A New Approach to the Decomposition of Boolean Functions by the Method of q-Partitions. I. Separating Decomposition of Full and Partial Functions

A new approach to the decomposition of Boolean functions of n variables is considered; the functions being decomposed can be represented in various forms. The approach is based on the method of q-partitions of minterms and on the introduced concept of a decomposition clone. The theorem on simple separating decomposition of full and partial functions is formulated. The approach proposed is illustrated by examples.     

A new approach to the decomposition of Boolean functions by the method of <Emphasis Type="Italic">q</Emphasis>-partitions. III. Joint decomposition of a function system

The paper considers the set-theoretical approach to the joint decomposition of systems of Boolean functions of variables specified in different representation forms. The approach is based on the method of q-partitions of conjuncterms and concept of decomposition clones. Theorems on joint decomposition of a system of full and partial functions are formulated. The approach is illustrated by examples. Parts I and II of this article are published in No. 5 (2001) and No. 1 (2002). __________ Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 39–58, March–April 2007.     

A New Method of Coding Minorants in Problems of Synthesis of Digital Devices from PLAs

A new method of coding minorants in decomposing a system of Boolean function is proposed. The method is based on the initial coding obtained from the minimal representation of the system in the so-called set-theoretical decomposition form obtained as a result of q-partitioning of conjuncterms of functions represented in DNF. A distinctive feature of the method is the preservation of the correspondence of the obtained codes of minorants to the block structure of the system being decomposed, which provides the minimum of informational capacity of PLAs.     

A new approach to the decomposition of boolean functions. 4. non-disjoint decomposition: the method of <Emphasis Type="Italic">p</Emphasis>, <Emphasis Type="Italic">q</Emphasis>-partitions

A set-theoretical approach to the non-disjoint decomposition of different forms of representation of Boolean functions of n variables is considered. This approach is based on the method of p,q-partitions of conjuncterms and on the concept of a decomposition clone. Two techniques of searching for some non-disjoint functional decomposition are described. Theorems on the non-disjoint decomposition of full and partial functions and their systems are formulated. The proposed approach is illustrated by examples. Parts I, II, and III of this article are published in No. 5 (2001), No. 1 (2002), and No. 2 (2007). Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 15-41, May–June 2009.     

Decomposition of Separable Concave Structuring Functions

This paper presents a decomposition scheme for a large class of greyscale concave structuring functions from mathematical morphology. In contrast with many existing decomposition schemes, our method is valid in the continuous domain. Conditions are given under which this continuous method can be properly discretized. The class of functions that can be decomposed with our method contains all concave 2D-functions that are separable in two 1D-functions. This class contains the class of quadratic functions, that are of major importance in, for instance, distance transforms and morphological scale space. In the continuous domain, the size of the structuring elements resulting from the decomposition, can be chosen arbitrarily small. Conditions under which the discrete version of the decomposition can be guaranteed are given. For functions from the mentioned class, that can be separated along the standard image axes, a discrete decomposition in elements of n × n pixels is always possible, with n 3. The parabola fall in the atter category.     

PARACONSISTENT LOGIC (in Italian), by Nicola Grana. Lofredo Editore,Naples, 1983. 134 pp.

Modified Reconstructibility Analysis (MRA), a novel decomposition within the framework of set-theoretic (crisp possibilistic) reconstructibility analysis, is presented. It is shown that in some cases, while three-variable NPN-classified Boolean functions are not decomposable using Conventional Reconstructibility Analysis (CRA), they are decomposable using MRA. Also, it is shown that whenever a decomposition of three-variable NPN-classified Boolean functions exists in both MRA and CRA, MRA yields simpler or equal complexity decompositions. A comparison of the corresponding complexities for Ashenhurst-Curtis decompositions and MRA is also presented. While both AC and MRA decompose some but not all NPN-classes, MRA decomposes more classes, and consequently more Boolean functions. MRA for many-valued functions is also presented, and algorithms using two different methods (intersection and union) are given. A many-valued case is presented where CRA fails to decompose but MRA decomposes.     

Structure-Texture Image Decomposition—Modeling, Algorithms, and Parameter Selection

This paper explores various aspects of the image decomposition problem using modern variational techniques. We aim at splitting an original image f into two components u and ρ, where u holds the geometrical information and ρ holds the textural information. The focus of this paper is to study different energy terms and functional spaces that suit various types of textures. Our modeling uses the total-variation energy for extracting the structural part and one of four of the following norms for the textural part: L², G, L¹ and a new tunable norm, suggested here for the first time, based on Gabor functions. Apart from the broad perspective and our suggestions when each model should be used, the paper contains three specific novelties: first we show that the correlation graph between u and ρ may serve as an efficient tool to select the splitting parameter, second we propose a new fast algorithm to solve the TV − L¹ minimization problem, and third we introduce the theory and design tools for the TV-Gabor model. First online version published in February, 2006     

基于generalized dominator的BDD布尔表达式优化分解

BDD是布尔函数的一种图形表示方式,可以直观地反映出布尔函数的逻辑结构,利用BDD可以实现对布尔函数的分解和优化。针对BDD的数据结构和一种以generalizeddominators为基础的布尔表达式的优化方法进行研究,并且着重对其中的一种方法:连接的BDD分解方法(ConjunctiveBDDDecomposition)进行了详细的分析。     

On Complexity of Computation of Partial Derivatives of Boolean Functions Realized by Zhegalkin Polynomials

The combinational complexity of a system of partial derivatives in the basis of linear functions is established for a Boolean function of n variables that is realized by a Zhegalkin polynomial. An algorithm whose complexity equals 3ⁿ – 2ⁿ modulo 2 additions is proposed for computation of all partial derivatives of such a function from the coefficients of its Zhegalkin polynomial.     

A General Decomposition Theorem for the k-Server Problem

The first general decomposition theorem for the k-server problem is presented. Whereas all previous theorems are for the case of a finite metric with k+1 points, the theorem given here allows an arbitrary number of points in the underlying metric space. This theorem implies O(polylog(k))-competitive randomized algorithms for certain metric spaces consisting of a polylogarithmic number of widely separated subspaces and takes a first step toward a general O(polylog(k))-competitive algorithm. The only other cases for which polylogarithmic competitive randomized algorithms are known are the uniform metric space and the weighted cache metric space with two weights.     

基于多次广义解调的多分量信号分解方法

在广义解调时频分析方法的基础上提出了基于多次广义解调的多分量信号分解方法。对于分量较多的多分量信号,经过一次广义解调后,在小波包时频空间中将某个分量的时频曲线与其他分量的时频曲线分开,然后将该分量分离出来,再循环进行广义解调直到所有分量都分离出来。采用多次广义解调方法对仿真信号进行了分析,并与其他的方法进行比较,结果表明了该方法的有效性。     

Bessel function of the first kind with complex argument

A new method of computing integral order Bessel functions of the first kind J_n(z) when either the absolute value of the real part or the imaginary part of the argument z = x + iy is small, is described. This method is based on computing the Bessel functions from asymptotic expressions when x∼ 0 (or y ∼ 0). These expansions are derived from the integral definition of Bessel functions. This method is necessary because some existing algorithms and methods fail to give correct results for small x small y. In addition, our overall method of computing Bessel functions of any order and argument is discussed and the logarithmic derivative is used in computing these functions. The starting point of the backward recurrence relations needed to evaluate the Bessel function and their logarithmic derivatives are investigated in order to obtain accurate numerical results. Our numerical method, together with established techniques of computing the Bessel functions, is easy to implement, efficient, and produces reliable results for all z.     

Canonical Decomposition of Steerable Functions

This paper describes a general mathematical formulation for the problem of constructing steerable functions. The formulation is based on Lie group theory and is thus applicable to transformations which are Lie groups, such as, rotation, translation, scaling, and affine transformation. For one-parameter and Abelian multi-parameter Lie transformation groups, a canonical decomposition of all possible steerable functions, derived using the Jordan decomposition of matrices, is developed. It is shown that any steerable function under Lie transformation groups can be described using this decomposition. Finally, a catalog of steerable functions for several common multi-parameter image transformation groups is also provided.     

Convex Hodge Decomposition and Regularization of Image Flows

The total variation (TV) measure is a key concept in the field of variational image analysis. In this paper, we focus on vector-valued data and derive from the Hodge decomposition of image flows a definition of TV regularization for vector-valued data that extends the standard componentwise definition in a natural way. We show that our approach leads to a convex decomposition of arbitrary vector fields, providing a richer decomposition into piecewise harmonic fields rather than piecewise constant ones, and motion texture. Furthermore, our regularizer provides a measure for motion boundaries of piecewise harmonic image flows in the same way, as the TV measure does for contours of scalar-valued piecewise constant images.

Information Preserving Time Decompositions of Time Stamped Documents*

Extraction of sequences of events from news and other documents based on the publication times of these documents has been shown to be extremely effective in tracking past events. This paper addresses the issue of constructing an optimal information preserving decomposition of the time period associated with a given document set, i.e., a decomposition with the smallest number of subintervals, subject to no loss of information. We introduce the notion of the compressed interval decomposition, where each subinterval consists of consecutive time points having identical information content. We define optimality, and show that any optimal information preserving decomposition of the time period is a refinement of the compressed interval decomposition. We define several special classes of measure functions (functions that measure the prevalence of keywords in the document set and assign them numeric values), based on their effect on the information computed as document sets are combined. We give algorithms, appropriate for different classes of measure functions, for computing an optimal information preserving decomposition of a given document set. We studied the effectiveness of these algorithms by computing several compressed interval and information preserving decompositions for a subset of the Reuters–21578 document set. The experiments support the obvious conclusion that the temporal information gleaned from a document set is strongly dependent on the measure function used and on other user-defined parameters.

基于参数分解的正则四边形插值细分曲面的快速求值

提出了两种正则四边形网格插值细分曲面的求值算法.算法基于参数m-进制分解和构造矩阵序列,通过参数分解数列对应的矩阵乘积得到基函数值,得到初始网格上对应控制点的权值,从而实现插值细分曲面求值.算法1 基于2D 细分掩模,算法2 基于张量积.数值实验表明,算法高效且低存储.     

An extension theorem for polynomials on triangles

We present an extension theorem for polynomial functions that proves a quasi-optimal bound for a lifting from L ² on edges onto a fractional-order Sobolev space on triangles. The extension is such that it can be further extended continuously by zero within the trace space of H ¹. Such an extension result is critical for the analysis of non-overlapping domain decomposition techniques applied to the p-and hp-versions of the finite and boundary element methods for elliptic problems of second order in three dimensions. Supported by the FONDAP Programme in Applied Mathematics, Chile.     

Query Decomposition for a Distributed Object-Oriented Mediator System

The mediator-wrapper approach to integrate data from heterogeneous data sources has usually been centralized in the sense that a single mediator system is placed between a number of data sources and applications. As the number of data sources increases, the centralized mediator architecture becomes an administrative and performance bottleneck. This paper presents a query decomposition algorithm for a distributed mediation architecture where the communication among the mediators is on a higher level than the communication between a mediator and a data source. Some of the salient features of the proposed approach are: (i) exploring query execution schedules that contain data flow to the sources, necessary when integrating object-oriented sources that provide services (programs) and not only data; (ii) handling of functions with multiple implementations at more than one mediator or source; (iii) multi-phase query decomposition using a combination of heuristics and cost-based strategies; (iv) query plan tree rebalancing by distributed query recompilation.     

The binomial decomposition of OWA functions,the 2‐additive and 3‐additive cases in n dimensions

In the context of the binomial decomposition of ordered weighted averaging (OWA) functions, we investigate the constraints associated with the 2‐additive and 3‐additive cases in n dimensions. The 2‐additive case depends on one coefficient whose feasible region does not depend on the dimension n. On the other hand, the feasible region of the 3‐additive case depends on two coefficients and is explicitly dependent on the dimension n. This feasible region is a convex polygon with n vertices and n edges, which is strictly expanding in the dimension n. The orness of the OWA functions within the feasible region is linear in the two coefficients, and the vertices associated with maximum and minimum orness are identified. Finally, we discuss the 3‐additive binomial decomposition in the asymptotic infinite dimensional limit.     

Image Decomposition into a Bounded Variation Component and an Oscillating Component

We construct an algorithm to split an image into a sum u + v of a bounded variation component and a component containing the textures and the noise. This decomposition is inspired from a recent work of Y. Meyer. We find this decomposition by minimizing a convex functional which depends on the two variables u and v, alternately in each variable. Each minimization is based on a projection algorithm to minimize the total variation. We carry out the mathematical study of our method. We present some numerical results. In particular, we show how the u component can be used in nontextured SAR image restoration.Jean-François Aujol graduated from 1 Ecole Normale Supérieure de Cachan in 2001. He was a PHD student in Mathematics at the University of Nice-Sophia-Antipolis (France). He was a member of the J.A. Dieudonné Laboratory at Nice, and also a member of the Ariana research group (CNRS/INRIA/UNSA) at Sophia-Antipolis (France). His research interests are calculus of variations, nonlinear partial differential equations, numerical analysis and mathematical image processing (and in particular classification, texture, decomposition model, restoration). He is Assistant Researcher at UCLA (Math Department).Gilles Aubert received the These dEtat es-sciences Mathematiques from the University of Paris 6, France, in 1986. He is currently professor of mathematics at the University of Nice-Sophia Antipolis and member of the J.A. Dieudonne Laboratory at Nice, France. His research interests are calculus of variations, nonlinear partial differential equations and numerical analysis; fields of applications including image processing and, in particular, restoration, segmentation, optical flow and reconstruction in medical imaging.Laure Blanc-Féraud received the Ph.D. degree in image restoration in 1989 and the Habilitation á Diriger des Recherches on inverse problems in image processing in 2000, from the University of Nice-Sophia Antipolis, France. She is currently director of research at CNRS in Sophia Antipolis. Her research interests are inverse problems in image processing by deterministic approach using calculus of variation and PDEs. She is also interested in stochastic models for parameter estimation and their relationship with the deterministic approach. She is currently working in the Ariana research group (I3S/INRIA) which is focussed on Earth observation.Antonin Chambolle studied mathematics and physics at the Ecole normale Supérieure in Paris and received the Ph.D. degree in applied mathematics from the Université de Paris-Dauphine in 1993. Since then he has been a CNRS researcher at the CEREMADE, Université de Paris-Dauphine, and, for a short period, a researcher at the SISSA, Trieste, Italy. His research interest include calculus of variations, with applications to shape optimization, mechanics and image processing.