A Measure of Q -convexity for Shape Analysis
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A Measure of Q-convexity for Shape Analysis Péter Balázs1 · Sara Brunetti2 Received: 23 April 2019 / Accepted: 7 May 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this paper, we study three basic novel measures of convexity for shape analysis. The convexity considered here is the so-called Q-convexity, that is, convexity by quadrants. The measures are based on the geometrical properties of Q-convex shapes and have the following features: (1) their values range from 0 to 1; (2) their values equal 1 if and only if the binary image is Q-convex; and (3) they are invariant by translation, reflection, and rotation by 90 degrees. We design a new algorithm for the computation of the measures whose time complexity is linear in the size of the binary image representation. We investigate the properties of our measures by solving object ranking problems and give an illustrative example of how these convexity descriptors can be utilized in classification problems. Keywords Shape descriptor · Shape analysis · Convexity measure · Q-convexity · Algorithms
1 Introduction Techniques of shape analysis are widely applied in various fields of computer vision, e.g., in object classification, image segmentation, and simplification. The use of shape descriptors and the development of new measures for descriptors in shape analysis is a current topic of broad interest [15]. Recent works develop sophisticated recognition methods and powerful machine learning approaches for dealing with the challenge of classification of deformation, occlusion and view variation of the images [1,26]. Differently from these approaches, in this paper, we propose a very easy shape representation by a scalar convexity measure. The advantage of this approach is that it is computationally extremely efficient. Our method can be used for classification of suitable datasets as well as a basic step of solving more complex issues. Convexity measures and their applications are studied in several papers which can be grouped into different categories: area-based measures form one popular category [8,24,25], while boundary-based ones [27] are also frequently used.
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Sara Brunetti [email protected] Péter Balázs [email protected]
1
Department of Image Processing and Computer Graphics, University of Szeged, Árpád tér 2., Szeged 6720, Hungary
2
Dipartimento di Ingegneria dell’Informazione e Scienze Matematiche, Via Roma, 56, 53100 Siena, Italy
Other methods use simplification of the contour [19], a probabilistic approach [22,23] or fuzzy set theory and mathematical morphology [21] to measure convexity. In discrete geometry, and especially in discrete tomography, a natural notion of convexity is provided by the horizontal and vertical convexity (shortly, hv-convexity), arising inherently from the pixel-based representation (and the notion of neighborhood) of the digital image (see, e.g., [6,12]). A first attempt to define a measure of directional (horizontal or vertical) convexity was studied in detail in [2]. Independe
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