X-ray CT Trajectory Function Evaluation for Numerical Stability
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Solution Overview
Problem
Existing x-ray computed tomography reconstruction methods face numerical instabilities and inaccuracies due to the need to derive the Crofton function, particularly in 3-dimensional reconstructions, leading to unsatisfactory reliability and accuracy.
Innovation Solution
A method that evaluates projection data by determining a trajectory function, transforming it into a frequency domain, and using it to reconstruct objects without needing the derivative of the Crofton function, allowing for flexible and precise evaluation of projection data along any dimension, including 2D, 3D, and 4D, and accommodating non-Tuy curve trajectories.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If the Crofton function derivative is used in reconstruction methods, then the reconstruction can be performed based on the 3-dimensional radon transform and x-ray transform relationship, but numerical instabilities occur leading to unsatisfactory reliability and accuracy
Solution Approach 1:
The patent extracts and eliminates the problematic derivative operation of the Crofton function from the reconstruction process. By formulating a new reconstruction method that does not require this derivative, the patent removes the source of numerical instabilities while maintaining the mathematical relationship between 3-dimensional radon transform and x-ray transform.
Solution Approach 2:
The patent changes the mathematical parameters and operations used in the reconstruction process. Instead of using the derivative of the Crofton function, the patent employs alternative mathematical formulations that achieve the same reconstruction goal without introducing numerical instabilities, thereby improving both reliability and accuracy.
2Reliability
If a weighting function is used to avoid the derivative of the Crofton function, then numerical stability is improved, but the establishment of the weighting function becomes laborious
Solution Approach 1:
The patent removes the need for complex weighting function establishment entirely by formulating a reconstruction method that naturally avoids the derivative of the Crofton function. This extraction of the problematic element simplifies the overall process while maintaining numerical stability.
Solution Approach 2:
The patent creates a self-service reconstruction method where the mathematical formulation itself inherently avoids numerical instabilities without requiring additional corrective measures like complex weighting functions. The method serves its own stability needs through its intrinsic mathematical structure.
3Productivity
If traditional reconstruction methods are used, then projection data can be evaluated, but the process requires derivative calculations that reduce precision and reliability
Solution Approach 1:
The patent changes the mathematical parameters and operations in the projection data evaluation process by eliminating derivative calculations. This parameter change maintains the productivity of evaluating projection data while significantly improving precision and reliability through more stable mathematical operations.
Applied Scientific Principles
This section explains which scientific principles are used to turn an abstract innovation direction into a practical engineering solution.
Function Achieved in This Case
The method provides stable, precise, and reliable evaluation of projection data, overcoming numerical instabilities and enabling flexible reconstruction of objects with improved accuracy and simplicity, especially for elongated objects and those with structural limitations.
Implementation Method 1
In x-ray computed tomography, the weakening intensity of the radiation along different projection beams through an object being examined is measured and used to deduce the density distribution in the object
Data Source
AI summary
In a method and an evaluation device for the evaluation of projection data of an object being examined, which are determined along a trajectory in a multiplicity of projection positions relative to a co-ordinate origin, a particular trajectory function is determined for the projection positions, for each of a multiplicity of positions from a reconstruction region of dimension n by establishing an offset (d) and a direction vector at the co-ordinate origin, establishing a hyperplane of dimension n−1 which runs perpendicular to the direction vector and has an offset to the co-ordinate origin, establishing a number of intersection points where the hyperplane intersects the trajectory, establishing a derivative vector of the trajectory according to its trajectory path and calculating the derivative vector in the projection position, and establishing an absolute value of a scalar product between the derivative vector and the position and dividing the absolute value by the number. The determined trajectory functions are transformed to a frequency domain of dimension n and the projection data are evaluated by means of the transformed trajectory functions.