Piecewise parametric surfaces have long been established as prevalent geometric representations; however, they often require surface refinement or sophisticated quadrangulation to accurately represent complex geometries. Geometric deep learning has shown that neural networks can provide greater representational power than conventional methods. Nevertheless, approaches using a single parametric surface for shape fitting struggle to capture fine-grained geometric details, while multi-patch methods fail to ensure seamless connections between adjacent patches. We present Neural Piecewise Parametric Surfaces ( NeuPPS ), the first piecewise neural surface representation that allows for coarse patch layouts composed of arbitrary n -sided surface patches to model complex surface geometries with high precision, offering enhanced flexibility compared with traditional parametric surfaces. This new surface representation guarantees, by construction, the continuity between adjacent patches, a property that other neural patch-based approaches cannot ensure. Two novel components are introduced: a learnable feature complex and a continuous mapping function approximated by multi-layer perceptrons (MLPs). We apply the proposed NeuPPS to surface fitting and shape space learning tasks. Extensive experiments demonstrate the advantages of NeuPPS over traditional parametric representations and existing patch-based learning approaches.

While Implicit Neural Representations (INRs) have demonstrated significant success in image representation, they are often hindered by large training memory and slow decoding speed. Recently, Gaussian Splatting (GS) has emerged as a promising solution in 3D reconstruction due to its highquality novel view synthesis and rapid rendering capabilities, positioning it as a valuable tool for a broad spectrum of applications. In particular, a GS-based representation, 2DGS, has shown potential for image fitting. In our work, we present Large Images are Gaussians (LIG), which delves deeper into the application of 2DGS for image representations, addressing the challenge of fitting large images with 2DGS in the situation of numerous Gaussian points, through two distinct modifications: 1) we adopt a variant of representation and optimization strategy, facilitating the fitting of a large number of Gaussian points; 2) we propose a Level-of-Gaussian approach for reconstructing both coarse low-frequency initialization and fine high-frequency details. Consequently, we successfully represent large images as Gaussian points and achieve high-quality large image representation, demonstrating its efficacy across various types of large images.

Neural implicit representations are increasingly used to depict three-dimensional (3D) shapes owing to their inherent smoothness and compactness, contrasting with traditional discrete representations. Yet, the multilayer perceptron–based neural representation, because of its smooth nature, rounds sharp corners or edges, rendering it unsuitable for representing objects with sharp features like computer-aided design (CAD) models. Moreover, neural implicit representations need long training times to fit 3D shapes. While previous works address these issues separately, we present a unified neural implicit representation called Patch-Grid , which efficiently fits complex shapes, preserves sharp features delineating different patches, and can also represent surfaces with open boundaries and thin geometric features. Patch-Grid learns a signed distance field (SDF) to approximate an encompassing surface patch of the shape with a learnable patch feature volume. To form sharp edges and corners in a CAD model, Patch-Grid merges the learned SDFs via the constructive solid geometry (CSG) approach. Core to the merging process is a novel merge grid design that organizes different patch feature volumes in a common octree structure. This design choice ensures robust merging of multiple learned SDFs by confining the CSG operations to localized regions. Additionally, it drastically reduces the complexity of the CSG operations in each merging cell, allowing the proposed method to be trained in seconds to fit a complex shape at high fidelity. Experimental results demonstrate that the proposed Patch-Grid representation is capable of accurately reconstructing shapes with complex sharp features, open boundaries, and thin geometric elements, achieving state-of-the-art reconstruction quality with high computational efficiency within seconds.

Neural implicit shape representation has drawn significant attention in recent years due to its smoothness, differentiability, and topological flexibility. However, directly modeling the shape of a neural implicit surface, especially as the zero-level set of a neural signed distance function (SDF), with sparse geometric control is still a challenging task. Sparse input shape control typically includes 3D curve networks or, more generally, 3D curve sketches, which are unstructured and cannot be connected to form a curve network, and therefore more difficult to deal with. While 3D curve networks or curve sketches provide intuitive shape control, their sparsity and varied topology pose challenges in generating high-quality surfaces to meet such curve constraints. In this paper, we propose NeuVAS, a variational approach to shape modeling using neural implicit surfaces constrained under sparse input shape control, including unstructured 3D curve sketches as well as connected 3D curve networks. Specifically, we introduce a smoothness term based on a functional of surface curvatures to minimize shape variation of the zero-level set surface of a neural SDF. We also develop a new technique to faithfully model G 0 sharp feature curves as specified in the input curve sketches. Comprehensive comparisons with the state-of-the-art methods demonstrate the significant advantages of our method.

Neural implicit fields, such as the neural signed distance field (SDF) of a shape, have emerged as a powerful representation for many applications, e.g., encoding a 3D shape and performing collision detection. Typically, implicit fields are encoded by Multi-layer Perceptrons (MLP) with positional encoding (PE) to capture high-frequency geometric details. However, a notable side effect of such PE-equipped MLPs is the noisy artifacts present in the learned implicit fields. While increasing the sampling rate could in general mitigate these artifacts, in this paper we aim to explain this adverse phenomenon through the lens of Fourier analysis. We devise a tool to determine the appropriate sampling rate for learning an accurate neural implicit field without undesirable side effects. Specifically, we propose a simple yet effective method to estimate the intrinsic frequency of a given network with randomized weights based on the Fourier analysis of the network's responses. It is observed that a PE-equipped MLP has an intrinsic frequency much higher than the highest frequency component in the PE layer. Sampling against this intrinsic frequency following the Nyquist-Sannon sampling theorem allows us to determine an appropriate training sampling rate. We empirically show in the setting of SDF fitting that this recommended sampling rate is sufficient to secure accurate fitting results, while further increasing the sampling rate would not further noticeably reduce the fitting error. Training PE-equipped MLPs simply with our sampling strategy leads to performances superior to the existing methods.

We propose a method, named DualMesh-UDF, to extract a surface from unsigned distance functions (UDFs), encoded by neural networks, or neural UDFs. Neural UDFs are becoming increasingly popular for surface representation because of their versatility in presenting surfaces with arbitrary topologies, as opposed to the signed distance function that is limited to representing a closed surface. However, the applications of neural UDFs are hindered by the notorious difficulty in extracting the target surfaces they represent. Recent methods for surface extraction from a neural UDF suffer from significant geometric errors or topological artifacts due to two main difficulties: (1) A UDF does not exhibit sign changes; and (2) A neural UDF typically has substantial approximation errors.DualMesh-UDF addresses these two difficulties. Specifically, given a neural UDF encoding a target surface $\bar S$ to be recovered, we first estimate the tangent planes of $\bar S$ at a set of sample points close to $\bar S$. Next, we organize these sample points into local clusters, and for each local cluster, solve a linear least squares problem to determine a final surface point. These surface points are then connected to create the output mesh surface, which approximates the target surface. The robust estimation of the tangent planes of the target surface and the subsequent minimization problem constitute our core strategy, which contributes to the favorable performance of DualMesh-UDF over other competing methods. To efficiently implement this strategy, we employ an adaptive Octree. Within this framework, we estimate the location of a surface point in each of the octree cells identified as containing part of the target surface. Extensive experiments show that our method outperforms existing methods in terms of surface reconstruction quality while maintaining comparable computational efficiency.