Fast finite Hilbert transform via DE quadrature scheme.

Area of Science:

• Numerical Analysis
• Signal Processing
• Medical Imaging

Background:

• The finite Hilbert transform (FHT) and inverse finite Hilbert transform (IFHT) are crucial for filtering back-projected data in chord-line based computerized tomography (CT) reconstruction algorithms.
• Efficient computation of FHT is essential for improving CT image reconstruction speed and accuracy.

Purpose of the Study:

• To implement, enhance, and validate a fast numerical solution for the FHT using a double exponential (DE) integration scheme.
• To develop a strategy for computing the inverse finite Hilbert transform (IFHT) using a similar approach.

Main Methods:

• A double exponential (DE) integration scheme was employed to address potential floating-point underflow issues.
• Variable transformation ranges and integration levels were optimized numerically by analyzing FHT error maps and contours to determine optimal parameters for fast FHT computation.
• Known analytical FHTs were used to validate the implemented DE scheme.

Main Results:

• The study successfully implemented and validated a fast numerical FHT via the DE scheme.
• Numerical methods were established to determine the optimal integration level and variable transformation range for a given precision and signal fluctuation.
• These optimized parameters allow for efficient FHT computation of signals with similar or lower fluctuation degrees.

Conclusions:

• The DE-based FHT and the associated numerical optimization method provide an efficient approach for FHT computation.
• This method is particularly applicable to data filtering tasks within chord-line based CT reconstruction algorithms, enhancing processing speed.