Connectivity on Complete Lattices

Classically, connectivity is a topological notion for sets, often introduced by means of arcs. A nontopological axiomatics has been proposed by Matheron and Serra. The present paper extends it to complete sup-generated lattices. A connection turns out to be characterized by a family of openings labelled by the sup-generators, which partition each element of the lattice into maximal terms, of zero infima. When combined with partition closings, these openings generate strong sequential alternating filters. Starting from a first connection several others may be designed by acting on some dilations or symmetrical operators. When applying this theory to function lattices, one interprets the so-called connected operators in terms of actual connections, as well as the watershed mappings. But the theory encompasses the numerical functions and extends, among others, to multivariate lattices.     

一种基于数学形态学的图象多尺度分析方法的研究

提出了一种基于数学形态学的图象多尺度分析方法。利用数学形态学在描述图象形态特征方面的优势，通过改变结构基元的尺度，提取图象在不同尺度下的有用信息。该方法将基于二值图象的集合处理的数学形态学方法，推广至基于灰度形态处理的数学形态学方法，提出了广义的数学形态学形态变换算子，拓宽了数学形态学理论在图象处理领域的应用范围。本文给出了人造的实验图象和实际的景物图象的实验结果，提取出图象在不同尺度下各个层次的目标和特征，证明该方法的有效性。     

Multiscale Connected Operators

Among the major developments in Mathematical Morphology in the last two decades are the interrelated subjects of connectivity classes and connected operators. Braga-Neto and Goutsias have proposed an extension of the theory of connectivity classes to a multiscale setting, whereby one can assign connectivity to an object observed at different scales. In this paper, we study connected operators in the context of multiscale connectivity. We propose the notion of a -connected operator, that is, an operator connected at scale . We devote some attention to the study of binary -grain operators. In particular, we show that families of -grain openings and -grain closings, indexed by the connectivity scale parameter, are granulometries and anti-granulometries, respectively. We demonstrate the use of multiscale connected operators with image analysis applications. The first is the scale-space representation of grayscale images using multiscale levelings, where the role of scale is played by the connectivity scale. Then we discuss the application of multiscale connected openings in granulometric analysis, where both size and connectivity information are summarized. Finally, we describe an application of multiscale connected operators to an automatic target recognition problem.Ulisses Braga-Neto received the Baccalaureate degree in Electrical Engineering from the Universidade Federal de Pernambuco (UFPE), Brazil, in 1992, the Masters degree in Electrical Engineering from the Universidade Estadual de Campinas, Brazil, in 1994, the M.S.E. degree in Electrical and Computer Engineering and the M.S.E. degree in Mathematical Sciences, both from The Johns Hopkins University, in 1998, and the Ph.D. degree in Electrical and Computer Engineering, from The Johns Hopkins University, in 2001. He was a Post-Doctoral Fellow at the University of Texas MD Anderson Cancer Center and a Visiting Scholar at Texas A&M University, from 2002 to 2004. He is currently an Associate Researcher at the Aggeu Magalhães Research Center of the Osvaldo Cruz Foundation, Brazilian Ministry of Health. His research interests include Bioinformatics, Pattern Recognition, Image Analysis, and Mathematical Morphology.     

A Theoretical Tour of Connectivity in Image Processing and Analysis

Connectivity is a concept of great relevance to image processing and analysis. It is extensively used in image filtering and segmentation, image compression and coding, motion analysis, pattern recognition, and other applications. In this paper, we provide a theoretical tour of connectivity, with emphasis on those concepts of connectivity that are relevant to image processing and analysis. We review several notions of connectivity, which include classical topological and graph-theoretic connectivity, fuzzy connectivity, and the theories of connectivity classes and of hyperconnectivity. It becomes clear in this paper that the theories of connectivity classes and of hyperconnectivity unify all relevant notions of connectivity, and provide a solid theoretical foundation for studying classical and fuzzy approaches to connectivity, as well as for constructing new examples of connectivity useful for image processing and analysis applications.     

Geometric Invariant Shape Representations Using Morphological Multiscale Analysis

In this paper, we present a new geometric invariant shape representation using morphological multiscale analysis. The geometric invariant is based on the area and perimeter evolution of the shape under the action of a morphological multiscale analysis. First, we present some theoretical results on the perimeter and area evolution across the scales of a shape. In the case of similarity transformations, the proposed geometric invariant is based on a scale-normalized evolution of the isoperimetric ratio of the shape. In the case of general affine geometric transformations the proposed geometric invariant is based on a scale-normalized evolution of the area. We present some numerical experiments to evaluate the performance of the proposed models. We present an application of this technique to the problem of shape classification on a real shape database and we study the well-posedness of the proposed models in the framework of viscosity solution theory.     

灰度图的峰分解:算法及应用

借助于形态学和地形学的概念,灰度图可以分解为一系列的峰,并且可以用树的结构来保存分解的结果。本文称之为灰度图的峰分解,给出其数学定义,并且基于数学形态学中的watershed变换,实现了峰分解的快速算法。该分解具有很强的直观性,本文将它应用于原子力显微镜照片中的量子点的统计,发现它的应用情况很广,分割性能也非常精确,对噪声也不敏感,有很强的鲁棒性。     

核不相关鉴别分析以及它在字符识别中的应用

核不相关鉴别分析是在线性不相关鉴别分析的基础上发展起来的．然而,由于核函数的运用,计算核不相关矢量集变得更加复杂．为了解决这个问题,提出一种解决核不相关鉴别分析的有效算法．该算法巧妙地利用了矩阵的分解,然后在一个矩阵对上进行广义奇异值分解．与此同时,提出了几个相关的定理．最重要的是,提出的算法能克服核不相关鉴别分析中矩阵的奇异问题．在某种意义上,提出的算法拓宽了已有的算法,即从线性问题到非线性问题．最后,用手写数字字符识别实验来验证提出的算法是可行和有效的．     

Multi-View Scene Capture by Surfel Sampling: From Video Streams to Non-Rigid 3D Motion,Shape and Reflectance

In this paper we study the problem of recovering the 3D shape, reflectance, and non-rigid motion properties of a dynamic 3D scene. Because these properties are completely unknown and because the scene's shape and motion may be non-smooth, our approach uses multiple views to build a piecewise-continuous geometric and radiometric representation of the scene's trace in space-time. A basic primitive of this representation is the dynamic surfel, which (1) encodes the instantaneous local shape, reflectance, and motion of a small and bounded region in the scene, and (2) enables accurate prediction of the region's dynamic appearance under known illumination conditions. We show that complete surfel-based reconstructions can be created by repeatedly applying an algorithm called Surfel Sampling that combines sampling and parameter estimation to fit a single surfel to a small, bounded region of space-time. Experimental results with the Phong reflectancemodel and complex real scenes (clothing, shiny objects, skin) illustrate our method's ability to explain pixels and pixel variations in terms of their underlying causes—shape, reflectance, motion, illumination, and visibility.