American Institute of Mathematical Sciences

Advanced
November  2007, 1(4): 673-690. doi: 10.3934/ipi.2007.1.673

On the application of projection methods for computing optical flow fields

Thomas Schuster 1, and  Joachim Weickert 2,

1. 

Fakultät für Maschinenbau, Helmut--Schmidt--Universität, Holstenhofweg 85, 22043 Hamburg, Germany
2. 

Fakultät für Mathematik und Informatik, Universität des Saarlandes, Geb. E1.1, 66041 Saarbrücken, Germany

Received  May 2007 Published  October 2007

Detecting optical flow means to find the apparent displacement field in a sequence of images. As starting point for many optical flow methods serves the so called optical flow constraint (OFC), that is the assumption that the gray value of a moving point does not change over time. Variational methods are amongst the most popular tools to compute the optical flow field. They compute the flow field as minimizer of an energy functional that consists of a data term to comply with the OFC and a smoothness term to obtain uniqueness of this underdetermined problem. In this article we replace the smoothness term by projecting the solution to a finite dimensional, affine subspace in the spatial variables which leads to a smoothing and gives a unique solution as well. We explain the mathematical details for the quadratic and nonquadratic minimization framework, and show how alternative model assumptions such as constancy of the brightness gradient can be incorporated. As basis functions we consider tensor products of B-splines. Under certain smoothness assumptions for the global minimizer in Sobolev scales, we prove optimal convergence rates in terms of the energy functional. Experiments are presented that demonstrate the feasibility of our approach.
Keywords: projection methods, variational methods, tensor product B-spline., Optical flow.
Mathematics Subject Classification: Primary: 65F22, 68T45; Secondary 90C2.
Citation: Thomas Schuster, Joachim Weickert. On the application of projection methods for computing optical flow fields. Inverse Problems and Imaging, 2007, 1 (4) : 673-690. doi: 10.3934/ipi.2007.1.673
[1]

Richard A. Norton, David I. McLaren, G. R. W. Quispel, Ari Stern, Antonella Zanna. Projection methods and discrete gradient methods for preserving first integrals of ODEs. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 2079-2098. doi: 10.3934/dcds.2015.35.2079
[2]

Dang Van Hieu. Projection methods for solving split equilibrium problems. Journal of Industrial and Management Optimization, 2020, 16 (5) : 2331-2349. doi: 10.3934/jimo.2019056
[3]

Jin Wang, Jun-E Feng, Hua-Lin Huang. Solvability of the matrix equation $ AX^{2} = B $ with semi-tensor product. Electronic Research Archive, 2021, 29 (3) : 2249-2267. doi: 10.3934/era.2020114
[4]

Jae-Hong Pyo, Jie Shen. Normal mode analysis of second-order projection methods for incompressible flows. Discrete and Continuous Dynamical Systems - B, 2005, 5 (3) : 817-840. doi: 10.3934/dcdsb.2005.5.817
[5]

Abdeslem Hafid Bentbib, Smahane El-Halouy, El Mostafa Sadek. Extended Krylov subspace methods for solving Sylvester and Stein tensor equations. Discrete and Continuous Dynamical Systems - S, 2022, 15 (1) : 41-56. doi: 10.3934/dcdss.2021026
[6]

Lori Badea. Multigrid methods for some quasi-variational inequalities. Discrete and Continuous Dynamical Systems - S, 2013, 6 (6) : 1457-1471. doi: 10.3934/dcdss.2013.6.1457
[7]

Zalman Balanov, Carlos García-Azpeitia, Wieslaw Krawcewicz. On variational and topological methods in nonlinear difference equations. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2813-2844. doi: 10.3934/cpaa.2018133
[8]

Xing-Bin Pan. Variational and operator methods for Maxwell-Stokes system. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3909-3955. doi: 10.3934/dcds.2020036
[9]

A. Pedas, G. Vainikko. Smoothing transformation and piecewise polynomial projection methods for weakly singular Fredholm integral equations. Communications on Pure and Applied Analysis, 2006, 5 (2) : 395-413. doi: 10.3934/cpaa.2006.5.395
[10]

Philippe Angot, Pierre Fabrie. Convergence results for the vector penalty-projection and two-step artificial compressibility methods. Discrete and Continuous Dynamical Systems - B, 2012, 17 (5) : 1383-1405. doi: 10.3934/dcdsb.2012.17.1383
[11]

Julian Koellermeier, Roman Pascal Schaerer, Manuel Torrilhon. A framework for hyperbolic approximation of kinetic equations using quadrature-based projection methods. Kinetic and Related Models, 2014, 7 (3) : 531-549. doi: 10.3934/krm.2014.7.531
[12]

O. Chadli, Z. Chbani, H. Riahi. Recession methods for equilibrium problems and applications to variational and hemivariational inequalities. Discrete and Continuous Dynamical Systems, 1999, 5 (1) : 185-196. doi: 10.3934/dcds.1999.5.185
[13]

Raffaella Servadei, Enrico Valdinoci. Variational methods for non-local operators of elliptic type. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 2105-2137. doi: 10.3934/dcds.2013.33.2105
[14]

Michał Jóźwikowski, Mikołaj Rotkiewicz. Bundle-theoretic methods for higher-order variational calculus. Journal of Geometric Mechanics, 2014, 6 (1) : 99-120. doi: 10.3934/jgm.2014.6.99
[15]

Dang Van Hieu, Le Dung Muu, Pham Kim Quy. New iterative regularization methods for solving split variational inclusion problems. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021185
[16]

Stefan Klus, Christof Schütte. Towards tensor-based methods for the numerical approximation of the Perron--Frobenius and Koopman operator. Journal of Computational Dynamics, 2016, 3 (2) : 139-161. doi: 10.3934/jcd.2016007
[17]

Zhangxin Chen. On the control volume finite element methods and their applications to multiphase flow. Networks and Heterogeneous Media, 2006, 1 (4) : 689-706. doi: 10.3934/nhm.2006.1.689
[18]

T. Diogo, N. B. Franco, P. Lima. High order product integration methods for a Volterra integral equation with logarithmic singular kernel. Communications on Pure and Applied Analysis, 2004, 3 (2) : 217-235. doi: 10.3934/cpaa.2004.3.217
[19]

Ömer Arslan, Selçuk Kürşat İşleyen. A model and two heuristic methods for The Multi-Product Inventory-Location-Routing Problem with heterogeneous fleet. Journal of Industrial and Management Optimization, 2022, 18 (2) : 897-932. doi: 10.3934/jimo.2021002
[20]

Lukas F. Lang, Otmar Scherzer. Optical flow on evolving sphere-like surfaces. Inverse Problems and Imaging, 2017, 11 (2) : 305-338. doi: 10.3934/ipi.2017015

2020 Impact Factor: 1.639

Tools

Metrics

  • PDF downloads (51)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy