Abstract
The Fourier convolution theorem (FCT) is used to filter images in k-space. The literature provides theoretical proof but not experimental evidence of the extension of FCT to Z-space. Using two-dimensional images, this research presents empirical evidence of: 1. the extension of the FCT to Z-space, and 2. the difference between k-space filtering and Z-space filtering. The following procedure is carried out using FCT and extension of FCT to Z-space. The convolving function is sampled in image space and on the Argand plane. Multiplication between convolving function and sinc function determines the so-called ‘sinc-shaped convolving function’ and precedes direct Fourier and direct Z transformations so to obtain a rectangular region in the frequency domain of the sinc-shaped convolving function. The spatial size of the rectangular region is determined by the numerical values of bandwidth and sampling rate. Pointwise multiplication between the frequency domain of sinc-shaped convolving function and frequency domain of departing image is inverse Fourier and inverse Z transformed so as to obtain k-space and Z-space filtered images. Frequency domain analysis of filtered images shows the difference between k-space filtering and Z-space filtering. Using the FCT as theoretical framework, the novelty of this research is to empirically extend FCT to Z-space and to report experimental evidence of the difference between k-space filtering and Z-space filtering.
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Software is freely available to the public upon request.
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The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.
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Acknowledgements
The authors express sincere gratitude for the MR images provided by Dr. Dimitar Veljanovski and Dr. Filip A. Risteski. The author is also very grateful to Professor Ustijana Rechkoska Shikoska because of the coordination of the human resources. Dr. Dimitar Veljanovski and Dr. Filip A. Risteski are affiliated with the Department of Radiology at the General Hospital 8-mi Septemvri located in Boulevard 8th September in the city of Skopje—Republic of North Macedonia. Professor Ustijana Rechkoska Shikoska is the Vice Rector of University of Information Science and Technology (UIST) located in Partizanska BB in the city of Ohrid—Republic of North Macedonia.
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CC involved in conceptualization, software, data collection, writing, and editing; IX involved in data collection, data curation, and writing.
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MRI acquisition was consented. Approval was obtained from subjects to use the images for research through the administration of written consent. The research protocol for MRI data acquisition was approved by the Department of Radiology at the General Hospital 8-mi Septemvri located in Boulevard 8th September in the city of Skopje—Republic of North Macedonia. This study was conducted according to principles stated in Helsinki declaration of the year 1964.
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Ciulla, C., Xhaferri, I. Empirical evidence of the extension of the Fourier convolution theorem to Z-space. SIViP 17, 2889–2896 (2023). https://doi.org/10.1007/s11760-023-02509-y
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DOI: https://doi.org/10.1007/s11760-023-02509-y