Observability-Aware Deformation Activation

• The paper presents a framework that uses Fisher information analysis and time-series priors to activate non-rigid deformations only when estimation is well-conditioned, significantly reducing drift.
• It integrates visual-inertial constraints and staged activation to separate rigid and non-rigid motions, achieving up to 80% reduction in trajectory error.
• The approach is validated across SLAM, VIO, and interactive perception pipelines, demonstrating state-of-the-art robustness in dynamic, deformable scenes.

Observability-aware deformation activation encompasses a family of strategies in estimation, SLAM, and robot perception for deformable scenes that explicitly monitor, manage, and recover the observability of both rigid and non-rigid state variables. This approach analyzes the rank properties of the relevant Fisher information matrix (FIM) or observability matrix under deformation models—most prominently the Embedded Deformation (ED) graph—and activates or constrains non-rigid degrees of freedom only when the joint estimation problem is sufficiently well-posed. The objective is to avoid entanglement between camera/camera pose and non-rigid scene deformations, thereby preventing spurious solutions and drift. Empirically validated in both SLAM back-ends and interactive perception pipelines, this paradigm has led to state-of-the-art robustness and accuracy in scenarios where unobservable deformation severely degrades conventional algorithms (Song et al., 2019, Cerezo et al., 2 Jan 2026, Weng et al., 2024).

1. Observability Challenges in Deformable Scene Estimation

Deformable scene estimation frameworks such as those employing the Embedded Deformation (ED) graph model each 3D point $v_i$ as a weighted combination of local affine node transformations, followed by a global camera pose transform. This parametrization introduces severe gauge freedoms: rigid body motions can be absorbed into the deformation field and vice versa. Specifically, for any solution $(R_c, T_c, \Lambda, T)$ there exists a continuous family of equivalent solutions obtained by composition with arbitrary $V_0 \in SO(3)$, i.e.

$$(R_c, T_c, \Lambda, T) \leftrightarrow (R_c V_0, T_c, V_0^T \Lambda, V_0^T T)\;,$$

demonstrating an inherent unobservability in the absence of further priors. The Fisher information matrix (FIM) of the system exhibits a 6-dimensional nullspace spanning both camera and deformation parameters unless additional independent constraints are introduced; this renders standard optimization-based back-ends incapable of uniquely separating rigid motion from non-rigid deformation (Song et al., 2019).

2. Time-Series Priors for Observability Recovery

One principled method to recover uniqueness is the incorporation of a time-series prior on the deformation trajectory: the scene shape at instant $t$ is constrained to reside within the linear span of shapes at the previous $k$ instants,

$$D_t^i = \sum_{\ell=1}^k \alpha_\ell D_{t-\ell}^i\,,$$

where the combination coefficients $\alpha_\ell$ become variables in joint optimization. This deformation-activation regularization yields an additional FIM block whose introduction removes the problematic gauge freedoms. The linear time-series model ensures that non-rigid scene evolution is explained strictly through known bases, while the residual motion is attributed to rigid pose change. The augmented Hessian becomes full-rank over the physically meaningful parameters, leaving only the auxiliary subspace of $\delta$-coefficients unobservable. Experimental evidence shows that this strategy achieves superior root mean squared error (RMSE) performance in both synthetic and real datasets, reducing state estimation drift by over 80% relative to baseline ED-only or rigid SLAM algorithms (Song et al., 2019).

3. Conditioning-Based Deformation Activation in Visual-Inertial Odometry

In fused visual-inertial frameworks for deformable scenes, such as DefVINS, observability-aware deformation activation leverages inertial constraints to anchor the rigid subspace, followed by selective activation of non-rigid state variables. The combined state is partitioned as $(\mathbf{x}_r, \mathbf{x}_{NR})$, where $\mathbf{x}_r$ encodes the IMU-anchored rigid variables and $\mathbf{x}_{NR}$ the deformation graph node positions across consecutive frames (Cerezo et al., 2 Jan 2026).

At each optimization step, the system computes the FIM and partitions it according to the rigid/non-rigid blocks. Nodes of the deformation graph are only activated (unfrozen) for estimation once the smallest singular value $\sigma_{\min}$ of their associated FIM block exceeds a fixed threshold $\epsilon_{\rm act}$, indicating well-conditioned estimation. Otherwise, poor excitation (e.g., lack of parallax or motion orthogonal to the imposed constraints) would allow the non-rigid warp to spuriously absorb unknown rigid drift. The IMU preintegration and gravity alignment constraints directly address the under-determination in scale and global orientation, while the activation schedule enforces a staged, observability-respecting optimization regime. Empirically, this yields 30–80% reductions in absolute trajectory error and dramatic improvements in frame tracking ratio under severe, non-rigid scene deformation (Cerezo et al., 2 Jan 2026).

4. Observability-Aware Activation in Interactive Perception

Interactive perception of deformable object manipulation adopts a sequential decision-theoretic framework with explicit observability-centric deformation activation via the Dynamic Active Vision Space (DAVS) construction (Weng et al., 2024). Here, the POMDP policy constrains camera actions to a time-varying submanifold $\mathcal{M}(t) \subset S^2(r)$, constructed on the basis of observed structure-of-interest (SOI) keypoints. The DAVS boundary is generated as a geodesic convex hull of keypoint projections onto an action sphere, guaranteeing that the camera's exploration is geometrically coordinated with the current state of the deforming object and the manipulator.

By restricting perception actions to $\mathcal{M}(t)$, the system ensures persistent visibility and regular excitation of the SOI loop, thereby actively managing local observability and preventing degeneracies due to occlusion or poor viewpoint allocation. Experimental data demonstrates that this approach reduces required episode steps by a factor of 2–3 and doubles task success rates, with robustness verified across both simulation and real-robot domains and across unseen elasticity-damping regimes (Weng et al., 2024).

5. Algorithmic Structures and Pipeline Integration

Observability-aware deformation activation is instantiated in multiple concrete algorithmic pipelines:

• Back-end Factor Graph SLAM: The optimization objective takes the form

$$\min_{\{R^j, p^j\}, B, c} E_{\rm obs} + E_f + E_{\rm ini}$$

where $E_{\rm obs}$ encodes measurement consistency, $E_f$ enforces the time-series prior, and $E_{\rm ini}$ regularizes initialization frames. All factors are iteratively linearized and solved using sparse nonlinear least squares frameworks (e.g., g2o, Ceres). The analytical Jacobians are constructed to respect the constraints advocated by the observability analysis (Song et al., 2019).

• Visual-Inertial Odometry Pipelines: The state is initialized under rigid-only estimation, transitioning to joint rigid-deformation optimization following the rank/conditioning test of the non-rigid FIM block. This step-wise activation is verified, and the transition is repeated as new keyframes enter the sliding window, providing both stability and adaptiveness in rapidly changing scenes (Cerezo et al., 2 Jan 2026).
• Sequential Policy-based Perception: For deformable object interactive manipulation tasks, action selection is constrained in real time by online re-computation of the DAVS at each step, with action exploration concentrated in highly observable regions. Optimization of the exploration policy via policy gradient is facilitated by efficiently computable submanifolds and routine re-alignment using the observed state (Weng et al., 2024).

6. Empirical Outcomes and Significance

Empirical results across multiple domains illustrate the impact of observability-aware deformation activation:

Method / Domain	Pose RMSE (m)	Success Rate (%)	Notable Outcome
Proposed Deformable SLAM (synthetic, (Song et al., 2019))	0.119	--	3–18× lower RMSE vs. ED
DefVINS AOA (real, (Cerezo et al., 2 Jan 2026))	9.4 mm	85–95	30–80% drift reduction
DAVS IP (sim/real, (Weng et al., 2024))	--	95–100	2× reward, 2–3× faster

In such tasks, classic ED-only or rigid SLAM shows persistent 6-DOF gauge indeterminacy, severe drift, or poor convergence. Time-series priors and/or FIM-conditioned activation restore full rank on the pose and deformation subspace, while empirical studies record substantial quantitative and qualitative performance improvements—including successful transfer from simulation to hardware in robotic manipulation scenarios.

7. Broader Implications and Connections

Observability-aware deformation activation provides a systematic route to robust estimation in high-dimensional, underdetermined non-rigid systems. Its principles extend to any free-form deformation framework where joint parametric entanglement of rigid and non-rigid motion impairs state recovery. The recurring paradigm is that local or global priors (time-series, inertial, geometric) supply external information sufficient to remove ambiguity, while activation and optimization schedules are devised so that new degrees of freedom are admitted only as they become sufficiently observable. This foundational strategy is now migrating into active sensing, multi-agent cooperative SLAM, and reinforcement learning-based perception for increasingly complex, dynamic environments, setting the stage for future work on uncertainty-aware structure learning, adaptive observability management, and task-driven deformation modeling (Song et al., 2019, Cerezo et al., 2 Jan 2026, Weng et al., 2024).