Sinc Functions for K-Space and Z-Space Adaptive Filtering of Magnetic Resonance Imaging of the Human Brain

Sampling at the Radon image grid determines sinusoidal waves that embed low-pass filtering characteristics. This paper describes and evaluates a newly proposed frequency domain adaptive filter by the name of Sinc Filter (SF). The Sinc Filter consists of an image whose pixels are the numerical values of the Sinc function computed at the Radon image space coordinates. The Sinc filter has two configurations. The first configuration is determined by numerical values of bandwidth and sampling rate that obey the sampling theorem. The second configuration is determined by the adaptive bandwidth and the value of the sampling rate. The pixel-local scaled k-space is the measure of the adaptive bandwidth, and it consists of the normalized k-space magnitude of the departing image. The Sinc Filter is then Fourier transformed or Z-transformed to obtain a frequency domain filter. Image smoothing is determined through: 1. The Fourier convolution theorem, which uses the frequency domain called k-space, and 2. The extension of the Fourier convolution theorem to Z-space, which uses the frequency domain called Z-space. For the two configurations, SF performance is compared to the performance of six frequency domain filters. Three filters make use of the k-space, and three more make use of the Z-space. The filters use Bessel, Butterworth, and Chebyshev based computed transfer functions. The Sinc Filter can smooth the images. The novelty of this paper is the adaptive frequency domain filtering procedure.