Sampling technique for Fourier convolution theorem based k‐space filtering

Abstract

This research presents a sampling technique for Fourier convolution theorem (FCT) based k‐space filtering. One polynomial function and three transfer functions were selected: (1) Gaussian, (2) Bessel, (3) Butterworth, and (4) Chebyshev. The functions were sampled on the image grid, and they are called filtering functions. Each filtering function was multiplied by the Sinc function to obtain the “<jats:italic>Sinc‐shaped convolving function</jats:italic>.” The k‐space of the Sinc‐shaped convolving function is calculated by direct Fourier transform (FT) and it is featured by a central region. This central region of the k‐space is rectangular in its shape because it is consequential to the direct FT of the product between the filtering function and the Sinc function. Low‐pass and high‐pass filtering is obtained by inverse FT of the pointwise multiplication between the k‐space of the departing image and the k‐space of the Sinc‐shaped convolving function. A variety of cut‐off frequencies, bandwidth, sampling rates, and numbers of poles of the filters were verified as effective to filter the images. Filtering strength can be modified by fine‐tuning the size of the central rectangular k‐space region of the Sinc‐shaped convolving functions. K‐space analysis of departing images and filtered images provide additional evidence of effective filtering. Moreover, k‐space filtering was compared to Z‐space filtering using the extension of the FCT to Z‐space. The novelty of this research is the sampling technique used to determine the Sinc‐shaped convolving function. The sampling technique uses fine‐tuning of bandwidth and sampling rate to determine the strength of the k‐space filter.