Insertion and Expansion Operations for n-Dimensional Generalized Maps
Hierarchical representations, such as irregular pyramids, are the bases of several applications in the field of discrete imagery. So, n-dimensional ”bottom-up” irregular pyramids can be defined as stacks of successively reduced n-dimensional generalized m
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SIC-XLIM, Universit´e de Poitiers, UMR CNRS 6172, 86962 Futuroscope Chasseneuil Cedex, France [email protected], [email protected] 2 LaBRI, Universit´e Bordeaux 1, UMR CNRS 5800, 33405 Talence Cedex, France [email protected] 3 Laboratoire d’Informatique Scientifique et Industrielle (LISI), ENSMA, France [email protected]
Abstract. Hierarchical representations, such as irregular pyramids, are the bases of several applications in the field of discrete imagery. So, ndimensional ”bottom-up” irregular pyramids can be defined as stacks of successively reduced n-dimensional generalized maps (n-G-maps) [11], each n-G-map being defined from the previous level by using removal and contraction operations defined in [8]. Our goal is to build a theoretical framework for defining and handling n-dimensional ”top-down” irregular pyramids. To do so, we propose in this paper to study the definition of both insertion and expansion operations that allow to conceive these kinds of pyramids.
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Introduction
Hierarchical representations form the bases of several applications in the field of discrete imagery. Our goal is the study of basic problems related to the definition of hierarchical structures. To achieve this goal, removal and contraction operations have been defined in [8]. In this paper, we define two others basic operations: insertion and expansion, which allow to define ”top-down” pyramids. Many works deal with regular or irregular image pyramids for multi-level analysis and treatments. The first ones are [7,12,16,17]. In irregular pyramids, each level represents a partition of the pixel set into cells, i.e. connected subsets of pixels. There are two ways to build an irregular pyramid: ”botom-up” and ”topdown”1 . In the first case, the number of cells increases between two contiguous levels of a pyramid, while in the second case this number of cells decreases. To manipulate these models, it is neccessary to handle a (topological) representation and some basic operations, for instance dual graphs [13] and removal and contraction operations [8]. 1
Partially supported by the ANR program ANR-06-MDCA-015/VORTISS. In the following sections, we use the terms ”top-down” and ”bottom-up” pyramids to refer to the pyramids built by using ”top-down” and ”bottom-up” approaches, respectively.
D. Coeurjolly et al. (Eds.): DGCI 2008, LNCS 4992, pp. 141–152, 2008. c Springer-Verlag Berlin Heidelberg 2008
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M. Baba-ali et al. removal
dual
inverse insertion
contraction inverse
dual
expansion
Fig. 1. Links between the basic operations which allow to handle irregular pyramids
Grasset-Simon and al [11] build a theoretical framework for defining and handling n-dimensional ”bottom-up” irregular pyramids. To do so, they use the removal and contraction operations, defined in [8], in order to get consistent definitions of data structures for any dimension. Our goal is the same as [11]: build a theoretical framework, but for defining and handling n-dimensional ”top-down” irregular pyramids. Therefore, we study the definition of
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