Polycube-maps play a critical role in computer graphics, especially for generating high-quality hexahedral meshes. Existing polycube validity conditions, primarily based on Steinitz and Eppstein’s approach, are limited to 3-connected graphs. Extending polycube-maps to handle higher connectivity graphs is crucial for practical applications. In this work, we introduce Validity-Augmented Topological Conditions (VAT conditions) based on the Gauss–Bonnet theorem. These conditions offer both global and local topological criteria, enabling the solvability of k-connected graphs, non-manifold structures, and meshes with voids. Our VAT conditions allow models that do not meet traditional polycube validity criteria but are still valid polycube polyhedra in practice. Additionally, we propose an Immune Genetic Algorithm (ImGA) tailored to our VAT conditions to enhance the robustness of polycube-map generation. We evaluate our method using the Thingi10k and ABC datasets. Results demonstrate that our VAT conditions expands the solvable space of polycubes and achieves higher quality all-hexahedral meshing for higher-connectivity or more complex models. Furthermore, we discuss the limitations associated with our proposed method.

Serre’s conjecture, stating that every finitely generated projective module over a polynomial ring is free, was proven by Quillen and Suslin independently in 1976. An equivalent form of the Quillen-Suslin theorem says, “Every unimodular row over a polynomial ring can be completed to a unimodular matrix.” In this paper, we generalize the Quillen-Suslin theorem to the parametric case and present an algorithm to construct the unimodular completion matrix system for any polynomial vector with parameters. Specifically, we first determine the conditions on the parameters under which the vector is unimodular using the comprehensive Gröbner system. Furthermore, we use a constructive method to find a finite partition of the parameter space such that, for each branch, the vector under specializations can be completed into a unimodular matrix in the same form. Since the method is constructive, we present an explicit algorithm to construct the unimodular completion matrix system for any polynomial vector with parameters. The correctness and termination of the algorithm have been proven, and an example is provided to demonstrate how the algorithm works.

A new algorithm for computing the parametric greatest common right divisor (GCRD) of a set of parametric Ore polynomials is presented in this paper. The algorithm is based on Gröbner bases for modules. Inspired by the resultant theory in Ore polynomial rings, the Sylvester matrix is defined for a set of Ore polynomials. In the case of non-parametric polynomials, the GCRD of Ore polynomials can be obtained by computing the row echelon form of the Sylvester matrix. For the parametric case, the parametric Sylvester matrix is also defined in the paper. Based on this, under the assumption that the specializations commute with the conjugate operator and derivation in the Ore polynomial ring, the parametric GCRD of parametric Ore polynomials can be obtained by computing the Gröbner basis for the module generated by rows of the parametric Sylvester matrix. As a consequence, the algorithm for computing the parametric GCRD is presented in detail and has been implemented in the computer algebra system Singular.

Adversarial attacks on deep neural networks keep raising security concerns in natural language processing research.Existing defenses focus on improving the robustness of the victim model in the training stage.However, they often neglect to proactively mitigate adversarial attacks during inference.Towards this overlooked aspect, we propose a defense framework that aims to mitigate attacks by confusing attackers and correcting adversarial contexts that are caused by malicious perturbations.Our framework comprises three components: (1) a synonym-based transformation to randomly corrupt adversarial contexts in the word level, (2) a developed BERT defender to correct abnormal contexts in the representation level, and (3) a simple detection method to filter out adversarial examples, any of which can be flexibly combined.Additionally, our framework helps improve the robustness of the victim model during training.Extensive experiments demonstrate the effectiveness of our framework in defending against word-level adversarial attacks.

Emotion cause analysis (ECA) aims to extract emotion clauses and find the corresponding cause of the emotion. Existing methods adopt fine-tuning paradigm to solve certain types of ECA tasks. These task-specific methods have a deficiency of universality. And the relations among multiple objectives in one task are not explicitly modeled. Moreover, the relative position information introduced in most existing methods may make the model suffer from dataset bias. To address the first two problems, this paper proposes a universal prompt tuning method to solve different ECA tasks in the unified framework. As for the third problem, this paper designs a directional constraint module and a sequential learning module to ease the bias. Considering the commonalities among different tasks, this paper proposes a cross-task training method to further explore the capability of the model. The experimental results show that our method achieves competitive performance on the ECA datasets.

This paper is concerned with a property of modules over a polynomial ring and its application in multivariate polynomial matrix factorizations. We construct a specific polynomial such that the product of the polynomial and a nonzero vector in a module over a polynomial ring can be represented by the elements in a maximum linearly independent vector set of the module over the polynomial ring. Based on this property, a relationship between a rank-deficient matrix and any of its full row rank submatrices is presented. By this result, we show that the problem for general factorizations of rank-deficient matrices can be translated into that of any of their full row rank submatrices in the regular case. Then many results on factorizations of full row rank matrices, such as zero prime factorizations, minor prime factorizations and factor prime factorizations, can be extended to the rank-deficient case. We implement the algorithm of general factorizations for rank-deficient matrices on the computer algebra system Maple, and two examples are given to illustrate the algorithm.

An algorithm for computing the rational univariate representation of zero-dimensional ideals with parameters is presented in the paper. Different from the rational univariate representation of zero-dimensional ideals without parameters, the number of zeros of zero-dimensional ideals with parameters under various specializations is different, which leads to choosing and checking the separating element, the key to computing the rational univariate representation, is difficult. In order to pick out the separating element, by partitioning the parameter space we can ensure that under each branch the ideal has the same number of zeros. Subsequently based on the extended subresultant theorem for parametric cases, the separating element corresponding to each branch is chosen with the further partition of parameter space. Finally, with the help of parametric greatest common divisor theory a finite set of the rational univariate representation of zero-dimensional ideals with parameters can be obtained.

Optimal transportation plays a fundamental role in many fields in engineering and medicine, including surface parameterization in graphics, registration in computer vision, and generative models in deep learning. For quadratic distance cost, optimal transportation map is the gradient of the Brenier potential, which can be obtained by solving the Monge-Ampère equation. Furthermore, it is induced to a geometric convex optimization problem. The Monge-Ampère equation is highly non-linear, and during the solving process, the intermediate solutions have to be strictly convex. Specifically, the accuracy of the discrete solution heavily depends on the sampling pattern of the target measure. In this work, we propose a self-adaptive sampling algorithm which greatly reduces the sampling bias and improves the accuracy and robustness of the discrete solutions. Experimental results demonstrate the efficiency and efficacy of our method.

Inclement weather, haze, and fog severely decrease the performance of outdoor imaging systems. Due to a large range of the depth-of-field, most image dehazing or enhancement methods suffer from color distortions and halo artifacts when applied to real-world hazy outdoor scenes, especially those with the sky. To effectively recover details in both distant and nearby regions as well as to preserve color fidelity of the sky, in this study, we propose a novel image defogging and enhancement approach based on a replaceable plug-in segmentation module and region-adaptive processing. First, regions of the grayish sky, pure white objects, and other parts are separated. Second, a luminance-inverted multi-scale Retinex with color restoration (MSRCR) and region-ratio-based adaptive Gamma correction are applied to non-grayish and non-white areas. Finally, the enhanced regions are stitched seamlessly by using a mean-filtered region mask. Extensive experiments show that the proposed approach not only outperforms several state-of-the-art defogging methods in terms of both visibility and color fidelity, but also provides enhanced outputs with fewer artifacts and halos, particularly in sky regions.

This work introduces a novel surface registration method based on foliation. A foliation decomposes the surface into a family of closed loops, such that the decomposition has local tensor product structure. By projecting each loop to a point, the surface is collapsed into a graph. Two homeomorphic surfaces with consistent foliations can be registered by first matching their foliation graphs, then matching the corresponding leaves. This foliation based method is capable of handling surfaces with complicated topologies and large non-isometric deformations, rigorous with solid theoretic foundation, easy to implement, robust to compute. The result mapping is diffeomorphic. Our experimental results show the efficiency and efficacy of the proposed method.