CLARK: Constrained Limited-Angle Reconstruction Kernel

• CLARK is a model-driven framework for limited-angle CT that integrates precomputed, data-adapted reconstruction kernels with spectral filtering and edge-preserving constraints to mitigate severe ill-posedness.
• It employs a truncated singular value expansion and controlled spectral filtering to suppress both streak and oscillatory artifacts, ensuring high-fidelity recovery even in angularly missing regions.
• Its semi-discrete implementation balances interpolation error with regularization, making it a robust solution for practical imaging challenges in industrial and medical applications.

A Constrained Limited-Angle Reconstruction Kernel (CLARK) is a model-driven regularization framework for addressing the severe ill-posedness of limited-angle computed tomography (CT) by combining precomputed, data-adapted reconstruction kernels with spectral filtering and edge-preserving constraints imposed directly on projection data. By integrating these steps, CLARK stabilizes the inversion of the limited-angle Radon transform and suppresses both classical streak and oscillatory (wave-type) artifacts, enabling high-fidelity recovery of features even in angularly missing regions without the need for extensive data-driven learning (Hahn et al., 5 Oct 2025).

1. Theoretical Basis: The Method of the Approximate Inverse and LARK Construction

The foundation of CLARK is the method of the approximate inverse, which reconstructs a smoothed representation of the object function $f$ by convolving with a mollifier $e_x^\gamma$ (with $\int e_x^\gamma(y) dy = 1$), yielding $f^\gamma(x) = \int f(y) e_x^\gamma(y) dy$. For the limited-angle Radon transform $R_\Phi$, one computes a reconstruction kernel $\psi_x^\gamma$ by solving

$R_\Phi^* \psi_x^\gamma = e_x^\gamma,$

so that the smoothed value $f^\gamma(x)$ can be reconstructed by duality:

$$f^\gamma(x) = \langle R_\Phi f, \psi_x^\gamma \rangle = S_\gamma g(x),$$

where $g = R_\Phi f$ and $S_\gamma g(x) = \langle g, \psi_x^\gamma \rangle$.

In the full-view case, analytic inversion is possible. For limited angles, one constructs $\psi_x^\gamma$ via a truncated and filtered singular value expansion using the SVD $(\sigma_{ml}, u_{ml}, v_{ml})$ of $R_\Phi$:

$$\psi_x^{(\gamma,\tau,n)} = \sum_{m=0}^n \sum_{l=0}^m \frac{F_\tau(\sigma_{ml})}{\sigma_{ml}} \langle e_x^\gamma, v_{ml} \rangle u_{ml}.$$

Here, $F_\tau(\sigma)$ denotes a spectral filter and $n$ is the truncation cutoff. This filtered reconstruction kernel—referred to as the Limited-Angle Reconstruction Kernel (LARK)—serves as the core of the model-driven operator, adapting to both the specific data geometry and smoothing requirements.

2. Regularization via Spectral Filtering and Edge-Preserving Data Constraints

The principal challenge with LARK in limited-angle tomography is the exponentially fast decay of singular values for $R_\Phi$, leading to strong amplification of measurement noise and ill-conditioning of the inverse. Direct (unfiltered) truncation readily discards directional information irrecoverably, so CLARK introduces a spectral filter $F_\tau(\sigma)$ (e.g., $F_\tau(\sigma) = \frac{\sigma^2}{\sigma^2+\tau}$ or $$F_\tau(\sigma) = (\sigma/\tau) \arctan(\tau/\sigma)$$, $\tau>0$) to gently attenuate, rather than hard-cut, the small singular values while ensuring $F_\tau(\sigma)/\sigma$ remains bounded.

After this spectral filtering and smoothing, CLARK addresses the residual noise and artifacts—particularly oscillatory (wave-type) features stemming from ill-posed inversion—via a constraint imposed on the measured data. Specifically, with noisy projection data $g^\delta$, a constrained variational denoising is performed:

$$D_\lambda(g^\delta) \in \arg\min_g \left\{ \frac{1}{2} \|g-g^\delta\|^2 + \lambda [P \circ S_\gamma](g) \right\},$$

where $P$ is a penalty functional (typically edge-preserving, e.g., total variation or nonlinear diffusion penalties), and $\lambda$ is a regularization parameter. The final reconstruction is thus realized as

$f_\gamma^\lambda := S_\gamma(D_\lambda(g^\delta)).$

This coupling—LARK-based approximate inversion plus a variational edge-preserving constraint on the data—defines the CLARK methodology.

3. Semi-Discrete Implementation and Error Estimates

In real experimental settings, measurement and object domains are semi-discrete. The unknown $f$ is approximated by interpolating sampled coefficients on a grid via a smooth basis function $\varphi$:

$\Pi_n f = \sum_i f_i \varphi(\cdot - x_i).$

The discrete forward operator is then

$A_\Phi f := \Xi_m R_\Phi \Pi_n f,$

where $\Xi_m$ applies the measurement (sampling) procedure. The discrete approximate inverse kernel $\Psi^(\gamma)$ is constructed by solving

$A_\Phi^T \Psi^\gamma = (E^\gamma)^T$

with $$E^\gamma_{ji} = \int \varphi(x-x_i) e_{z_j}^\gamma(x) dx$$. The filtered kernel is then

$$\Psi^{(\gamma,\tau)} = U \Sigma_\tau V^T (E^\gamma)^T,$$

using the SVD of $A_\Phi^T$.

The reconstruction error at sampled points is governed by both the interpolation error $P_\varphi(h)$ (with $h$ the fill distance) and the norm of the reconstruction kernel:

$$\| (\Psi_h^\gamma)^T g - f_z^\gamma \|_{\mathbb{R}^r} \leq C \|f\|_\varphi [2\sqrt{m} \|\Psi_h^\gamma\|_2 + \sqrt{r}] P_\varphi(h),$$

where $\|f\|_\varphi$ is the native space norm. As the grid is refined ($h \rightarrow 0$), $P_\varphi(h)$ decreases, but $\|\Psi_h^\gamma\|_2$ increases because of ill-conditioning, highlighting the necessity of balancing resolution and regularization.

4. Artefact Suppression and Stabilization Mechanisms

CLARK targets two main artifact sources:

• Streak artifacts: Arising in standard approaches (e.g., filtered backprojection) due to missing angular data.
• Oscillatory (wave-type) artifacts: Emerging in unregularized LARK-type inversions, due to inversion of extremely small singular values.

The spectral filter $F_\tau$ suppresses streaks without discarding all the missing information, and the data constraint—typically implemented through edge-preserving denoising (e.g., strong TV or nonlinear diffusion regularizers)—attenuates rapid oscillations and confines the solution to piecewise-smooth, physically plausible functions. The combination produces reconstructions that preserve sharp features, stabilize against noise amplification, and mitigate direction-dependent artifacts inherent to limited angular coverage.

5. Numerical Validation and Application Scenarios

Extensive validation was performed on both synthetic phantoms (e.g., Shepp–Logan) and real measured data (silicon/aluminum objects; Helsinki Tomography Challenge datasets) (Hahn et al., 5 Oct 2025):

• Standard FBP reconstructions fail in heavily ill-posed scenarios (severe streaking, unrecoverable features in missing angle sectors).
• Unconstrained LARK is capable of filling in the missing regions but introduces spurious oscillatory artifacts.
• CLARK, by combining spectral filtering with an edge-preserving variational constraint, suppresses both types of artifacts and enables recovery of structure even for moderate-to-large missing angles (e.g., 30–50° of missing data).
• In the semi-discrete setting, use of smooth interpolation functions (e.g., radial basis functions) further regularizes the inversion and aligns discrete implementation with the continuous model.

6. Practical Implications and Limitations

CLARK provides a mathematically well-founded, interpretable approach to limited-angle CT in industrial and medical settings where acquisition geometry is fundamentally constrained:

• Exact knowledge of measurement geometry is required for kernel precomputation.
• The ill-conditioning imposed by limited angular coverage cannot be eliminated, but can be mitigated to a practical extent by the coordinated use of spectral filtering and constraints.
• The residual artifacts depend sensitively on the trade-off between denoising strength and information loss due to aggressive regularization.
• Explicit data-driven learning is not central to CLARK, but the framework is conducive to hybridization with learned priors or data-consistent artifact correction as in some follow-up literature (Huang et al., 2019, Gao et al., 2022).

7. Significance, Generalization, and Future Directions

CLARK represents an overview of analytic inversion, spectral regularization, and constraint-based stabilization in the context of severely ill-posed, angularly limited tomographic problems. Its principled structure, capacity for uncertainty quantification, and ability to deliver artifact-suppressed reconstructions in challenging data regimes make it a robust alternative to conventional and purely data-driven methods. Moreover, its modular construction facilitates integration with machine learning for adaptive regularization or for improved edge localization via data-driven microlocal priors (Rautio et al., 2021).