A primal-dual adaptive finite element method for total variation minimization
Alkämper, Martin ; Hilb, Stephan and Langer, Andreas LU
(2025)
In Advances in Computational Mathematics
51.
- Abstract
- Based on previous work, we extend a primal-dual semi-smooth Newton method for minimizing a general L^1-L^2-TV functional over the space of functions of bounded variations by adaptivity in a finite element setting. For automatically generating an adaptive grid, we introduce indicators based on a-posteriori error estimates. Further, we discuss data interpolation methods on unstructured grids in the context of image processing and present a pixel-based interpolation method. The efficiency of our derived adaptive finite element scheme is demonstrated on image inpainting and the task of computing the optical flow in image sequences. In particular, for optical flow estimation, we derive an adaptive finite element coarse-to-fine scheme which... (More)
- Based on previous work, we extend a primal-dual semi-smooth Newton method for minimizing a general L^1-L^2-TV functional over the space of functions of bounded variations by adaptivity in a finite element setting. For automatically generating an adaptive grid, we introduce indicators based on a-posteriori error estimates. Further, we discuss data interpolation methods on unstructured grids in the context of image processing and present a pixel-based interpolation method. The efficiency of our derived adaptive finite element scheme is demonstrated on image inpainting and the task of computing the optical flow in image sequences. In particular, for optical flow estimation, we derive an adaptive finite element coarse-to-fine scheme which allows resolving large displacements and speeds up the computing time significantly. (Less)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/2a7782c8-91c8-465e-84df-989d6d5884ad
- author
- Alkämper, Martin
; Hilb, Stephan
and Langer, Andreas
LU
- organization
- publishing date
- 2025-08-21
- type
- Contribution to journal
- publication status
- published
- subject
- keywords
- A-posteriori error estimate, Adaptive finite element discretization, Non-smooth optimisation, Combined L^1/L^2 data-fidelity, Total variation, Optical flow estimation, Inpainting
- in
- Advances in Computational Mathematics
- volume
- 51
- article number
- 42
- pages
- 35 pages
- publisher
- Springer
- external identifiers
-
- scopus:105014022356
- ISSN
- 1572-9044
- DOI
- 10.1007/s10444-025-10254-8
- project
- Lokalt adaptiva metoder för fria diskontinuitetsproblem
- language
- English
- LU publication?
- yes
- id
- 2a7782c8-91c8-465e-84df-989d6d5884ad
- date added to LUP
- 2023-10-10 18:03:30
- date last changed
- 2025-10-17 04:01:28
@article{2a7782c8-91c8-465e-84df-989d6d5884ad,
abstract = {{Based on previous work, we extend a primal-dual semi-smooth Newton method for minimizing a general L^1-L^2-TV functional over the space of functions of bounded variations by adaptivity in a finite element setting. For automatically generating an adaptive grid, we introduce indicators based on a-posteriori error estimates. Further, we discuss data interpolation methods on unstructured grids in the context of image processing and present a pixel-based interpolation method. The efficiency of our derived adaptive finite element scheme is demonstrated on image inpainting and the task of computing the optical flow in image sequences. In particular, for optical flow estimation, we derive an adaptive finite element coarse-to-fine scheme which allows resolving large displacements and speeds up the computing time significantly.}},
author = {{Alkämper, Martin and Hilb, Stephan and Langer, Andreas}},
issn = {{1572-9044}},
keywords = {{A-posteriori error estimate; Adaptive finite element discretization; Non-smooth optimisation; Combined L^1/L^2 data-fidelity; Total variation; Optical flow estimation; Inpainting}},
language = {{eng}},
month = {{08}},
publisher = {{Springer}},
series = {{Advances in Computational Mathematics}},
title = {{A primal-dual adaptive finite element method for total variation minimization}},
url = {{http://dx.doi.org/10.1007/s10444-025-10254-8}},
doi = {{10.1007/s10444-025-10254-8}},
volume = {{51}},
year = {{2025}},
}