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Hao Pan

Hao Pan appears in the imported research catalog. Authorship, coauthor and topic links are available while profile ownership is still unclaimed.

ResearcherUnclaimed source record
math.NTmath.COComputer VisionGraphicsMachine Learningmath.DSArtificial IntelligenceComputational Geometryeess.SYHuman-Computer Interactionmath.GRquant-phSystems and Control

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Published work

42 published item(s)

preprint2026arXiv

Sketch2Arti: Sketch-based Articulation Modeling of CAD Objects

Articulation modeling aims to infer movable parts and their motion parameters for a 3D object, enabling interactive animation, simulation, and shape editing. In this paper, we present Sketch2Arti, the first sketch-based articulation modeling system for CAD objects. Our key observation is that designers naturally communicate articulation intent through lightweight sketches (e.g., arrows and strokes) that indicate how parts should move, yet translating such sketches into articulated 3D models remains largely manual. Sketch2Arti bridges this gap by enabling users to specify articulation through simple 2D sketches drawn from a chosen viewpoint. Given a CAD model and user sketches, our approach automatically discovers the corresponding movable parts and predicts their motion parameters, allowing iterative modeling of multiple articulations on complex objects with fine-grained control. Importantly, Sketch2Arti is trained in a category-agnostic manner without requiring object category information, leading to strong generalization to diverse objects beyond existing articulation datasets. Moreover, for shell models lacking interior structures, Sketch2Arti supports controllable internal completion guided by user sketches, generating plausible internal components consistent with the existing geometry and predicted motion constraints. Comprehensive experiments and user evaluations demonstrate the effectiveness, controllability, and generalization of Sketch2Arti. The code, dataset, and the prototype system are at https://arlo-yang.github.io/Sketch2Arti.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

ComplexGen: CAD Reconstruction by B-Rep Chain Complex Generation

We view the reconstruction of CAD models in the boundary representation (B-Rep) as the detection of geometric primitives of different orders, i.e. vertices, edges and surface patches, and the correspondence of primitives, which are holistically modeled as a chain complex, and show that by modeling such comprehensive structures more complete and regularized reconstructions can be achieved. We solve the complex generation problem in two steps. First, we propose a novel neural framework that consists of a sparse CNN encoder for input point cloud processing and a tri-path transformer decoder for generating geometric primitives and their mutual relationships with estimated probabilities. Second, given the probabilistic structure predicted by the neural network, we recover a definite B-Rep chain complex by solving a global optimization maximizing the likelihood under structural validness constraints and applying geometric refinements. Extensive tests on large scale CAD datasets demonstrate that the modeling of B-Rep chain complex structure enables more accurate detection for learning and more constrained reconstruction for optimization, leading to structurally more faithful and complete CAD B-Rep models than previous results.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

On the supercongruences involving harmonic numbers of order 2

We prove several supercongruences involving the harmonic number of order two $H_n^{(2)}:=\sum_{k=1}^n1/k^2$. For example, if $p>5$ is prime and $α$ is $p$-integral, then we can completely determine $$ \sum_{k=0}^{p-1}\frac{H_k^{(2)}}{k}\cdot\binomα{k}\binom{-1-α}{k}\quad\text{and}\quad \sum_{k=0}^{\frac{p-1}{2}}\frac{H_k^{(2)}}{k}\cdot\binomα{k}\binom{-1-α}{k} $$ modulo $p^3$. In particular, by setting $α=-1/2$, we confirm two conjectured congruences of Z.-W. Sun.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Partitions of finite nonnegative integer sets with identical representation functions

Let $\mathbb{N}$ be the set of all nonnegative integers. For $S\subseteq \mathbb{N}$ and $n\in \mathbb{N}$, let the representation function $R_{S}(n)$ denote the number of solutions of the equation $n=s+s'$ with $s, s'\in S$ and $s<s'$. In this paper, we determine the structure of $C, D\subseteq \mathbb{N}$ with $C\cup D=[0, m]$ and $|C\cap D|=2$ such that $R_{C}(n)=R_{D}(n)$ for any nonnegative integer $n$.

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preprint2022arXiv

Partitions of nonnegative integers with identical representation functions

Let $\mathbb{N}$ be the set of all nonnegative integers. For any integer $r$ and $m$, let $r+m\mathbb{N}=\{r+mk: k\in\mathbb{N}\}$. For $S\subseteq \mathbb{N}$ and $n\in \mathbb{N}$, let $R_{S}(n)$ denote the number of solutions of the equation $n=s+s'$ with $s, s'\in S$ and $s<s'$. Let $r_{1}, r_{2}, m$ be integers with $0<r_{1}<r_{2}<m$ and $2\mid r_{1}$. In this paper, we prove that there exist two sets $C$ and $D$ with $C\cup D=\mathbb{N}$ and $C\cap D=(r_{1}+m\mathbb{N})\cup (r_{2}+m\mathbb{N})$ such that $R_{C}(n)=R_{D}(n)$ for all $n\in\mathbb{N}$ if and only if there exists a positive integer $l$ such that $r_{1}=2^{2l+1}-2, r_{2}=2^{2l+1}-1, m=2^{2l+2}-2$.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Safe, efficient and socially-compatible decision of automated vehicles: a case study of unsignalized intersection driving

Safe and smooth interacting with other vehicles is one of the ultimate goals of driving automation. However, recent reports of demonstrative deployments of automated vehicles (AVs) indicate that AVs are still difficult to meet the expectation of other interacting drivers, which leads to several AV accidents involving human-driven vehicles (HVs). This is most likely due to the lack of understanding about the dynamic interaction process, especially about the human drivers. By investigating the causes of 4,300 video clips of traffic accidents, we find that the limited dynamic visual field of drivers is one leading factor in inter-vehicle interaction accidents, especially in those involving trucks. A game-theoretic decision algorithm considering social compatibility is proposed to handle the interaction with a human-driven truck at an unsignalized intersection. Starting from a probabilistic model for the visual field characteristics of truck drivers, social fitness and reciprocal altruism in the decision are incorporated in the game payoff design. Human-in-the-loop experiments are carried out, in which 24 subjects are invited to drive and interact with AVs deployed with the proposed algorithm and two comparison algorithms. Totally 207 cases of intersection interactions are obtained and analyzed, which shows that the proposed decision-making algorithm can not only improve both safety and time efficiency, but also make AV decisions more in line with the expectation of interacting human drivers. These findings can help inform the design of automated driving decision algorithms, to ensure that AVs can be safely and efficiently integrated into the human-dominated traffic.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Self-Supervised Image Representation Learning with Geometric Set Consistency

We propose a method for self-supervised image representation learning under the guidance of 3D geometric consistency. Our intuition is that 3D geometric consistency priors such as smooth regions and surface discontinuities may imply consistent semantics or object boundaries, and can act as strong cues to guide the learning of 2D image representations without semantic labels. Specifically, we introduce 3D geometric consistency into a contrastive learning framework to enforce the feature consistency within image views. We propose to use geometric consistency sets as constraints and adapt the InfoNCE loss accordingly. We show that our learned image representations are general. By fine-tuning our pre-trained representations for various 2D image-based downstream tasks, including semantic segmentation, object detection, and instance segmentation on real-world indoor scene datasets, we achieve superior performance compared with state-of-the-art methods.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Sketch2PQ: Freeform Planar Quadrilateral Mesh Design via a Single Sketch

The freeform architectural modeling process often involves two important stages: concept design and digital modeling. In the first stage, architects usually sketch the overall 3D shape and the panel layout on a physical or digital paper briefly. In the second stage, a digital 3D model is created using the sketch as a reference. The digital model needs to incorporate geometric requirements for its components, such as the planarity of panels due to consideration of construction costs, which can make the modeling process more challenging. In this work, we present a novel sketch-based system to bridge the concept design and digital modeling of freeform roof-like shapes represented as planar quadrilateral (PQ) meshes. Our system allows the user to sketch the surface boundary and contour lines under axonometric projection and supports the sketching of occluded regions. In addition, the user can sketch feature lines to provide directional guidance to the PQ mesh layout. Given the 2D sketch input, we propose a deep neural network to infer in real-time the underlying surface shape along with a dense conjugate direction field, both of which are used to extract the final PQ mesh. To train and validate our network, we generate a large synthetic dataset that mimics architect sketching of freeform quadrilateral patches. The effectiveness and usability of our system are demonstrated with quantitative and qualitative evaluation as well as user studies.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

PFCNN: Convolutional Neural Networks on 3D Surfaces Using Parallel Frames

Surface meshes are widely used shape representations and capture finer geometry data than point clouds or volumetric grids, but are challenging to apply CNNs directly due to their non-Euclidean structure. We use parallel frames on surface to define PFCNNs that enable effective feature learning on surface meshes by mimicking standard convolutions faithfully. In particular, the convolution of PFCNN not only maps local surface patches onto flat tangent planes, but also aligns the tangent planes such that they locally form a flat Euclidean structure, thus enabling recovery of standard convolutions. The alignment is achieved by the tool of locally flat connections borrowed from discrete differential geometry, which can be efficiently encoded and computed by parallel frame fields. In addition, the lack of canonical axis on surface is handled by sampling with the frame directions. Experiments show that for tasks including classification, segmentation and registration on deformable geometric domains, as well as semantic scene segmentation on rigid domains, PFCNNs achieve robust and superior performances without using sophisticated input features than state-of-the-art surface based CNNs.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Sketch2CAD: Sequential CAD Modeling by Sketching in Context

We present a sketch-based CAD modeling system, where users create objects incrementally by sketching the desired shape edits, which our system automatically translates to CAD operations. Our approach is motivated by the close similarities between the steps industrial designers follow to draw 3D shapes, and the operations CAD modeling systems offer to create similar shapes. To overcome the strong ambiguity with parsing 2D sketches, we observe that in a sketching sequence, each step makes sense and can be interpreted in the \emph{context} of what has been drawn before. In our system, this context corresponds to a partial CAD model, inferred in the previous steps, which we feed along with the input sketch to a deep neural network in charge of interpreting how the model should be modified by that sketch. Our deep network architecture then recognizes the intended CAD operation and segments the sketch accordingly, such that a subsequent optimization estimates the parameters of the operation that best fit the segmented sketch strokes. Since there exists no datasets of paired sketching and CAD modeling sequences, we train our system by generating synthetic sequences of CAD operations that we render as line drawings. We present a proof of concept realization of our algorithm supporting four frequently used CAD operations. Using our system, participants are able to quickly model a large and diverse set of objects, demonstrating Sketch2CAD to be an alternate way of interacting with current CAD modeling systems.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

Ergodic recurrence and bounded gaps between primes

Let $(X,B_X,μ,T)$ be a measure-preserving probability system with $T$ is invertible. Suppose that $A\in B_X$ with $μ(A)>0$ and $ε>0$. For any $m\geq 1$, there exist infinitely many primes $p_0,p_1,\ldots,p_m$ with $p_0<\cdots<p_m$ such that $$ μ(A\cap T^{-(p_i-1)}A)\geq μ(A)^2-ε$$ for each $0\leq i\leq m$ and $$ p_m-p_0<C_m, $$ where $C_m>0$ is a constant only depending on $m$, $A$ and $ε$.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

Identities involving weighted Catalan, Schroder and Motzkin paths

In this paper, we investigate the weighted Catalan, Motzkin and Schröder numbers together with the corresponding weighted paths. The relation between these numbers is illustrated by three equations, which also lead to some known and new interesting identities. To show these three equations, we provide combinatorial proofs. One byproduct is to find a bijection between two sets of Catalan paths: one consisting of those with $k$ valleys, and the other consisting of $k$ $\mathbf{N}$ steps in even positions.

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preprint2016arXiv

On the generalized restricted sumsets in abelian groups

Suppose that $A$, $B$ and $S$ are non-empty subsets of a finite abelian group $G$. Then the generalized restricted sumset $$ A\stackrel{S}+B:=\{a+b:\,a\in A,\ b\in B,\ a-b\not\in S\} $$ contains at least $$ \min\{|A|+|B|-3|S|,p(G)\} $$ elements, where $p(G)$ is the least prime factor of $|G|$. Further, we also have $$ |A\stackrel{S}+B|\geq \min\{|A|+|B|-|S|-2,p(G)\}, $$ provided that both $|A|$ and $|B|$ are large with respect to $|S|$.

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preprint2016arXiv

On the number of three-term arithmetic progressions in a dense subset of $F_q^n$

Let $q$ be an odd prime power. Combining the discussion of Varnavides and a recent theorem of Ellenberg and Gijswijt, we show that a subset $A\subset{\mathbb F}_q^n$ will contain many non-trivial three-term arithmetic progressions, whenever $|A|\geq (c_q q)^n$ for some constant $c_q>0$. After the first version of our manuscript was uploaded in the arXiv, we learned from Professors Jacob Fox and Terence Tao that our result is a special case of a result of Fox and Lovasz [1, Theorem 3]. In fact, [1, Theorem 3] gives a much better bound than ours. For example, when $q=3$, the lower bound given by Fox and Lovasz is $|A|^{2}\cdot (|A|q^{-n})^{11.901}$, while our bound is $|A|^{2}\cdot (|A|q^{-n})^{25.803}$. We thank Professors Jacob Fox and Terence Tao for their helpful comments on our manuscript. [1] Jacob Fox, László Miklós Lovász, A tight bound for Green's arithmetic triangle removal lemma in vector spaces, preprint, arXiv:1606.01230.

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preprint2015arXiv

Proof of a conjecture involving Sun polynomials

The Sun polynomials $g_n(x)$ are defined by \begin{align*} g_n(x)=\sum_{k=0}^n{n\choose k}^2{2k\choose k}x^k. \end{align*} We prove that, for any positive integer $n$, there hold \begin{align*} &\frac{1}{n}\sum_{k=0}^{n-1}(4k+3)g_k(x) \in\mathbb{Z}[x],\quad\text{and}\\ &\sum_{k=0}^{n-1}(8k^2+12k+5)g_k(-1)\equiv 0\pmod{n}. \end{align*} The first one confirms a recent conjecture of Z.-W. Sun, while the second one partially answers another conjecture of Z.-W. Sun. We give three different proofs of the former. One of them depends on the following congruence: $$ {m+n-2\choose m-1}{n\choose m}{2n\choose n}\equiv 0\pmod{m+n}\quad\text{for $m,n\geqslant 1$.} $$

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preprint2014arXiv

Proof of three conjectures on congruences

In this paper we prove three conjectures on congruences involving central binomial coefficients or Lucas sequences. Let $p$ be an odd prime and let $a$ be a positive integer. We show that if $p\equiv 1\pmod{4}$ or $a>1$ then $$ \sum_{k=0}^{\lfloor\frac34p^a\rfloor}\binom{-1/2}k\equiv\left(\frac{2}{p^a}\right)\pmod{p^2}, $$ where $(-)$ denotes the Jacobi symbol. This confirms a conjecture of the second author. We also confirm a conjecture of R. Tauraso by showing that $$\sum_{k=1}^{p-1}\frac{L_k}{k^2}\equiv0\pmod{p}\quad {\rm provided}\ \ p>5,$$ where the Lucas numbers $L_0,L_1,L_2,\ldots$ are defined by $L_0=2,\ L_1=1$ and $L_{n+1}=L_n+L_{n-1}\ (n=1,2,3,\ldots)$. Our third theorem states that if $p\not=5$ then we can determine $F_{p^a-(\frac{p^a}5)}$ mod $p^3$ in the following way: $$\sum_{k=0}^{p^a-1}(-1)^k\binom{2k}k\equiv\left(\frac{p^a}5\right)\left(1-2F_{p^a-(\frac{p^a}5)}\right)\ \pmod{p^3},$$ which appeared as a conjecture in a paper of Sun and Tauraso in 2010.

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preprint2010arXiv

On the Capability of Measurement-Based Quantum Feedback

As a key method in dealing with uncertainties, feedback has been understood fairly well in classical control theory. But for quantum control systems, the capability of measurement-based feedback control (MFC) has not been investigated systematically. In contrast to the control of classical systems where the measurement effect is negligible, the quantum measurement will cause a quantum state to collapse, which will inevitably introduce additional uncertainties besides the system initial uncertainty. Therefore, there is a complicated tradeoff between the uncertainty introduced and the information gained by the measurement, and thus a theoretical investigation of the capability of MFC is of fundamental importance. In this paper, inspired by both the Heisenberg uncertainty principle for quantum systems and the investigation of the feedback capability for classical systems, we try to answer the following three basic questions: (i) How to choose the measurement channel appropriately? (ii) Is the MFC still superior to the open loop control in dealing with the system information uncertainties? and (iii) What is the maximum capability or limitation of the MFC? These questions will be answered theoretically by establishing several theorems concerning the asymptotic reachability of eigenstates of a typical class of Hamiltonian control mode.

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