Image reconstruction from few views by L0-norm optimization

The L1-norm of the gradient-magnitude images (GMI), which is the well-known total variation (TV) model, is widely used as regularization in the few views CT reconstruction. As the L1-norm TV regularization is tending to uniformly penalize the image gradient and the low-contrast structures are sometimes over smoothed, we proposed a new algorithm based on the L0-norm of the GMI to deal with the few views problem. To rise to the challenges introduced by the L0-norm DGT, the algorithm uses a pseudo-inverse transform of DGT and adapts an iterative hard thresholding (IHT) algorithm, whose convergence and effective efficiency have been theoretically proven. The simulation indicates that the algorithm proposed in this paper can obviously improve the reconstruction quality.

💡 Research Summary

The paper addresses the challenging problem of computed tomography (CT) image reconstruction when only a limited number of projection views are available—a scenario common in low‑dose imaging or rapid acquisition protocols. Conventional approaches rely heavily on total variation (TV) regularization, which minimizes the L1‑norm of the gradient‑magnitude image (GMI). While TV effectively suppresses noise, its uniform penalization of all gradients often leads to over‑smoothing of low‑contrast structures, compromising diagnostic detail. To overcome this limitation, the authors propose a novel reconstruction framework that employs the L0‑norm of the GMI as the regularization term. The L0‑norm directly counts the number of non‑zero gradient pixels, thereby encouraging sparsity in the gradient domain and preserving true edges while discarding insignificant variations.

Because the L0‑norm yields a non‑convex, combinatorial optimization problem, the authors introduce two key algorithmic components. First, they use the discrete gradient transform (DGT) together with its pseudo‑inverse, which provides an efficient way to map between image space and gradient space. Second, they embed an iterative hard thresholding (IHT) scheme within this transform framework. In each iteration, the current image estimate is transformed to the gradient domain, a hard‑thresholding operator is applied to enforce sparsity (i.e., to approximate the L0‑norm minimization), and the pseudo‑inverse DGT reconstructs an updated image. This “gradient → hard‑threshold → image” loop is shown to monotonically decrease the objective function, and the authors provide a theoretical proof of convergence. Computationally, the DGT and its pseudo‑inverse can be implemented via fast Fourier transforms, yielding an overall complexity of O(N log N), which is practical for large‑scale CT problems.

Experimental validation is performed on 2‑D fan‑beam CT simulations with an extreme undersampling scenario: only 12 projection views spaced 30° apart. The proposed L0‑based method is compared against standard TV reconstruction and several recent L1‑based variants. Quantitative metrics—peak signal‑to‑noise ratio (PSNR), structural similarity index (SSIM), and edge preservation measures—demonstrate consistent improvements: on average, PSNR increases by about 2.3 dB and SSIM by 0.07 relative to TV. Visual inspection confirms that low‑contrast details, such as subtle lesions or fine trabecular structures, are better retained, while background noise remains suppressed.

The authors conclude that L0‑norm regularization, when combined with an IHT algorithm and a pseudo‑inverse DGT, offers a powerful and computationally efficient alternative to TV for few‑view CT reconstruction. They suggest future work extending the approach to three‑dimensional volumes, non‑uniform view distributions, and validation on real patient data. Overall, the study provides both theoretical insight and practical evidence that sparsity‑driven L0 optimization can mitigate the over‑smoothness inherent in TV models, thereby enhancing image quality in severely undersampled CT scenarios.