Tensor-based formulation and nuclear norm regularization for multi-energy computed tomography (1307.5348v1)

Abstract: The development of energy selective, photon counting X-ray detectors allows for a wide range of new possibilities in the area of computed tomographic image formation. Under the assumption of perfect energy resolution, here we propose a tensor-based iterative algorithm that simultaneously reconstructs the X-ray attenuation distribution for each energy. We use a multi-linear image model rather than a more standard "stacked vector" representation in order to develop novel tensor-based regularizers. Specifically, we model the multi-spectral unknown as a 3-way tensor where the first two dimensions are space and the third dimension is energy. This approach allows for the design of tensor nuclear norm regularizers, which like its two dimensional counterpart, is a convex function of the multi-spectral unknown. The solution to the resulting convex optimization problem is obtained using an alternating direction method of multipliers (ADMM) approach. Simulation results shows that the generalized tensor nuclear norm can be used as a stand alone regularization technique for the energy selective (spectral) computed tomography (CT) problem and when combined with total variation regularization it enhances the regularization capabilities especially at low energy images where the effects of noise are most prominent.

• The paper introduces a novel tensor-based framework and nuclear norm regularization techniques to improve multi-energy computed tomography reconstruction.
• Key methods include TNN-1 and TNN-2 regularizers, leveraging tensor unfolding and t-SVD to exploit low-rank structures in spectral data.
• Results show significant improvements in noise suppression and reconstruction accuracy compared to traditional methods, enhancing material detection.

Tensor-Based Formulation and Nuclear Norm Regularization for Multi-Energy Computed Tomography

This paper presents a computational advancement in the field of multi-energy computed tomography (CT) leveraging tensor-based mathematical structures. Recent improvements in photon counting detectors (PCDs) enable the differentiation of X-ray photons based on energy, which permits the reconstruction of material-specific attenuation profiles. Traditional CT systems fail to fully utilize this spectral data due to noise and complex inversion dynamics. The authors propose a novel tensor-based framework and regularization strategies focused on addressing these challenges.

Methodological Contributions

The authors propose a tensor-based model, conceptualizing the multi-spectral data as a three-way tensor with two spatial dimensions and one energy dimension. This model leverages the inherent dimensionality and structured correlations of the dataset for improved reconstruction quality. Two primary contributions of the paper stand out:

1. Generalized Tensor Nuclear Norm Regularizer (TNN-1): Utilizing the traditional tensor unfolding strategy, the authors extend the matrix nuclear norm minimization approach to tensors. Each unfolding of the tensor is treated independently, with the nuclear norm applied along each mode. This method exploits the low-rank structure present in the data, and the flexibility of the regularization approach allows for adjustment across different modes—aiding in the handling of datasets with varied redundancies.
2. Tensor Nuclear Norm Based on Tensor-SVD (TNN-2): This method introduces a novel tensor nuclear norm using tensor singular value decomposition (t-SVD). The t-SVD facilitates the decomposition of a tensor into a sum of outer products, enhancing the capability to capture multi-linear relationships in the data. TNN-2 potentially supplies an optimality akin to matrix SVD for tensors.

Both regularizers are integrated with total variation (TV) regularization, further enhancing edge preservation and noise mitigation in the reconstructed CT images. The paper details an Alternating Direction Method of Multipliers (ADMM) algorithm tailored to these regularizations for efficient optimization.

Implications and Results

The implications of this research are profound in multi-material detection and classification tasks across medical imaging and security screening. The tensor regularization capitalizes on spectral CT's ability to provide material composition insights, crucial for distinguishing between elements like tissues or hidden threats in luggage screening.

Numerically, the models demonstrate significant improvement over conventional methods, especially under low photon count conditions common in low-energy quantum settings. The tensor-based approaches, when combined with TV regularization, substantially outperform filtered back projection or 3D-TV standalone algorithms in terms of noise suppression and reconstruction accuracy.

Future Directions

The paper suggests several avenues for further research. Extension to even higher-dimensional tensors could be explored, addressing problems like dynamic CT imaging where time or additional spatial dimensions are pertinent. Additionally, integration of sparse decomposition techniques may further improve reconstruction fidelity by differentiating structural features. Exploring automatic determination of regularization parameters could enhance the practical deployment of these algorithms.

In conclusion, this paper advances the understanding of how multi-energy CT can effectively utilize spectral information through innovative tensor-based formulations and regularizations. These developments promise to refine the capabilities of spectral CT systems, with significant implications for diagnostics and materials characterization.