Total Variation Regularization in Computed Tomography Image Reconstruction

Total variation (TV) regularization serves as a robust mathematical framework in computed tomography (CT) reconstruction, effectively suppressing noise while maintaining critical edge information. In CT systems, reconstructed images frequently exhibit artifacts and noise patterns due to sparse projection data or low-dose acquisition protocols. TV regularization mitigates these issues by imposing a penalty on high-frequency variations through gradient-based minimization, preserving structural integrity through L1-norm optimization.

The fundamental principle involves minimizing the total variation functional, calculated as the integral of absolute gradient magnitudes across the image domain. This formulation promotes piecewise-constant solutions through gradient descent algorithms or compressed sensing techniques. Implementation typically involves alternating between data fidelity updates (e.g., filtered backprojection) and TV-proximal operations using split-Bregman or primal-dual optimization methods. The core TV minimization problem can be expressed as: argmin_x ||Ax-b||₂² + λTV(x), where A represents the system matrix and λ controls regularization strength.

A significant advantage of TV regularization in CT is its capability to achieve diagnostic-quality reconstructions with substantially reduced projection counts, enabling lower radiation doses. Computational challenges include convergence speed and potential texture loss in homogeneous regions. Enhanced variants like anisotropic TV (directional gradient weighting) and patch-based TV have been developed to preserve fine structures while maintaining noise suppression capabilities. Algorithm implementation often utilizes GPU acceleration for practical reconstruction times.

Integrating TV regularization into iterative reconstruction pipelines establishes an optimal balance between noise reduction and feature preservation, proving particularly valuable for low-dose CT, sparse-view CT, and limited-angle tomography applications where conventional filtered backprojection methods exhibit significant limitations.