A Theoretical Analysis of a Bilevel Optimization Formulation for Combined Local-Global Optical Flow Models

El Mostafa Kalmoun

math.OCNov 5, 2026

A Theoretical Analysis of a Bilevel Optimization Formulation for Combined Local-Global Optical Flow Models

Authors: El Mostafa Kalmoun

Organizations: School of Science and Engineering, Al Akhawayn University in Ifrane, Ifrane, 53000, Morocco.

Abstract

This paper analyzes a bilevel formulation for learning the regularization parameter of combined local-global (CLG) optical flow models. The lower-level problem computes the flow from a Lucas--Kanade data term and a quadratic or total variation (TV) regularizer, and the upper-level problem selects the parameter that minimizes the squared distance to a ground-truth flow. The setting is finite-dimensional, and a full-rank assumption makes the lower-level solution unique. For the quadratic model we derive sufficient conditions for the existence of an optimal parameter, for every optimal parameter to be positive, and for strict local optimality. For TV regularization an optimal parameter always exists, because the lower-level solution vanishes for large parameters, and we give a sufficient condition for positivity that accounts for the coupling of the two flow components through the local structure tensors. Synthetic examples illustrate the conditions, including cases in which they fail.

Figures & tables

Quantity	Value
$Q(0)<\\|w_{g}\\|_{2}^{2}$ (Dirichlet existence, ( 3.8 ))	Yes
$\langle\mathcal{M}_{\mathrm{LK}}^{-1}Lw_{0},\,w_{0}-w_{g}\rangle$	$0.110$
Condition ( 3.9 ) satisfied	Yes
Optimal $\alpha^{*}$	$0.585$
$Q(0)=\\|w_{0}-w_{g}\\|_{2}^{2}$	$0.648$
$\\|w_{g}\\|_{2}^{2}$	$2.79$

Table 1: Key numerical values for Example 1 .

Figure 1: Objective $Q(\alpha)=\|w_{\alpha}-w_{g}\|_{2}^{2}$ versus the regularization parameter $\alpha$ for the $3\times 3$ quadratic example. The curve decreases for small $\alpha>0$ , which confirms the prediction $\alpha^{*}>0$ from condition ( 3.9 )

Quantity	Value
$Q(0)=\\|w_{0}-w_{g}\\|_{2}^{2}$	$1294.8$
$\\|w_{g}\\|_{2}^{2}=Q_{\infty}$	$1450.0$
$Q(0)<\\|w_{g}\\|_{2}^{2}$ (Dirichlet existence, ( 3.8 ))	Yes
$\langle\mathcal{M}_{\mathrm{LK}}^{-1}Lw_{0},\,w_{0}-w_{g}\rangle$ (positivity, ( 3.9 ))	$1.24\times 10^{4}$
$\alpha^{*}$ (interior minimum, guaranteed by Prop. 2 and Thm. 3 )	$0.532$
$Q(\alpha^{*})$	$896.4$

Table 2: Key numerical values for Example 2 .

Figure 2: Example 2 : (a) Frame $I_{1}$ (smooth random texture). (b) Noisy warped frame $I_{2}$ ( $\sigma=0.10$ ). (c) Ground‑truth flow magnitude $|w_{g}|$ (uniform: global translation $(0.7,-0.3)$ , magnitude $\approx 0.76$ ). (d) $Q(\alpha)$ on a log scale, with the minimizer $\alpha^{*}\approx 0.532$ (red marker). (e) Pointwise error $|w_{0}-w_{g}|$ before regularization. (f) Pointwise error $|w(\alpha^{*})-w_{g}|$ after regularization ( $30.8\%$ reduction in $Q$ )

Figure 3: Noise-sensitivity of the bilevel improvement for Example 2 ( $50\times 50$ textured global-translation scene, quadratic regularization, five random seeds). (a) Median $Q(0)$ and $Q(\alpha^{*})$ versus noise level $\sigma$ (log $y$ -scale). (b) Median improvement ratio $(Q(0)-Q(\alpha^{*}))/Q(0)$ with min/max band over seeds. The band width remains below $11$ percentage points over the five seeds

Quantity	Value
$Q(0)<\\|w_{g}\\|_{2}^{2}$ (Dirichlet existence, ( 3.8 ))	Yes
Condition ( 3.9 ) satisfied	Yes
Prescribed $\alpha^{*}$	$1.000$
$Q(0)=\\|w_{0}-w_{g}\\|_{2}^{2}$	$0.166$
$\\|w_{g}\\|_{2}^{2}$	$1.764$
$Q(\alpha^{*})$	$0.075$

Table 3: Key numerical values for the auxiliary Example 3 .

Quantity	Value
$Q(0)=\\|w_{0}-w_{g}\\|_{2}^{2}$	$6.00$
$\\|w_{g}\\|_{2}^{2}$	$8.00$
$Q(0)<\\|w_{g}\\|_{2}^{2}$ (improvement over zero flow, ( 4.5 ))	Yes
Condition ( a ) satisfied	Yes
$\mathcal{R}_{\mathrm{var}}=0$ , $\mathcal{R}_{\mathrm{mix}}=0$ ( $M_{p}=I$ )	—
Alignment $S_{u}+S_{v}$ in ( b )	$17.91$

Table 4: Key numerical values for Example 4 .

Figure 4: Objective $Q(\alpha)=\|w_{\alpha}-w_{g}\|_{2}^{2}$ versus $\alpha$ for the $3\times 3$ TV example. The minimum is at $\alpha^{*}\approx 0.211$ . For $\alpha\geq 0.72$ the computed flow is zero and $Q(\alpha)=\|w_{g}\|_{2}^{2}=8$

Quantity	Value
$Q(0)=\\|w_{0}-w_{g}\\|_{2}^{2}$	$381.9$
$\\|w_{g}\\|_{2}^{2}$	$58.0$
$Q(0)<\\|w_{g}\\|_{2}^{2}$ (improvement over zero flow, ( 4.5 ))	No
Condition ( a ) satisfied	Yes
$\|\Omega_{11}\|/\|\Omega_{10}\|/\|\Omega_{01}\|/\|\Omega_{00}\|$	$2500/0/0/0$
Condition ( b ) verified	Yes

Table 5: Key numerical values for Example 5 .

Figure 5: Example 5 : (a) Frame $I_{1}$ : centered $10\times 10$ square of intensity $1$ (white) on a zero (black) background. (b) Noisy translated frame $I_{2}$ ( $\sigma=0.10$ ). (c) Ground-truth flow magnitude $|w_{g}|$ : displacement $(0.7,-0.3)$ on the square, zero elsewhere (magnitude $\approx 0.76$ ). (d) $Q(\alpha)$ on log–log axes: $Q(0)\approx 381.9>\|w_{g}\|_{2}^{2}=58.0$ (condition ( 4.5 ) fails). The minimum $Q(\alpha^{*})\approx 29.3$ at $\alpha^{*}\approx 1.68$ (red marker) lies below $\|w_{g}\|_{2}^{2}$ . For $\alpha\geq 3.161$ the lower-level solution is zero, by a dual certificate, and $Q(\alpha)=\|w_{g}\|_{2}^{2}$ . (e) Pointwise error $|w_{0}-w_{g}|$ before regularization: large spurious errors in the static background. (f) Pointwise error $|w_{\alpha^{*}}-w_{g}|$ after TV regularization ( $92.3\%$ reduction in $Q$ ): the background error is largely removed, and the remaining error is a nearly uniform block over the square and a narrow band around it, which reflects the loss of contrast of TV

Quantity	Value
$M_{p}$ (spatially uniform)	$\bigl[\begin{smallmatrix}1&-\frac{1}{2}\\ -\frac{1}{2}&1\end{smallmatrix}\bigr]$
$Q(0)=\\|w_{0}-w_{g}\\|_{2}^{2}$	$2$
$\\|w_{g}\\|_{2}^{2}$	$8$
$Q(0)<\\|w_{g}\\|_{2}^{2}$ (improvement over zero flow, ( 4.5 ))	Yes
Condition ( a ) satisfied	Yes
$\|\Omega_{11}\|/\|\Omega_{10}\|/\|\Omega_{01}\|/\|\Omega_{00}\|$	$0/9/0/7$

Table 6: Key numerical values for Example 6 .

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We present FlowIt, a novel architecture for optical flow estimation that combines global matching with confidence and occlusion-guided refinement. At its core, FlowIt leverages a hierarchical transformer architecture that captures extensive global context, enabling the model to effectively model long-range correspondences. To overcome the limitations of localized matching, we formulate the flow initialization as an optimal transport problem. This formulation yields a highly robust initial flow field, alongside explicitly derived occlusion and confidence maps. These cues are then seamlessly integrated into a guided refinement stage, where the network actively propagates reliable motion estimates from high-confidence regions into ambiguous, low-confidence areas. Extensive experiments across the Sintel, KITTI, Spring, and LayeredFlow datasets validate the effectiveness of our approach. FlowIt achieves state-of-the-art results on the competitive Sintel benchmark and establishes new state-of-the-art cross-dataset zero-shot generalization performance on Sintel, Spring, and LayeredFlow, while also delivering competitive performance on both the KITTI benchmark and KITTI zero-shot generalization settings.

Abstract

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