Fast computation of 3D radon transform via a direct Fourier method

Area of Science:

• Structural biology
• Biomedicine
• Medical imaging
• Electron tomography

Background:

• Three-dimensional (3D) data arrays are essential in structural biology, biomedicine, and clinical imaging.
• The Radon transform is crucial for manipulating 3D data, particularly for solving inverse tomographic problems where data are collected as projections.
• Efficient algorithms are needed for the Radon transform in applications like electron tomography.

Purpose of the Study:

• To introduce a direct Fourier method (DFM) for computing the 3D Radon transform.
• To demonstrate the DFM's implementation using coordinate transformations in 2D Fourier space.
• To highlight the algorithm's utility in electron tomography and related fields.

Main Methods:

• A direct Fourier method (DFM) is proposed to compute the 3D Radon transform.
• The algorithm utilizes a two-step process involving coordinate transformations in 2D Fourier space.
• The DFM is invertible, allowing density distribution retrieval from the Radon transform.

Main Results:

• The DFM provides a fast and accurate computation of the 3D Radon transform for sampled functions with compact support.
• The algorithm facilitates rapid retrieval of any projection from a structure's Radon transform, crucial for electron tomography angular refinement.
• Computational times for projection computation using the DFM are favorably compared against other algorithms.

Conclusions:

• The proposed DFM offers an efficient and accurate approach for calculating the 3D Radon transform.
• This method has significant implications for electron tomography, particularly in angular refinement processes.
• Potential applications extend to 'projection onto convex sets' (POCS) methods.