Recovering missing slices of the discrete Fourier transform using Ghosts

Area of Science:

• Image reconstruction
• Signal processing
• Applied mathematics

Background:

• The discrete Fourier transform (DFT) is crucial for solving inverse problems with incomplete frequency information.
• Missing data in Fourier space leads to systematic artifacts known as Ghosts.
• Existing methods for Ghost artifact removal can be computationally intensive or iterative.

Purpose of the Study:

• To present a fast and exact method for deconvolving cyclic artifacts (Ghosts) in DFT.
• To introduce a new cyclic theory of Ghosts that unifies previous work.
• To demonstrate the method's application in image reconstruction from asymmetric projections.

Main Methods:

• Utilizes redundant image regions to deconvolve cyclic artifacts.
• Based on the discrete Fourier slice theorem and projective Discrete Radon Transform.
• Employs a novel cyclic theory of Ghosts for artifact analysis.

Main Results:

• Achieves a computational complexity of O(n log(2) n) for an n=N×N image.
• Provides an exact, non-iterative method for Ghost artifact removal.
• Successfully applied to image reconstruction from sparse, asymmetric projection data.

Conclusions:

• The proposed method offers an efficient and accurate solution for handling Ghosts in DFT.
• The new cyclic theory of Ghosts provides a unified framework for understanding these artifacts.
• The technique enables fast and exact image reconstruction in challenging scenarios with sparse data.