The fast discrete Radon transform. I. Theory

Area of Science:

• Image processing
• Computational mathematics
• Medical imaging

Background:

• The discrete Radon transform (DRT) is crucial for image reconstruction from projections.
• Existing inversion algorithms, like Beylkin's (1987), struggle with steep slopes and nonlinear variations.
• These limitations hinder accurate image reconstruction in various scientific fields.

Purpose of the Study:

• To present a novel inversion scheme for image reconstruction using the slope-intercept form of the DRT.
• To overcome the dispersion and inversion limitations of prior DRT algorithms.
• To develop fast, generalized algorithms for both forward and inverse Radon transforms.

Main Methods:

• Formulating a discrete computation of the continuous Radon transform.
• Deriving generalized inversion methods that address shortcomings of previous algorithms.
• Developing generalized forward (FRT) and inverse (IFRT) algorithms.

Main Results:

• The proposed generalized inversion methods effectively overcome limitations of earlier DRT algorithms.
• The new algorithms eliminate the need for interpolation calculations.
• Direct conversion between raster scan grids and rectangular/polar grids is achieved in a single step.

Conclusions:

• The developed generalized FRT and IFRT algorithms offer a significant advancement in image reconstruction.
• These methods provide faster, more accurate, and versatile image reconstruction capabilities.
• The approach is applicable to various fields requiring image reconstruction from projections.