Low-rank tensor completion with fractional-Jacobian-extended tensor regularization for multi-component visual data inpainting

Several low-rank tensor completion methods have been integrated with total variation (TV) regularization to retain edge information and promote piecewise smoothness. In this paper, we first construct a fractional Jacobian matrix to nonlocally couple the structural correlations across components and propose a fractional-Jacobian-extended tensor regularization model, whose energy functional was designed proportional to the mixed norm of the fractional Jacobian matrix. Consistent regularization could thereby be performed on each component, avoiding band-by-band TV regularization and enabling effective handling of the contaminated fine-grained and complex details due to the introduction of a fractional differential. Since the proposed spatial regularization is linear convex, we further produced a novel fractional generalization of the classical primal-dual resolvent to develop its solver efficiently. We then combined the proposed tensor regularization model with low-rank constraints for tensor completion and addressed the problem by employing the augmented Lagrange multiplier method, which provides a splitting scheme. Several experiments were conducted to illustrate the performance of the proposed method for RGB and multispectral image restoration, especially its abilities to recover complex structures and the details of multi-component visual data effectively.     

体数据的多尺度张量表达与可视化

图形硬件的发展为实时体数据可视化提供了硬件保证,然而随着扫描技术的发展,大数据可视化仍然面临显存不足问题,因此研究保持数据特征的压缩表达方法就非常重要。应用张量近似思想建立了体数据的多尺度表达与可视化方法,一方面多尺度张量近似实现了数据压缩,解决了大数据的绘制问题;另一方面,张量近似的自适应压缩基保持了体数据的尺度特征。实验结果表明,该方法是有效的。     

植物运动的层次式简化技术

利用力学预计算方法生成植物运动的真实感动画需要对形变数据作有效简化和压缩。针对植物及其运动天然具有层级性的特点,提出一种基于视觉感知层次的动态几何简化算法。依据植物的树型层级结构,将各级“子树”的运动作高低频分解;“低频运动”表征“子树”的主体运动,“高频运动”即为运动细节。按“广度优先”遍历植物层级结构,在某一“深度”（层级）上将所有“子树”的“低频运动”合成即获得原始运动在此层次的逼近。遍历深度递增将构成植物运动的渐进逼近;递减则为植物运动的逐步抽象化,这实质上实现了植物运动的细节层次控制。实验表明该算法可实现植物运动的高效简化和压缩,利于生成大规模植物场景的真实感动画。     

面向多维混合型数据分布的混合多维直方图初探

现实世界中,多维数据分布常常不是单一一种类型,而是在不同的数据区域中呈现不同类型的数据分布.提出了一种面向多维混合型数据分布的混合多维直方图COCA*-Hist方法.这种方法在给定的空间预算下,根据数据分布空间不同的区域中的数据分布类型,可以包含多种不同类型的直方桶,从总体上提高直方图的准确性.由于需要对创建多维直方图的树结构进行二次遍历,以识别不同类型的数据分布区域并进行空间预算的重分配,COCA*-Hist时间效率略低于MHist算法,但对因此获得的准确性的提高和面对不同数据分布类型的通用性来说,是可以接受的.     

A note on tensor chain approximation

This paper deals with the approximation of \(d\) -dimensional tensors, as discrete representations of arbitrary functions \(f(x_1,\ldots ,x_d)\) on \([0,1]^d\) , in the so-called tensor chain format. The main goal of this paper is to show that the construction of a tensor chain approximation is possible using skeleton/cross approximation type methods. The complete algorithm is described, computational issues are discussed in detail and the complexity of the algorithm is shown to be linear in \(d\) . Some numerical examples are given to validate the theoretical results.     

Hierarchical classification of images by sparse approximation

Using image hierarchies for visual categorization has been shown to have a number of important benefits. Doing so enables a significant gain in efficiency (e.g., logarithmic with the number of categories [16,12]) or the construction of a more meaningful distance metric for image classification [17]. A critical question, however, still remains controversial: would structuring data in a hierarchical sense also help classification accuracy? In this paper we address this question and show that the hierarchical structure of a database can be indeed successfully used to enhance classification accuracy using a sparse approximation framework. We propose a new formulation for sparse approximation where the goal is to discover the sparsest path within the hierarchical data structure that best represents the query object. Extensive quantitative and qualitative experimental evaluation on a number of branches of the Imagenet database [7] as well as on the Caltech-256 [12] demonstrate our theoretical claims and show that our approach produces better hierarchical categorization results than competing techniques.     

Fast similarity join for multi-dimensional data

The efficient processing of multidimensional similarity joins is important for a large class of applications. The dimensionality of the data for these applications ranges from low to high. Most existing methods have focused on the execution of high-dimensional joins over large amounts of disk-based data. The increasing sizes of main memory available on current computers, and the need for efficient processing of spatial joins suggest that spatial joins for a large class of problems can be processed in main memory. In this paper, we develop two new in-memory spatial join algorithms, the Grid-join and EGO*-join, and study their performance. Through evaluation, we explore the domain of applicability of each approach and provide recommendations for the choice of a join algorithm depending upon the dimensionality of the data as well as the expected selectivity of the join. We show that the two new proposed join techniques substantially outperform the state-of-the-art join algorithm, the EGO-join.     

Compact implicit surface reconstruction via low-rank tensor approximation

Implicit representations have gained an increasing popularity in geometric modeling and computer graphics due to their ability to represent shapes with complicated geometry and topology. However, the storage requirement, e.g. memory or disk usage, for implicit representations of complex models is relatively large. In this paper, we propose a compact representation for multilevel rational algebraic spline (MRAS) surfaces using low-rank tensor approximation technique, and exploit its applications in surface reconstruction. Given a set of 3D points equipped with oriented normals, we first fit them with an algebraic spline surface defined on a box that bounds the point cloud. We split the bounding box into eight sub-cells if the fitting error is greater than a given threshold. Then for each sub-cell over which the fitting error is greater than the threshold, an offset function represented by an algebraic spline function of low rank is computed by locally solving a convex optimization problem. An algorithm is presented to solve the optimization problem based on the alternating direction method of multipliers (ADMM) and the CANDECOMP/PARAFAC (CP) decomposition of tensors. The procedure is recursively performed until a certain accuracy is achieved. To ensure the global continuity of the MRAS surface, quadratic B-spline weight functions are used to blend the offset functions. Numerous experiments show that our approach can greatly reduce the storage of the reconstructed implicit surface while preserve the fitting accuracy compared with the state-of-the-art methods. Furthermore, our method has good adaptability and is able to produce reconstruction results with high quality.     

On tensor approximation of Green iterations for Kohn-Sham equations

In the present paper we discuss efficient rank-structured tensor approximation methods for 3D integral transforms representing the Green iterations for the Kohn-Sham equation. We analyse the local convergence of the Newton iteration to solve the Green’s function integral formulation of the Kohn-Sham model in electronic structure calculations. We prove the low-separation rank approximations for the arising discrete convolving kernels given by the Coulomb and Yukawa potentials 1/|x|, and e ^?λ|x|/|x|, respectively, with $x \in {\mathbb{R}}^{d} $ . Complexity analysis of the nonlinear iteration with truncation to the fixed Kronecker tensor-product format is presented. Our method has linear scaling in the univariate problem size. Numerical illustrations demostrate uniform exponential convergence of tensor approximations in the orthogonal Tucker and canonical formats.     

Nonparametric profile monitoring in multi-dimensional data spaces

Profile monitoring has received increasingly attention in a wide range of applications in statistical process control (SPC). In this work, we propose a framework for monitoring nonparametric profiles in multi-dimensional data spaces. The framework has the following important features: (i) a flexible and computationally efficient smoothing technique, called Support Vector Regression, is employed to describe the relationship between the response variable and the explanatory variables; (ii) the usual structural assumptions on the residuals are not required; and (iii) the dependence structure for the within-profile observations is appropriately accommodated. Finally, real AIDS data collected from hospitals in Taiwan are used to illustrate and evaluate our proposed framework.     

质量度量指标驱动的数据聚合与多维数据可视化

以多维数据可视化为研究对象,在质量度量模型下,采用数据聚合为基本手段,来提高多维数据可视化的图像质量.在质量度量指标驱动的框架下提出了均分 K-means++数据聚合算法,在传统 K-means算法的基础上,专门以数据可视化为目的对算法进行了改进,使得算法聚合得到的数据既能够较好地保持原数据的大部分特性,又能显著地提高可视化后的图像质量.仿真实验证明,在不同的数据抽象级别DAL下,无论是图像质量指标还是质量度量指标HDM（直方图差值度量）、NNM（最近邻距离度量）,算法都表现出了较好的仿真结果.     

多维数据的复杂查询聚集算法研究

对空间多维数据的复杂查询是多维数据研究的重点和难点,目前研究的结论相对较少。在传统算法的基础上,进行了几个方面的改进：按分组属性值进行数据分块;对分组数据进行有效的排序;在聚集函数的应用上进行优化。模拟数据的试验表明：改进算法较大地提高了查询效率。     

Hierarchical TS fuzzy system and its universal approximation

An efficient tool to deal with the ‘rule explosion’ problem is the hierarchical system by which a fuzzy system can be decomposed into a number of hierarchically connected low-dimensional systems. In this paper a generalized hierarchical Tagaki–Sugeno (TS) system is built. It is shown that the input–output (I/O) relationship of this generalized hierarchical system can be represented as one of a standard TS fuzzy system. And the system approximation capability is analyzed by taking piecewise linear functions as a bridge. By constructive method it is proven that the hierarchical fuzzy systems (HFS’s) can be universal approximators. For the given approximation accuracy, an estimation formula about the number of the rules needed in the HFS is established. Finally some simulation examples confirm that the HFS’s with smaller size rule base can approximate the given functions with high accuracy. The results obtained here provide us with the theoretical basis for various applications of HFS’s.     

Nonlinear model order reduction with low rank tensor approximation

In this article, two methods of model order reduction based on the low rank approximation of tensor are introduced for the large scale nonlinear problem. We first introduce some definitions and results on tensor extended from matrix theory. Then we show how the general nonlinear system can be converted into the low rank form we treated in this research. We put the model order reduction of it in two frameworks, that is, polynomial framework and moment‐matching framework. In these two frameworks we construct the algorithms correspondingly, and analyze properties of these algorithms, including the preservation of stability, and moment‐matching properties. Next the priorities of these algorithms are presented. Finally we setup several numerical experiments to validate the effectiveness of the algorithms.     

Neighborhood graphs for indexing and retrieving multi-dimensional data

We propose a methodology based on a structure called neighborhood graphs for indexing and retrieving multi-dimensional data. In accordance with the increase of the quantity of data, it gets more and more important to process multi-dimensional data. Processing of data includes various tasks, for instance, mining, classifying, clustering, to name a few. However, to enable the effective processing of such multi-dimensional data, it is often necessary to locate each data precisely in the multi-dimensional space where the data reside so that each data can be effectively retrieved for processing. This amounts to solving the point location problem (neighborhood search) for multi-dimensional space. In this paper, in order to utilize the structure of neighborhood graphs as an indexing structure for multi-dimensional data, we propose the following: i) a local insertion and deletion method, and ii) an incremental neighborhood graph construction method. The first method enables to cope with the problem incurred from the updating of the graph. The second method realizes fast neighborhood graph construction from scratch, through the recursive application of the first method. Several experiments are conducted to evaluate the proposed approach, and the results indicate the effectiveness of our approach.     

Compressed hierarchical binary histograms for summarizing multi-dimensional data

Hierarchical binary partitions of multi-dimensional data are investigated as a basis for the construction of effective histograms. Specifically, the impact of adopting lossless compression techniques for representing the histogram on both the accuracy and the efficiency of query answering is investigated. Compression is obtained by exploiting the hierarchical partition scheme underlying the histogram, and then introducing further restrictions on the partitioning which enable a more compact representation of bucket boundaries. Basically, these restrictions consist of constraining the splits of the partition to be laid onto regular grids defined on the buckets. Several heuristics guiding the histogram construction are also proposed, and a thorough experimental analysis comparing the accuracy of histograms resulting from combining different heuristics with different representation models (both the new compression-based and the traditional ones) is provided. The best accuracy turns out from combining our grid-constrained partitioning scheme with one of the new heuristics. Histograms resulting from this combination are compared with state-of-the-art summarization techniques, showing that the proposed approach yields lower error rates and is much less sensitive to dimensionality, and that adopting our compression scheme results in improving the efficiency of query estimation.     

Random weighting estimation for fusion of multi-dimensional position data

This paper adopts the concept of random weighting estimation to multi-sensor data fusion. It presents a new random weighting estimation methodology for optimal fusion of multi-dimensional position data. A multi-sensor observation model is constructed for multi-dimensional position. Based on this observation model, a random weighting estimation algorithm is developed for estimation of position data from single sensors. Using the random weighting estimations from each single sensor, an optimization theory is established for optimal fusion of multi-sensor position data. Experimental results demonstrate that the proposed methodology can effectively fuse multi-sensor dimensional position data, and the fusion accuracy is much higher than that of the Kalman fusion method.