Time-Dependent Tomography

• Time-dependent tomograms are imaging representations that incorporate time as an independent variable into inversion models to reconstruct dynamic processes.
• They employ spatial and temporal regularization strategies, such as penalized least-squares and transform-domain methods, to stabilize underdetermined inverse problems.
• These methods enable applications across astrophysics, quantum state analysis, and biomedical imaging with optimized computational techniques for efficient dynamic reconstruction.

A time dependent tomogram is a mathematical or physical representation that encodes the spatiotemporal evolution of an object, field, or signal by combining the principles of tomographic inversion with explicit modeling of time as an independent variable or coupling parameter. Unlike conventional (static) tomograms—which assume objects are unchanged during data acquisition—time dependent tomograms reconstruct dynamic processes by integrating temporal evolution into forward and inverse models, regularization strategies, and reconstruction algorithms. Such methods are crucial for disciplines where the object's morphology, state, or function changes on timescales comparable to—or faster than—acquisition, including astrophysics (solar corona), condensed matter (phase transitions), biomedical tomography, signal processing, and quantum state characterization.

1. Mathematical Frameworks for Time Dependent Tomography

Time dependent tomograms generalize the static tomographic problem by explicitly modeling temporal evolution within the forward operator and the inversion procedure. In the most general setting, the measurement process is described as

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

where $\mathbf{x}_t$ represents the time-dependent unknown (e.g., 3D emissivity or attenuation at time $t$); $\mathbf{A}_t$ is the projection operator for time $t$ (possibly including system geometry, data acquisition parameters, or the physics of projection); and $\mathbf{y}_t$ is the dataset acquired at $t$.

In dynamic scenarios, the full dataset is stacked: $$\bar{\mathbf{y}} = \operatorname{diag}(\mathbf{A}_1, \ldots, \mathbf{A}_T) \bar{\mathbf{x}} + \bar{\mathbf{n}},$$ with $$\bar{\mathbf{x}} = [\mathbf{x}_1^\top, \ldots, \mathbf{x}_T^\top]^\top$$ and $T$ the number of time steps.

Solving for $\bar{\mathbf{x}}$ is severely underdetermined ($T$ times as many unknowns as in the static case), and standard approaches incorporate temporal and spatial regularization through penalized least-squares or variational frameworks: $$\hat{\bar{\mathbf{x}}} = \underset{\bar{\mathbf{x}}\ge 0}{\operatorname{argmin}} ~ \left\| \bar{\mathbf{y}} - \bar{\mathbf{A}} \bar{\mathbf{x}} \right\|^2_2 + \lambda_s^2 \left\| \bar{S} \bar{\mathbf{x}} \right\|^2_2 + \lambda_t^2 \left\| \bar{T} \bar{\mathbf{x}} \right\|^2_2,$$ where $\bar{S}$ and $\bar{T}$ implement spatial and temporal smoothness constraints, and $\lambda_s$, $\lambda_t$ are regularization parameters (Barbey et al., 2011, Vibert et al., 2016).

Alternative frameworks decompose the time-dependent object as a sum over basis functions in time: $$f(x, y, z, t) \approx \sum_{j=0}^{M-1} f_j(x, y, z) \phi_j(t),$$ reducing the number of unknowns (and effectively compressing the temporal dimension), especially when dynamics are smooth or can be sparsely represented (Nikitin et al., 2018).

Wavelet-domain (Bubba et al., 2021) and shearlet-domain (Bubba et al., 2020) regularizations further exploit spatiotemporal sparsity, enforcing joint space–time priors and effective edge or motion preservation.

2. Regularization and Stabilization of the Inverse Problem

The dynamic tomography inverse problem is often highly ill-posed. Regularization is essential, both for suppressing noise and ensuring physically plausible temporal evolution:

• Spatial and temporal smoothness: Penalizing gradients or higher-order derivatives in space and time suppresses unphysical oscillations or abrupt changes, as in

$$\|\sqrt{ (\partial_x f)^2 + (\partial_y f)^2 + (\partial_z f)^2 + (\lambda_2 \partial_t f)^2 } \|_1$$

used with L1 (total variation) penalties for edge preservation (Nikitin et al., 2018).

• Basis compression: Expanding the temporal dependence in a basis (e.g., DCT, Fourier, wavelets) reduces the effective dimension, as discontinuous or piecewise-smooth dynamics can be represented sparsely (Kadu et al., 2023).
• Model-driven priors: Incorporating explicit physical constraints—such as coherence of coronal structures (solar corona), a single per-voxel event time (DYRECT (Goethals et al., 15 Nov 2024)), or motion priors for rigid, non-rigid, or piecewise-homogeneous objects—enhances interpretability and stabilizes the inversion.
• Advanced transforms: High-dimensional transforms (3D/4D wavelets, 3D shearlets) provide shift-invariance, directional selectivity, and optimal sparsity for "cartoon-like" evolving objects (Bubba et al., 2021, Bubba et al., 2020).
• Co-rotating regularization: In astrophysical applications, penalizing corotating components suppresses artifacts that track the observer's plane (Vibert et al., 2016).

3. Computational Techniques and Algorithmic Strategies

Time-dependent reconstructions typically require significant computational resources due to increased data and unknowns. Algorithms are optimized to handle these challenges via:

• Parallelized and high-performance implementations: Projection and back-projection are often parallelized on multicore CPUs (OpenMP (Barbey et al., 2011)) and/or GPUs (CUDA (Nikitin et al., 2018)) to achieve tractable inversion times with large numbers of time steps and parameters.
• Iterative Primal–Dual Algorithms: Non-differentiable penalties (e.g., L1 or TV norms) are efficiently solved with primal–dual algorithms such as Chambolle–Pock or Primal–Dual Fixed Point (PDFP), with proximity operators handling constraints and regularization terms.
• Variable Projection and Bilinear Models: Projection-domain separable modeling (e.g., ProSep (Iskender et al., 2022)) exploits bilinear factorization of dynamic tomograms, enabling efficient estimation of spatial and temporal components via variable projection, with theoretical guarantees of uniqueness and stable recovery under appropriate sampling.
• Self-supervised deep learning: For extreme-angular-sparsity settings, 4D deep networks with tensorial factorization and small MLP regressors reconstruct high-fidelity dynamic tomograms using only a few degrees of angular coverage, enabling unprecedented temporal resolution (see STRT (Hu et al., 15 Apr 2025)).
• Physics-guided forward models: In fields such as solar or cloud tomography, detailed physics-based forward operators (e.g., inclusive of scattering or absorption) are combined with spatiotemporal regularization, and inversion is achieved via gradient-based optimization (Ronen et al., 2020).

4. Application Domains and Case Studies

Astrophysics: Solar Corona

Time-dependent tomograms reconstruct 3D coronal emissivity by incorporating the temporal evolution of the corona, as seen in STEREO/EUVI and LASCO-C2 white-light data (Barbey et al., 2011, Vibert et al., 2016). Spatiotemporal and co-rotating regularization suppress plane-of-sky artifacts, achieving RMS error and artifact reductions over static models.

Quantum State Tomography

In quantum optics, the time-dependent tomogram is a rotated quadrature distribution directly accessible via homodyne detection, with time evolution tracked through linear differential equations on moments, providing a window into decoherence and the dynamics of nonclassical states (Filippov et al., 2011, Rohith et al., 2015, Rohith, 2016).

Biomedical and Materials Tomography

Methods such as 4D shearlet-based variational regularization (Bubba et al., 2020) or the event-based DYRECT model (Goethals et al., 15 Nov 2024) enable accurate temporal localization of dynamic events (e.g., phase transitions, fluid flow) with minimal data, reducing both radiation dose and memory requirements.

Signal Processing

Time-dependent tomograms also generalize to strictly positive bilinear time–frequency transforms, providing robust, artifact-free component separation and denoising for nonstationary signals, especially when operator pairs are adapted to the features of interest (Aguirre et al., 2012).

5. Temporal Resolution, Sampling Strategies, and Performance Considerations

The temporal resolution achievable by time dependent tomograms depends on the interplay between acquisition protocols, model assumptions, and algorithmic advances:

• Small angular range per frame: Approaches like STRT (Hu et al., 15 Apr 2025) show that using ~10x smaller angular ranges per reconstructed frame (e.g., 18° instead of 180°) leads to a commensurate increase in temporal fidelity, provided the underlying object is sufficiently static within these windows and spatial–temporal priors (via 4D deep learning or tensor decomposition) can fill in missing information.
• Event-based models: Event-based representations (DYRECT) can resolve events at sub-frame timescales (less than a tenth of a standard CT rotation), reducing data storage and making real-time analysis tractable.
• Sampling design: Theoretical guarantees (ProSep) tie reconstruction stability to the number and distribution of projection angles and to the partial separability (low-rankness) of spatiotemporal variations (Iskender et al., 2022).
• Resource requirements: High-dimensional inversion is alleviated by basis decomposition, spatial–temporal regularization, and GPU acceleration, allowing for tractable run times (e.g., ≪ 8 hours for 132 million parameters in solar dynamic inversion (Barbey et al., 2011), or 7–11x GPU/CPU acceleration (Nikitin et al., 2018)).

6. Limitations, Extensions, and Future Research Directions

Although time dependent tomograms have achieved significant progress, several active areas remain:

• Generality vs. efficiency: Highly specialized (e.g., single-transition, event-based) models are compact and efficient but may not capture complex, continuous, or repeating dynamics without further extension.
• Integration of spatial priors: Many methods operate voxelwise; incorporating spatial constraints (TV, wavelets, shearlets, deep generative priors) could enhance robustness and denoising, especially in high-noise or data-sparse regimes.
• Nonlinear and diffractive tomography: Extending time-dependent modeling to nonlinear or diffractive regimes (e.g., phase retrieval under time-dependent rotation (Beinert et al., 2022)) increases analytical and algorithmic complexity.
• Sampling limitations and artifact control: Extremely small angular coverage may still induce blurring, especially in regions with rapid changes; balancing acquisition speed and data completeness is an open area (see periodic artifact observation in STRT (Hu et al., 15 Apr 2025)).
• Multimodal fusion and open datasets: As dynamic imaging extends to more challenging domains (e.g., atmospheric cloud evolution (Ronen et al., 2020), dynamic materials under stress), fusion with other measurement modalities and large-scale benchmark datasets will further advance the field.
• Automated hyperparameter and basis selection: Adaptive schemes for choosing regularization strength, basis dimension, and network architecture (in deep learning approaches) will be critical, especially as complexity and data diversity increase.

7. Summary Table: Key Methodological Components

Domain / Application	Temporal Model	Regularization	Computational Strategy
Solar corona (STEREO, LASCO)	Time-sliced emissivity maps	Spatio-temporal, co-rotating	Parallelized projection (OpenMP)
Dynamic CT, plant studies	Temporal basis expansion (Fourier, DCT, wavelet, shearlet)	3D Shearlet, TV, L1, 4D wavelet	GPU-accelerated primal–dual
Event-based dynamics	Per-voxel transition times	Implicit via projection fitting	Modified SIRT iterations
Quantum state evolution	Quadrature/moment evolution	N/A (analytic/linear models)	Analytic, direct
Deep learning (STRT, X-Hexplane)	Tensorial factorization (feature planes)	Spatial–temporal TV, deep priors	Self-supervised, 4D deep networks

Time dependent tomograms thus form a foundational technology for high-resolution, temporally-accurate imaging, enabling the quantitative paper of dynamic phenomena across domains. Their continued methodological development and integration with physics-based, machine learning, and computational science tools are expected to further enhance their utility in both academic and industrial contexts.