Theoretical foundations of spatially-variant mathematical morphology part ii: gray-level images

In this paper, we develop a spatially-variant (SV) mathematical morphology theory for gray-level signals and images in the Euclidean space. The proposed theory preserves the geometrical concept of the structuring function, which provides the foundation of classical morphology and is essential in signal and image processing applications. We define the basic SV gray-level morphological operators (i.e., SV gray-level erosion, dilation, opening, and closing) and investigate their properties. We demonstrate the ubiquity of SV gray-level morphological systems by deriving a kernel representation for a large class of systems, called V-systems, in terms of the basic SV graylevel morphological operators. A V-system is defined to be a gray-level operator, which is invariant under gray-level (vertical) translations. Particular attention is focused on the class of SV flat gray-level operators. The kernel representation for increasing V-systems is a generalization of Maragos' kernel representation for increasing and translation-invariant function-processing systems. A representation of V-systems in terms of their kernel elements is established for increasing and upper-semi-continuous V-systems. This representation unifies a large class of spatially-variant linear and non-linear systems under the same mathematical framework. Finally, simulation results show the potential power of the general theory of gray-level spatially-variant mathematical morphology in several image analysis and computer vision applications.     

Theoretical foundations of spatially-variant mathematical morphology part I: binary images

We develop a general theory of spatially-variant (SV) mathematical morphology for binary images in the Euclidean space. The basic SV morphological operators (i.e., SV erosion, SV dilation, SV opening and SV closing) are defined. We demonstrate the ubiquity of SV morphological operators by providing a SV kernel representation of increasing operators. The latter representation is a generalization of Matheron's representation theorem of increasing and translation-invariant operators. The SV kernel representation is redundant, in the sense that a smaller subset of the SV kernel is sufficient for the representation of increasing operators. We provide sufficient conditions for the existence of the minimal basis representation in terms of upper-semi-continuity in the hit-or-miss topology. The latter minimal basis representation is a generalization of Maragos' minimal basis representation for increasing and translation-invariant operators. Moreover, we investigate the upper-semi-continuity property of the basic SV morphological operators. Several examples are used to demonstrate that the theory of spatially-variant mathematical morphology provides a general framework for the unification of various morphological schemes based on spatiallyvariant geometrical structuring elements (e.g., circular, affine and motion morphology). Simulation results illustrate the theory of the proposed spatially-variant morphological framework and show its potential power in various image processing applications.     

Self-dual morphological operators and filters

The median operator is a nonlinear image transformation celebrated for its noise cleaning capacities. It treats the foreground and background of an image identically, i.e., it is self-dual. Unfortunately, the median operator has one major drawback: it is not idempotent. Even worse, subsequent iterations of a given image may lead to oscillations. This paper describes a general method for the construction of morphological operators which are self-dual. This construction is based upon the concept of a switch operator. Subsequently, the paper treats a class of operators, the so-called activity-extensive operators, which have the intriguing property that every sequence of iterates of a given image is pointwise monotone and therefore convergent. The underlying concept is that of the activity ordering. Every increasing, self-dual operator can be modified in such a way that it becomes activity-extensive. The sequence of iterates of this modification converges to a self-dual morphological filter.     

Inf-Semilattice Approach to Self-Dual Morphology

Today, the theoretical framework of mathematical morphology is phrased in terms of complete lattices and operators defined on them. The characterization of a particular class of operators, such as erosions or openings, depends almost entirely upon the choice of the underlying partial ordering. This is not so strange if one realizes that the partial ordering formalizes the notions of foreground and background of an image. The duality principle for partially ordered sets, which says that the opposite of a partial ordering is also a partial ordering, gives rise to the fact that all morphological operators occur in pairs, e.g., dilation and erosion, opening and closing, etc. This phenomenon often prohibits the construction of tools that treat foreground and background of signals in exactly the same way. In this paper we discuss an alternative framework for morphological image processing that gives rise to image operators which are intrinsically self-dual. As one might expect, this alternative framework is entirely based upon the definition of a new self-dual partial ordering.     

卷积形态滤波与图像去噪

本文提出的卷积形态变换是一种新的形态变换形式,具有线性卷积的结构和形态变换的性质。这种新的形态变换以乘性结构元素为特征,它不同于具有加性结构元素的普通形态变换,对于它们的性质和结构的研究也是本文的主要工作之一。另一方面的工作是针对卷积形态核提出了一种结构化的自动生成算法。研究表明,卷积形态滤波与卷积积分变换一样,对于图像具有去噪和平滑作用,且在实验效果上具有较通常的形态滤波和线性卷积变换更优的去噪和平滑功能。     

A general framework for tree-based morphology and its applications to self-dual filtering

This paper presents a tree-based framework for producing self-dual morphological operators, based on a tree-representation complete inf-semilattice (CISL). The idea is to use a self-dual tree transform to map a given image into the above CISL, perform one or more morphological operations there, and map the result back to the image domain using the inverse tree transform. We also present a particular case of this general framework, involving a new tree transform, the Extrema-Watershed Tree (EWT). The operators obtained by using the EWT in the above framework behave like classical morphological operators, but in addition are self-dual. Some application examples are provided: pre-processing for OCR and dust and scratch removal algorithms, and image denoising. We also explore first steps towards obtaining tree transforms that induce a CISL on the image domain as well.     

矢量自对偶形态学滤波算子

自对偶形态学算子不依赖形态学腐蚀、膨胀算子的先后次序, 是一种等同处理图像背景和前景的形态学算子. 而将自对偶形态学算子拓展到多通道图像处理是一个难题. 为了解决该问题, 提出了基于极值约束的矢量自对偶形态学滤波算子(EC-VSDMF). 首先根据对称矢量排序算法构建满足对偶性的矢量形态学算子, 然后依据形态学算子中的极值原理优化矢量集合, 从而有效抑制矢量集合中包含单通道极值的矢量作为输出结果, 最终实现了具有约束功能的矢量自对偶形态学滤波算子(VSDMF). 实验结果表明, EC-VSDMF继承了传统自对偶形态学滤波算子的性质, 将其应用于彩色图像滤波可以改善现有矢量形态学滤波算子导致滤波后图像亮度和色度发生偏移的问题. 滤波后的图像在有效抑制噪声的同时较好地保留了图像细节, 滤波性能甚至超过了多种现有的矢量中值滤波算子.     

Morphological representation of nonlinear filters

Morphological operators provide very efficient algorithms for signal (image) processing. The efficiency of morphological operators has been captured by using them as approximations of nonlinear operators in numerous applications (e.g., image restoration). Our approach to the approximation of nonlinear operators is the construction of morphological bounds on them. We present a general theory on the morphological bounds on nonlinear operators, propose conditions for the existence of these bounds, and derive several fundamental morphological bounds.We also derive morphological bounds on iterations of nonlinear operators, which are superior to the original nonlinear operator in some applications. Because obtaining the results of the convergence of iterations of a nonlinear operator is often particularly desirable, we provide morphological bounds on the convergence of such iterations, and propose conditions for their convergence based on morphological properties. Finally, we propose several criteria for the morphological characterization of roots of nonlinear operators.This work was supported by U.S. Office of Naval Research award N00014-91-J-1725.     

On the Decomposition of Interval-Valued Fuzzy Morphological Operators

Interval-valued fuzzy mathematical morphology is an extension of classical fuzzy mathematical morphology, which is in turn one of the extensions of binary morphology to greyscale morphology. The uncertainty that may exist concerning the grey value of a pixel due to technical limitations or bad recording circumstances, is taken into account by mapping the pixels in the image domain onto an interval to which the pixel’s grey value is expected to belong instead of one specific value. Such image representation corresponds to the representation of an interval-valued fuzzy set and thus techniques from interval-valued fuzzy set theory can be applied to extend greyscale mathematical morphology. In this paper, we study the decomposition of the interval-valued fuzzy morphological operators. We investigate in which cases the [α ₁,α ₂]-cuts of these operators can be written or approximated in terms of the corresponding binary operators. Such conversion into binary operators results in a reduction of the computation time and is further also theoretically interesting since it provides us a link between interval-valued fuzzy and binary morphology.     

Mathematical morphology on hypergraphs,application to similarity and positive kernel

The focus of this article is to develop mathematical morphology on hypergraphs. To this aim, we define lattice structures on hypergraphs on which we build mathematical morphology operators. We show some relations between these operators and the hypergraph structure, considering in particular transversals and duality notions. Then, as another contribution, we show how mathematical morphology can be used for classification or matching problems on data represented by hypergraphs: thanks to dilation operators, we define a similarity measure between hypergraphs, and we show that it is a kernel. A distance is finally introduced using this similarity notion.     

Computational Gray-scale Mathematical Morphology on Lattices (A Comparator-based Image Algebra) Part 1: Architecture

Computational mathematical-morphology has been developed to provide a directly computable alternative to classical gray-scale morphology that is range preserving and compatible with the design of statistically optimal filters based on morphological representation. It serves as an image algebra because of the expressive capability of its image-operator representations. Because representations are based on binary comparators used in conjunction with AND and OR operations, it provides a low-level, efficient computational environment that is a direct extension of the finite Boolean operational environment. The paper focuses on development of the comparator-based representation, providing the relevant representation theory for lattice operators and lattice-vector operators. Computational lattice-operator theory represents a comparator-based alternative to the classical morphological lattice theory, one that is directly implementable in logic. For totally ordered valuation spaces, the lattice theory reduces to a simplified form appropriate to straightforward optimization. The present paper treats architectural considerations and a second part considers the implications for latticevalued image operators.     

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.     

Generalized modal satisfiability

It is well known that modal satisfiability is PSPACE-complete (Ladner (1977) [21]). However, the complexity may decrease if we restrict the set of propositional operators used. Note that there exist an infinite number of propositional operators, since a propositional operator is simply a Boolean function. We completely classify the complexity of modal satisfiability for every finite set of propositional operators, i.e., in contrast to previous work, we classify an infinite number of problems. We show that, depending on the set of propositional operators, modal satisfiability is PSPACE-complete, coNP-complete, or in P. We obtain this trichotomy not only for modal formulas, but also for their more succinct representation using modal circuits. We consider both the uni-modal and the multi-modal cases, and study the dual problem of validity as well.     

For real! XCS with continuous-valued inputs

Many real-world problems are not conveniently expressed using the ternary representation typically used by Learning Classifier Systems and for such problems an interval-based representation is preferable. We analyse two interval-based representations recently proposed for XCS, together with their associated operators and find evidence of considerable representational and operator bias. We propose a new interval-based representation that is more straightforward than the previous ones and analyse its bias. The representations presented and their analysis are also applicable to other Learning Classifier System architectures. We discuss limitations of the real multiplexer problem, a benchmark problem used for Learning Classifier Systems that have a continuous-valued representation, and propose a new test problem, the checkerboard problem, that matches many classes of real-world problem more closely than the real multiplexer. Representations and operators are compared using both the real multiplexer and checkerboard problems and we find that representational, operator and sampling bias all affect the performance of XCS in continuous-valued environments.     

一类拓广的形态举算子

本文初步研究了一类拓广的形态学基本算子，并对其进行了一定的分析。结合仿真实例，比较了拓广的形态学算子与传统形态学算子对噪声图像处理的性能，结果表明，这种新形态学算子比传统形态学算子在噪声抑制方面效果更好。     

General sweep mathematical morphology

General sweep mathematical morphology provides a new class of morphological operations, which allow one to select varying shapes and orientations of structuring elements during the sweeping process. Such a class holds syntactic characteristics similar to algebraic morphology as well as sweep geometric modeling. The conventional morphology is a subclass of the general sweep morphology. The sweep morphological dilation/erosion provides a natural representation of sweep motion in the manufacturing processes, and the sweep opening/closing provides variant degrees of smoothing in image filtering. The theoretical framework for representation, computation and analysis of sweep morphology is presented in this paper. Its applications to the sweeping with deformations, image enhancement, edge linking, and shortest path planning for rotating objects are also discussed.     

Realization theory in Hilbert space

A representation theorem for infinite-dimensional, linear control systems is proved in the context of strongly continuous semigroups in Hilbert spaces. The result allows for unbounded input and output operators and is used to derive necessary and sufficient conditions for the realizability in a Hilbert space of a time-invariant, causal input-output operator ℐ. The relation between input-output stability and stability of the realization is discussed. In the case of finite-dimensional input and output spaces the boundedness of the output operator is related to the existence of a convolution kernel representing the operator ℐ. This research has been supported by the Nuffields Foundation.     

Toggle and top-hat based morphological contrast operators

Top-hat based contrast operator and toggle contrast operator are two important strategies for image contrast enhancement in mathematical morphology. Top-hat based contrast operator enhances image regions and toggle contrast operator sharpens image. However, top-hat based contrast operator could not well sharpen image and toggle contrast operator could not well enhance image regions. Therefore, composing them together may produce efficient contrast operators which are both efficient for image region enhancement and image sharpening. Two ways of composing of top-hat based contrast operator and toggle contrast operator are discussed in this paper, which produce sequential contrast operators and combined contrast operators. Moreover, the multi scale extensions of these morphological contrast operators are discussed. Experiments have verified that, because the efficiencies of top-hat based contrast operator and toggle contrast operator are composed in the sequential contrast operators and combined contrast operators, the composed operators could both enhance image regions and sharpen image, which enhances the contrast of image and makes image clearer.     

Operator theoretic approach to the optimal two-disk problem

The nonstandard two-disk problem plays a fundamental role in robust feedback optimization. Here, it is shown via Banach space duality theory that its solutions satisfy an extremal identity, and may be viewed as a dual extremal kernel of a particular L/sup 1/-optimization problem. A novel operator theoretic framework to characterize explicitly its solutions is developed, in particular, the two-disk optimization is shown to be equal to the induced norm of a specific operator defined on a projective tensor product space involving a non-Hilbert version of a vector valued H/sup 2/ space. Moreover, this operator is shown to be a combination of multiplication and Toeplitz operators. Under certain conditions, existence of maximal vectors is established leading to an explicit formula for the optimal controller. An "infinite matrix" representation with respect to a canonical basis is derived, together with an algorithm to compute it. The norm of the relevant operator is approximated by special finite dimensional optimizations whose solutions lead to solving semi-definite programming problems involving the computation of a matrix projective tensor norm.