thereby replacing the priority queue with a contention-free ring buffer, which transforms the execution model from a state-dependent serial search into a high-throughput, asynchronous batch-processing paradigm. The framework consists of four key components: (1) a parallel priority-queue-free B&B paradigm; (2) a hierarchical GPU execution architecture combining batched depth-first scheduling with fused collaborative kernels; (3) a geometrically robust seven-stage culling pipeline featuring novel tests for challenging geometries; and (4) a compact 29-byte procedural task descriptor that achieves an 83.9% memory reduction. Evaluations demonstrate substantial speedups: a median of 71.7× over the state-of-the-art CPU algorithm on general benchmarks, and exceeding 10,000× on challenging CAD models with dense planar structures. The throughput advantage scales super-linearly with problem complexity. We showcase practical value by building a strictly Hausdorff-distance-bounded mesh simplification tool entirely on the GPU. Our work provides a new method for high-throughput, tolerance-controllable B&B-based geometric queries on GPUs. Code and data are available at https://github.com/huzhihao2001/pqfree-hd.

Reconstructing large-scale 3D scenes remains challenging due to the need to balance photorealistic quality, real-time rendering, and compact storage. Recent progress in 3D Gaussian Splatting (3DGS) has achieved impressive fidelity and speed, yet its large-scale application suffers from excessive primitive counts, leading to prohibitive storage and rendering costs. To overcome this inefficiency, we introduce a novel semantic-guided hybrid representation that unifies textured meshes and 3D Gaussians in a differentiable framework. The key idea is to leverage meshes for geometrically regular regions such as roads and building facades, while reserving Gaussians for fine, complex details like vegetation. Our method is realized through three key technical contributions. First, we develop a semantic-guided adaptive modeling pipeline that fuses multi-view segmentation onto the scene mesh to robustly partition the scene and prune redundant Gaussians. Second, we introduce a high-performance CUDA-based hybrid renderer that seamlessly combines mesh rasterization with Gaussian splatting, enabling correct occlusion handling and joint optimization of both representations. Finally, we propose a mesh-guided sampling strategy that adaptively adds Gaussians to recover fine details in under-reconstructed areas. Extensive experiments on diverse large-scale datasets demonstrate that our approach significantly reduces storage requirements and accelerates rendering performance while maintaining comparable or superior visual quality.

Generating high-quality novel view renderings of 3D Gaussian Splatting (3DGS) in scenes featuring transient objects is challenging. We propose a novel hybrid representation, termed as HybridGS, using 2D Gaussians for transient objects per image and maintaining traditional 3D Gaussians for the whole static scenes. 3DGS is suited for modeling static scenes that assume multi-view consistency, but the transient objects appear occasionally and do not adhere to the assumption, thus we model them as planar objects from a single view by 2D Gaussians. Our novel representation decomposes the scene from the perspective of fundamental viewpoint consistency. Additionally, we present a multi-view supervision method for 3DGS that leverages information from co-visible regions, further enhancing the distinctions between the transients and statics. Then, we propose a straightforward yet effective multi-stage training strategy to ensure robust training and view synthesis. Experiments on benchmarks show our state-of-the-art performance of novel view synthesis in indoor and outdoor scenes, even in the presence of distracting elements. Project page: https://gujiaqivadin.github.io/hybridgs/

Abstract In this paper, we introduce NoPe‐NeRF++, a novel local‐to‐global optimization algorithm for training Neural Radiance Fields (NeRF) without requiring pose priors. Existing methods, particularly NoPe‐NeRF, which focus solely on the local relationships within images, often struggle to recover accurate camera poses in complex scenarios. To overcome the challenges, our approach begins with a relative pose initialization with explicit feature matching, followed by a local joint optimization to enhance the pose estimation for training a more robust NeRF representation. This method significantly improves the quality of initial poses. Additionally, we introduce global optimization phase that incorporates geometric consistency constraints through bundle adjustment, which integrates feature trajectories to further refine poses and collectively boost the quality of NeRF. Notably, our method is the first work that seamlessly combines the local and global cues with NeRF, and outperforms state‐of‐the‐art methods in both pose estimation accuracy and novel view synthesis. Extensive evaluations on benchmark datasets demonstrate our superior performance and robustness, even in challenging scenes, thus validating our design choices.

We introduce a novel class of G 2 continuous splines constructed using an innovative blending method, which guarantees precise interpolation of given control points. These splines are designed to achieve local curvature maxima specifically at these control points and possess compact local support, thereby eliminating the need for global optimization processes. The formulation ensures the splines are free from cusps and self-intersections and, notably, prevents adjacent segments from intersecting—a significant improvement over prior blending-based curve techniques. This framework utilizes quadratic Bézier splines in conjunction with quartic Bézier blending functions. A constructive algorithm is presented that generates these curvature-controlled curves without relying on global optimization. Through parametric adjustments of curvatures, the curve's geometry near control points can be tuned to create features ranging from smooth to sharp, thus broadening the design possibilities. Rigorous mathematical proofs and visual demonstrations validate all claimed properties of the framework.

Point cloud denoising is a fundamental and challenging problem in geometry processing. Existing methods typically involve direct denoising of noisy input or filtering raw normals followed by point position updates. Recognizing the crucial relationship between point cloud denoising and normal filtering, we re-examine this problem from a multitask perspective and propose an end-to-end network called PCDNF for joint normal filtering-based point cloud denoising. We introduce an auxiliary normal filtering task to enhance the network's ability to remove noise while preserving geometric features more accurately. Our network incorporates two novel modules. First, we design a shape-aware selector to improve noise removal performance by constructing latent tangent space representations for specific points, taking into account learned point and normal features as well as geometric priors. Second, we develop a feature refinement module to fuse point and normal features, capitalizing on the strengths of point features in describing geometric details and normal features in representing geometric structures, such as sharp edges and corners. This combination overcomes the limitations of each feature type and better recovers geometric information. Extensive evaluations, comparisons, and ablation studies demonstrate that the proposed method outperforms state-of-the-art approaches in both point cloud denoising and normal filtering.

Abstract Shape interpolation is a fundamental problem in computer graphics. Recently, there have been some interpolation methods developed which guarantee that the results are of bounded amount of geometric distortion, hence ensure high quality interpolation. However, none of these methods is applicable to shapes within the multiply‐connected domains. In this work, we develop an interpolation scheme for harmonic mappings, that specifically addresses this limitation. We opt to interpolate the pullback metric of the input harmonic maps as proposed by Chen et al. [CWKBC13]. However, the interpolated metric does not correspond to any planar mapping, which is the main challenge in the interpolation problem for multiply‐connected domains. We propose to solve this by projecting the interpolated metric into the planar harmonic mapping space. Specifically, we develop a Newton iteration to minimize the isometric distortion of the intermediate mapping, with respect to the interpolated metric. For more efficient Newton iteration, we further derived a simple analytic formula for the positive semidefinite (PSD) projection of the Hessian matrix of our distortion energy. Through extensive experiments and comparisons with the state‐of‐the‐art, we demonstrate the efficacy and robustness of our method for various inputs.

In the mirror cup and saucer art created by artists Yul Cho and Sang-Ha Cho, part of the saucer is directly visible to the viewer, while the other part of the saucer is occluded and can only be seen as a reflection through a mirror cup. Thus, viewers see an image directly on the saucer and another image on the mirror cup; however, the existing art design is limited to wavelike saucers. In this work, we propose a general computational framework for mirror cup and saucer art design. As input, we take from the user one image for the direct view, one image for the reflected view, and the base shape of the saucer. Our algorithm then generates a suitable saucer shape by deforming the input shape. We formulate this problem as a constrained optimization for the saucer surface. Our framework solves for the fine geometry details on the base shape along with its texture, such that when a mirror cup is placed on the saucer, the user-specified images are observed as direct and reflected views. Through extensive experiments, we demonstrate the effectiveness of our framework and the great design flexibility that it offers to users. We further validate the produced art pieces by fabricating the colored saucers using three-dimensional printing.

Recent studies have shown that the Total Generalized Variation (TGV) is highly effective in preserving sharp features as well as smooth transition variations for image processing tasks. However, currently there is no existing work that is suitable for applying TGV to 3D data, in particular, triangular meshes. In this article, we develop a novel framework for discretizing second-order TGV on triangular meshes. Further, we propose a TGV-based variational method for the denoising of face normal fields on triangular meshes. The TGV regularizer in our method is composed of a first-order term and a second-order term, which are automatically balanced. The first-order term allows our TGV regularizer to locate and preserve sharp features, while the second-order term allows our regularizer to recognize and recover smoothly curved regions. To solve the optimization problem, we introduce an efficient iterative algorithm based on variable-splitting and augmented Lagrangian method. Extensive results and comparisons on synthetic and real scanning data validate that the proposed method outperforms the state-of-the-art visually and numerically.

In the age of real-time online traffic information and GPS-enabled devices, fastest-path computations between two points in a road network modeled as a directed graph, where each directed edge is weighted by a “travel time” value, are becoming a standard feature of many navigation-related applications. To support this, very efficient computation of these paths in very large road networks is critical. Fastest paths may be computed as minimal-cost paths in a weighted directed graph, but traditional minimal-cost path algorithms based on variants of the classical Dijkstra algorithm do not scale well, as in the worst case they may traverse the entire graph. A common improvement, which can dramatically reduce the number of graph vertices traversed, is the A* algorithm, which requires a good heuristic lower bound on the minimal cost. We introduce a simple, but very effective, heuristic function based on a small number of values assigned to each graph vertex. The values are based on graph separators and are computed efficiently in a preprocessing stage. We present experimental results demonstrating that our heuristic provides estimates of the minimal cost superior to those of other heuristics. Our experiments show that when used in the A* algorithm, this heuristic can reduce the number of vertices traversed by an order of magnitude compared to other heuristics.

We present a highly efficient method for interactive volumetric meshless shape deformation. Our method operates within a low dimensional sub-space of shape-aware C ∞ harmonic maps, and is the first method that is guaranteed to produce a smooth locally injective deformation in 3D. Unlike mesh-based methods in which local injectivity is enforced on tetrahedral elements, our method enforces injectivity on a sparse set of domain samples. The main difficulty is then to certify the map as locally injective throughout the entire domain. This is done by utilizing the Lipschitz continuity property of the harmonic basis functions. We show a surprising relation between the Lipschitz constant of the smallest singular value of the map Jacobian and the norm of the Hessian. We further carefully derive a Lipschitz constant for the Hessian, and develop a sufficient condition for the injectivity certification. This is done by utilizing the special structure of the harmonic basis functions combined with a novel regularization term that pushes the Lipschitz constants further down. As a result, the injectivity analysis can be performed on a relatively sparse set of samples. Combined with a parallel GPU-based implementation, our method can produce superior deformations with unique quality guarantees at real-time rates which were possible only in 2D so far.

We present a method for volumetric shape interpolation with unique shape preserving features. The input to our algorithm are two or more 3-manifolds, immersed into R 3 and discretized as tetrahedral meshes with shared connectivity. The output is a continuum of shapes that naturally blends the input shapes, while striving to preserve the geometric character of the input. The basis of our approach relies on the fact that the space of metrics with bounded isometric and angular distortion is convex [Chien et al. 2016b]. We show that for high dimensional manifolds, the bounded distortion metrics form a positive semidefinite cone product space. Our method can be seen as a generalization of the bounded distortion interpolation technique of [Chen et al. 2013] from planar shapes immersed in R 2 to solids in R 3 . The convexity of the space implies that a linear blend of the (squared) edge lengths of the input tetrahedral meshes is a simple yet powerful-and-natural choice. Linearly blending flat metrics results in a new metric which is, in general, not flat, and cannot be immersed into three-dimensional space. Nonetheless, the amount of curvature that is introduced in the process tends to be very low in practical settings. We further design an extremely robust nonconvex optimization procedure that efficiently flattens the metric. The flattening procedure strives to preserve the low distortion exhibited in the blended metric while guaranteeing the validity of the metric, resulting in a locally injective map with bounded distortion. Our method leads to volumetric interpolation with superb quality, demonstrating significant improvement over the state-of-the-art and qualitative properties which were obtained so far only in interpolating manifolds of lower dimensions.

Abstract Path generation is an important problem in many fields, especially robotics. One way to create a path between a source point z and a target point y inside a complex planar domain Ω is to define a non‐negative distance function d(y, z), such that following the negative gradient of d (by z) traces out such a path. This presents two challenges: (1) The mathematical challenge of defining d, such that d(y, z) has a single minimum at z = y for any fixed y, because the gradient‐descent path may otherwise terminate at a local minimum before reaching y; (2) The computational challenge of defining d, such that it can be computed efficiently. Using the concepts of harmonic measure and f‐divergence, we show how to assign a set of reduced coordinates to each point in Ω and to define a family of distance functions based on these coordinates, such that both the mathematical and the computational challenge are met. Since in practice, especially in robotics applications, the path is often restricted to follow the edges of a discrete network defined on a finite set of sites sampled from Ω, any method that works well in the continuous setting must be discretized appropriately to preserve the important properties of the continuous case. We show how to define a network connecting a finite set of sites, such that a greedy routing algorithm, which is the discrete equivalent of continuous gradient descent, based on our reduced coordinates is guaranteed to generate a path in the network between any two sites. In many cases, this network is close to a planar graph, especially if the set of sites is dense. Guaranteeing the existence of a greedy route between any two points in the graph is a significant advantage in practical applications, avoiding the complexity of other path‐planning methods, such as the shortest‐path and A* algorithms. While the paths generated by our algorithm are not the shortest possible, in practice we found that they are close to that.

Abstract We present an efficient modified Newton iteration for the optimization of nonlinear energies on triangle meshes. Noting that the linear mapping between any pair of triangles is a special case of harmonic mapping, we build upon the results of Chen and Weber [ CW17 ]. Based on the complex view of the linear mapping, we show that the Hessian of the isometric energies has a simple and compact analytic expression. This allows us to analytically project the per‐element Hessians to positive semidefinite matrices for efficient Newton iteration. We show that our method outperforms state‐of‐the‐art methods on 2D deformation and parameterization. Further, we inspect the spectra of the per triangle energy Hessians and show that given an initial mapping, simple global scaling can shift the energy towards a more convex state. This allows Newton iteration to converge faster than starting from the given initial state. Additionally, our formulations support adding an energy smoothness term to the optimization with little additional effort, which improves the mapping results such that concentrated distortions are reduced.

Abstract Continuous conformal maps are typically approximated numerically using a triangle mesh which discretizes the plane. Computing a conformal map subject to user‐provided constraints then reduces to a sparse linear system, minimizing a quadratic ‘conformal energy’. We address the more general case of non‐triangular elements, and provide a complete analysis of the case where the plane is discretized using a mesh of regular polygons, e.g. equilateral triangles, squares and hexagons, whose interiors are mapped using barycentric coordinate functions. We demonstrate experimentally that faster convergence to continuous conformal maps may be obtained this way. We provide a formulation of the problem and its solution using complex number algebra, significantly simplifying the notation. We examine a number of common barycentric coordinate functions and demonstrate that superior approximation to harmonic coordinates of a polygon are achieved by the Moving Least Squares coordinates. We also provide a simple iterative algorithm to invert barycentric maps of regular polygon meshes, allowing to apply them in practical applications, e.g. for texture mapping.

We present a highly efficient planar meshless shape deformation algorithm. Our method is based on an unconstrained minimization of isometric energies, and is guaranteed to produce C ∞ locally injective maps by operating within a reduced dimensional subspace of harmonic maps. We extend the harmonic subspace of [Chen and Weber 2015] to support multiply-connected domains, and further provide a generalization of the bounded distortion theorem that appeared in that paper. Our harmonic map, as well as the gradient and the Hessian of our isometric energies possess closed-form expressions. A key result is a simple-and-fast analytic modification of the Hessian of the energy such that it is positive definite, which is crucial for the successful operation of a Newton solver. The method is straightforward to implement and is specifically designed to harness the processing power of modern graphics hardware. Our modified Newton iterations are shown to be extremely effective, leading to fast convergence after a handful of iterations, while each iteration is fast due to a combination of a number of factors, such as the smoothness and the low dimensionality of the subspace, the closed-form expressions for the differentials, and the avoidance of expensive strategies to ensure positive definiteness. The entire pipeline is carried out on the GPU, leading to deformations that are significantly faster to compute than the state-of-the-art.

Abstract Discrete conformal mappings of planar triangle meshes, also known as the As‐Similar‐As‐Possible (ASAP) mapping, involve the minimization of a quadratic energy function, thus are very easy to generate and are popular in image warping scenarios. We generalize this classical mapping to the case of quad meshes, taking into account the mapping of the interior of the quad, and analyze in detail the most common case ‐ the unit grid mesh. We show that the generalization, when combined with barycentric coordinate mappings between the source and target polygons, spawns an entire family of new mappings governed by quadratic energy functions, which allow to control quite precisely various effects of the mapping. This approach is quite general and applies also to arbitrary planar polygon meshes. As an application of generalized ASAP mappings of the unit grid mesh, we demonstrate how they can be used to warp digital photographs to achieve a variety of effects. One such effect is modifying the perspective of the camera that took a given photograph (without moving the camera). A related, but more challenging, effect is re‐photography ‐ warping a contemporary photograph in order to reproduce the camera view present in a vintage photograph of the same scene ‐ taken many years before with a different camera from a different viewpoint. We apply the generalized ASAP mapping to these images, discretized to a unit grid. Using a quad mesh (as opposed to a triangle mesh) permits biasing towards affine maps of the unit squares. This allows the introduction of an As‐Affine‐As‐Possible (AAAP) mapping for a good approximation of the homographies present in these warps, achieving quite accurate results. We demonstrate the advantages of the AAAP mapping on a variety of synthetic and real‐world examples.

Abstract Transfinite barycentric kernels are the continuous version of traditional barycentric coordinates and are used to define interpolants of values given on a smooth planar contour. When the data is two‐dimensional, i.e. the boundary of a planar map, these kernels may be conveniently expressed using complex number algebra, simplifying much of the notation and results. In this paper we develop some of the basic complex‐valued algebra needed to describe these planar maps, and use it to define similarity kernels, a natural alternative to the usual barycentric kernels. We develop the theory behind similarity kernels, explore their properties, and show that the transfinite versions of the popular three‐point barycentric coordinates (Laplace, mean value and Wachspress) have surprisingly simple similarity kernels. We furthermore show how similarity kernels may be used to invert injective transfinite barycentric mappings using an iterative algorithm which converges quite rapidly. This is useful for rendering images deformed by planar barycentric mappings.

Planar shape interpolation is a classic problem in computer graphics. We present a novel shape interpolation method that blends C ∞ planar harmonic mappings represented in closed-form. The intermediate mappings in the blending are guaranteed to be locally injective C ∞ harmonic mappings, with conformal and isometric distortion bounded by that of the input mappings. The key to the success of our method is the fact that the blended differentials of our interpolated mapping have a simple closed-form expression, so they can be evaluated with unprecedented efficiency and accuracy. Moreover, in contrast to previous approaches, these differentials are integrable, and result in an actual mapping without further modification. Our algorithm is embarrassingly parallel and is orders of magnitude faster than state-of-the-art methods due to its simplicity, yet it still produces mappings that are superior to those of existing techniques due to its guaranteed bounds on geometric distortion.

We present a framework for the computation of harmonic and conformal mappings in the plane with mathematical guarantees that the computed mappings are C ∞ , locally injective and satisfy strict bounds on the conformal and isometric distortion. Such mappings are very desirable in many computer graphics and geometry processing applications. We establish the sufficient and necessary conditions for a harmonic planar mapping to have bounded distortion. Our key observation is that these conditions relate solely to the boundary behavior of the mapping. This leads to an efficient and accurate algorithm that supports handle-based interactive shape-and-image deformation and is demonstrated to outperform other state-of-the-art methods.

Polyhedral meshes (PM)-meshes having planar faces-have enjoyed a rise in popularity in recent years due to their importance in architectural and industrial design. However, they are also notoriously difficult to generate and manipulate. Previous methods start with a smooth surface and then apply elaborate meshing schemes to create polyhedral meshes approximating the surface. In this paper, we describe a reverse approach: given the topology of a mesh, we explore the space of possible planar meshes having that topology. Our approach is based on a complete characterization of the maximal linear spaces of polyhedral meshes contained in the curved manifold of polyhedral meshes with a given topology. We show that these linear spaces can be described as nullspaces of differential operators, much like harmonic functions are nullspaces of the Laplacian operator. An analysis of this operator provides tools for global and local design of a polyhedral mesh, which fully expose the geometric possibilities and limitations of the given topology.

Planar shape interpolation is widely used in computer graphics applications. Despite a wealth of interpolation methods, there is currently no approach that produces shapes with a bounded amount of distortion with respect to the input. As a result, existing interpolation methods may produce shapes that are significantly different than the input and can suffer from fold-overs and other visual artifacts, making them less useful in many practical scenarios. We introduce a novel shape interpolation scheme designed specifically to produce results with a bounded amount of conformal (angular) distortion. Our method is based on an elegant continuous mathematical formulation and provides several appealing properties such as existence and uniqueness of the solution as well as smoothness in space and time domains. We further present a discretization and an efficient practical algorithm to compute the interpolant and demonstrate its usability and good convergence behavior on a wide variety of input shapes. The method is simple to implement and understand. We compare our method to state-of-the-art interpolation methods and demonstrate its superiority in various cases.

Abstract We describe a fast sampling algorithm for generating uniformly‐distributed point patterns with good blue noise characteristics. The method, based on constrained farthest point optimization, is provably optimal and may be easily parallelized, resulting in an algorithm whose performance/quality tradeoff is superior to other state‐of‐the‐art approaches.

Abstract Aesthetic images evoke an emotional response that transcends mere visual appreciation. In this work we develop a novel computational means for evaluating the composition aesthetics of a given image based on measuring several well‐grounded composition guidelines. A compound operator of crop‐and‐retarget is employed to change the relative position of salient regions in the image and thus to modify the composition aesthetics of the image. We propose an optimization method for automatically producing a maximally‐aesthetic version of the input image. We validate the performance of the method and show its effectiveness in a variety of experiments.