This paper presents PCDreamer, a novel method for point cloud completion. Traditional methods typically extract features from partial point clouds to predict missing regions, but the large solution space often leads to unsatisfactory results. More recent approaches have started to use images as extra guidance, effectively improving performance, but obtaining paired data of images and partial point clouds is challenging in practice. To overcome these limitations, we harness the relatively view-consistent multi-view diffusion priors within large models, to generate novel views of the desired shape. The resulting image set encodes both global and local shape cues, which are especially beneficial for shape completion. To fully exploit the priors, we have designed a shape fusion module for producing an initial complete shape from multi-modality input (i.e., images and point clouds), and a follow-up shape consolidation module to obtain the final complete shape by discarding unreliable points introduced by the inconsistency from diffusion priors. Extensive experimental results demonstrate our superior performance, especially in recovering fine details.

As commonly used implicit geometry representations, the signed distance function (SDF) is limited to modeling watertight shapes, while the unsigned distance function (UDF) is capable of representing various surfaces. However, its inherent theoretical shortcoming, i.e., the non-differentiability at the zero-level set, would result in sub-optimal reconstruction quality. In this paper, we propose the scaled-squared distance function (S2DF), a novel implicit surface representation for modeling arbitrary surface types. S2DF does not distinguish between inside and outside regions while effectively addressing the non-differentiability issue of UDF at the zero-level set. We demonstrate that S2DF satisfies a second-order partial differential equation of Monge-Ampere-type, allowing us to develop a learning pipeline that leverages a novel MongeAmpere regularization to directly learn S2DF from raw unoriented point clouds without supervision from ground-truth S2DF values. Extensive experiments across multiple datasets show that our method significantly outperforms state-of-the-art supervised approaches that require ground-truth surface information as supervision for training. The code will be publicly available at https://github.com/chuanxiang-yang/S2DF.

Extracting wireframes from CAD models represented by point cloud remains a significant challenge in computer graphics. This difficulty arises from two main factors: first, imperfections in the point cloud data, such as lack of orientation, noise, and sparsity; and second, the inherent complexity of geometric shapes, which often feature a high density of sharp edges in close proximity. In this paper, we propose D-FRAME, a multi-stage wireframe extraction framework that incorporates a novel direction field to improve edge detection quality and connectivity, a refinement strategy to address sparse or noisy edge points, and a final coarse-to-fine connection module to extract a robust wireframe. The direction field not only facilitates connectivity but also enhances the precision of extracted edges by mitigating the impact of misclassified points. By combining the Restricted Voronoi Diagram (RVD) with the extracted wireframes and the original point cloud, our approach also achieves highly faithful reconstruction of CAD model. Experiments conducted on synthetic and real-world scanned CAD datasets demonstrate that D-FRAME effectively manages noise, sparsity, and complex geometries, yielding high-fidelity wireframes.

We propose a new method for computing smooth and integrable cross fields on 2D and 3D surfaces. our approach first computes smooth cross fields by minimizing the Dirichlet energy. Unlike existing optimization-based methods, our technique determines the singularity configuration-i.e., the number, locations, and indices of singularities-by iteratively adjusting them. Singularities can move, merge and split, akin to the behavior of like charges repelling and unlike charges attracting. Once all singularities stop moving, we obtain a cross field with (locally) the lowest Dirichlet energy. In simply connected domains, this cross field is guaranteed to be integrable. However, this property does not hold in multiply connected domains. To make a smooth cross field integrable, we construct a vector field $\bf c$c that characterizes the deviation of the cross field from a curl-free field. We then optimize the locations of singularities by moving them along the field lines of $\bf c$c. Our method is fundamentally different from existing integer programming-based approaches, as it avoids combinatorial optimization. It is fully automatic and includes a parameter to control the number of singularities. Our method is well suited for smooth models where exact boundary alignment and sparse hard directional constraints are desired, and can guide seamless conformal parameterization and T-junction-free quadrangulation.

The article presents a new method for constructing self-supporting surfaces using arch beams that are designed to convert their thrust into supporting force, thereby eliminating shear stress and bending moments. Our method allows for the placement of the arch beams on the boundary or within a surface and partitions the surface into multiple self-supporting parts. The use of arch beams enhances stability and durability, adds aesthetic appeal, and allows for greater flexibility in the design process. We develop an iterative algorithm for designing self-supporting surfaces with arch beams that enables the user to control the shape of the beams and surface through intuitive parameters and specify the desired location of the arch beams. We verify the physical stability of the structure using finite element analysis. Experimental results show that our method can produce visually pleasing self-supporting surfaces that satisfy the equilibrium equation with high accuracy.

We present a simple yet effective method for constructing 3D self-supporting surfaces with planar quadrilateral (PQ) elements. Starting with a triangular discretization of a self-supporting surface, we first compute the principal curvatures and directions of each triangular face using a new discrete differential geometry approach, yielding more accurate results than existing methods. Then, we smooth the principal direction field to reduce the number of singularities. Next, we partition all faces into two groups in terms of principal curvature difference. For each face with small curvature difference, we compute a stretch matrix that turns the principal directions into a pair of conjugate directions. For the remaining triangular faces, we simply keep their smoothed principal directions. Finally, applying a mixed-integer programming solver to the mixed principal and conjugate direction field, we obtain a planar quadrilateral mesh. Experimental results show that our method is computationally efficient and can yield high-quality PQ meshes that well approximate the geometry of the input surfaces and maintain their self-supporting properties.

This article presents a new computational framework for constructing 3D self-supporting surfaces with isotropic stress. Inspired by the self-supporting property of catenary and the fact that catenoid (the surface of revolution of the catenary curve) is a minimal surface, we discover the relation between 3D self-supporting surfaces and 4D minimal hypersurfaces (which are 3-manifolds). Lifting the problem into 4D allows us to convert gravitational forces into tensions and reformulate the equilibrium problem to total potential energy minimization, which can be solved using a variational method. We prove that the hyper-generatrix of a 4D minimal hyper-surface of revolution is a 3D self-supporting surface, implying that constructing a 3D self-supporting surface is equivalent to volume minimization. We show that the energy functional is simply the surface’s gravitational potential energy, which in turn can be converted into a surface reconstruction problem with mean curvature constraint. Armed with our theoretical findings, we develop an iterative algorithm to construct 3D self-supporting surfaces from triangle meshes. Our method guarantees convergence and can produce near-regular triangle meshes, thanks to a local mesh refinement strategy similar to centroidal Voronoi tessellation. It also allows users to tune the geometry via specifying either the zero potential surface or its desired volume. We also develop a finite element method to verify the equilibrium condition on 3D triangle meshes. The existing thrust network analysis methods discretize both geometry and material by approximating the continuous stress field through uniaxial singular stresses, making them an ideal tool for analysis and design of beam structures. In contrast, our method works on piecewise linear surfaces with continuous material. Moreover, our method does not require the 3D-to-2D projection, therefore it also works for both height and non-height fields.