Multicolor well-composed pictures

doi:10.1016/0167-8655(94)00104-B

Pattern Recognition Letters

Volume 16, Issue 4, April 1995, Pages 425-431

https://doi.org/10.1016/0167-8655(94)00104-B Get rights and content

Abstract

As was noted early in the history of computer vision, using the same adjacency relation for the entire digital picture leads to so-called “paradoxes” related to the Jordan Curve Theorem. The most popular idea to avoid these paradoxes in binary images was using different adjacency relations for the foreground and the background: 8-adjacency for black points and 4-adjacency for white points, or vice versa. This idea cannot be extended in a straightforward way to multicolor pictures.

In this paper a solution is presented which guarantees us to avoid the connectivity paradoxes related to the Jordan Curve Theorem for all multicolor pictures. Only one connectedness relation will be used for the entire digital picture, i.e. for every component of every color. The idea is not to allow a certain “critical configuration” which can be detected locally to occur in digital pictures; such pictures will be called “well-composed”. Well-composed pictures have very nice topological properties. For example the Jordan Curve Theorem holds and the Euler characteristic is locally computable. This implies that properties of algorithms used in computer vision can be stated and proved in a clear way, and that the algorithms themselves become simpler and faster. Moreover, if a digitization process is guaranteed to preserve topology, then the obtained digital pictures must be well-composed.

References (11)

T.Y. Kong et al.
If we use 4- or 8-connectedness for both the objects and the background, the Euler characteristic is not locally computable
Pattern Recognition Lett.
(1990)
T.Y. Kong et al.
Digital topology: Introduction and survey
Computer Vision, Graphics, and Image Processing
(1989)
A. Nakamura et al.
Some results concerning connected fuzzy digital pictures
Pattern Recognition Lett.
(1991)
A. Rosenfeld
Fuzzy digital topology
Inform. and Control
(1979)
T. Boult et al.
G-neighbors

There are more references available in the full text version of this article.

Cited by (16)

Thinning grayscale well-composed images
2004, Pattern Recognition Letters
Citation Excerpt :
It has led to the development of an efficient watershed algorithm (Couprie and Bertrand, 1997). A definition of well-composed multicolor images has also been proposed (Latecki, 1995). In this section, we introduce well-composed grayscale images (Latecki, 2000), which is compatible with the cross section topology, and we extend the notion of simple points in such a framework.
Usual approaches for constructing topological maps on discrete structures are based on cellular complexes topology. This paper aims to construct a coherent topological map defined on a square grid from a watershed transformation. The main idea behind the proposed approach is to impose some constraints on the original image in order to obtain good properties of the resulting watershed. We propose a definition of well-composed grayscale images based on the well-composed set theory and the cross-section topology. Properties of two different thinning algorithms are then studied and we show how to obtain a thin crest network. We derive an efficient algorithm that permits the construction of a meaningful topological map, resulting in a topological segmentation, i.e. a segmentation that describes in a coherent framework faces and contours. Finally, we demonstrate the usefulness of this algorithm for multilevel image segmentation.
A cellular model for multi-objects multi-dimensional homotopic deformations
2001, Pattern Recognition
We deal in this paper with complex three-dimensional scenes, partitioned into several objects of any shape, and with varying local dimensions. We propose a consistent topological modeling of such scenes, based on a cellular model. We introduce a definition of simple elements for any adjacency graph-based geometrical structure. Then we prove a local characterization of simple cells of a cellular model, in the case of two objects, and of any number of objects. This characterization allows to define homotopic deformations of a complex scene.
Well-composed sets
2000, Advances in Imaging and Electron Physics
This chapter discusses the concept of well-composed sets, which was first introduced in Latecki et al. with the goal of distinguishing a special class of subsets of two-dimensional (2D) binary digital pictures called “well-composed pictures.” The 2D well-composed pictures have the following topological properties: (1) the Jordan curve theorem holds for them, (2) their Euler characteristic is locally computable, and (3) they have only one connectedness relation because 4- and 8-connectedness are equivalent. The class of well-composed sets seems to be general enough; it is possible to determine the digital sets in digital images obtained in practical applications in such a way that they are well-composed sets. The chapter shows an example of application of well-composed sets to determine an optimal threshold for gray-level document images. The simpler adjacency structure of a well-composed image suggests that algorithms used in image processing should become simpler when restricted to well-composed images. The chapter discusses this conjecture on thinning algorithms. The resulting skeletons have a very simple structure if the input set is well composed. Thinning algorithms can be defined for well-composed sets that generate really thin skeletons that are one point thick.
Digital Connectivity and Extended Well-Composed Sets for Gray Images
1997, Computer Vision and Image Understanding
In this paper, we propose a new definition of digital connectivity for gray images on a 2D space for arbitrary grid systems. We extend a digital version of the Jordan curve theorem and its converse proved earlier by Rosenfeld for the rectangular grid system. We also extend in two directions the concept of well-composed sets introduced by Lateckiet al.(1995,Comput. Vision Image Understanding61,70–83). First, we extend the definition of well-composed sets from the quadratic grid system to an arbitrary grid system. Then, by using the concept of parameter-dependent connected components introduced by us in a previous work, we allow the pixels in a connected component of a well-composed set to have different gray values so that the connectivity of connected components accommodates a wider meaning.
3D Well-composed Pictures
1997, Graphical Models and Image Processing
By asegmentedimage, we mean a digital image in which each point is assigned a unique label that indicates the object to which it belongs. By the foreground (objects) of a segmented image, we mean the objects whose properties we want to analyze and by the background, all the other objects of a digital image. If one adjacency relation is used for the foreground of a 3D segmented image (e.g., 6-adjacency) and a different relation for the background (e.g., 26-adjacency), then interchanging the foreground and the background can change the connected components of the digital picture. Hence, the choice of foreground and background is critical for the results of the subsequent analysis (like object grouping), especially in cases where it is not clear at the beginning of the analysis what constitutes the foreground and what the background, since this choice immediately determines the connected components of the digital picture. A special class of segmented digital 3D pictures called “well-composed pictures” will be defined. Well-composed pictures have very nice topical and geometrical properties; in particular, the boundary of every connected component is a Jordan surface and there is only one type of connected component in a well-composed picture, since 6-, 14-, 18-, and 26-connected components are equal. This implies that for a well-composed digital picture, the choice of the foreground and the background is not critical for the results of the subsequent analysis. Moreover, a very natural definition of a continuous analog for well-composed digital pictures leads to regular properties of surfaces. This allows us to give a simple proof of a digital version of the 3D Jordan–Brouwer separation theorem.
Image Analysis and Computer Vision: 1995
1996, Computer Vision and Image Understanding
This paper presents a bibliography of over 1550 references related to computer vision and image analysis, arranged by subject matter. The topics covered include computational techniques; feature detection and segmentation; image and scene analysis; two-dimensional shape; pattern; color and texture; matching and stereo; three-dimensional recovery and analysis; three-dimensional shape; and motion. A few references are also given on related topics, including geometry and graphics, compression and processing, sensors and optics, visual perception, neural networks, artificial intelligence and pattern recognition, as well as on applications.

View all citing articles on Scopus

View full text

Multicolor well-composed pictures

Abstract

Pattern Recognition Lett.

Computer Vision, Graphics, and Image Processing

Pattern Recognition Lett.

Inform. and Control

G-neighbors