Using Randomized Nyström Preconditioners to Accelerate Variational Image Reconstruction.

Area of Science:

• Computational imaging
• Applied mathematics
• Scientific computing

Background:

• Model-based iterative reconstruction is crucial for inverse problems but faces challenges with large-scale, nonsmooth, and nonconvex minimization.
• Efficient iterative solvers are needed, and preconditioning methods can accelerate convergence.
• Forward models in image reconstruction are often operators, lacking explicit matrices, complicating preconditioner design.

Purpose of the Study:

• To develop computationally inexpensive and effective preconditioners for accelerating image reconstruction.
• To address the challenge of unavailable explicit matrices for forward models.
• To enable on-the-fly computation and application of preconditioners using modern hardware.

Main Methods:

• Adaptation of the randomized Nyström approximation for computing preconditioners.
• Leveraging GPU computational platforms for on-the-fly preconditioner computation.
• Development of efficient application approaches for nonsmooth regularizers (wavelet, total variation, Hessian Schatten-norm).

Main Results:

• Demonstrated acceleration of image reconstruction convergence.
• Effective preconditioners computed without requiring explicit forward model matrices.
• Successful application on image deblurring, super-resolution with impulsive noise, and 2D computed tomography.

Conclusions:

• The proposed randomized Nyström-based preconditioner is efficient and effective for accelerating image reconstruction.
• On-the-fly GPU computation makes the method practical for real-world applications.
• The approach successfully handles various nonsmooth regularizers and reconstruction tasks.