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 "Sinc-shaped convolving function." 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 highpass 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 Zspace. 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.