Affiliations: The City Key Lab of Medical Physics and Engineering,
Peking University, Beijing 100871, China | School of Telecommunication Engineering, Beijing
University of Posts and Telecommunications, Beijing 100876, China
Abstract: Purpose: The finite Hilbert transform (FHT) or inverse finite
Hilbert transform (IFHT) is recently found to have some important applications
in computerized tomography (CT) arena [1-6], where they are used
to filter the derivatives of back-projected data in the chord-line based CT
reconstruction algorithms. In this paper, we implemented, improved and
validated a fast numerical solution to the FHT via a double exponential (DE)
integration scheme. A same strategy can be used to compute IFHT. Methods: To overcome the underflow of floating-point numbers, we first determined the
range of variable transformation from the minimum positive value of single or
double precision floating point number, the integration step can be further
determined by the range of variable transformation and the integration level.
Two functions with their known analytical FHTs are used to validate the
implementation of the FHT via DE scheme. The surface map and 2D contour of the
FHT transformation error with respect to integration level and the range of the
variable transformation are used to numerically determine the optimal numbers
for a fast FHT. Results: Given a specific precision, the lowest
integration level and the optimal range of variable transformation, which are
used to transform a signal with a certain degree of fluctuation, can be
numerically determined by the surface map and 2D contour of the standard
deviation of transformation error. These two numbers can then be taken to
efficiently compute the FHT for other signals with the same or less degree of
fluctuation. Conclusions: The FHT via DE scheme and the numerical method
to determine the integration level and the range of transformation can be used
for fast FHT in certain applications, such as data filtering in chord-line
based CT reconstruction algorithms.
