Differential and Integral Geometry of Linear Scale-Spaces

Linear scale-space theory provides a useful framework to quantify the differential and integral geometry of spatio-temporal input images. In this paper that geometry comes about by constructing connections on the basis of the similarity jets of the linear scale-spaces and by deriving related systems of Cartan structure equations. A linear scale-space is generated by convolving an input image with Green's functions that are consistent with an appropriate Cauchy problem. The similarity jet consists of those geometric objects of the linear scale-space that are invariant under the similarity group. The constructed connection is assumed to be invariant under the group of Euclidean movements as well as under the similarity group. This connection subsequently determines a system of Cartan structure equations specifying a torsion two-form, a curvature two-form and Bianchi identities. The connection and the covariant derivatives of the curvature and torsion tensor then completely describe a particular local differential geometry of a similarity jet. The integral geometry obtained on the basis of the chosen connection is quantified by the affine translation vector and the affine rotation vectors, which are intimately related to the torsion two-form and the curvature two-form, respectively. Furthermore, conservation laws for these vectors form integral versions of the Bianchi identities. Close relations between these differential geometric identities and integral geometric conservation laws encountered in defect theory and gauge field theories are pointed out. Examples of differential and integral geometries of similarity jets of spatio-temporal input images are treated extensively.     

2-D常系数线性离散系统可分性判定

针对二维常系数线性离散系统一般状态空间模型(2-DGM)的可分性判定问题进行探讨,给出了系统可分的若干判定法则,这些法则有助于进一步完善2-D系统现有的理论体系.     

Non-Linear Scale-Spaces Isomorphic to the Linear Case with Applications to Scalar,Vector and Multispectral Images

A basic requirement of scale-space representations in general is that of scale causality, which states that local extrema in the image should not be enhanced when resolution is diminished. We consider a special class of nonlinear scale-spaces consistent with this constraint, which can be linearised by a suitable isomorphism in the grey-scale domain so as to reproduce the familiar Gaussian scale-space. We consider instances in which nonlinear representations may be the preferred choice, as well as instances in which they enter by necessity. We also establish their relation to morphological scale-space representations based on a quadratic structuring function.     

Adjunctions in Pyramids,Curve Evolution and Scale-Spaces

We have been witnessing lately a convergence among mathematical morphology and other nonlinear fields, such as curve evolution, PDE-based geometrical image processing, and scale-spaces. An obvious benefit of such a convergence is a cross-fertilization of concepts and techniques among these fields. The concept of adjunction however, so fundamental in mathematical morphology, is not yet shared by other disciplines. The aim of this paper is to show that other areas in image processing can possibly benefit from the use of adjunctions. In particular, a strong relationship between pyramids and adjunctions is presented. We show how this relationship may help in analyzing existing pyramids, and construct new pyramids. Moreover, it will be explained that adjunctions based on a curve evolution scheme can provide idempotent shape filters. This idea is illustrated in this paper by means of a simple affine-invariant polygonal flow. Finally, the use of adjunctions in scale-space theory is also addressed.     

Timescale Analysis with an Entropy-Based Shift-Invariant Discrete Wavelet Transform

This paper presents an invariant discrete wavelet transform that enables point-to-point (aligned) comparison among all scales, contains no phase shifts, relaxes the strict assumption of a dyadic-length time series, deals effectively with boundary effects and is asymptotically efficient. It also introduces a new entropy-based methodology for the determination of the optimal level of the multiresolution decomposition, as opposed to subjective or ad-hoc approaches used hitherto. As an empirical application, the paper relies on wavelet analysis to reveal the complex dynamics across different timescales for one of the most widely traded foreign exchange rates, namely the Great Britain Pound. The examined period covers the global financial crisis and the Eurozone debt crisis. The timescale analysis attempts to explore the micro-dynamics of across-scale heterogeneity in the second moment (volatility) on the basis of market agent behavior with different trading preferences and information flows across scales. New stylized properties emerge in the volatility structure and the implications for the flow of information across scales are inferred.     

Algebraic and PDE Approaches for Lattice Scale-Spaces with Global Constraints

This paper begins with analyzing the theoretical connections between levelings on lattices and scale-space erosions on reference semilattices. They both represent large classes of self-dual morphological operators that exhibit both local computation and global constraints. Such operators are useful in numerous image analysis and vision tasks including edge-preserving multiscale smoothing, image simplification, feature and object detection, segmentation, shape and motion analysis. Previous definitions and constructions of levelings were either discrete or continuous using a PDE. We bridge this gap by introducing generalized levelings based on triphase operators that switch among three phases, one of which is a global constraint. The triphase operators include as special cases useful classes of semilattice erosions. Algebraically, levelings are created as limits of iterated or multiscale triphase operators. The subclass of multiscale geodesic triphase operators obeys a semigroup, which we exploit to find PDEs that can generate geodesic levelings and continuous-scale semilattice erosions. We discuss theoretical aspects of these PDEs, propose discrete algorithms for their numerical solution which converge as iterations of triphase operators, and provide insights via image experiments.     

Morphological and Linear Scale Spaces for Fiber Enhancement in DW-MRI

We consider morphological and linear scale spaces on the space ?³?S ² of 3D positions and orientations naturally embedded in the group SE(3) of 3D rigid body movements. The general motivation for these (convection-)diffusions and erosions is to obtain crossing-preserving fiber enhancement on probability densities defined on the space of positions and orientations. The strength of these enhancements is that they are expressed in a moving frame of reference attached to fiber fragments, allowing us to diffuse along the fibers and to erode orthogonal to them. The linear scale spaces are described by forward Kolmogorov equations of Brownian motions on ?³?S ² and can be solved by convolution with the corresponding Green’s functions. The morphological scale spaces are Bellman equations of cost processes on ?³?S ² and we show that their viscosity solutions are given by a morphological convolution with the corresponding morphological Green’s function. For theoretical underpinning of our scale spaces on ?³?S ² we introduce Lagrangians and Hamiltonians on ?³?S ² indexed by a parameter η∈[1,∞). The Hamiltonian induces a Hamilton-Jacobi-Bellman system that coincides with our morphological scale spaces on ?³?S ². By means of the logarithm on SE(3) we provide tangible estimates for both the linear- and the morphological Green’s functions. We also discuss numerical finite difference upwind schemes for morphological scale spaces (erosions) of Diffusion-Weighted Magnetic Resonance Imaging (DW-MRI), which allow extensions to data-adaptive erosions of DW-MRI. We apply our theory to the enhancement of (crossing) fibres in DW-MRI for imaging water diffusion processes in brain white matter.     

An Explanation for the Logarithmic Connection between Linear and Morphological System Theory

Dorst/van den Boomgaard and Maragos introduced the slope transform as the morphological equivalent of the Fourier transform. Generalising the conjugacy operation from convex analysis it formed the basis of a morphological system theory that bears an almost logarithmic relation to linear system theory; a connection that has not been fully understood so far. Our article provides an explanation by disclosing that morphology in essence is linear system theory in specific algebras. While linear system theory uses the standard plus-prod algebra, morphological system theory is based on the max-plus algebra and the min-plus algebra. We identify the nonlinear operations of erosion and dilation as linear convolutions in the latter algebras. The logarithmic Laplace transform makes a natural appearance as it corresponds to the conjugacy operation in the max-plus algebra. Its conjugate is given by the so-called Cramer transform. Originating from stochastics, the Cramer transform maps Gaussians to quadratic functions and relates standard convolution to erosion. This fundamental transform relies on the logarithm and constitutes the direct link between linear and morphological system theory. Many numerical examples are presented that illustrate the convexifying and smoothing properties of the Cramer transform.First online version published in June, 2005     

关于大学生就业是否专业对口的调查研究

近年来,大学生的就业问题一直是社会公众广泛关注的焦点。为了清楚了解社会、学校以及大学应届毕业生在就业时所面临专业是否对口的问题,该文以问卷形式展开调查,并对调查结果进行分析,对大学生就业提出了一些宝贵的参考意见。     

高职对口升学计算机应用技术专业课程体系开发研究

对口生是高职生源的重要组成部分,而目前的高职课程体系多以普招生作为教学对象和起点,很少有专门面向对口招生学生的人才培养方案和课程体系,造成对口生学习困难,积极性不高。采用基于工作过程系统化的方法,综合考虑对口生的认知特点,构建了全新的计算机应用技术专业课程体系,并进一步探讨了保证措施。     

Mathematical Morphology Based on Linear Combined Metric Spaces on Z2 (Part II): Constant Time Morphological Operations

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.In this second part, the concepts developed in part I (see Díaz de León and Sossa Azuela, Mathematical morphology based on linear combined metric spaces on Z¹ (part I): Fast distance transforms, Journal of Mathematical Imaging and Vision, Vol. 12, No. 2, pp. 137–154, 2000) are used to prove that it is possible to reduce the complexity of the morphological operators to zero complexity (constant time algorithms) for any regular discrete metric space and to eliminate the use of the structural element. In particular, this is done for an infinite family of metric spaces further defined. The use of the distance transformation is proposed for it comprises the information concerning all the discs included in a region to obtain fast morphological operators: erosions, dilations, openings and closings (of zero complexity) for an infinite (countable) family of regular metric spaces. New constant time, in contrast with the polynomial time algorithms, for the computation of these basics operators for any structural element are next derived by using this background. Practical examples showing the efficiency of the proposed algorithms, final comments and present research are also given here.     

Linear Problems and Linear Algorithms

Using predicate logic, the concept of a linear problem is formalized. The class of linear problems is huge, diverse, complex, and important. Linear and randomized linear algorithms are formalized. For each linear problem, a linear algorithm is constructed that solves the problem and a randomized linear algorithm is constructed that completely solves it, that is, for any data of the problem, the output set of the randomized linear algorithm is identical to the solution set of the problem. We obtain a single machine, referred to as the Universal (Randomized) Linear Machine, which (completely) solves every instance of every linear problem. Conversely, for each randomized linear algorithm, a linear problem is constructed that the algorithm completely solves. These constructions establish a one-to-one and onto correspondence from equivalence classes of linear problems to equivalence classes of randomized linear algorithms.Our construction of (randomized) linear algorithms to (completely) solve linear problems as well as the algorithms themselves are based on Fourier Elimination and have superexponential complexity. However, there is no evidence that the inefficiency of our methods is unavoidable relative to the difficulty of the problem.     

Linear Time Logic Control of Discrete-Time Linear Systems

The control of complex systems poses new challenges that fall beyond the traditional methods of control theory. One of these challenges is given by the need to control, coordinate and synchronize the operation of several interacting submodules within a system. The desired objectives are no longer captured by usual control specifications such as stabilization or output regulation. Instead, we consider specifications given by linear temporal logic (LTL) formulas. We show that existence of controllers for discrete-time controllable linear systems and LTL specifications can be decided and that such controllers can be effectively computed. The closed-loop system is of hybrid nature, combining the original continuous dynamics with the automatically synthesized switching logic required to enforce the specification     

Morphological image compositing

Image mosaicking can be defined as the registration of two or more images that are then combined into a single image. Once the images have been registered to a common coordinate system, the problem amounts to the definition of a selection rule to output a unique value for all those pixels that are present in more than one image. This process is known as image compositing. In this paper, we propose a compositing procedure based on mathematical morphology and its marker-controlled segmentation paradigm. Its scope is to position seams along salient image structures so as to diminish their visibility in the output mosaic even in the absence of radiometric corrections or blending procedures. We also show that it is suited to the seamless minimization of undesirable transient objects occurring in the regions where two or more images overlap. The proposed methodology and algorithms are illustrated for the composition of satellite images minimizing cloud cover.     

Morphological associative memories

The theory of artificial neural networks has been successfully applied to a wide variety of pattern recognition problems. In this theory, the first step in computing the next state of a neuron or in performing the next layer neural network computation involves the linear operation of multiplying neural values by their synaptic strengths and adding the results. A nonlinear activation function usually follows the linear operation in order to provide for nonlinearity of the network and set the next state of the neuron. In this paper we introduce a novel class of artificial neural networks, called morphological neural networks, in which the operations of multiplication and addition are replaced by addition and maximum (or minimum), respectively. By taking the maximum (or minimum) of sums instead of the sum of products, morphological network computation is nonlinear before possible application of a nonlinear activation function. As a consequence, the properties of morphological neural networks are drastically different than those of traditional neural network models. The main emphasis of the research presented here is on morphological associative memories. We examine the computing and storage capabilities of morphological associative memories and discuss differences between morphological models and traditional semilinear models such as the Hopfield net.     

线性霍尔传感器在直线位移中的应用

为了实现实时的无接触直线位移测量,通过线性霍尔传感器获取磁场强度信号,采用定时中断方式完成数据采集,并以MSP430F169单片机作为运算及控制单元。考虑到普通的永磁无法形成梯度、线性的磁场分布,因此,在改进磁铁结构的基础上,提出了线性霍尔传感器非线性方法的测量方案,实现了对位置信号的无接触实时精确测量。实验结果表明,该霍尔变送器系统能实现对位置值的准确测量。     

齿轮齿条直线运动的线性测量与分析

在齿轮传动机构的直线运动研究中,为了测量直线误差,设计了齿轮齿条的测量平台;采用伺服电机直接驱动齿轮在齿条上运动,并以基于运动控制点位运动模式来实现测量间距和待机等控制要求;在齿轮齿条传动机构直线运动的过程中,用激光干涉仪对与齿轮同轴运动的工作台实际运动位置测量,获得测量数据;用最小二乘法求解线性矛盾方程的方法来构造拟合函数,并将其推广至任意次和任意多个变量的拟合函数,使用MATLAB编程求解,获得与设计数据曲线对应的实际数据曲线,得出齿轮齿条的线性定位精度和重复精度;这项技术可以推广到对大多数的直线运动机构来进行线性测量及数据分析;也能成为直线运动控制中线性参数自动补偿算法的依据。     

Learning Linear and Nonlinear PCA with Linear Programming

An SVM-like framework provides a novel way to learn linear principal component analysis (PCA). Actually it is a weighted PCA and leads to a semi-definite optimization problem (SDP). In this paper, we learn linear and nonlinear PCA with linear programming problems, which are easy to be solved and can obtain the unique global solution. Moreover, two algorithms for learning linear and nonlinear PCA are constructed, and all principal components can be obtained. To verify the performance of the proposed method, a series of experiments on artificial datasets and UCI benchmark datasets are accomplished. Simulation results demonstrate that the proposed method can compete with or outperform the standard PCA and kernel PCA (KPCA) in generalization ability but with much less memory and time consuming.