Smooth multiple B-spline surface fitting with Catmull%ndash;Clark subdivision surfaces for extraordinary corner patches

This paper presents an algorithm for simultaneously fitting smoothly connected multiple surfaces from unorganized measured data. A hybrid mathematical model of B-spline surfaces and Catmull–Clark subdivision surfaces is introduced to represent objects with general quadrilateral topology. The interconnected multiple surfaces are G ² continuous across all surface boundaries except at a finite number of extraordinary corner points where G ¹ continuity is obtained. The algorithm is purely a linear least-squares fitting procedure without any constraint for maintaining the required geometric continuity. In case of general uniform knots for all surfaces, the final fitted multiple surfaces can also be exported as a set of Catmull–Clark subdivision surfaces with global C ² continuity and local C ¹ continuity at extraordinary corner points. Published online: 14 May 2002 Correspondence to: W. Ma     

A novel region-based active contour model based on kernel function for image segmentation

It is a difficult task to accurately segment images with intensity inhomogeneity, because most of existing algorithms are based upon the assumption of the homogeneity of image intensity. In this paper, we propose a novel region-based active contour model, referred to as the K-GLIF, which utilizes both global and local image intensity fittings with kernel functions. The model consists of an intensity fitting term and a new regularization term. The intensity fitting term of the level set function is the gradient descent flow that minimizes the global binary fitting energy functional. The local intensity fitting value based on the generalized Gaussian kernel function is then incorporated into the global intensity fitting value to form the weighted intensity fitting value on the two sides of the contour. Owing to the kernel function, the intensity information in local regions is extracted to guide the motion of the contour, which enables the model to effectively segment images with intensity inhomogeneity and smooth noise. A new regularization term is used to control the smoothness of the level set function and avoid complicated re-initialization. Experimental results and comparisons with other models of inhomogeneous images, synthetic images, medical images, multi-object images, natural and infrared images show that the proposed K-GLIF model improves the quality of image segmentation in terms of accuracy and robustness of initial contours.

     

Efficient Exact Algorithms on Planar Graphs: Exploiting Sphere Cut Decompositions

We present a general framework for designing fast subexponential exact and parameterized algorithms on planar graphs. Our approach is based on geometric properties of planar branch decompositions obtained by Seymour and Thomas, combined with refined techniques of dynamic programming on planar graphs based on properties of non-crossing partitions. To exemplify our approach we show how to obtain an  $O(2^{6.903\sqrt{n}})We present a general framework for designing fast subexponential exact and parameterized algorithms on planar graphs. Our approach is based on geometric properties of planar branch decompositions obtained by Seymour and Thomas, combined with refined techniques of dynamic programming on planar graphs based on properties of non-crossing partitions. To exemplify our approach we show how to obtain an  O(2^6.903?n)O(2^{6.903\sqrt{n}}) time algorithm solving weighted Hamiltonian Cycle on an n-vertex planar graph. Similar technique solves Planar Graph Travelling Salesman Problem with n cities in time O(2^9.8594?n)O(2^{9.8594\sqrt{n}}) . Our approach can be used to design parameterized algorithms as well. For example, we give an algorithm that for a given k decides if a planar graph on n vertices has a cycle of length at least k in time O(2^13.6?kn+n³)O(2^{13.6\sqrt{k}}n+n^{3}) .     

特征自表达和图正则化的鲁棒无监督特征选择

为了在揭示数据全局结构的同时保留其局部结构,本文将特征自表达和图正则化统一到同一框架中,给出了一种新的无监督特征选择(unsupervised feature selection,UFS)模型与方法。模型使用特征自表达,用其余特征线性表示每一个特征,以保持特征的局部结构;用基于 ${L_{2, 1}}$ 范数的图正则化项,在保留数据的局部几何结构的同时可以降低噪声数据对特征选择的影响;除此之外,在权重矩阵上施加了低秩约束,保留数据的全局结构。在6个不同的公开数据集上的实验表明,所给算法明显优于其他5个对比算法,表明了所提出的UFS框架的有效性。     

Hybrid geodesic region-based active contours for image segmentation

In this paper, we propose novel hybrid edge and region based active contour models. First, we consider geodesic curve and region-based model, and evolve contours based on global information to segment images with intensity homogeneity. Second, we extend the global model to the local intensity fitting energy for segmenting the images with intensity inhomogeneity. Moreover, the level set regularization term is added to the energy functional to ensure accurate computation and avoid expensive re-initialization of the evolving level set function. Experimental results indicate the proposed method has advantage over the geodesic active contour (GAC) model, the Chan–Vese (C–V) model, the Lankton’s method and the local binary fitting (LBF) model in terms of efficiency and robustness.     

Local Histogram Based Segmentation Using the Wasserstein Distance

We propose and analyze a nonparametric region-based active contour model for segmenting cluttered scenes. The proposed model is unsupervised and assumes pixel intensity is independently identically distributed. Our proposed energy functional consists of a geometric regularization term that penalizes the length of the partition boundaries and a region-based image term that uses histograms of pixel intensity to distinguish different regions. More specifically, the region data encourages segmentation so that local histograms within each region are approximately homogeneous. An advantage of using local histograms in the data term is that histogram differentiation is not required to solve the energy minimization problem. We use Wasserstein distance with exponent 1 to determine the dissimilarity between two histograms. The Wasserstein distance is a metric and is able to faithfully measure the distance between two histograms, compared to many pointwise distances. Moreover, it is insensitive to oscillations, and therefore our model is robust to noise. A fast global minimization method based on (Chan et al. in SIAM J. Appl. Math. 66(5):1632–1648, 2006; Bresson et al. in J. Math. Imaging Vis. 28(2):151–167, 2007) is employed to solve the proposed model. The advantages of using this method are two-fold. First, the computational time is less than that of the method by gradient descent of the associated Euler-Lagrange equation (Chan et al. in Proc. of SSVM, pp. 697–708, 2007). Second, it is able to find a global minimizer. Finally, we propose a variant of our model that is able to properly segment a cluttered scene with local illumination changes. This research is supported by ONR grant N00014-09-1-0105 and NSF grant DMS-0610079.     

Feature‐Preserving Surface Reconstruction From Unoriented,Noisy Point Data

We propose a robust method for surface mesh reconstruction from unorganized, unoriented, noisy and outlier‐ridden 3D point data. A kernel‐based scale estimator is introduced to estimate the scale of inliers of the input data. The best tangent planes are computed for all points based on mean shift clustering and adaptive scale sample consensus, followed by detecting and removing outliers. Subsequently, we estimate the normals for the remaining points and smooth the noise using a surface fitting and projection strategy. As a result, the outliers and noise are removed and filtered, while the original sharp features are well preserved. We then adopt an existing method to reconstruct surface meshes from the processed point data. To preserve sharp features of the generated meshes that are often blurred during reconstruction, we describe a two‐step approach to effectively recover original sharp features. A number of examples are presented to demonstrate the effectiveness and robustness of our method.     

正则化和交叉验证在组合预测模型中的应用

组合预测模型的权重确定方式对于提高模型精度至关重要,为研究正则化与交叉验证是否能改善组合预测模型的预测效果,提出将正则化和交叉验证应用于基于最小二乘法的组合预测模型.通过在组合模型的最优化求解中分别加入L₁、L₂范数正则化项,并对数据集进行留一交叉验证后发现：L₁、L₂范数正则化都对组合模型的预测精度具有改善效果,且L₁范数正则化比L₂范数正则化对组合预测模型的改善效果更好,并且参与组合预测的单项预测模型越多,正则化的改善效果越好,交叉验证对组合预测模型的改善效果则与给定实验数据量呈现正相关.     

Robust fitting of implicitly defined surfaces using Gauss–Newton-type techniques

We describe Gauss–Newton-type methods for fitting implicitly defined curves and surfaces to given unorganized data points. The methods are suitable not only for least-squares approximation, but they can also deal with general error functions, such as approximations to the ℓ ₁ or ℓ _∞ norm of the vector of residuals. Two different definitions of the residuals will be discussed, which lead to two different classes of methods: direct methods and data-based ones. In addition we discuss the continuous versions of the methods, which furnish geometric interpretations as evolution processes. It is shown that the data-based methods—which are less costly, as they work without the computation of the closest points—can efficiently deal with error functions that are adapted to noisy and uncertain data. In addition, we observe that the interpretation as evolution process allows to deal with the issues of regularization and with additional constraints.

图像恢复中的凸能量泛函正则化模型综述

目的凸能量泛函正则化模型(EFRM)的综述论文在国内外还少有报道,为使即将进入该领域的研究者全面了解发展现状,结合图像恢复,对该领域国内外研究现状进行综述。方法在参考大量文献的基础上,从凸EFRM的起因、组成、处理和发展趋势等方面加以总结和比较。首先,给定反问题,无法获得可行解,解决此问题的有效方法是建立EFRM。其次,从能量泛函的组成,分析拟合项和正则项的适用条件,给出引起图像模糊的5种点扩散函数,阐述权重的重要性及确定方法。再次,将能量泛函的拟合项和正则项分为整体处理、单独处理,分析空域、变换域和混合域正则化模型求解算法,评述模型和算法的优缺点。最后,指出图像恢复EFRM的发展趋势及存在的问题。结果一般说来,无法直接求解由拟合项、正则项和权重组成的原始凸EFRM,然而,通过转化模型、对偶模型和原始—对偶模型,利用数值代数、矩阵论和优化理论对转化模型进行整体处理、分裂处理,可以设计出高效、快速求解算法。结论图像恢复中的EFRM研究虽然取得了很多有意义的理论与应用成果,但随着大规模数据处理问题的不断涌现,建立准确的数学模型,设计高效快速的求解算法以及分析算法的收敛性等理论问题有待进一步深入研究。     

Multi-instance clustering with applications to multi-instance prediction

In the setting of multi-instance learning, each object is represented by a bag composed of multiple instances instead of by a single instance in a traditional learning setting. Previous works in this area only concern multi-instance prediction problems where each bag is associated with a binary (classification) or real-valued (regression) label. However, unsupervised multi-instance learning where bags are without labels has not been studied. In this paper, the problem of unsupervised multi-instance learning is addressed where a multi-instance clustering algorithm named Bamic is proposed. Briefly, by regarding bags as atomic data items and using some form of distance metric to measure distances between bags, Bamic adapts the popular k -Medoids algorithm to partition the unlabeled training bags into k disjoint groups of bags. Furthermore, based on the clustering results, a novel multi-instance prediction algorithm named Bartmip is developed. Firstly, each bag is re-represented by a k-dimensional feature vector, where the value of the i-th feature is set to be the distance between the bag and the medoid of the i-th group. After that, bags are transformed into feature vectors so that common supervised learners are used to learn from the transformed feature vectors each associated with the original bag’s label. Extensive experiments show that Bamic could effectively discover the underlying structure of the data set and Bartmip works quite well on various kinds of multi-instance prediction problems.     

Multiphase Soft Segmentation with Total Variation and <Emphasis Type="Italic">H</Emphasis><Superscript>1</Superscript> Regularization

In this paper, we propose a variational soft segmentation framework inspired by the level set formulation of multiphase Chan-Vese model. We use soft membership functions valued in [0,1] to replace the Heaviside functions of level sets (or characteristic functions) such that we get a representation of regions by soft membership functions which automatically satisfies the sum to one constraint. We give general formulas for arbitrary N-phase segmentation, in contrast to Chan-Vese’s level set method only 2 ^m -phase are studied. To ensure smoothness on membership functions, both total variation (TV) regularization and H ¹ regularization used as two choices for the definition of regularization term. TV regularization has geometric meaning which requires that the segmentation curve length as short as possible, while H ¹ regularization has no explicit geometric meaning but is easier to implement with less parameters and has higher tolerance to noise. Fast numerical schemes are designed for both of the regularization methods. By changing the distance function, the proposed segmentation framework can be easily extended to the segmentation of other types of images. Numerical results on cartoon images, piecewise smooth images and texture images demonstrate that our methods are effective in multiphase image segmentation.     

Local and Nonlocal Discrete Regularization on Weighted Graphs for Image and Mesh Processing

We propose a discrete regularization framework on weighted graphs of arbitrary topology, which unifies local and nonlocal processing of images, meshes, and more generally discrete data. The approach considers the problem as a variational one, which consists in minimizing a weighted sum of two energy terms: a regularization one that uses the discrete p-Dirichlet form, and an approximation one. The proposed model is parametrized by the degree p of regularity, by the graph structure and by the weight function. The minimization solution leads to a family of simple linear and nonlinear processing methods. In particular, this family includes the exact expression or the discrete version of several neighborhood filters, such as the bilateral and the nonlocal means filter. In the context of images, local and nonlocal regularizations, based on the total variation models, are the continuous analog of the proposed model. Indirectly and naturally, it provides a discrete extension of these regularization methods for any discrete data or functions.     

A Novel Sparsity Reconstruction Method from Poisson Data for 3D Bioluminescence Tomography

In this paper, we consider 3D Bioluminescence tomography (BLT) source reconstruction from Poisson data in three dimensional space. With a priori information of sources sparsity and MAP estimation of Poisson distribution, we study the minimization of Kullback-Leihbler divergence with ℓ ¹ and ℓ ⁰ regularization. We show numerically that although several ℓ ¹ minimization algorithms are efficient for compressive sensing, they fail for BLT reconstruction due to the high coherence of the measurement matrix columns and high nonlinearity of Poisson fitting term. Instead, we propose a novel greedy algorithm for ℓ ⁰ regularization to reconstruct sparse solutions for BLT problem. Numerical experiments on synthetic data obtained by the finite element methods and Monte-Carlo methods show the accuracy and efficiency of the proposed method.     

Gene Expression Data to Mouse Atlas Registration Using a Nonlinear Elasticity Smoother and Landmark Points Constraints

This paper proposes a numerical algorithm for image registration using energy minimization and nonlinear elasticity regularization. Application to the registration of gene expression data to a neuroanatomical mouse atlas in two dimensions is shown. We apply a nonlinear elasticity regularization to allow larger and smoother deformations, and further enforce optimality constraints on the landmark points distance for better feature matching. To overcome the difficulty of minimizing the nonlinear elasticity functional due to the nonlinearity in the derivatives of the displacement vector field, we introduce a matrix variable to approximate the Jacobian matrix and solve for the simplified Euler-Lagrange equations. By comparison with image registration using linear regularization, experimental results show that the proposed nonlinear elasticity model also needs fewer numerical corrections such as regridding steps for binary image registration, it renders better ground truth, and produces larger mutual information; most importantly, the landmark points distance and L ² dissimilarity measure between the gene expression data and corresponding mouse atlas are smaller compared with the registration model with biharmonic regularization.     

Statistical outlier detection using direct density ratio estimation

We propose a new statistical approach to the problem of inlier-based outlier detection, i.e., finding outliers in the test set based on the training set consisting only of inliers. Our key idea is to use the ratio of training and test data densities as an outlier score. This approach is expected to have better performance even in high-dimensional problems since methods for directly estimating the density ratio without going through density estimation are available. Among various density ratio estimation methods, we employ the method called unconstrained least-squares importance fitting (uLSIF) since it is equipped with natural cross-validation procedures, allowing us to objectively optimize the value of tuning parameters such as the regularization parameter and the kernel width. Furthermore, uLSIF offers a closed-form solution as well as a closed-form formula for the leave-one-out error, so it is computationally very efficient and is scalable to massive datasets. Simulations with benchmark and real-world datasets illustrate the usefulness of the proposed approach.     

Deriving Proof Rules from Continuation Semantics

We claim that a continuation style semantics of a programming language can provide a starting point for constructing its proof system. The basic idea is to see weakest preconditions as a particular instance of continuation style semantics, hence to interpret correctness assertions (e.g. Hoare triples {p} C {r}) as inequalities over continuations. This approach also shows a correspondence between labels in a program and annotations. Received July 1997 / Accepted in revised form August 1999     

Outliers, inliers and the generalized least trimmed squares estimator in system identification

The least trimmed squares estimator (LTS) is a well known robust estinaator in terms of protecting the estimatefrom the outliers. Its high computational complexity is however a problem in practice. We show that the LTS estimate can be obtained by a simple algorithm with the complexity O( N In N) for large N, where N is the number of measurements. We also showthat though the LTS is robust in terms of the outliers, it is sensitive to the inliers. The concept of the inliers is introduced. Moreover, the Generalized Least Trimmed Squares estimator (GLTS) together with its solution are presented that reduces the effect of both the outliers and the inliers.     

分裂合并运动分割的多运动视觉里程计方法

目的 视觉里程计（visual odometry,VO）仅需要普通相机即可实现精度可观的自主定位,已经成为计算机视觉和机器人领域的研究热点,但是当前研究及应用大多基于场景为静态的假设,即场景中只有相机运动这一个运动模型,无法处理多个运动模型,因此本文提出一种基于分裂合并运动分割的多运动视觉里程计方法,获得场景中除相机运动外多个运动目标的运动状态。方法 基于传统的视觉里程计框架,引入多模型拟合的方法分割出动态场景中的多个运动模型,采用RANSAC（random sample consensus）方法估计出多个运动模型的运动参数实例;接着将相机运动信息以及各个运动目标的运动信息转换到统一的坐标系中,获得相机的视觉里程计结果,以及场景中各个运动目标对应各个时刻的位姿信息;最后采用局部窗口光束法平差直接对相机的姿态以及计算出来的相机相对于各个运动目标的姿态进行校正,利用相机运动模型的内点和各个时刻获得的相机相对于运动目标的运动参数,对多个运动模型的轨迹进行优化。结果 本文所构建的连续帧运动分割方法能够达到较好的分割结果,具有较好的鲁棒性,连续帧的分割精度均能达到近100%,充分保证后续估计各个运动模型参数的准确性。本文方法不仅能够有效估计出相机的位姿,还能估计出场景中存在的显著移动目标的位姿,在各个分段路径中相机自定位与移动目标的定位结果位置平均误差均小于6%。结论 本文方法能够同时分割出动态场景中的相机自身运动模型和不同运动的动态物体运动模型,进而同时估计出相机和各个动态物体的绝对运动轨迹,构建出多运动视觉里程计过程。