A gravitational approach to edge detection based on triangular norms
Introduction
In the early years of digital image processing, many methods were introduced for edge detection on images. Even if speaking of edge detection as a single operation, it is usually considered a multi-stage process. Bezdek et al. [1] divide the edge extraction process in four different phases: conditioning, feature extracting, blending and scaling. This work focuses on the feature extraction process, that is, how to associate to every position in the image information about the intensity change. In this work, this information comes in the shape of a vector.
As to the feature extraction phase, most of the pioneering methods for edge detection are based on a convolution of the image with a given operator, typically implemented as a mask. The most famous ones are due to Sobel [2] and Prewitt [3]. Then, both Marr and Hildreth [4] and Canny [5], [6] introduced more complex methods, based on analytical considerations. Those works were followed by many other researchers, who developed methods based on the assumption that the image could be treated as a continuous function [7], [8].
In recent years, completely new approaches have been explored. Feature extraction has been proposed based on wavelet-like filtering [9], [10], [11], neural networks [12], [13], statistics [14], [15], rule bases [1], [16], fuzzy morphological concepts [17], [18], interval-valued fuzzy sets [19], [20] or fuzzy pattern matching [21], [22]. These recent methods are usually claimed to perform better than the old ones, but are not frequently applied. Most common implementations still concern the early, simple methods except for some very specific situations. Consequently, simplicity has been one of the aims when developing this method.
The method we introduce is based on the original use of gravitational forces by Sun et al. [23]. We will refer to this method as the gravitational method. We have developed a theoretical background in order to allow the product, used to combine the masses, to be substituted by other triangular norms (t-norms) in the calculation of the gravitational forces. This method, called gravitational edge detector based on a t-norm T (GED-T), is as simple as a mask-based algorithm, but can offer great flexibility and competitive results.
The method has been tested on the Berkeley Segmentation Dataset (BSDS), using its test images. This test set consists of 100 natural images, along with the human-labeled ground truth edges. GED-T shows to be competitive, obtaining better results than different Canny method parameterizations on a significant portion of the test images.
We have structured the remainder of this paper as follows. Section 2 describes the basic gravitational method as introduced by Sun et al. [23] with some considerations about its performance. The new proposal is presented in Section 3. Section 4 is devoted to the analysis of the behavior of different t-norms when used in the method. We present the experimental results of the application of the method on the BSDS in Section 5. To finish, conclusions and future lines of research are presented in Section 6.
Section snippets
Some considerations on the gravitational approach
As stated by Newton in the law of universal gravity (LUG) [24], any body attracts every other body by a force proportional to the product of their masses. More concretely, having a situation as in Fig. 1, the force is given bywhere m1 and m2 are the masses of the bodies, is the vector connecting the positions of the masses and G the gravitational constant.
Forces are represented in this paper as vectors. Besides, the gravitational forces produced by two bodies are
Construction of a fuzzy set from gravitational forces
The purpose of our method is to create a fuzzy set representing the edges, so that every position in the image is assigned a membership degree proportional to the estimated resulting force acting on it. First, we present a generalization of the LUG (Section 3.1). However, we have to be sure that the resulting forces are going to have magnitudes in the range [0,1], so that they can be used as valid membership degrees. This problem is tackled in Section 3.2. Section 3.3 presents a method for
On the behavior of different t-norms
The GED-T allows the use of any t-norm T, giving rise to an infinite number of possible implementations. We should therefore understand the behavior of each implementation, i.e. the impact of the t-norm on the edge detection and the practical influence of its features on the resulting edges.
The study of this influence relates to the study of the function shape. We have chosen to study 4 prototypical t-norms: , , and . The first analysis is based on the response to the parameterized
Experimental validation
In this section we compare the results of the proposed edge detectors to those produced by different versions of the Canny method. In Section 5.1 we cover the comparison of the results, using Baddeley's error metric(BEM) [39]. In Section 5.2 we introduce the dataset used in the experiments. Section 5.3 introduces the methodology of the experiments. The actual results are comprised in the Section 5.4.
Conclusions and future research
In this paper we have developed the theoretical grounds for a simple, customizable edge detector based on gravitational forces. Besides, we have pointed out its problems, along with some possible solutions. We have carried out a study relating the properties of the different t-norms used in the detection to the way they detect edges. Finally, we have applied the technique on a well-known image database, with results similar, and sometimes better, than those of the Canny method. Still, the
Acknowledgments
This work has been partially founded by the Spanish Ministry of Science, Project TIN2007-65981 and by the Research Services of the Public University of Navarre.
Carlos Lopez-Molina received the M.S. degree in Computer Sciences from the Public University of Navarre, Spain, in 2008. He currently holds a research position at the Department of Automatics and Computation. His research interest are edge detection, image processing, aggregation operators and fuzzy logic applications.
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Carlos Lopez-Molina received the M.S. degree in Computer Sciences from the Public University of Navarre, Spain, in 2008. He currently holds a research position at the Department of Automatics and Computation. His research interest are edge detection, image processing, aggregation operators and fuzzy logic applications.
Humberto Bustince is Full Professor at the Department of Automatics and Computation, Public University of Navarre, Spain. He holds a Ph.D. degree in Mathematics from Public University of Navarre from 1994. His research interests are fuzzy logic theory, extensions of fuzzy sets (type-2 fuzzy sets, interval-valued fuzzy sets, Atanassov's intuitionistic fuzzy sets), fuzzy measures, aggregation functions, and fuzzy techniques for image processing. He is author of over 50 published original articles and involved in teaching Artificial Intelligence for students of Computer Sciences.
Javier Fernandez is an Associate Lecturer at the Department of Automatics and Computation, Public University of Navarre, Spain. M.Sc in Mathematics at the University of Zaragoza in 1999, he got his Ph.D. in Mathematics at the University of the Basque Country in 2003. His research interest are fuzzy techniques for image processing, fuzzy sets theory, interval-valued fuzzy sets theory, aggregation functions, fuzzy measures, and stability. He is author of around 10 published original articles and involved in teaching Artificial Intelligence and Computational Mathematics for students of Computer Sciences.
Pedro Couto received is B.S. in Electrical Engineering from the University of Trás-os-Montes and Alto Douro (UTAD), Portugal in 1999. He obtained the M.Sc. degree in Engineering Technologies in 2003 and the Ph.D. degree in Electrical Engineering (Computer Vision) in 2007 from UTAD, Portugal. Currently, he is an Assistant Professor in the Engineering Department at UTAD and a Researcher in the Biosystems Engineering Group at the Centre for the Research and Technology of Agro-Environment and Biological Sciences (CITAB). His current research activities are focused on computer vision, pattern recognition and movement analysis using non-conventional techniques, namely fuzzy sets and their extensions.
Bernard de Baets holds an M.Sc. in Maths (1988), a Postgraduate degree in Knowledge Technology (1991) and a Ph.D. in Maths (1995), all summa cum laude from Ghent University (Belgium), and is a Government of Canada Award holder (1988). He is a Full Professor in Applied Maths (1999) at Ghent University, where he is leading KERMIT, the research unit Knowledge-Based Systems. He is an Honorary Professor of Budapest Tech (2006). His publications comprise more than 170 papers in international journals and about 45 book chapters. He serves on the Editorial Boards of various international journals, in particular as co-editor-in-chief of Fuzzy Sets and Systems. B. De Baets coordinates EUROFUSE, the EURO Working Group on Fuzzy Sets, and is member of the Board of Directors of EUSFLAT, the Technical Committee on Artificial Intelligence and Expert Systems of IASTED, and of the Administrative Board of the Belgian OR Society.