A bisector Line Field Approach to Interpolation of Orientation Fields
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A bisector Line Field Approach to Interpolation of Orientation Fields Nicolas Boizot1
· Ludovic Sacchelli2
Received: 29 July 2019 / Accepted: 31 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We propose an approach to the problem of global reconstruction of an orientation field. The method is based on a geometric model called bisector line fields, which maps a pair of vector fields to an orientation field, effectively generalizing the notion of doubling phase vector fields. Endowed with a well-chosen energy minimization problem, we provide a polynomial interpolation of a target orientation field while bypassing the doubling phase step. The procedure is then illustrated with examples from fingerprint analysis. Keywords Orientation fields · bisector line fields · polynomial interpolation · fingerprint analysis · singularities
1 Introduction The present article deals with the question of global reconstruction of orientation fields on the basis of a discrete dataset. The aim is to present an alternative way of modeling orientation fields that allows to use a natural energy. As continuous mathematical objects, orientation fields adequately model texture patterns predominantly displaying orientation information. They provide a unifying framework for various patterns observed in nature such as fingerprints [22,28,33,49], liquid crystals arrangements in their nematic phase [10,13,17,30] or the pinwheel structure of the visual cortex V1 of mammals [6,7,11,25,34,35]. The model we want to discuss ties in with classical techniques used in the field of fingerprint reconstruction and authentication; therefore, the problem of the estimation of fingerprint ridge topologies is used to illustrate this approach. Indeed, as it is emphasized in [22,27,29,33,48,49], the estiThis research has been partially supported by the ANR SRGI (reference ANR-15-CE40-0018).
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Nicolas Boizot [email protected] Ludovic Sacchelli [email protected]
1
Laboratoire d’Informatique et des Systèmes, CNRS UMR 7020, LIS, Université de Toulon, Aix Marseille Univ, Marseille, France
2
Department of Mathematics, Lehigh University, Bethlehem, PA, USA
mation of fingerprint ridge topologies can be a necessary step before the use of high-level classification algorithms. In the review article [5], the authors proposed a classification of estimation methods into three broad categories: gradient-based methods, mathematical model-based methods and learningbased methods. The matter discussed in the present paper falls into the second one, in particular, in the subclass of methods that do not require prior heuristic knowledge. The classical procedure consists in first obtaining a coarse estimation of the orientation field in the form of a discrete dataset. This step can be achieved with gradient-based methods. Then, modeling choices are made and an optimization algorithm is applied to fit a model to the dataset. Finally, the orientation field is reconstructed with the help of the fully identified mathematical model. On th
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