Fuzzy vector structures for transient phenomenon representation

This paper deals with data structures within GIS. Continuous phenomena are usually represented by raster structures for simplicity reasons. With such structures, spatial repartitions of the data are not easily interpretable. Moreover, in an overlapping/clustering context, these structures remove the links between the data and the algorithms. We propose a vector representation of such data based on non-regular multi-ring polygons. The structure requires multi-part nested polygons and new set operations. We present the formalism based on belief theory and uncertainty reasoning. We also detail the implementation of the structures and the set operations. The structures and the set operations are illustrated in the context of forest classi ﬁ cation having diffuse transitions.


Introduction
The content of paper lies at the crossroads of knowledge representation [16], fuzzy modeling [1,6] and geographic information systems.Indeed, we propose a new way to represent uncertain knowledge within a GIS [2,7,8].
The fuzzy concept is used to represent uncertainty [3,5].A fuzzy set is based on a belief function f (see Fig. 1) [10][11][12][13], which gives for each point P of the space a confidence coefficient (C 2 0; 1 ½ ) regarding the membership of P to a given hypothesis or characteristics.
Within a GIS there are two ways to represent data [15]: a vector layer [4]o ra raster one.Without loss of generality, let us consider spatial data in the rest of the article [i.e., two dimensions Px ; y ðÞ ].Spatial data can be classified in two categories: discrete and continuous.Discrete data-such as buildings, roads, or administrative E. Grandchamp (&) LAMIA, Université des Antilles, Campus de Fouillole, 97157 Pointe-à-Pitre, Guadeloupe, France e-mail: enguerran.grandchamp@univ-antilles.fr limits-naturally find a representation with the vector formalism because they have, by definition, a well-defined and localized border.In contrast, continuous data, often resulting from a phenomenon (such as temperature, humidity, ecosystem ground occupation, forests, etc.), find a better representation in a raster.The space is then divided in regular cells using a grid having a resolution depending on the required precision, the sources, or any other criterion such as memory limits to load or store the data.
In the GIS context, fuzzy structures are represented in a raster way (Figs. 1 and 2).We propose in this paper a vector structure for fuzzy sets as well as implementation and coding inspired by the standard notation WKT as defined by the OGC.
The rest of this article is organized as follows: Sect. 2 presents the standard notation for classical structures as defined by the OGC.Section 3 gives our formalism based on previous ones.Section 4 presents the constraints on the topology when building the fuzzy polygons, the fuzzy set operation, and the fusion of two fuzzy sets.Section 5 gives the conclusion and perspectives of the study.

OGC-SFS Formalism
In this section, we give a brief and partial description of WKT encoding, which is more detailed for polygons in order to prepare fuzzy description.For a complete description of WKT notation, readers are referred to [14].

Wkt
WKT (Well-Known Text) encoding allows the description of geometries with a simple string.Described points, lines and polygons can be the following: simple or complex: A polygon is complex if it has an interior ring.single or multiple (MULTI): composed of distinct objects in two or three (Z) dimensions with or without a measure (M) for each coordinate.
Table 1 gives some encoding examples, and Fig. 3 gives some illustrations.

Fuzzy Formalism
In this paper, we only deal with simple polygons (i.e., those without interior rings).The extension to complex polygons will we presented in another article.The main particularities with such polygons is the notion of interior and exterior, which is inverted for fuzzy rings starting from the interior ring of the fuzzy polygon.

Fuzzy Polygon
A fuzzy polygon splits the space into two subspaces: The Exterior ring has a confidence value = 0.The ExteriorBorderFuzzyRing = FuzzyRing0.The rings are listed in an ascendant order of confidence value.It is possible to have several rings with the same confidence (see Fig. 4).The extended WKT notation corresponding to Fig. 4 is as follows: Fig. 5 Fuzzy polygon for confidence >0 and >0.5 4 Constraints on Topology and Fuzzy Set Operations

Constraints
The building of a fuzzy set is governed by some rules and constraints on the topology in order to guarantee the well-formed structure.
The main constraint is that there is no intersection (neither point nor lines) between the fuzzy rings.
This leads to a succession of rings (or polygons) inclusion.They represent a followed suit with the following rule:

Fuzzy Set Operation
A fuzzy set is a set of regions labeled with a confidence.The regions represent a partition of the space from the highest confidence (1) to the lowest (0).The region with confidence between C 1 and C 2 is obtained with the set difference between FP C1 and FP C2 (Fig. 6).

Fuzzy polygon fusion
When two or more fuzzy polygons split the same geographical area, they must be merged.The set operation sequence allowing merging the fuzzy polygons is as follows: The resulting fuzzy polygons have two confidence values.Depending on the nature of FP 1 and FP 2 , the meaning of the confidence of these two values can be ordered in a two-objectives way or within a single expression.

Conclusion and Perspectives
We present in this paper an overview offuzzy-vector structures within GIS based on the OGC standard WKT encoding.It also presents constraints and basic operation such as sub-region extraction and fuzzy-polygon fusion.A complete description of the encoding of complex fuzzy polygons, as well as all of the operators, will be given in a future paper.Management scripts are developed in JAVA and Python for a better integration within a GIS tool.Currently, we are working on a better representation of fuzzy polygons.Indeed, when a two-fuzzy polygon model shows two structures evolving close to each other (such as two types of forest or other ecosystems), and having a transition that spreads over a long distance, the diffuse border involves an overlap of the corresponding fuzzy polygons.We must develop a tool allowing us to automatically visualize the two fuzzy polygons at the same time.
Finally, in a classification perspective based on the fuzzy point of view, this approach raises the question of such a transition [9].Belonging to one of the original classes or to another new one.This question will be discussed in another issue and will deal with multi-label classification, emerging class, and overlap classification.

Table 1
WKT examples