A shape-from-shading method of polyhedral objects using prior information

We propose a new method for recovering the 3D shape of a polyhedral object from its single 2D image using the shading information contained in the image and the prior information on the object. In a strict sense, we cannot recover the shape of a polyhedron from an incorrect line drawing, even if it is practically almost correct. In order to overcome this problem, we propose a flexible face positioning method that can permit inconsistencies in the recovered shape that arise from vertex-position errors contained in incorrect line drawings. Also, we propose to use prior information about the horizontality and verticality of special faces and the convex and concave properties of the edges in order to attain good solutions and present a method of formulating such prior information as physical constraints. The shape-from-shading method is formulated as a minimization problem of a nonlinear cost function with the nonlinear constraints and its solution is searched by a global optimization algorithm. In the experiments with a synthetic image and three kinds of real images, shapes that are similar to those of the actual objects were recovered in all cases. As a result, the proposed method has proven to be effective in the shape recovery of simple-shape polyhedral objects.     

Proximity Queries Between Convex Objects: An Interior Point Approach for Implicit Surfaces

This paper presents a general method for exact distance computation between convex objects represented as intersections of implicit surfaces. Exact distance computation algorithms are particularly important for applications involving objects that make intermittent contact, such as in dynamic simulations and in haptic interactions. They can also be used in the narrow phase of hierarchical collision detection. In contrast to geometric approaches developed for polyhedral objects, we formulate the distance computation problem as a convex optimization problem. We use an interior point method to solve the optimization problem and demonstrate that, for general convex objects represented as implicit surfaces, interior point approaches are globally convergent, and fast in practice. Further, they provide polynomial-time guarantees for implicit surface objects when the implicit surfaces have self-concordant barrier functions. We use a primal-dual interior point algorithm that solves the Karush-Kuhn-Tucker (KKT) conditions obtained from the convex programming formulation. For the case of polyhedra and quadrics, we establish a theoretical time complexity of O(n^1.5), where n is the number of constraints. We present implementation results for example implicit surface objects, including polyhedra, quadrics, and generalizations of quadrics such as superquadrics and hyperquadrics, as well as intersections of these surfaces. We demonstrate that in practice, the algorithm takes time linear in the number of constraints, and that distance computation rates of about 1 kHz can be achieved. We also extend the approach to proximity queries between deforming convex objects. Finally, we show that continuous collision detection for linearly translating objects can be performed by solving two related convex optimization problems. For polyhedra and quadrics, we establish that the computational complexity of this problem is also O(n^1.5).     

Polyhedral object recognition by indexing

In computer vision, the indexing problem is the problem of recognizing a few objects in a large database of objects while avoiding the help of the classical image-feature-to-object-feature matching paradigm. In this paper we address the problem of recognizing three-dimensional (3-D) polyhedral objects from 2-D images by indexing. Both the objects to be recognized and the images are represented by weighted graphs. The indexing problem is therefore the problem of determining whether a graph extracted from the image is present or absent in a database of model graphs. We introduce a novel method for performing this graph indexing process which is based both on polynomial characterization of binary and weighted graphs and on hashing. We describe in detail this polynomial characterization and then we show how it can be used in the context of polyhedral object recognition. Next we describe a practical recognition-by-indexing system that includes the organization of the database, the representation of polyhedral objects in terms of 2-D characteristic views, the representation of this views in terms of weighted graphs and the associated image processing. Finally, some experimental results allow the evaluation of the system performance.     

适合多观测样本的基于LS-SVM的新分类算法

针对多观测样本的二分类问题,提出适合多观测样本的基于LS-SVM的新分类算法。每次分类中,待分类的模式使用多观测样本集进行表示,首先对多观测样本集的标签进行假设,将此假设条件作为LS-SVM中优化问题的约束条件,由此得到分类误差,通过比较两次假设下的分类误差确定多观测样本的类别。该方法无需提前训练获得分类器,而是同时利用已知标签样本和多观测样本集,充分利用同类样本在特征空间中连续分布的特点。最后通过三组实验验证了所提方法的有效性。     

Polyhedral Feasible Set Computation of MPC-Based Optimal Control Problems

Feasible sets play an important role in model predictive control (MPC) optimal control problems (OCPs). This paper proposes a multi-parametric programming-based algorithm to compute the feasible set for OCP derived from MPC-based algorithms involving both spectrahedron (represented by linear matrix inequalities) and polyhedral (represented by a set of inequalities) constraints. According to the geometrical meaning of the inner product of vectors, the maximum length of the projection vector from the feasible set to a unit spherical coordinates vector is computed and the optimal solution has been proved to be one of the vertices of the feasible set. After computing the vertices, the convex hull of these vertices is determined which equals the feasible set. The simulation results show that the proposed method is especially efficient for low dimensional feasible set computation and avoids the non-unicity problem of optimizers as well as the memory consumption problem that encountered by projection algorithms.      

奥运调度问题模型转换及优化研究

本文研究了奥运会调度问题的模型转换和优化. (1)时间区间约束是奥运会调度问题的关键约束, 本文建立了一种时间区间模型语言以描述这个调度问题. (2)奥运会调度问题是一个约束满足问题, 考虑其本质复杂性, 本文通过柔化决赛时间约束将约束满足问题转化为约束优化问题. (3)约束优化模型中, 项由场地约束关联起来, 如果去掉场地约束, 各项则是相互独立的. 因而本文通过松弛场地约束将约束优化问题分解为若干子问题. 全局优化解通过调整拉格朗日乘子获得. (4)为了调整拉格朗日乘子, 本文研究了变直径次梯度投影算法, 此算法不依赖于任何先验知识收敛, 本文给出了收敛效率. 仿真结果说明了算法的收敛性, 显示出变直径次梯度投影算法与简化算法在性能上的差别, 并且表明原约束满足问题的相变现象可以通过变直径次梯度投影算法获得正的对偶值的概率和首次获得正的对偶值的时间来识别.     

Distributed robust MPC for nonholonomic robots with obstacle and collision avoidance

Considering that the inevitable disturbances and coupled constraints pose an ongoing challenge to distributed control algorithms, this paper proposes a distributed robust model predictive control (MPC) algorithm for a multi-agent system with additive external disturbances and obstacle and collision avoidance constraints. In particular, all the agents are allowed to solve optimization problems simultaneously at each time step to obtain their control inputs, and the obstacle and collision avoidance are accomplished in the context of full-dimensional controlled objects and obstacles. To achieve the collision avoidance between agents in the distributed framework, an assumed state trajectory is introduced for each agent which is transmitted to its neighbors to construct the polyhedral over-approximations of it. Then the polyhedral over-approximations of the agent and the obstacles are used to smoothly reformulate the original nonconvex obstacle and collision avoidance constraints. And a compatibility constraint is designed to restrict the deviation between the predicted and assumed trajectories. Moreover, recursive feasibility of each local MPC optimization problem with all these constraints derived and input-to-state stability of the closed-loop system can be ensured through a sufficient condition on controller parameters. Finally, simulations with four agents and two obstacles demonstrate the efficiency of the proposed algorithm.     

Boundary finding with parametrically deformable models

Segmentation using boundary finding is enhanced both by considering the boundary as a whole and by using model-based global shape information. The authors apply flexible constraints, in the form of a probabilistic deformable model, to the problem of segmenting natural 2-D objects whose diversity and irregularity of shape make them poorly represented in terms of fixed features or form. The parametric model is based on the elliptic Fourier decomposition of the boundary. Probability distributions on the parameters of the representation bias the model to a particular overall shape while allowing for deformations. Boundary finding is formulated as an optimization problem using a maximum a posteriori objective function. Results of the method applied to real and synthetic images are presented, including an evaluation of the dependence of the method on prior information and image quality     

The auto‐geometric correction of multi‐projector for cylindrical surface using Bézier patches

Geometric calibration to projection images is an indispensable operation for projection‐based spatial display. In this paper, we propose a new method for correcting images generated in a computer onto a cylindrical surface accurately, which can project a high‐resolution projection image with pixels matching avoiding too much manual operation. Images waiting to be projected are pre‐warped according to the rough correspondence between projectors and physical surface. To solve the errors resulting from unexpected pixel shifts in overlap projection area, we fit the Bézier interpolation to the images and apply the optimization theory with added constraints to correct the projection image accurately. This optimization process, by taking the pixels with specific significance on the images as the basis of calculation, avoids the traditional ways of translating the control points of the Bézier surface directly. The final results achieve a completely accurate projection picture even if the projection surface shape is inaccurate and irregular. We present the details of the proposed accurate calibration algorithm and illustrate our method, which, with its scalability, can achieve perfect projection efficiently and accurately with experiments.     

A Bayesian approach to simultaneously recover camera pose and non-rigid shape from monocular images

In this paper we bring the tools of the Simultaneous Localization and Map Building (SLAM) problem from a rigid to a deformable domain and use them to simultaneously recover the 3D shape of non-rigid surfaces and the sequence of poses of a moving camera. Under the assumption that the surface shape may be represented as a weighted sum of deformation modes, we show that the problem of estimating the modal weights along with the camera poses, can be probabilistically formulated as a maximum a posteriori estimate and solved using an iterative least squares optimization. In addition, the probabilistic formulation we propose is very general and allows introducing different constraints without requiring any extra complexity. As a proof of concept, we show that local inextensibility constraints that prevent the surface from stretching can be easily integrated.An extensive evaluation on synthetic and real data, demonstrates that our method has several advantages over current non-rigid shape from motion approaches. In particular, we show that our solution is robust to large amounts of noise and outliers and that it does not need to track points over the whole sequence nor to use an initialization close from the ground truth.     

基于面域理解的多面体三维重建

已有的三维重建算法一般是从点线或体的角度来理解视图,实际上,视图中的环包含了 大量的形体面信息,可以直接从中间层次的面理解形体的投影。文中根据实体构成和平面投影的约束和规则重建显式表达的面,并进一步推导出隐式表达的面,从而得到完整的形体边界模型,本算法适用于平面多面体,通过面域作为中介,容易实现与模型引导方法的融合,从而拓展到二次曲面体的重建。     

基于梯度投影法与随机优化算法的约束优化方法

针对带有线性等式和不等式约束的无确定函数形式的约束优化问题,提出一种利用梯度投影法与遗传算法、同时扰动随机逼近等随机算法相结合的优化方法。该方法利用遗传算法进行全局搜索,利用同时扰动随机逼近算法进行局部搜索,算法在每次进化时根据线性约束计算父个体处的梯度投影方向,以产生新个体,从而能够严格保证新个体满足全部约束条件。将上述约束优化算法应用于典型约束优化问题,其仿真结果表明了所提出算法的可行性和收敛性。     

Enhanced POD projection basis with application to shape optimization of car engine intake port

In this paper we present a rigorous method for the construction of enhanced Proper Orthogonal Decomposition (POD) projection bases for the development of efficient Reduced Order Models (ROM). The resulting ROMs are seen to exactly interpolate global quantities by design, such as the objective function(s) and nonlinear constraints involved in the optimization problem, thus narrowing the search space by limiting the number of constraints that need to be explicitly included in the statement of the optimization problem. We decompose the basis into two subsets of orthogonal vectors, one for the representation of constraints and the other one, in a complementary space, for the minimization of the projection errors. An explicit algorithm is presented for the case of linear objective functions. The proposed method is then implemented within a bi-level ROM and is illustrated with an application to the multi-objective shape optimization of a car engine intake port for two competing objectives: CO₂ emissions and engine power. We show that optimization using the proposed method produces Pareto dominant and realistic solutions for the flow fields within the combustion chamber, providing further insight into the flow properties.     

Iterative constrained weighted least squares estimator for TDOA and FDOA positioning of multiple disjoint sources in the presence of sensor position and velocity uncertainties

Sensor position and velocity uncertainties are known to be able to degrade the source localization accuracy significantly. This paper focuses on the problem of locating multiple disjoint sources using time differences of arrival (TDOAs) and frequency differences of arrival (FDOAs) in the presence of sensor position and velocity errors. First, the explicit Cramér–Rao bound (CRB) expression for joint estimation of source and sensor positions and velocities is derived under the Gaussian noise assumption. Subsequently, we compare the localization accuracy when multiple-source positions and velocities are determined jointly and individually based on the obtained CRB results. The performance gain resulted from multiple-target cooperative positioning is also quantified using the orthogonal projection matrix. Next, the paper proposes a new estimator that formulates the localization problem as a quadratic programming with some indefinite quadratic equality constraints. Due to the non-convex nature of the optimization problem, an iterative constrained weighted least squares (ICWLS) method is developed based on matrix QR decomposition, which can be achieved through some simple and efficient numerical algorithms. The newly proposed iterative method uses a set of linear equality constraints instead of the quadratic constraints to produce a closed-form solution in each iteration. Theoretical analysis demonstrates that the proposed method, if converges, can provide the optimal solution of the formulated non-convex minimization problem. Moreover, its estimation mean-square-error (MSE) is able to reach the corresponding CRB under moderate noise level. Simulations are included to corroborate and support the theoretical development in this paper.     

二阶多面体网格中关键特征控制的表面重建技术

针对基于二阶多节点多面体网格的表面重建过程中存在的准确拓扑及绘制、传输代价等问题,提出了一种基于关键特征控制的表面重建技术.研究并分析了二阶多节点多面体单元等参插值函数的性质特征,在网格单元棱边插值计算曲面轮廓点,在网格表面及体内提取曲面的几何特征关键点;根据3类插值关键点间的逻辑关系制定了令拓扑准确唯一的面片三角化规则及修复策略,设计了基于关键点的三角面片压缩索引结构.实验结果表明,该方法可准确计算并描述基于二阶多节点多面体网格单元的曲面几何拓扑结构,反映网格单元内部面片的真实凹凸性质,克服了拓扑二义性,具备对不同精度要求的适应性,并有效降低了绘制与传输代价.     

Evolutionary approach on connectivity-based sensor network localization

The sensor network localization based on connectivity can be modeled as a non-convex optimization problem. It can be argued that the actual problem should be represented as an optimization problem with both convex and non-convex constraints. A two-objective evolutionary algorithm is proposed which utilizes the result of all convex constraints to provide a starting point on the location of the unknown nodes and then searches for a solution to satisfy all the convex and non-convex constraints of the problem. The final solution can reach the most suitable configuration of the unknown nodes because all the information on the constraints (convex and non-convex) related to connectivity have been used. Compared with current models that only consider the nodes that have connections, this method considers not only the connection constraints, but also the disconnection constraints. As a MOEA (Multi-Objective Evolution Algorithm), PAES (Pareto Archived Evolution Strategy) is used to solve the problem. Simulation results have shown that better solution can be obtained through the use of this method when compared with those produced by other methods.     

几个多面体网格剖分问题的NP难度证明

主要讨论了两类多面体网格剖分问题——网格表面单调剖分和地形多面体剖分.首先研究了判定一个多面体表面能否被剖分成k个单调片的问题,通过构造与SAT问题(satisfiability problem)相应的几何模型,证明出该判定问题是NP完全的,而与之对应的最优剖分问题是NP-hard的.然后将证明方法推广到地形多面体剖分的问题:将一个带洞多面体或者简单多面体剖分成最小数量的地形多面体,这两个问题都被证明是NP-hard的.