Deep functional map frameworks (DFM) for shape correspondence are powerful, yet fundamentally limited by their reliance on end-to-end differentiability. This constraint prevents the integration of highly accurate, non-differentiable refinement techniques, capping their overall performance, especially on challenging non-isometric shapes. To overcome this, we introduce MDND, a novel DFM paradigm built on the principle of merging differentiable and non-differentiable components. Our framework facilitates unsupervised learning guided by an internal, non-differentiable refinement. Specifically, MDND employs a dual-branch architecture: a non-differentiable refinement branch leverages a novel, multiscale iterative solver to produce highly robust correspondences, acting as a refined target. Concurrently, a fully differentiable branch learns to predict correspondences from features. The entire system is trained end-to-end without supervision by enforcing a consistency loss that compels the differentiable branch to learn from the superior, refined results of the non-differentiable branch. Extensive experiments show that MDND sets a new state-of-the-art, demonstrating remarkable robustness on shapes with non-isometric deformations and topological noise.

Abstract Horizontal and vertical lines hold significant aesthetic and psychological importance, providing a sense of order, stability, and security. This paper presents an image stylization method that quickly generates non‐self‐intersecting and regular continuous lines based on the Hilbert curve, a well‐known space‐filling curve consisting of only horizontal and vertical segments. We first calculate the grayscale threshold based on gray quantization for the original image and recursively subdivide the cells according to the density in each cell. To avoid generating new feature curves due to limited gray quantization, a recursive subdivision with probability is designed to smooth the density. Then, we utilize the rule of Hilbert curve to generate continuous lines connecting all the cells. Between different degrees of Hilbert curves, bridge curves composed of horizontal and vertical lines are constructed, which are also intersection‐free, instead of a straight line linking them directly. There are two parameters provided for feasibly adjusting variate effects. The image stylization framework could be generalized to other space‐filling curves like the Peano curve. Compared to existing methods, our approach can generate pleasing results quickly and is fully automated. Many results show our method is robust and effective.

In recent years, deep functional maps (DFM) have emerged as a leading learning-based framework for non-rigid shape-matching problems, offering diverse network architectures for this domain. This richness also makes exploring better and novel design beliefs for existing powerful DFM components to promote performance meaningful and engaging. This paper delves into this problem and successfully produces the SEDFMNet, a simple yet highly efficient DFM pipeline. To achieve this, we systematically deconstruct the core modules of the general DFM framework and analyze key design choices in existing approaches to identify the most critical components through extensive experiments. By reassembling these crucial components, we culminate in developing our SEDFMNet, which features a simpler structure than conventional DFM pipelines while delivering superior performance. Our approach is rigorously validated through comprehensive experiments on diverse datasets, where the SEDFMNet consistently achieves state-of-the-art results, even in challenging scenarios such as non-isometric shape matching and shape matching with topological noise. Our work offers fresh insights into DFM research and opens new avenues for advancing this field.

A large number of 3D spectral descriptors have been proposed in the literature, which act as an essential component for 3D deformable shape matching and related applications. An outstanding descriptor should have desirable natures including high-level descriptive capacity, cheap storage, and robustness to a set of nuisances. It is, however, unclear which descriptors are more suitable for a particular application. This paper fills the gap by comprehensively evaluating nine state-of-the-art spectral descriptors on ten popular deformable shape datasets as well as perturbations such as mesh discretization, geometric noise, scale transformation, non-isometric setting, partiality, and topological noise. Our evaluated terms for a spectral descriptor cover four major concerns, i.e., distinctiveness, robustness, compactness, and computational efficiency. In the end, we present a summary of the overall performance and several interesting findings that can serve as guidance for the following researchers to construct a new spectral descriptor and choose an appropriate spectral feature in a particular application.

Traditional deep functional map frameworks are widely used for 3D shape matching; however, many methods fail to adaptively capture the relevant frequency information required for functional map estimation in complex scenarios, leading to poor performance, especially under significant deformations. To address these challenges, we propose a novel unsupervised learning-based framework, Deep Frequency Awareness Functional Maps (DFAFM), specifically designed to tackle diverse shape-matching problems. Our approach introduces the Spectral Filter Operator Preservation constraint, which ensures the preservation of critical frequency information. These constraints promote frequency awareness by learning a set of spectral filters and incorporating them as a loss function to jointly supervise the functional maps, pointwise maps, and spectral filters. The spectral filters are constructed using orthonormal Jacobi polynomials with learnable coefficients, enabling adaptive and efficient frequency representation. Furthermore, we propose a refinement strategy that leverages the learned spectral filters and constraints to enhance the accuracy of the final pointwise map. Extensive experiments conducted on multiple benchmark datasets demonstrate that our method outperforms state-of-the-art approaches, particularly in challenging scenarios involving non-isometric deformations and inconsistent topology.

Abstract In deep functional maps, the regularizer computing the functional map is especially crucial for ensuring the global consistency of the computed pointwise map. As the regularizers integrated into deep learning should be differentiable, it is not trivial to incorporate informative axiomatic structural constraints into the deep functional map, such as the orientation‐preserving term. Although commonly used regularizers include the Laplacian‐commutativity term and the resolvent Laplacian commutativity term, these are limited to single‐scale analysis for capturing geometric information. To this end, we propose a novel and theoretically well‐justified regularizer commuting the functional map with the multiscale spectral manifold wavelet operator. This regularizer enhances the isometric constraints of the functional map and is conducive to providing it with better structural properties with multiscale analysis. Furthermore, we design an unsupervised deep functional map with the regularizer in a fully differentiable way. The quantitative and qualitative comparisons with several existing techniques on the (near‐)isometric and non‐isometric datasets show our method's superior accuracy and generalization capabilities. Additionally, we illustrate that our regularizer can be easily inserted into other functional map methods and improve their accuracy.

In traditional deep functional maps for non-rigid shape correspondence, estimating a functional map including high-frequency information requires enough linearly independent features via the least square method, which is prone to be violated in practice, especially at an early stage of training, or costly post-processing, e.g. ZoomOut. In this paper, we propose a novel method called RFMNet (Robust Deep Functional Map Networks), which jointly considers training stability and more geometric shape features than previous works. We directly first produce a pointwise map by resorting to optimal transport and then convert it to an initial functional map. Such a mechanism mitigates the requirements for the descriptor and avoids the training instabilities resulting from the least square solver. Benefitting from the novel strategy, we successfully integrate a state-of-the-art geometric regularization for further optimizing the functional map, which substantially filters the initial functional map. We show our novel computing functional map module brings more stable training even under encoding the functional map with high-frequency information and faster convergence speed. Considering the pointwise and functional maps, an unsupervised loss is presented for penalizing the correspondence distortion of Delta functions between shapes. To catch discretization-resistant and orientation-aware shape features with our network, we utilize DiffusionNet as a feature extractor. Experimental results demonstrate our apparent superiority in correspondence quality and generalization across various shape discretizations and different datasets compared to the state-of-the-art learning methods.

Shape descriptors have recently gained popularity in shape matching, statistical shape modeling, etc. Their discriminative ability and efficiency play a decisive role in these tasks. In this paper, we first propose a novel handcrafted anisotropic spectral descriptor using Chebyshev polynomials, called the anisotropic Chebyshev descriptor (ACD); it can effectively capture shape features in multiple directions. The ACD inherits many good characteristics of spectral descriptors, such as being intrinsic, robust to changes in surface discretization, etc. Furthermore, due to the orthogonality of Chebyshev polynomials, the ACD is compact and can disambiguate intrinsic symmetry since several directions are considered. To improve the ACD’s discrimination ability, we construct a Chebyshev spectral manifold convolutional neural network (CSMCNN) that optimizes the ACD and produces a learned ACD. Our experimental results show that the ACD outperforms existing state-of-the-art handcrafted descriptors. The combination of the ACD and the CSMCNN is better than other state-of-the-art learned descriptors in terms of discrimination, efficiency, and robustness to changes in shape resolution and discretization.

Abstract We propose a novel unsupervised learning approach for computing correspondences between non‐rigid 3D shapes. The core idea is that we integrate a novel structural constraint into the deep functional map pipeline, a recently dominant learning framework for shape correspondence, via a powerful spectral manifold wavelet transform (SMWT). As SMWT is isometrically invariant operator and can analyze features from multiple frequency bands, we use the multiscale SMWT results of the learned features as function preservation constraints to optimize the functional map by assuming each frequency‐band information of the descriptors should be correspondingly preserved by the functional map. Such a strategy allows extracting significantly more deep feature information than existing approaches which only use the learned descriptors to estimate the functional map. And our formula strongly ensure the isometric properties of the underlying map. We also prove that our computation of the functional map amounts to filtering processes only referring to matrix multiplication. Then, we leverage the alignment errors of intrinsic embedding between shapes as a loss function and solve it in an unsupervised way using the Sinkhorn algorithm. Finally, we utilize DiffusionNet as a feature extractor to ensure that discretization‐resistant and directional shape features are produced. Experiments on multiple challenging datasets prove that our method can achieve state‐of‐the‐art correspondence quality. Furthermore, our method yields significant improvements in robustness to shape discretization and generalization across the different datasets. The source code and trained models will be available at https://github.com/HJ-Xu/WTFM-Layer .

The functional map framework has proven to be extremely effective for representing dense correspondences between deformable shapes. A key step in this framework is to formulate suitable preservation constraints to encode the geometric information that must be preserved by the unknown map. For this issue, we construct novel and powerful constraints to determine the functional map, where multiscale spectral manifold wavelets are required to be preserved at each scale correspondingly. Such constraints allow us to extract significantly more information than previous methods, especially those based on descriptor preservation constraints, and strongly ensure the isometric property of the map. In addition, we also propose a remarkable efficient iterative method to alternatively update the functional maps and pointwise maps. Moreover, when we use the tight wavelet frames in iterations, the computation of the functional maps boils down to a simple filtering procedure with low-pass and various band-pass filters, which avoids time-consuming solving large systems of linear equations commonly presented in functional maps. We demonstrate on a wide variety of experiments with different datasets that our approach achieves significant improvements both in the shape correspondence quality and the computing efficiency.

Contour detection plays an important role in both academic research and real-world applications. As the basic building block of many applications, its accuracy and efficiency highly influence the subsequent stages. In this work, we propose a novel lightweight system for contour detection that achieves state-of-the-art performance while keeps ultra-slim model size. The proposed method is built on an efficient encoder in a bottom-up/top-down fashion. Specially, we propose a novel decoder that compresses side features from an encoder and effectively decodes compact contextual information for high-accurate boundary localization. Besides, we propose a novel loss function that is able to assist a model to produce crisp object boundaries.

Abstract In this paper, we present a powerful spectral shape descriptor for shape analysis, named Anisotropic Spectral Manifold Wavelet Descriptor (ASMWD). We proposed a novel manifold harmonic signal processing tool termed Anisotropic Spectral Manifold Wavelet Transform (ASMWT) first. ASMWT allows to comprehensively analyse signals from multiple wavelet diffusion directions on local manifold regions of the shape with a series of low‐pass and band‐pass frequency filters in each direction. Based on the ASMWT coefficients of a very simple signal, the ASMWD is efficiently constructed as a localizable and discriminative multi‐scale point descriptor. Since the wavelets used in our descriptor are direction‐sensitive and able to robustly reconstruct the signals with a finite number of scales, it makes our descriptor compact, efficient, and unambiguous under intrinsic symmetry. The extensive experiments demonstrate that our descriptor achieves significantly better performance than the state‐of‐the‐art descriptors and can greatly improve the performance of shape matching methods including both handcrafted and learning‐based methods.

Establishing correspondence between shapes is a very important and active research topic in many domains. Due to the powerful ability of deep learning on geometric data, lots of attractive results have been achieved by convolutional neural networks (CNNs). In this paper, we propose a novel architecture for shape correspondence, termed Anisotropic Chebyshev spectral CNNs (ACSCNNs), based on a new extension of the manifold convolution operator. The extended convolution operators aggregate the local features of signals by a set of oriented kernels around each point, which allows to much more comprehensively capture the intrinsic signal information. Rather than using fixed oriented kernels in the spatial domain in previous CNNs, in our framework, the kernels are learned by spectral filtering, based on the eigen-decompositions of multiple Anisotropic Laplace-Beltrami Operators. To reduce the computational complexity, we employ an explicit expansion of the Chebyshev polynomial basis to represent the spectral filters whose expansion coefficients are trainable. Through the benchmark experiments of shape correspondence, our architecture is demonstrated to be efficient and be able to provide better than the state-of-the-art results in several datasets even if using constant functions as inputs.

Object contour detection is the fundamental and preprocessing step for multimedia applications such as icon generation, object segmentation, and tracking. The quality of contour prediction is of great importance in these applications since it affects the subsequent process. In this work, we aim to develop a high-performance contour detection system. We first propose a novel yet very effective loss function for contour detection. The proposed loss function is capable of penalizing the distance of contour-structure similarity between each pair of prediction and ground-truth. Moreover, to better distinguishing object contours and background textures, we introduce a novel convolutional encoder-decoder network. Within the network, we present a hyper module that captures dense connections among high-level features and produces effective semantic information. Then the information is progressively propagated and fused with low-level features. We conduct extensive experiments on the BSDS500 and Multi-Cue datasets, the results show significant improvement against the state-of-the-art competitors. We further demonstrate the benefit of our DSCD method for crowd counting.

In this paper, quasi-interpolation based on compactly supported radial basis functions (CSRBFs) is presented for more accurate and efficient data fitting compared with global RBFs. Firstly, a CSRBF-based quasi-interpolator is constructed considering only the positions of the given data and their values. Then we make use of the first derivatives to propose a new quasi-interpolator which can achieve higher approximate order and better shape-preserving. Numerical examples demonstrate that the proposed CSRBF-based quasi-interpolation schemes are valid.

Abstract The emergence of laser/LiDAR sensors, reliable multi‐view stereo techniques and more recently consumer depth cameras have brought point clouds to the forefront as a data format useful for a number of applications. Unfortunately, the point data from those channels often incur imperfection, frequently contaminated with severe outliers and noise. This paper presents a robust consolidation algorithm for low‐quality point data from outdoor scenes, which essentially consists of two steps: 1) outliers filtering and 2) noise smoothing. We first design a connectivity‐based scheme to evaluate outlierness and thereby detect sparse outliers. Meanwhile, a clustering method is used to further remove small dense outliers. Both outlier removal methods are insensitive to the choice of the neighborhood size and the levels of outliers. Subsequently, we propose a novel approach to estimate normals for noisy points based on robust partial rankings, which is the basis of noise smoothing. Accordingly, a fast approach is exploited to smooth noise, while preserving sharp features. We evaluate the effectiveness of the proposed method on the point clouds from a variety of outdoor scenes.

A fast offset surface generation approach is presented in this paper to construct intersection-free offset surfaces, which preserve sharp features, from freeform triangular mesh surfaces. The basic spirit of our algorithm is to sample a narrowband signed distance-field from the input model on a uniform grid and then employ a contouring algorithm to build the resultant offset mesh surface from the signed distance-field. Four filters are conducted to generate the narrowband signed distance-field around the offset surface in a very efficient way by alleviating computation redundancies in the regions far from the offset surfaces. The resultant mesh surfaces are generated by a modified dual contouring algorithm which relies on accurate intersections between the grid edges and the isosurfaces. A hybrid method is developed to prevent the expensive bisection search in the configurations that the analytical solutions exist. Our modified intersection-free dual contouring algorithm is based on convex-concave analysis, which is more robust and efficient. The quality and performance of our approach are demonstrated with a number of experimental tests on various examples.

This paper presents an algorithm to approximate a solid model by a hierarchical set of bounding ellipsoids having optimal shape and volume approximation errors. The ellipsoid-tree is constructed in a top-down splitting framework. Starting from the root of hierarchy the volume occupied by a given model is divided into k sub-volumes where each is approximated by a volume bounding ellipsoid and will be later subdivided into k ellipsoids for the next level in hierarchy. The difficulty for implementing this algorithm comes from how to evaluate the volume of an ellipsoid outside the given model effectively and efficiently (i.e., the outside-volume-error). A new method - analytical computation based - is presented in this paper to compute the outside-volume-error. One application of ellipsoid-tree approximation has also been given at the end of the paper.

As ellipsoids have been employed in the collision handling of many applications in physical simulation and robotics systems, we present a novel algorithm for generating a bounding volume hierarchy (BVH) from a given model with ellipsoids as primitives. Our algorithm approximates the given model by a hierarchical set of optimized bounding ellipsoids. The ellipsoid-tree is constructed by a top-down splitting. Starting from the root of hierarchy, the volume occupied by a given model is divided into k sub-volumes where each is approximated by a volume bounding ellipsoid. Recursively, each sub-volume is then subdivided into ellipsoids for the next level in the hierarchy. The k ellipsoids at each hierarchy level for a sub-volume bounding is generated by a bottom-up algorithm - simply, the sub-volume is initially approximated by m spheres (m » k), which will be iteratively merged into k volume bounding ellipsoids and globally optimized to minimize the approximation error. Benefited from the anisotropic shape of primitives, the ellipsoid-tree constructed in our approach gives tighter volume bound and higher shape fidelity than another widely used BVH, sphere-tree.

We present a new high quality tessellation method for implicit surfaces in this paper. The approach can handle arbitrary implicit functions and dynamic implicit surfaces based on skeletal primitives. We first samples the implicit surface uniformly using particle fission and floating, then reconstructs a triangular mesh from the sample points using ball pivoting algorithm (BPA). Finally, we subdivide the reconstructed surface using a 1 to 4 subdivision scheme to obtain the high quality implicit surface tessellation.

Abstract In this paper, we propose a novel practical method for blob‐based liquid 3D morphing. Firstly, blobby objects are employed to approximate a given polygonal surface. The primitives in the medial axis sphere‐tree of a polygonal model are utilized as initial blobs—this greatly improves the robustness and efficiency of the blob‐based approximation. Secondly, we establish the blob correspondences between two models by sphere cellular matching and hierarchical matching. Finally, we interpolate the parameters of the implicit representation to get the intermediate shapes. Experiments show our method can produce visually pleasing liquid morphing effects. Copyright © 2005 John Wiley & Sons, Ltd.