Gradient Descent Provably Solves Nonlinear Tomographic Reconstruction

Area of Science:

• Medical Imaging
• Computational Imaging
• Applied Mathematics

Background:

• Computed tomography (CT) reconstruction typically involves linearizing the Beer-Lambert Law, which is sensitive to high-density materials like metal.
• This nonlinear preprocessing step can lead to numerical instability and artifacts, particularly near metal implants.
• Existing methods struggle with metal artifacts, impacting diagnostic accuracy in CT.

Purpose of the Study:

• To develop and validate a direct nonlinear CT reconstruction method that avoids problematic preprocessing.
• To demonstrate the provable convergence and efficiency of this direct nonlinear approach.
• To reduce metal artifacts in CT reconstruction and improve image quality.

Main Methods:

• Directly reconstructing CT data using the nonlinear forward model without intermediate linearization.
• Applying gradient descent optimization to a nonconvex objective function, proving convergence to the global optimum.
• Incorporating prior structural information via constraints for under-determined signal reconstruction.

Main Results:

• Gradient descent provably converges to the global optimum at a geometric rate for direct nonlinear CT reconstruction.
• The method achieves near-perfect signal reconstruction with a minimal number of measurements, even in under-determined cases.
• Cone-beam CT experiments show significant reduction in metal artifacts compared to standard linear methods.

Conclusions:

• Direct nonlinear CT reconstruction offers a more robust and artifact-free alternative to conventional methods.
• The proposed technique effectively mitigates metal artifacts, enhancing the utility of CT imaging.
• Logarithmic preprocessing itself can introduce artifacts, highlighting the benefits of direct nonlinear reconstruction.