Trajectory-Controllable Unfolding Framework

• The trajectory-controllable unfolding framework explicitly designs the path of intermediate reconstructions to ensure a gradual transition from noisy input to high-quality output.
• It uses controlled interpolative updates and back-projection to maintain measurement consistency and avoid abrupt stage-to-stage changes.
• Empirical results demonstrating improved PSNR and SSIM in hyperspectral imaging validate the benefits of explicit stage supervision and hybrid architectural design.

Searching arXiv for the explicitly trajectory-controllable unfolding paper and closely related trajectory-control work.

Searching arXiv for "trajectory-controllable unfolding framework" and "Progressive Flow-inspired Unfolding".

A trajectory-controllable unfolding framework is an unfolding architecture in which the path of intermediate states is treated as an explicit design object rather than a by-product of iterative inference. The phrase is used explicitly by FLoUNet for coded aperture snapshot spectral imaging (CASSI), where the central objective is to make reconstruction evolve along a smooth, progressively improving trajectory from a noisy initial estimate to a high-quality hyperspectral image, instead of allowing abrupt stage-to-stage jumps and poorly regulated intermediate outputs (Wang et al., 15 Sep 2025). A broader reading is also suggested by adjacent trajectory-conditioned generative systems, in which latent or physical state is unfolded under explicit motion, structural, or behavioral constraints rather than only endpoint supervision.

1. Canonical formulation in deep unfolding

In its canonical usage, the framework addresses severely ill-posed inverse problems in which a latent state must be reconstructed through a finite sequence of learned updates. For CASSI, the unknown hyperspectral image $\mathbf{x}\in\mathbb{R}^n$ and compressed measurement $\mathbf{y}\in\mathbb{R}^m$ satisfy

$\mathbf{y}=\mathbf{A}\mathbf{x}+\mathbf{n},$

with $m\ll n$. Reconstruction is posed as a MAP problem,

$$\hat{\mathbf{x}}=\arg\min_{\mathbf{x}} \frac{1}{2}\|\mathbf{y}-\mathbf{A}\mathbf{x}\|_2^2-\log p(\mathbf{x}),$$

and standard deep unfolding alternates a data-fidelity update with a learned proximal or denoising step. The generic form given is

$$\mathbf{z}_k \leftarrow \mathbf{x}_k-\gamma \frac{\partial}{\partial \mathbf{x}_k}\left\|\mathbf{W}(\mathbf{y}-\mathbf{H}\mathbf{x}_k)\right\|_2^2,\qquad \mathbf{x}_{k+1}\leftarrow \operatorname{prox}_{\sigma}(\mathbf{z}_k),$$

with the proximal operator

$$\operatorname{prox}_{\lambda h}(\mathbf{v})=\arg\min_{\mathbf{x}}\left\{\frac{1}{2\lambda}\|\mathbf{x}-\mathbf{v}\|_2^2+h(\mathbf{x})\right\}.$$

The defining claim of the trajectory-controllable variant is that conventional unfolding controls the endpoint but not the intermediate reconstruction trajectory. Intermediate stages may improve erratically, later stages may dominate disproportionately, and optimization may become sensitive to initialization. In this formulation, trajectory controllability means that each stage should correspond to a meaningful partial restoration step and that the entire stage sequence should approximate a gradual transport from measurement-corrupted input to clean output, not merely a stack of denoisers (Wang et al., 15 Sep 2025).

2. Flow-inspired stage dynamics and trajectory supervision

FLoUNet instantiates this idea by replacing abrupt denoiser overwrites with a controlled interpolative update. The proposed stage transition is

$$x_{k+1}=\left(1-\frac{s}{K}\right)\mathcal{D}_\sigma(x_k)+\frac{s}{K}x_k,$$

where $x_k$ is the current estimate, $\mathcal{D}_\sigma(x_k)$ is the denoiser output, $\mathbf{y}\in\mathbb{R}^m$0 is the number of unfolding stages, and $\mathbf{y}\in\mathbb{R}^m$1 is a learned progression coefficient. The next state is therefore a convex interpolation between the current estimate and its denoised target. This replaces unconstrained stage replacement with a bounded transport step.

After denoising, measurement consistency is restored by back-projection: $\mathbf{y}\in\mathbb{R}^m$2 whose closed-form update is

$\mathbf{y}\in\mathbb{R}^m$3

with

$\mathbf{y}\in\mathbb{R}^m$4

The paper notes that in CASSI, $\mathbf{y}\in\mathbb{R}^m$5 is diagonal, making this step efficient.

Trajectory control is reinforced during training by an explicit intermediate-stage loss. The manuscript writes

$\mathbf{y}\in\mathbb{R}^m$6

with stage weighting

$\mathbf{y}\in\mathbb{R}^m$7

The manuscript also notes that several equations contain typographical or indexing inconsistencies, but its intended meaning is clear: intermediate outputs are explicitly supervised to lie on a progressive path between degraded and clean states (Wang et al., 15 Sep 2025).

3. Architectural realization

The trajectory-controllable update rule is paired with a stage architecture in which each unfolding block contains a proximal network and a back-projection step. The proximal network is built around an efficient spatial-spectral Transformer tailored to hyperspectral data. For a local window feature

$\mathbf{y}\in\mathbb{R}^m$8

spectral attention treats channels as tokens: $\mathbf{y}\in\mathbb{R}^m$9

$\mathbf{y}=\mathbf{A}\mathbf{x}+\mathbf{n},$0

Deeper layers use low-rank spatial attention with $\mathbf{y}=\mathbf{A}\mathbf{x}+\mathbf{n},$1: $\mathbf{y}=\mathbf{A}\mathbf{x}+\mathbf{n},$2

$\mathbf{y}=\mathbf{A}\mathbf{x}+\mathbf{n},$3

The efficiency rationale is local windows, spectral attention in shallow layers, and low-rank spatial attention in deeper layers.

A second architectural component is the frequency-aware fusion module inserted into U-Net skip connections. The fusion is written as

$\mathbf{y}=\mathbf{A}\mathbf{x}+\mathbf{n},$4

with

$\mathbf{y}=\mathbf{A}\mathbf{x}+\mathbf{n},$5

Decoder features are channel-aligned by

$\mathbf{y}=\mathbf{A}\mathbf{x}+\mathbf{n},$6

The low-pass branch promotes coarse semantic consistency; the high-pass branch enhances edges and fine textures. The paper explicitly ties skip-connection design to trajectory shaping, arguing that stable inter-stage information flow is part of controllable unfolding rather than an orthogonal architectural detail (Wang et al., 15 Sep 2025).

4. What is controlled

The control mechanism has two coupled parts: interpolative stage transition and intermediate trajectory supervision. The first bounds how far any one stage may move from the current state; the second makes each stage accountable to a progressive restoration target. The resulting process is deterministic rather than stochastic. There is no noise injection during inference, and the method is described as closer to a discrete flow or residual transport than to standard stochastic diffusion sampling.

This control is not a guarantee of monotonic objective decrease. The paper does not prove monotonic PSNR improvement or derive a neural ODE limit. Instead, it imposes smoothness, bounded stage transitions, learned progression coefficients, and measurement consistency after each denoising step. In that sense, “trajectory-controllable” denotes regulation of the optimization path itself.

The empirical support is stage-centric rather than endpoint-only. The trajectory-focused ablation shows that removing skip connections causes the process to become “targetless” and the trajectory to be lost, especially at the second stage; weight sharing across stages regularizes convergence; and with the proposed design, stage losses decrease more gradually. This suggests that trajectory controllability, in this framework, is not reducible to a single interpolation scalar but depends on update rule, supervision, and internal feature transport jointly (Wang et al., 15 Sep 2025).

5. Relation to adjacent trajectory-controlled systems

A broader editorial usage of the term is suggested by contemporaneous work in video generation, choreography, and trajectory prediction, although these papers do not all use “unfolding” in the same strict sense. In image-to-video generation, FlexTraj turns motion control into a unified structured conditioning problem over time through point trajectories carrying segmentation ID, temporally consistent trajectory ID, and optional color cues, then injects these controls into a pretrained video diffusion transformer by efficient sequence concatenation (Zhang et al., 9 Oct 2025). ATI likewise treats camera motion, object translation, and local deformation as point-wise trajectories, transporting latent image features along user-defined paths with Gaussian spatialization before patchification in DiT-based image-to-video models (Wang et al., 28 May 2025). MagicMotion organizes controllable video generation as a dense-to-sparse curriculum over masks, boxes, and sparse boxes, with a Trajectory ControlNet branch and latent segmentation loss to preserve object structure under weakening control signals (Li et al., 20 Mar 2025). FreeTraj demonstrates that pretrained video diffusion models already expose motion sensitivity through low-frequency initial noise and attention routing, allowing tuning-free trajectory steering by modifying noise construction and early attention computation (Qiu et al., 2024). FlashMotion shows that few-step controllable generation requires not only generator distillation but explicit re-alignment of the control adapter to the compressed denoising dynamics, using hybrid diffusion and adversarial finetuning (Li et al., 12 Mar 2026).

Outside video synthesis, related formulations use trajectory control in different state spaces. TCDiff separates multi-dancer root trajectories from detailed group motion generation, keeping trajectories fixed during diffusion and correcting lower-body realization with a footwork adaptor and relative forward-kinematic loss (Dai et al., 2024). TAE factorizes vehicle-trajectory latents into intention, aggressiveness, and residual components within a semi-supervised adversarial autoencoder, making trajectory generation and prediction behavior-aware but not explicitly stage-unfolded (Jiao et al., 2022). PrefCVAE aligns a semantic latent with average future velocity through weak preference pairs, enabling controllably diverse trajectory prediction via ordered latent traversal (Cao et al., 12 Oct 2025). Topology-Driven Trajectory Optimization uses a differentiable local homotopy invariant to specify clockwise, counterclockwise, or non-interacting multi-vehicle passing patterns as optimization constraints (Ma et al., 7 Mar 2025). Taken together, these works suggest that a broader trajectory-controllable unfolding framework can mean explicit control over latent evolution, structured motion memory, semantic trajectory latents, or topological interaction modes, even when the domain is not inverse reconstruction.

6. Empirical profile, limitations, and significance

On simulated hyperspectral reconstruction, FLoUNet-9stg reports the best average PSNR among the compared methods, with 40.27 PSNR and 0.975 SSIM, compared with 40.09 / 0.979 for LADE-9stg and 39.57 PSNR for RDLUF-MixS$\mathbf{y}=\mathbf{A}\mathbf{x}+\mathbf{n},$7-9stg. The 3-stage model is also strong, at 39.17 PSNR and 0.970 SSIM with 1.35M parameters and 26.18G FLOPs, supporting the claim that explicit trajectory supervision can make shallow unfolding models effective (Wang et al., 15 Sep 2025).

The ablation table isolates the trajectory-oriented contributions. The paper reports: Base-1 at 36.77 / 0.949, then + HS2Former at 38.76 / 0.972, + FreqFusion at 39.01 / 0.970, and + Trajectory Loss at 39.17 / 0.971. This indicates that the hybrid Transformer gives the largest boost, frequency-aware skip fusion adds further gains, and explicit trajectory supervision improves PSNR again beyond backbone changes alone. The significance of the framework therefore lies less in a single module than in the coupling of controlled update dynamics, architecture, and supervision.

Its limitations are equally explicit. The paper notes typographical inconsistencies in several formal expressions, especially the MAP equation, conditional-expectation proposition, and trajectory loss; it does not specify in detail whether the learned interpolation coefficient is a scalar, vector, or data-dependent quantity; it gives no rigorous monotonicity proof; and real-data evidence is qualitative, with some blur and artifacts remaining. A plausible implication is that the framework establishes a practical and conceptually clear notion of trajectory-controlled unfolding without yet closing the gap between suggestive flow-inspired intuition and a fully developed theory of stage-wise constrained dynamics.

In that sense, the trajectory-controllable unfolding framework is best understood as a design principle for iterative models: control not only the final state but also the route taken to reach it, supervise intermediate states against a meaningful progression, and build the architecture so that information transfer across stages is itself trajectory-aware. FLoUNet provides the clearest explicit formulation of that principle (Wang et al., 15 Sep 2025).