An anisotropic interaction model for simulating fingerprints

Düring, Bertram; Gottschlich, Carsten; Huckemann, Stephan; Kreußer, Lisa Maria; Schönlieb, Carola-Bibiane

doi:10.1007/s00285-019-01338-3

Cited by 15 publications

(42 citation statements)

References 64 publications

(183 reference statements)

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“…This summary is shown in Table 1. [17,36] we consider the unit square with periodic boundary conditions as the domain for our numerical simulations if not stated otherwise. The particle system (1.3) is solved by either the simple explicit Euler scheme or higher order methods such as the Runge-Kutta-Dormand-Prince method, all resulting in very similar simulation results.…”

Section: Exponential Force Coefficientsmentioning

confidence: 99%

“…Note that the time step has to be adjusted depending on the value of the cutoff radius R c . For efficient numerical simulation we consider cell lists as outlined in [36]. for different values of a s , b s is shown in Figure 4.…”

Section: Exponential Force Coefficientsmentioning

confidence: 99%

“…The particle model in its general form (1.3) has been studied in [17,36]. Here, the positions of each of the N particles at time t are denoted by x j = x j (t) ∈ T 2 , j = 1, .…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Stability Analysis of Line Patterns of an Anisotropic Interaction Model

Carrillo¹,

Düring²,

Kreußer

et al. 2019

SIAM J. Appl. Dyn. Syst.

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Motivated by the formation of fingerprint patterns we consider a class of interacting particle models with anisotropic, repulsive-attractive interaction forces whose orientations depend on an underlying tensor field. This class of models can be regarded as a generalization of a gradient flow of a nonlocal interaction potential which has a local repulsion and a long-range attraction structure. In addition, the underlying tensor field introduces an anisotropy leading to complex patterns which do not occur in isotropic models. Central to this pattern formation are straight line patterns. We show that there exists a preferred direction of straight lines, i.e. straight vertical lines can be stable for sufficiently many particles, while other rotations of the straight lines are unstable steady states, both for a sufficiently large number of particles and in the continuum limit. For straight vertical lines we consider specific force coefficients for the stability analysis of steady states, show that stability can be achieved for exponentially decaying force coefficients for a sufficiently large number of particles and relate these results to the Kücken-Champod model for simulating fingerprint patterns. The mathematical analysis of the steady states is completed with numerical results.AMSC: 35B36, 35Q92, 70F10, 70F45, 82C22

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Section: Exponential Force Coefficientsmentioning

confidence: 99%

Section: Exponential Force Coefficientsmentioning

confidence: 99%

See 1 more Smart Citation

Stability Analysis of Line Patterns of an Anisotropic Interaction Model

Carrillo¹,

Düring²,

Kreußer

et al. 2019

SIAM J. Appl. Dyn. Syst.

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“…This stability result is important for understanding the robustness of patterns in applications such as, for instance, for fingerprint simulations. Based on the theoretical results in [4,6] we simulate fingerprints with variable ridge distances as steady states to the particle model (1) in [7]. However, as the number of particles N tends to infinity, the particle simulations of (1) become very inefficient.…”

Section: Resultsmentioning

confidence: 99%

Analysis of Anisotropic Interaction Equations for Fingerprint Simulations

Kreußer

2018

Proc Appl Math and Mech

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Motivated by the simulation of fingerprints we consider a class of interaction models with anisotropic, repulsive-attractive interaction forces whose orientations depend on an underlying tensor field. This class of models can be regarded as a generalization of a gradient flow of a nonlocal interaction potential which has a local repulsion and a long-range attraction structure. These anisotropic forces in our class of models cannot be derived from a potential and the underlying tensor field introduces an anisotropy leading to complex patterns which do not occur in isotropic models. This anisotropy is characterized by one parameter in the model. We study the resulting stationary solutions both analytically and numerically by considering the particle model and its continuum counterpart.

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“…Moreover, it is straight forward to include interaction with obstacles, which can be represented at fixed positions with artificial velocities. The approach is inspired by [11,16,30], where the formation of fingerprint patterns is studied with the help of an anisotropic force field. Further studies may consider the anisotropy parameter to be density or obstacle dependent.…”

mentioning

confidence: 99%

An anisotropic interaction model with collision avoidance

Totzeck¹

2020

Kinetic &Amp; Related Models

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In this article an anisotropic interaction model avoiding collisions is proposed. Starting point is a general isotropic interacting particle system, as used for swarming or follower-leader dynamics. An anisotropy is induced by rotation of the force vector resulting from the interaction of two agents. In this way the anisotropy is leading to a smooth evasion behaviour. In fact, the proposed model generalizes the standard models, and compensates their drawback of not being able to avoid collisions. Moreover, the model allows for formal passage to the limit 'number of particles to infinity', leading to a mesoscopic description in the mean-field sense. Possible applications are autonomous traffic, swarming or pedestrian motion. Here, we focus on the latter, as the model is validated numerically using two scenarios in pedestrian dynamics. The first one investigates the pattern formation in a channel, where two groups of pedestrians are walking in opposite directions. The second experiment considers a crossing with one group walking from left to right and the other one from bottom to top. The well-known pattern of lanes in the channel and travelling waves at the crossing can be reproduced with the help of this anisotropic model at both, the microscopic and the mesoscopic level. In addition, the 'right-before-left' and 'left-before-right' rule appear intrinsically for different anisotropy parameters.

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An anisotropic interaction model for simulating fingerprints

Cited by 15 publications

References 64 publications

Stability Analysis of Line Patterns of an Anisotropic Interaction Model

Stability Analysis of Line Patterns of an Anisotropic Interaction Model

Analysis of Anisotropic Interaction Equations for Fingerprint Simulations

An anisotropic interaction model with collision avoidance

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