Cross-Directional Model

• Cross-Directional Model is a framework that formalizes multidimensional, non-separable dependencies across spatial, modal, or angular dimensions in complex systems.
• It employs advanced mathematical formulations, modal decompositions, and fusion architectures to accurately model and control coupled behaviors.
• Applications span process control, wireless communications, computer vision, and multi-modal learning, demonstrating improved prediction accuracy and robust generalization.

A cross-directional model formalizes multidimensional dependencies or couplings that occur across spatial or modality axes in diverse engineering, physical, and data-intensive systems. Such models can appear in industrial process control, large-scale feedback systems, communications, computer vision, multi-modal deep learning, and statistical modeling. The core unifying principle is encoding, inferring, or controlling relationships that are non-separable and potentially strongly coupled across distinct "directional" dimensions—which can be spatial (e.g., web width in roll-to-roll processes), beam angle (multi-directional wireless channels), feature streams (vision, remote sensing), or even modality (RGB/NIR/TIR for re-ID). Multiple mathematical, architectural, and statistical frameworks have been developed to exploit these cross-directional structures for modeling accuracy, controllability, learning performance, or statistical generality.

1. Mathematical Formulations and Physical Derivations

Cross-directional models in process control and physical systems are typically introduced to capture the multidimensional, spatially coupled relationships between control inputs and system outputs. In large-scale "cross-directional" (CD) processes (e.g., slot-die coating, synchrotron orbit correction), the plant is described as

$y(s) = R\,g(s)\,u(s) + d(s)$

where $R$ is a (potentially ill-conditioned) spatial response matrix, $g(s)$ a common actuator dynamic, and $y,u,d$ are the profile, input, and disturbances, respectively. The cross-directional nature is encoded via the full non-diagonal matrix $R$, which couples all actuator and measurement channels spatially. Reduced-order state-space models are identified by projecting high-fidelity dynamics (e.g., Navier–Stokes for slot-die coating) onto low-dimensional bases, yielding delay-free or delayed state-space systems, typically coupled in space through a DC gain matrix $H$: $$\dot\zeta(t) = A\zeta(t) + B\delta q(t), \quad \delta h(x_s,t) = C\zeta(t)$$ with collective output relation for all cross-directions

$\mathbf{Y}(s) = G(s)\,H\,\mathbf{Q}(s)$

where $G(s)$ models convective/relaxational dynamics and $H$ is empirically identified via CFD or experimental data (Kim et al., 4 Jan 2026).

Control-oriented cross-directional model approximations further leverage symmetries: $$X \in \mathcal{S} \subset \mathbb{R}^{n_{b_y}n \times n_{b_u}n}: \quad X(\Pi \otimes I) = (\Pi \otimes I)X$$ where block-circulant or block-centrosymmetric symmetries allow for decoupled modal controllers (e.g., via FFT) when $R$ (or its approximation $R_\mathcal{S}$) commutes with the symmetry (Kempf et al., 2023). When exact symmetries are unavailable, semidefinite programs are constructed to approximate the response while trading off nominal stability, performance, and robustness metrics.

2. Identification and Inference Methodologies

Cross-directional system identification can rely on data-driven or physics-informed techniques, depending on the context:

• Process Control/Physical Systems: System matrices (e.g., $H$ in coating, $R$ in synchrotrons) are identified through pseudo-random binary sequence excitations on actuators and least-squares fitting using measured outputs and internal model reductions (Kim et al., 4 Jan 2026, Stuart et al., 2024). Modal decompositions (SVD) let one design identification and control procedures mode-by-mode, mitigating ill-conditioning and strong directionality.
• Deep Learning and Vision: Cross-directional relationships are operationalized through network modules that fuse or attend to feature information across spatial, temporal, or modal axes. In remote sensing or damage assessment, the cross-directional feature fusion strategies comprise sequential channel-wise and spatial-wise recalibration of paired streams (e.g., pre- and post-disaster image features), with exchange operations:
  • Channel-wise: $$U^\text{cha}_\text{pre} = I_\text{cha} \odot U_\text{post} + U_\text{pre}$$
  • Spatial-wise: $$U^\text{spa}_\text{post} = I_\text{spa} \odot U^\text{cha}_\text{pre} + U_\text{post}$$
  • after extracting global statistics and learning cross-directional attention weights (Shen et al., 2020, Shen et al., 2021).
• Communications: In cross-band, multi-directional signal prediction for 5G/6G coverage, data-driven models (e.g., CommUNext) predict high-frequency, multi-beam RSS maps by integrating cross-band and directional priors within deep encoder–decoder architectures, parameterizing the input tensor as a stack of low-frequency context and sparse high-frequency directional anchors (Sung et al., 8 Nov 2025).

3. Cross-Directional Fusion and Attention Architectures

In neural architectures, cross-directional mechanisms are specifically designed to exploit paired structures:

• Feature Fusion Blocks: Cross-directional fusion blocks perform paired exchange at every scale in a shared-encoder network. They execute a sequence of channel-wise and spatial-wise fusion by exchanging recalibration weights across two (or more) parallel streams (e.g., pre- and post-disaster, or different sensor modalities). Both (Shen et al., 2020) and (Shen et al., 2021) formalize the fusion as a two-stage scheme:
  1. Global pooling and FC/sigmoid "squeeze" for channel-attention, applied cross-stream.
  2. 1×1 convolution and sigmoid for spatial mask, again applied cross-stream.
  3. Residual addition, retaining original branch context. These mechanisms improve change detection and fine-grained localization.

• Cross-Directional (Sample/Modality) Center Loss: In multi-modal representation learning (e.g., vehicle re-ID), cross-directional center loss aligns features not only within each modality's sample batch, but simultaneously brings the sample centers and modality centers closer for each identity label. This joint alignment encourages compactness across all cross-modality axes, improving generalizability in heterogeneous environments (Zheng et al., 2022).

• Bi-Directional Transformers: BiXT introduces a bi-directional cross-attention module where latent and input token feature sets attend to one another via a shared, efficiently computed attention matrix, supporting bidirectional (semantic ↔ spatial) information flow. This architecture enables scalable long-sequence modeling while maintaining expressivity for dense prediction tasks (Hiller et al., 2024).

4. Applications Across Scientific and Engineering Domains

Cross-directional models are found in diverse application settings:

• Industrial Web and Sheet Processes: Slot-die battery electrode coating and basis-weight or thickness control in papermaking and metal rolling are canonical domains. Spatial coupling is critical to accurately model, predict, and control the material profile (Kim et al., 4 Jan 2026, Kempf et al., 2023, Stuart et al., 2024).
• Large-Scale Feedback Control: Synchrotron orbit correction systems employ cross-directional modeling, SVD-based mode decoupling, and closed-loop sensitivity identification to ensure robust, precise beam stabilization despite high-dimensional ill-conditioning (Kempf et al., 2023, Stuart et al., 2024).
• Wireless Communications: Cross-directional deep models predict multi-directional signal strength maps, essential for beam management and coverage modeling in future cellular networks (Sung et al., 8 Nov 2025).
• Computer Vision/Remote Sensing: Change detection and building damage assessment use cross-directional fusion and attention to optimize difference-based segmentation in pre/post pairs (Shen et al., 2020, Shen et al., 2021).
• Multi-Modal Representation Learning: Multi-spectral re-ID leverages cross-directional consistency principles to structure learning over heterogeneous visual streams, improving embedding quality and retrieval metrics (Zheng et al., 2022).
• Human Dynamics and Pedestrian Flow: Directional statistics-based cross-directional models of the pedestrian fundamental diagram use angular variance statistics $\nu_1, \nu_2$ to encode the effects of flow directionality on capacity penalties, unifying uni-, bi-, and crossing flows in a single analytic framework (Nagasaki et al., 2024).

5. Control and Optimization Strategies

For control in cross-directional systems, standard decoupled SISO strategies are inadequate due to strong directional couplings:

• Modal Decoupling and Feedback: SVD or symmetry-based transform decouples the plant into independent or approximately independent modes, making SISO feedback (with scalar or diagonalized controllers) feasible and computationally tractable. Proportional or IMC-based controllers are designed in modal space, with explicit calculation of gains to ensure tracking, stability, and robustness:

$K_P = \beta H^{-1}, \quad \beta > 0$

yielding steady-state response decoupled across modes (Kim et al., 4 Jan 2026, Kempf et al., 2023).

• Symmetry-Aware Model Reduction: When symmetries are approximate, model approximation is performed via SDPs that jointly constrain spectral radius, performance, and robustness, yielding better closed-loop properties than pure Frobenius-norm minimization (Kempf et al., 2023).
• Closed-Loop Identification: Modal reference injection with amplitude bounds provides safe, accurate closed-loop system identification, essential for performance assessment in feedback-saturated, ill-conditioned plants (Stuart et al., 2024).
• Deep Learning Optimization: Multi-task loss functions balance reconstruction, discrimination, and segmentation objectives, regularizing cross-directional model training for robustness against measurement gaps and data sparsity (Sung et al., 8 Nov 2025, Zheng et al., 2022).

6. Empirical Performance and Validation

Cross-directional models generally yield improved empirical results over non-cross-directional baselines:

• Process Control: Cross-directional models achieve validation RMSE near 3.7–4 μm (≈4–4.5% of nominal thickness), coefficients of determination $R^2$ ≈ 0.98, and enable closed-loop convergence and uniformity within 1–2 μm for slot-die coating (Kim et al., 4 Jan 2026). SDP-based symmetric approximations reduce controller gain pathologies and maintain robust stability (Kempf et al., 2023).
• Wireless Prediction: CommUNext achieves median MAE ≈ 1.49–1.80 dB and RMSE ≈ 3.01–3.37 dB for multi-directional 7 GHz coverage maps, with millisecond inference and significant measurement overhead reduction (Sung et al., 8 Nov 2025).
• Vision/Remote Sensing: Cross-directional attention and fusion modules yield up to +0.6 F1 point improvement over state-of-the-art on large-scale building damage datasets; ablations confirm superior performance, especially in difficult classes (Shen et al., 2020, Shen et al., 2021).
• Re-ID: Multi-spectral vehicle identification achieves mAP gain of +10.8 points, rank-1 gain of +15.8%, and overall best-in-class metrics across several datasets using cross-directional center loss and adaptive normalization (Zheng et al., 2022).
• Pedestrian Dynamics: R² improvements for the cross-directional fundamental diagram model over traditional FDs range from ≈0.45–0.48 to ≈0.65 (train) and 0.62 (test), capturing capacity recovery and penalty effects across all flow regimes (Nagasaki et al., 2024).

7. Conceptual Significance and Generalization

Cross-directional models encode and exploit latent structures that are inherently multidimensional, coupled, and non-separable. By leveraging spatial, modal, angular, or feature-space correlations, these models provide a principled means to enhance prediction, control, interpretability, and generalization in complex systems. The architectural motifs—decoupling via symmetry or SVD, cross-stream fusion, joint representation learning—have demonstrated broad transferability across engineering, physical, and data-driven domains. The principled balance between computational tractability, robustness, and empirical accuracy is a defining feature, enabled by discipline-specific mathematical abstractions and optimization strategies.

Key references include (Sung et al., 8 Nov 2025, Shen et al., 2021, Kim et al., 4 Jan 2026, Shen et al., 2020, Nagasaki et al., 2024, Zheng et al., 2022, Hiller et al., 2024, Kempf et al., 2023), and (Stuart et al., 2024).