Can a Fingerprint be Modelled by a Differential Equation ?

TL;DR

This paper explores the possibility of modeling fingerprint patterns using differential equations within a dynamical systems framework, aiming to establish a rigorous mathematical foundation for fingerprint analysis.

Contribution

It introduces a novel approach linking fingerprint singularities to phase portraits of differential systems and discusses the challenges in representing fingerprint flow patterns as smooth deformations.

Findings

Identifies limitations of canonical phase portraits in modeling fingerprint singularities.

Proposes a normal form approach for connecting local phase portraits.

Lays groundwork for future fingerprint matching via large deformation flows.

Abstract

Some new directions to lay a rigorous mathematical foundation for the phase-portrait-based modelling of fingerprints are discussed in the present work. Couched in the language of dynamical systems, and preparing to a preliminary modelling, a back-to-basics analogy between Poincar\'{e}'s categories of equilibria of planar differential systems and the basic fingerprint singularities according to Purkyn\v{e}-Galton's standards is first investigated. Then, the problem of the global representation of a fingerprint's flow-like pattern as a smooth deformation of the phase portrait of a differential system is addressed. Unlike visualisation in fluid dynamics, where similarity between integral curves of smooth vector fields and flow streamline patterns is eye-catching, the case of an oriented texture like a fingerprint's stream of ridges proved to be a hard problem since, on the one hand, not all fingerprint singularities and nearby orientational behaviour can be modelled by canonical phase portraits on the plane, and on the other hand, even if it were the case, this should lead to a perplexing geometrical problem of connecting local phase portraits, a question which will be formulated within Poincar\'{e}'s index theory and addressed via a normal form approach as a bivariate Hermite interpolation problem. To a certain extent, the material presented herein is self-contained and provides a baseline for future work where, starting from a normal form as a source image, a transport via large deformation flows is envisaged to match the fingerprint as a target image.