Iterative CT Reconstruction Using Preconditioned Hessian Conditioning
Find Innovative SolutionsGenerate Solutions
Solution Overview
Problem
Conventional iterative tomographic reconstruction methods, particularly those using statistical weighting matrices, face challenges in maintaining the convergence rate due to poorly conditioned Hessian matrices, leading to increased iterations for achieving desired accuracy in image reconstruction.
Innovation Solution
The introduction of an auxiliary variable and a weighting operator or filter that causes the Hessian with respect to the image of the cost function to be well-conditioned, allowing for efficient iterative reconstruction without degrading the ramp filter approximation, thus maintaining mathematical equivalence with the original update process.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If statistical weighting matrices are used in iterative reconstruction update steps, then image quality and data fidelity are improved, but the Hessian matrix becomes poorly conditioned, degrading convergence rate
Solution Approach 1:
The patent introduces a preconditioner matrix P as an intermediary component in the iterative update equation: x_{n+1} = x_n + P^{-1} * (data fidelity gradient + regularization gradient). This preconditioner matrix serves as a mediator that transforms the poorly conditioned Hessian into a well-conditioned system, enabling statistical weighting matrices to be used while maintaining fast convergence. The preconditioner effectively decouples the conflict between using statistical weighting for image quality and maintaining convergence rate.
Solution Approach 2:
The patent modifies the optimization parameters by introducing a preconditioning matrix P that changes the metric space in which the optimization occurs. By transforming the Hessian matrix through this parameter change, the system achieves well-conditioned convergence while preserving the beneficial effects of statistical weighting matrices for image quality reconstruction.
2Object-affected harmful factors
If iterative reconstruction methods are used to reduce x-ray dose, then radiation exposure is reduced, but image reconstruction quality deteriorates due to under sampling
Solution Approach 1:
The patent implements an iterative feedback process where the reconstructed image is continuously compared with the measured data through the data fidelity term. The gradient of this difference drives the iterative updates, allowing the system to progressively improve the reconstruction quality while operating at reduced dose levels. The feedback mechanism enables the system to compensate for the reduced amount of data by systematically correcting errors in the reconstruction.
Solution Approach 2:
The patent combines multiple functional components into a composite reconstruction system: statistical weighting matrices for data fidelity, edge-preserving regularization for quality maintenance, and preconditioning for computational efficiency. This composite approach allows the system to achieve high reconstruction quality at reduced doses by integrating the strengths of each component while mitigating their individual weaknesses.
3Measurement precision
If edge-preserving regularization is applied to maintain image resolution, then boundary sharpness is improved, but computational complexity increases
Solution Approach 1:
The patent introduces a preconditioner matrix P as an intermediary that simplifies the computational structure of the iterative updates. By incorporating the preconditioner, the system can apply edge-preserving regularization more efficiently, reducing the computational burden while maintaining the resolution-enhancing effects. The preconditioner acts as a mediator that enables complex regularization operations to be performed with reduced computational cost.
Data Source
AI summary
Methods and systems for iterative computed tomography add an auxiliary variable to the reconstruction process, while retaining all variables in the original formulation, A weighting operator or filter can be applied that causes the Hessian with respect to an image of the cost function to be well-conditioned. An auxiliary sinogram variable distinct from both a set of actual image measurements and from the set of projections computed based on an image can be applied to iteratively update during the statistical iterative image reconstruction, with applied conditions that causes the Hessian with respect to an image of the cost function to be well-conditioned.