Reconstruction algorithm for incomplete projections in the framework of linear operators in normed linear spaces

Area of Science:

• Medical Imaging
• Applied Mathematics

Background:

• Computed tomography (CT) reconstruction relies on complete projection data.
• Incomplete data leads to ill-posed inverse problems, degrading image quality.

Purpose of the Study:

• Introduce a novel linear-vector space notation for computed tomography.
• Develop a consistent algorithm for completing incomplete projection data.
• Enhance the accuracy of CT reconstructions from limited or missing data.

Main Methods:

• Utilized the linearity of the Radon transform and convolution-backprojection.
• Defined a consistency condition for projection data completion.
• Applied regularization methods to solve ill-conditioned linear equations.
• Developed an algorithm exploiting data symmetries for completion.

Main Results:

• The proposed notation is broadly applicable in CT.
• The consistency condition identifies issues with incomplete data.
• The regularization algorithm effectively completes missing projection data.
• Demonstrated quantitative improvements in CT reconstructions using simulated and real data.

Conclusions:

• The new algorithm consistently completes arbitrary missing projection regions.
• This method enhances CT image quality in the presence of realistic data limitations and noise.
• The approach offers a valuable tool for improving CT reconstruction accuracy.