Fast finite Hilbert transform via DE quadrature scheme

The finite Hilbert transform (FHT) or inverse finite Hilbert transform (IFHT) is recently found to have some important applications in computerized tomography (CT) arena, 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.

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.

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.

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.