	\begin{algorithmic}[1]
		\State Set $\lambda^{(0)},\rho^{(0)}$. Choose $\epsilon_1^{(0)},\epsilon_2^{(0)}$.
		\State Obtain initial HS optical flow $\textbf{u}^{(0)}$
		\For{$n = 1,2,\dots$ until convergence \textbf{do}}
		\State update $\textbf{u}^{(n)}, d^{(n)}$
		\If $\|B\textbf{u}^{(n)}-c\|_{\mathcal{H}}\le \max\{\epsilon_1^{(n)},2\delta_{\text{HS}}\}$ 
		\If $\|fd^{(n)}\|_{\mathcal{H}}\le  \epsilon_2^{(n)}$
		\State break;
		\Else
		\State update $\lambda_1^{(n)}$ by (\ref{uzawa})
		\State $\rho^{(n+1)}\gets \rho^{(n)}$
		\State tighten tolerances $\epsilon_1^{(n+1)},\epsilon_2^{(n+1)}$
		\EndIf
		\Else 
		\State update $B\textbf{u}^{(n)}-c$
		\State $\lambda^{(n+1)}\gets\lambda^{(n)}$
		\State $\rho^{(n+1)}\gets 100\rho^{(n)}$
		\State tighten tolerances $\epsilon_1^{(n+1)},\epsilon_2^{(n+1)}$
		\EndIf
		\EndFor
	\end{algorithmic}

\begin{algorithmic}
[1]
\State Set $\lambda^{(0)},\rho^{(0)}$. Choose $\epsilon_1^{(0)},\epsilon_2^{(0)}$.
\State Obtain initial HS optical flow $\textbf{u}^{(0)}$
\For{$n = 1,2,\dots$ until convergence \textbf{do}}
\State update $\textbf{u}^{(n)}, d^{(n)}$
\If $\|B\textbf{u}^{(n)}-c\|_{\mathcal{H}}\le \max\{\epsilon_1^{(n)},2\delta_{\text{HS}}\}$
\If $\|fd^{(n)}\|_{\mathcal{H}}\le \epsilon_2^{(n)}$
\State break;
\Else
\State update $\lambda_1^{(n)}$ by (\ref{uzawa})
\State $\rho^{(n+1)}\gets \rho^{(n)}$
\State tighten tolerances $\epsilon_1^{(n+1)},\epsilon_2^{(n+1)}$
\EndIf
\Else
\State update $B\textbf{u}^{(n)}-c$
\State $\lambda^{(n+1)}\gets\lambda^{(n)}$
\State $\rho^{(n+1)}\gets 100\rho^{(n)}$
\State tighten tolerances $\epsilon_1^{(n+1)},\epsilon_2^{(n+1)}$
\EndIf
\EndFor
\end{algorithmic}
{
    "title": "a framework for fluid motion estimation using a constraint-based   refinement approach",
    "id": "2011.12267",
    "abstract": "physics-based optical flow models have been successful in capturing the deformities in fluid motion arising from digital imagery. however, a common theoretical framework analyzing several physics-based models is missing. in this regard, we formulate a general framework for fluid motion estimation using a constraint-based refinement approach. we demonstrate that for a particular choice of constraint, our results closely approximate the classical continuity equation-based method for fluid flow. this closeness is theoretically justified by augmented lagrangian method in a novel way. the convergence of uzawa iterates is shown using a modified bounded constraint algorithm. the mathematical wellposedness is studied in a hilbert space setting. further, we observe a surprising connection to the cauchy-riemann operator that diagonalizes the system leading to a diffusive phenomenon involving the divergence and the curl of the flow. several numerical experiments are performed and the results are shown on different datasets. additionally, we demonstrate that a flow-driven refinement process involving the curl of the flow outperforms the classical physics-based optical flow method without any additional assumptions on the image data.",
    "categories": "cs.cv math.ap",
    "doi": "",
    "created": "2020-11-24",
    "updated": "2024-02-21",
    "authors": [
        "hirak doshi",
        "n. uday kiran"
    ],
    "affiliation": [],
    "url": "https://arxiv.org/abs/2011.12267"
}
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