By N. Benvenuto, et al.,
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Additional info for Algorithms for Comm. Systs and Their Applns
The goal is to obtained a smoothed version of the underlying direction ﬁeld. Since SO(2) is isomorphic to S 1 , the proposed regularization scheme can be 32 G. Rosman et al. used for regularizing directions as well, as we demonstrate. A reasonable choice for a data term would try to align the rotated ﬁrst coordinate axis with the motion directions in the area, ⎝ ⎞ ⎬ U1,1 (vj )x + U1,2 (vj )y , (58) EP M D (U ) = (xj ,yj )∈N (i) ⎝ ⎞ where xj , yj , (vj )x , (vj )y represent a sampled motion particle  in the video sequence, and Ui,j represent elements of the solution u.
Yet another possibility is to further decouple the TV and data term, using separate variables and constraining them using an augmented Lagrangian approach. Of the above techniques, we have tried the ﬁrst one. The reconstruction obtained is the spatially-regularized version of the linear-least-squares (LLS) method. One can incorporate a weighted least-squares (WLS, ), or nonlinearleast-squares (NLS)  data term instead. Combining such data terms and exploring the interaction between the regularization and nonlinear terms is beyond the scope of this work.
An obvious choice is taking the total variation of second order [14,19] which can also be expressed in a dual formulation using symmetric matrix ﬁelds v : Γ ∪ S d×d : TV2 (u) = d|⊂2 u| = sup Ω Ω u div2 v dx v → Cc2 (Γ, S d×d ), ∈v∈∈ ≤ 1 . Here, the derivative ⊂u is modeled to be piecewise constant which has the consequence that it is itself a regular function and can thus not have any discontinuities. Therefore, solutions of variational problems with TV2 penalty cannot have jump discontinuities and object boundaries become inevitably blurry.
Algorithms for Comm. Systs and Their Applns by N. Benvenuto, et al.,