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5. Convective regularization for optical flow

  • José A. Iglesias and Clemens Kirisits
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Variational Methods
This chapter is in the book Variational Methods

Abstract

We argue that the time derivative in a fixed coordinate frame may not be the most appropriate measure of time regularity of an optical flow field. Instead, for a given velocity field v we consider the convective acceleration vt +∇vv which describes the acceleration of objects moving according to v. Consequently, we investigate the suitability of the nonconvex functional ‖vt + ∇vv‖2L 2 as a regularization term for optical flow. We demonstrate that this term acts as both a spatial and a temporal regularizer and has an intrinsic edge-preserving property. We incorporate it into a contrast invariant and time-regularized variant of the Horn-Schunck functional, prove the existence of minimizers and verify experimentally that it addresses someof the problems of basic quadratic models. For the minimization we use an iterative scheme that approximates the original nonlinear problem with a sequence of linear ones.We believe that the convective acceleration may be gainfully introduced in a variety of optical flow models.

Abstract

We argue that the time derivative in a fixed coordinate frame may not be the most appropriate measure of time regularity of an optical flow field. Instead, for a given velocity field v we consider the convective acceleration vt +∇vv which describes the acceleration of objects moving according to v. Consequently, we investigate the suitability of the nonconvex functional ‖vt + ∇vv‖2L 2 as a regularization term for optical flow. We demonstrate that this term acts as both a spatial and a temporal regularizer and has an intrinsic edge-preserving property. We incorporate it into a contrast invariant and time-regularized variant of the Horn-Schunck functional, prove the existence of minimizers and verify experimentally that it addresses someof the problems of basic quadratic models. For the minimization we use an iterative scheme that approximates the original nonlinear problem with a sequence of linear ones.We believe that the convective acceleration may be gainfully introduced in a variety of optical flow models.

Chapters in this book

  1. Frontmatter I
  2. Contents V
  3. Part I
  4. 1. Second-order decomposition model for image processing: numerical experimentation 5
  5. 2. Optimizing spatial and tonal data for PDE-based inpainting 35
  6. 3. Image registration using phase–amplitude separation 84
  7. 4. Rotation invariance in exemplar-based image inpainting 108
  8. 5. Convective regularization for optical flow 184
  9. 6. A variational method for quantitative photoacoustic tomography with piecewise constant coefficients 202
  10. 7. On optical flow models for variational motion estimation 225
  11. 8. Bilevel approaches for learning of variational imaging models 252
  12. Part II
  13. 9. Non-degenerate forms of the generalized Euler–Lagrange condition for state-constrained optimal control problems 295
  14. 10 The Purcell three-link swimmer: some geometric and numerical aspects related to periodic optimal controls 314
  15. 11. Controllability of Keplerian motion with low-thrust control systems 344
  16. 12. Higher variational equation techniques for the integrability of homogeneous potentials 365
  17. 13. Introduction to KAM theory with a view to celestial mechanics 387
  18. 14. Invariants of contact sub-pseudo-Riemannian structures and Einstein–Weyl geometry 434
  19. 15. Time-optimal control for a perturbed Brockett integrator 454
  20. 16. Twist maps and Arnold diffusion for diffeomorphisms 473
  21. 17. A Hamiltonian approach to sufficiency in optimal control with minimal regularity conditions: Part I 496
  22. Index 517
https://doi.org/10.1515/9783110430394-005

Chapters in this book

  1. Frontmatter I
  2. Contents V
  3. Part I
  4. 1. Second-order decomposition model for image processing: numerical experimentation 5
  5. 2. Optimizing spatial and tonal data for PDE-based inpainting 35
  6. 3. Image registration using phase–amplitude separation 84
  7. 4. Rotation invariance in exemplar-based image inpainting 108
  8. 5. Convective regularization for optical flow 184
  9. 6. A variational method for quantitative photoacoustic tomography with piecewise constant coefficients 202
  10. 7. On optical flow models for variational motion estimation 225
  11. 8. Bilevel approaches for learning of variational imaging models 252
  12. Part II
  13. 9. Non-degenerate forms of the generalized Euler–Lagrange condition for state-constrained optimal control problems 295
  14. 10 The Purcell three-link swimmer: some geometric and numerical aspects related to periodic optimal controls 314
  15. 11. Controllability of Keplerian motion with low-thrust control systems 344
  16. 12. Higher variational equation techniques for the integrability of homogeneous potentials 365
  17. 13. Introduction to KAM theory with a view to celestial mechanics 387
  18. 14. Invariants of contact sub-pseudo-Riemannian structures and Einstein–Weyl geometry 434
  19. 15. Time-optimal control for a perturbed Brockett integrator 454
  20. 16. Twist maps and Arnold diffusion for diffeomorphisms 473
  21. 17. A Hamiltonian approach to sufficiency in optimal control with minimal regularity conditions: Part I 496
  22. Index 517
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