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A proper generalized decomposition approach for optical flow estimation
Author(s) -
El Hamidi Abdallah,
Saleh Marwan,
Papadakis Nicolas,
Senneville B. Denis
Publication year - 2020
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.6275
Subject(s) - regularization (linguistics) , sobolev space , computation , mathematics , convergence (economics) , algorithm , fidelity , quadratic equation , computer science , mathematical analysis , artificial intelligence , telecommunications , geometry , economics , economic growth
This paper introduces the use of the proper generalized decomposition (PGD) method for the optical flow (OF) problem in a classical framework of Sobolev spaces, ie, optical flow methods including a robust energy for the data fidelity term together with a quadratic penalizer for the regularization term. A mathematical study of PGD methods is first presented for general regularization problems in the framework of (Hilbert) Sobolev spaces, and their convergence is then illustrated on OF computation. The convergence study is divided in two parts: (a) the weak convergence based on the Brézis‐Lieb decomposition and (b) the strong convergence based on a growth result on the sequence of descent directions. A practical PGD‐based OF implementation is then proposed and evaluated on freely available OF data sets. The proposed PGD‐based OF approach outperforms the corresponding non‐PGD implementation in terms of both accuracy and computation time for images containing a weak level of information, namely, low image resolution and/or low signal‐to‐noise ratio (SNR).