Using Metric Spaces in Optimum Scheduling

The solution of an optimum problem of scheduling with n workpieces and m machine tools represents an optimum schedule of putting pieces on machines. In turn, the schedule is defined by an optimum collection of m permutations out of n objects, i.e., the vector permutation = (₁, ..., _m ), where each permutation _i (1 i m) points up the sequence of working of all pieces on the ith machine. In this case, to each admissible schedule there must correspond an integral point from the m-dimensional Euclidean space of permutations (or, which is practically the same, the permutation out of numbers {1, 2, ..., mn}. In an effort to seek an optimum schedule, use is made of the notion of a metric space in the set of admissible schedules and the justified methodology of the search for an optimum schedule. A few metric spaces are described and analyzed and their comparative effectiveness is investigated for the solution of a different-route problem of scheduling.     

快速搜索任意形状二维目标质心策略

快速搜索任意形状二维目标的质心,一直是模式识别,目标跟踪等领域中的关键问题,通过对矩方法的进一步分析,提出基于目标质心与目标上各点间蹭之和取得最小值这的一特性的质心快速搜索策略。     

Reverse Nearest Neighbor Search in Metric Spaces

Given a set {cal D} of objects, a reverse nearest neighbor (RNN) query returns the objects o in {cal D} such that o is closer to a query object q than to any other object in {cal D}, according to a certain similarity metric. The existing RNN solutions are not sufficient because they either 1) rely on precomputed information that is expensive to maintain in the presence of updates or 2) are applicable only when the data consists of "Euclidean objects” and similarity is measured using the L_2 norm. In this paper, we present the first algorithms for efficient RNN search in generic metric spaces. Our techniques require no detailed representations of objects, and can be applied as long as their mutual distances can be computed and the distance metric satisfies the triangle inequality. We confirm the effectiveness of the proposed methods with extensive experiments.     

A Shape Metric for Blum Ribbons

A Blum ribbon is a figure whose boundary is the envelope of a family of circles the centers of which lie along a single unbranched curve called its medial axis. Define a Blum ribbon to be simple if its medial axis is the line segment joining points (0,0) and (1,0). Any Blum ribbon can be made simple by flexing the medial axis, rotating, then dilating, all of which are basic transformations in Blum's geometry of shape. The Lie group SL(2, R) acts on circles centered on the x-axis by linear fractional transformations. By means of this action it is possible to associate to any simple Blum ribbon a curve in SL(2, R). A distance between corresponding points lying on such curves is defined using the bi-invariant metric on SL(2, R). Resulting scale-invariant metrics on the set of figures defined as Blum ribbons are described and it is shown that these metrics can provide effective measures of shape difference.     

On Metric Dimension of Nonbinary Hamming Spaces

For q-ary Hamming spaces we address the problem of the minimum number of points such that any point of the space is uniquely determined by its (Hamming) distances to them. It is conjectured that for a fixed q and growing dimension n of the Hamming space this number asymptotically behaves as 2n/ log _q n. We prove this conjecture for q = 3 and q = 4; for q = 2 its validity has been known for half a century.     

度量空间中高维索引结构回顾

1 引言近年来,高维数据库的应用得到快速的发展,如海量的多媒体数据库、大规模的文本数据以及生物信息学中庞大的DNA数据库等,这些信息一般使用特征抽取等方法映射为高维数据,然后通过计算这些高维数据之间距离实现相似性查询。例如,对于图像数据,往往采用颜色直方图来表征一幅图像,当需要从数据集查找与给定图像相似的图像时,通过计算     

Shape Matching via Quotient Spaces

We introduce a novel method for non‐rigid shape matching, designed to address the symmetric ambiguity problem present when matching shapes with intrinsic symmetries. Unlike the majority of existing methods which try to overcome this ambiguity by sampling a set of landmark correspondences, we address this problem directly by performing shape matching in an appropriate quotient space, where the symmetry has been identified and factored out. This allows us to both simplify the shape matching problem by matching between subspaces, and to return multiple solutions with equally good dense correspondences. Remarkably, both symmetry detection and shape matching are done without establishing any landmark correspondences between either points or parts of the shapes. This allows us to avoid an expensive combinatorial search present in most intrinsic symmetry detection and shape matching methods. We compare our technique with state‐of‐the‐art methods and show that superior performance can be achieved both when the symmetry on each shape is known and when it needs to be estimated.     

A Part-aware Surface Metric for Shape Analysis

The notion of parts in a shape plays an important role in many geometry problems, including segmentation, correspondence, recognition, editing, and animation. As the fundamental geometric representation of 3D objects in computer graphics is surface-based, solutions of many such problems utilize a surface metric, a distance function defined over pairs of points on the surface, to assist shape analysis and understanding. The main contribution of our work is to bring together these two fundamental concepts: shape parts and surface metric. Specifically, we develop a surface metric that is part-aware. To encode part information at a point on a shape, we model its volumetric context – called the volumetric shape image (VSI) – inside the shape's enclosed volume, to capture relevant visibility information. We then define the part-aware metric by combining an appropriate VSI distance with geodesic distance and normal variation. We show how the volumetric view on part separation addresses certain limitations of the surface view, which relies on concavity measures over a surface as implied by the well-known minima rule. We demonstrate how the new metric can be effectively utilized in various applications including mesh segmentation, shape registration, part-aware sampling and shape retrieval.     

D-Index: Distance Searching Index for Metric Data Sets

In order to speedup retrieval in large collections of data, index structures partition the data into subsets so that query requests can be evaluated without examining the entire collection. As the complexity of modern data types grows, metric spaces have become a popular paradigm for similarity retrieval. We propose a new index structure, called D-Index, that combines a novel clustering technique and the pivot-based distance searching strategy to speed up execution of similarity range and nearest neighbor queries for large files with objects stored in disk memories. We have qualitatively analyzed D-Index and verified its properties on actual implementation. We have also compared D-Index with other index structures and demonstrated its superiority on several real-life data sets. Contrary to tree organizations, the D-Index structure is suitable for dynamic environments with a high rate of delete/insert operations.     

Mathematical Morphology Based on Linear Combined Metric Spaces on Z2 (Part I): Fast Distance Transforms

Mathematical Morphology (MM) is a general method for image processing based on set theory. The two basic morphological operators are dilation and erosion. From these, several non linear filters have been developed usually with polynomial complexity, and this because the two basic operators depend strongly on the definition of the structural element. Most efforts to improve the algorithm's speed for each operator are based on structural element decomposition and/or efficient codification.A new framework and a theoretical basis toward the construction of fast morphological operators (of zero complexity) for an infinite (countable) family of regular metric spaces are presented in work. The framework is completely defined by the three axioms of metric. The theoretical basis here developed points out properties of some metric spaces and relationships between metric spaces in the same family, just in terms of the properties of the four basic metrics stated in this work. Concepts such as bounds, neighborhoods and contours are also related by the same framework.The presented results, are general in the sense that they cover the most commonly used metrics such as the chamfer, the city block and the chess board metrics. Generalizations and new results related with distances and distance transforms, which in turn are used to develop the morphologic operations in constant time, in contrast with the polynomial time algorithms are also given.     

基于支撑向量机的收敛问题研究

在文章中提出了一种基于支持向量机思想的对任意距离空间求解最大分类间隔的方法,其优化问题可以用输入空间的距离来表示。首先将输入空间等距嵌入到Hilbert空间,在线性的Hilbert空间对优化问题进行线性处理,但是这种方法只适用于特定的距离空间。在原方案的基础上扩展研究了对任意距离空间求解最大分类间隔的方法。     

Fast Range Searching with Delaunay Triangulations

This paper studies the idea of answering range searching queries using simple data structures. The only data structure we need is the Delaunay Triangulation of the input points. The idea is to first locate a vertex of the (arbitrary) query polygon and walk along the boundary of the polygon in the Delaunay Triangulation and report all the points enclosed by the query polygon. For a set of uniformly distributed random points in 2-D and a query polygon the expected query time of this algorithm is O(n ^1/3 + Q + E K + L _r n ^1/2), where Q is the size of the query polygon , {\bf E}K = O(n\bcdot area is the expected number of output points, L _r is a parameter related to the shape of the query polygon and n, and L _r is always bounded by the sum of the edge lengths of . Theoretically, when L _r = O(1/n^1/6) the expected query time is O(n^1/3 + Q + E K), which improves the best known average query time for general range searching. Besides the theoretical meaning, the good property of this algorithm is that once the Delaunay Triangulation is given, no additional preprocessing is needed. In order to obtain empirical results, we design a new algorithm for generating random simple polygons within a given domain. Our empirical results show that the constant coefficient of the algorithm is small, at least for the special (practical) cases when the query polygon is either a triangle (simplex range searching) or an axis-parallel box (orthogonal range searching) and for the general case when the query polygons are generated by our new polygon-generating algorithms and their sizes are relatively small.     

大规模高维向量空间的快速范围查询

金字塔技术是目前针对高维空间范围查询的有效方法之一,但是随着数据量的增加,检索过程由于引入过多的误中点而导致不必要的高维距离计算,为此本文提出改进的金字塔技术.引入向量排序、活性维等概念,利用分段处理思想,将不包含候选点的误中分段剪枝,并通过逐维距离累加法过滤剩余分段内的误中点,从而快速排除所有的误中点,尽可能减少距离计算次数,实现大规模高维向量空间的快速范围查询.利用模拟数据和真实数据,实验验证了OPT方法的正确性和有效性.     

格子模型的快速序列搜索算法

一种快速序列穷举搜索蛋白质构像空间的算法。该算法利用二分技术将HP序列逐次分解,保存分解过程的中间结果,使搜索算法中所需的计算量大大减少。     

一种面向协同过滤的快速最近邻居搜索方法

针对协同过滤模型中寻找邻居集耗时,且部分邻居信息未能有效用于预测计算的问题,提出了一种快速搜寻最近邻居的方法.该方法改变了评分矩阵中数据组织方式,通过构建项目的用户评分列表和用户的项目评分列表,以此来筛选出对预测评分值产生影响的用户或项目,进而得到目标用户或项目的邻居集.该方法排除了不必要的相似性计算,提高了运算效率;...     

Composite Distance Transformation for Indexing and k-Nearest-Neighbor Searching in High-Dimensional Spaces

Due to the famous dimensionality curse problem, search in a high-dimensional space is considered as a "hard" problem. In this paper, a novel composite distance transformation method, which is called CDT, is proposed to support a fast k-nearest-neighbor (k-NN) search in high-dimensional spaces. In CDT, all (n) data points are first grouped into some clusters by a k-Means clustering algorithm. Then a composite distance key of each data point is computed. Finally, these index keys of such n data points are inserted by a partition-based B -tree. Thus, given a query point, its k-NN search in high-dimensional spaces is transformed into the search in the single dimensional space with the aid of CDT index. Extensive performance studies are conducted to evaluate the effectiveness and efficiency of the proposed scheme. Our results show-that this method outperforms the state-of-the-art high-dimensional search techniques, such as the X-Tree, VA-file, iDistance and NB-Tree.     

GWCNN: A Metric Alignment Layer for Deep Shape Analysis

Deep neural networks provide a promising tool for incorporating semantic information in geometry processing applications. Unlike image and video processing, however, geometry processing requires handling unstructured geometric data, and thus data representation becomes an important challenge in this framework. Existing approaches tackle this challenge by converting point clouds, meshes, or polygon soups into regular representations using, e.g., multi‐view images, volumetric grids or planar parameterizations. In each of these cases, geometric data representation is treated as a fixed pre‐process that is largely disconnected from the machine learning tool. In contrast, we propose to optimize for the geometric representation during the network learning process using a novel metric alignment layer. Our approach maps unstructured geometric data to a regular domain by minimizing the metric distortion of the map using the regularized Gromov–Wasserstein objective. This objective is parameterized by the metric of the target domain and is differentiable; thus, it can be easily incorporated into a deep network framework. Furthermore, the objective aims to align the metrics of the input and output domains, promoting consistent output for similar shapes. We show the effectiveness of our layer within a deep network trained for shape classification, demonstrating state‐of‐the‐art performance for nonrigid shapes.     

基于形状模板匹配的图像拼接算法

文章提出了一种基于形状模板匹配的图像自动拼接方法。提取图像的角点作为特征点,利用归一化梯度模板对其进行预匹配,然后利用形状模板在四个方向对模板内图像的边缘点与模板边界的最短距离进行统计,获取模板图像的结构特征向量以实现对特征点的精确匹配。实验结果表明该算法具有较好的实用价值。