Dual Uncertainty Optimization (DUO)

• DUO is a test-time adaptation method that jointly minimizes semantic (classification) and geometric (depth) uncertainties in monocular 3D object detection.
• It employs a dual-branch design using Conjugate Focal Loss for semantic optimization and a semantic-aware normal field constraint for stable geometric predictions.
• Empirical evaluations on datasets like KITTI-C and nuScenes demonstrate that DUO significantly improves detection performance and robustness across various corruption scenarios.

Dual Uncertainty Optimization (DUO) is a fully test-time adaptation (TTA) framework for monocular 3D object detection (M3OD) that is designed to jointly minimize two coupled uncertainty sources under test-time shifts: semantic uncertainty, which concerns ambiguity in category prediction, and geometric uncertainty, which concerns instability in 3D localization, especially depth and surface structure. DUO addresses the semantic branch with a Conjugate Focal Loss (CFL) and the geometric branch with a semantic-aware normal field constraint, yielding a dual-branch mechanism in which improved spatial perception sharpens semantic classification and more reliable semantic predictions refine spatial understanding (Hu et al., 28 Aug 2025).

1. Problem setting and motivating failure modes

DUO is formulated for M3OD under environmental and sensor shifts, including noise, blur, weather, and real-world day/night/rain changes. The setting is distinctive because M3OD does not merely assign a class label; it simultaneously predicts category and 3D geometry. The paper therefore argues that adaptation procedures built around a single uncertainty notion are insufficient, since both semantic uncertainty and geometric uncertainty rise as corruption severity increases (Hu et al., 28 Aug 2025).

The semantic component is measured by prediction entropy over classes, while the geometric component is measured by average uncertainty from the multi-head depth estimator. This dual rise under corruption motivates the term “dual uncertainty” in DUO. The method is presented as the first TTA framework designed to jointly minimize both uncertainties for robust M3OD.

The paper isolates two failure modes in prior entropy-centered TTA methods. The first is low-score object neglect: vanilla entropy minimization tends to improve already-confident predictions while providing weak supervision to hard, low-confidence objects, and foreground-background imbalance can amplify that effect. The second is spatial perception collapse: directly minimizing uncertainty in a multi-head depth estimator can collapse multiple heads into a single deterministic predictor, undermining the geometric diversity needed for robust spatial reasoning. DUO is explicitly constructed to avoid both behaviors.

2. Semantic uncertainty optimization through Conjugate Focal Loss

The semantic branch of DUO is based on a convex-optimization reinterpretation of focal loss. The starting point is the standard focal loss

$$\mathcal{L}_{\text{FL}(x, y) }=-\alpha (1-p)^\gamma\cdot y\log p,$$

where $p=\text{softmax}(h)$, $y$ is the one-hot label, and $\alpha,\gamma$ are focal-loss hyperparameters. The paper rewrites focal loss as

$\mathcal{L}_{\text{FL} }= f(h)-y^\top g(h),$

with

$$f(h)=\alpha\log s, \qquad g(h)= \alpha h+\alpha((1-p)^\gamma -1) \log p,$$

and $s=e^{h_1}+\cdots+e^{h_c}$. Under local invertibility of $g$,

$$\min_h \{f(h)-y^\top g(h)\} = \min_{z=g(h)} \{f\circ g^{-1}(z)-y^\top z\} = f^{*}(y).$$

This convex-conjugate formulation matters because standard focal loss requires the ground-truth label $y$, which is unavailable at test time. DUO therefore derives a surrogate target $p=\text{softmax}(h)$0 through a higher-order approximation: $p=\text{softmax}(h)$1 Substituting this into the conjugate form yields the Conjugate Focal Loss

$p=\text{softmax}(h)$2

The paper attributes four properties to this construction: it remains class-balanced through the $p=\text{softmax}(h)$3 factor, it is label-independent, it dynamically weights all classes through the inverse Jacobian term, and it can be used directly during TTA without labels. The appendix further states that the Jacobian of $p=\text{softmax}(h)$4 is locally non-singular under the default $p=\text{softmax}(h)$5, which supports the conjugate derivation (Hu et al., 28 Aug 2025).

3. Geometric uncertainty optimization through a semantic-aware normal field

The geometric branch does not directly minimize depth uncertainty. The paper argues that such a strategy can induce collapse in the multi-head geometry predictor. Instead, DUO regularizes the normal field derived from the predicted depth map, thereby imposing geometric coherence without forcing the depth heads toward a single deterministic estimate (Hu et al., 28 Aug 2025).

Given a depth map $p=\text{softmax}(h)$6, DUO resizes it to image resolution and applies Sobel operators: $p=\text{softmax}(h)$7 From these derivatives it constructs a surface normal field $p=\text{softmax}(h)$8. Local smoothness is enforced through

$p=\text{softmax}(h)$9

$y$0

The Normal Consistency Loss is then

$y$1

The factor

$y$2

is edge-aware: it preserves discontinuities at object boundaries while enforcing normal consistency in more homogeneous regions. This makes the regularizer spatially structured rather than globally smoothing.

A second safeguard is semantic guidance. The loss is applied only in regions with reliable semantic predictions. The paper defines low-semantic-uncertainty boxes

$y$3

with $y$4 by default, and then constructs the mask

$y$5

This means geometric regularization is restricted to confident, semantically stable boxes. The paper’s central claim here is that semantic guidance prevents the geometric branch from destabilizing ambiguous regions.

4. Joint objective, adaptation dynamics, and the complementary loop

DUO combines the semantic and geometric branches in a single test-time objective,

$y$6

where $y$7 is the semantic term, $y$8 is the geometric term, $y$9 is the semantic-guided region mask, and $\alpha,\gamma$0 by default (Hu et al., 28 Aug 2025).

This is a source-free and label-free TTA formulation. Model parameters are adapted online using unlabeled test images. The reported optimization details are: learning rate same as MonoTTA, momentum $\alpha,\gamma$1, batch size $\alpha,\gamma$2 for KITTI and $\alpha,\gamma$3 for nuScenes, and defaults $\alpha,\gamma$4, $\alpha,\gamma$5, $\alpha,\gamma$6.

The paper emphasizes that DUO is not merely the sum of two losses. It is a complementary loop:

• better semantic confidence identifies reliable regions and boxes;
• those regions constrain the normal field and improve geometry;
• improved geometry sharpens localization and spatial understanding;
• improved spatial understanding reduces semantic ambiguity.

The reported uncertainty trajectories align with this description. CFL primarily decreases semantic uncertainty but also slightly lowers geometric uncertainty; NCL primarily decreases geometric uncertainty but also slightly lowers semantic uncertainty; and the full method decreases both fastest. This mutual reinforcement is a defining feature of DUO rather than a secondary by-product.

5. Empirical evaluation, ablations, and operational profile

DUO is evaluated on KITTI-C and nuScenes using MonoFlex and MonoGround as base models, and compared against the TTA baselines TENT, EATA, DeYO, and MonoTTA (Hu et al., 28 Aug 2025). KITTI-C uses 13 corruption types—Gaussian noise, shot noise, impulse noise, defocus blur, glass blur, motion blur, snow, frost, fog, brightness, contrast, pixelation, saturation—across five severity levels, with the main paper reporting severity level 5. The metric is $\alpha,\gamma$7, averaged over Easy/Moderate/Hard. The nuScenes evaluation uses real-world shifts across Daytime, Night, Sunny, and Rainy scenes.

Setting	Comparison	Result
KITTI-C severity 5, MonoFlex, Car	DUO vs MonoTTA	$\alpha,\gamma$8 average AP vs $\alpha,\gamma$9
KITTI-C severity 5, MonoGround	DUO vs MonoTTA	$\mathcal{L}_{\text{FL} }= f(h)-y^\top g(h),$0 vs $\mathcal{L}_{\text{FL} }= f(h)-y^\top g(h),$1
nuScenes Daytime $\mathcal{L}_{\text{FL} }= f(h)-y^\top g(h),$2 Night, MonoFlex	DUO vs MonoTTA	$\mathcal{L}_{\text{FL} }= f(h)-y^\top g(h),$3 vs $\mathcal{L}_{\text{FL} }= f(h)-y^\top g(h),$4
nuScenes Night $\mathcal{L}_{\text{FL} }= f(h)-y^\top g(h),$5 Daytime, MonoFlex	DUO vs MonoTTA	$\mathcal{L}_{\text{FL} }= f(h)-y^\top g(h),$6 vs $\mathcal{L}_{\text{FL} }= f(h)-y^\top g(h),$7
nuScenes Sunny $\mathcal{L}_{\text{FL} }= f(h)-y^\top g(h),$8 Rainy average, MonoFlex	DUO vs MonoTTA	$\mathcal{L}_{\text{FL} }= f(h)-y^\top g(h),$9 vs $$f(h)=\alpha\log s, \qquad g(h)= \alpha h+\alpha((1-p)^\gamma -1) \log p,$$0

For MonoGround on nuScenes, the reported DUO values are 15.70 for Daytime $$f(h)=\alpha\log s, \qquad g(h)= \alpha h+\alpha((1-p)^\gamma -1) \log p,$$1 Night, 1.91 for Night $$f(h)=\alpha\log s, \qquad g(h)= \alpha h+\alpha((1-p)^\gamma -1) \log p,$$2 Daytime, and 11.21 for Sunny $$f(h)=\alpha\log s, \qquad g(h)= \alpha h+\alpha((1-p)^\gamma -1) \log p,$$3 Rainy average. The paper also reports an average improvement of roughly +18% over existing methods in the real-world shift setting.

The ablation study on MonoFlex under Gaussian corruption provides direct evidence for the two-branch design:

• source model: 2.08 avg
• CFL only: 10.63 avg
• NCL only: 6.61 avg
• CFL + NCL: 11.29 avg
• NCL + mask: 9.20 avg
• full DUO: 11.75 avg

These numbers support three specific conclusions reported by the paper. First, CFL is effective at rescuing low-score objects. Second, NCL alone is unstable unless guided semantically. Third, the full method is best. In operational terms, DUO is reported as efficient: processing 1k Gaussian-corrupted images takes 32s, compared with 26s for source inference and 87s for DeYO.

6. Terminological boundaries and relation to adjacent research

The term Dual Uncertainty Optimization (DUO) in this context refers specifically to the M3OD test-time adaptation framework described above, not to a generic family of dual-variable or uncertainty-aware methods. This distinction matters because several papers are conceptually adjacent while addressing different problem classes.

UDuo is a Universal Dual optimization framework for online matching under budget constraints. It formulates online allocation as a primal-dual resource allocation problem, introduces a temporal user arrival representation $$f(h)=\alpha\log s, \qquad g(h)= \alpha h+\alpha((1-p)^\gamma -1) \log p,$$4, uses a resource pacing learner, and relies on online time-series forecasting with MiFormer. Its uncertainty is non-stationary user arrivals and changing resource consumption dynamics, not semantic and geometric uncertainty in M3OD (Li et al., 28 May 2025).

Twin-Bootstrap Gradient Descent (Twin-Boot) is an uncertainty-aware optimization method that trains two identical models on independent bootstrap samples and uses their within-basin parameter divergence as an online uncertainty signal. Its uncertainty is bootstrap-based and local-basin constrained, and its regularization is adaptive weight noise. The paper explicitly notes that it is conceptually related to DUO only in the broad sense that both use uncertainty during optimization, while the mechanisms are different (Brito, 20 Aug 2025).

Optimistic robust linear quadratic dual control addresses parametric uncertainty in uncertain linear systems. Its “dual” character lies in balancing regulation with targeted exploration so that the controller both stabilizes the system robustly and reduces uncertainty in the parameters that matter most for control performance. This is a control-theoretic dual-control formulation, not a TTA method for visual detection (Umenberger et al., 2019).

Diffusion-Guided Uncertainty-Aware Delayed Policy Optimization (DUPO) studies stochastic delayed MDPs. It uses a conditional diffusion model to represent the posterior $$f(h)=\alpha\log s, \qquad g(h)= \alpha h+\alpha((1-p)^\gamma -1) \log p,$$5 over the current state given delayed observations and then reweights policy optimization by critic uncertainty. Here the uncertainty is delay-induced state ambiguity, not semantic-geometric uncertainty in 3D detection (Tu et al., 6 Jul 2026).

A common misconception is therefore to read “DUO” as a generic label for any dual or uncertainty-aware optimizer. In the literature summarized here, DUO denotes a specific source-free, label-free TTA framework for M3OD whose defining innovation is the joint treatment of semantic and geometric uncertainty through CFL and a semantic-aware normal field constraint (Hu et al., 28 Aug 2025).