Space-Variant Total Variation boosted by learning techniques in few-view tomographic imaging

TL;DR

This work addresses reconstruction from highly undersampled tomographic data by introducing space-variant Total Variation (TV) regularization, where pixelwise weights are derived from gradient magnitudes of a pre-estimate image. The principal method combines a data fidelity term with a weighted TV prior $TV_{\mathbf{w}}(\mathbf{x}) = ||\mathbf{w} \odot |\mathbf{D}\mathbf{x}|||_1$, with weights computed via TpV-inspired reweighting and a reconstructor $\Psi$ that supplies the pre-estimate, possibly including neural network enhancements. Theoretical results establish uniqueness of the TV-regularized solution in underdetermined settings and extend to the space-variant case, while the algorithmic framework (Chambolle-Pock) enables efficient optimization. Numerical experiments on synthetic and real chest CT data demonstrate superior edge preservation and reduced streaking artifacts compared with global TV, with the best results obtained when weights are derived from high-quality gradient information, including gradient-focused neural reconstructions. The framework is flexible and extensible to other regularizers and learning-based pre-estimates, offering a practical pathway to high-quality reconstructions from sparse tomographic measurements.

Abstract

This paper focuses on the development of a space-variant regularization model for solving an under-determined linear inverse problem. The case study is a medical image reconstruction from few-view tomographic noisy data. The primary objective of the proposed optimization model is to achieve a good balance between denoising and the preservation of fine details and edges, overcoming the performance of the popular and largely used Total Variation (TV) regularization through the application of appropriate pixel-dependent weights. The proposed strategy leverages the role of gradient approximations for the computation of the space-variant TV weights. For this reason, a convolutional neural network is designed, to approximate both the ground truth image and its gradient using an elastic loss function in its training. Additionally, the paper provides a theoretical analysis of the proposed model, showing the uniqueness of its solution, and illustrates a Chambolle-Pock algorithm tailored to address the specific problem at hand. This comprehensive framework integrates innovative regularization techniques with advanced neural network capabilities, demonstrating promising results in achieving high-quality reconstructions from low-sampled tomographic data.

Figures (6)

• Figure 1: A plot of $\left(\boldsymbol{w}_\eta(\tilde{\boldsymbol{x}})\right)_i$ for different values of $\eta$, over $| \boldsymbol{D} \tilde{\boldsymbol{x}} |_i$.
• Figure 2: Visual depictions of the proposed $\Psi$-W$\ell_1$ method, illustrating the key computational steps of our proposal, spanning from the sinogram to the final reconstruction. On the top, the scenario where $\Psi$ is implemented by a single solver; on the bottom, the case where it is formed through a composition of two functions.
• Figure 3: Experiment on the synthetic image with low noise ($\nu=0.005$). In the first row, from left to right: $\tilde{\boldsymbol{x}}^{GT}$, $\tilde{\boldsymbol{x}}^{FBP}$, $\tilde{\boldsymbol{x}}^{TV}$. In the second row: the corresponding images of weights $\boldsymbol{w}_\eta(\tilde{\boldsymbol{x}})$.
• Figure 4: Results of the experiment on the synthetic image with high noise ($\nu=0.02$). In the first row, from left to right: zooms on the lower-right angle of the reconstructions GT-$W\ell_1$, FBP-$W\ell_1$, TV-$W\ell_1$, global TV. In the second row: plot of the objective function (left) and of the Relative Error (right) over the iterations.
• Figure 5: From left to right: ground truth image from Mayo data set, a cropped portion of it, a plot of the RE over iterations for the reconstructions by FBP-$W\ell_1$ and NN-$W\ell_1$ with $\alpha=0, 0.5, 1$ ($\nu=0.005$).

Theorems & Definitions (32)

• Proposition 1
• proof
• Proposition 2
• proof
• Proposition 3
• proof
• Proposition 4
• proof
• Proposition 5
• proof