Empirical evidence of the extension of the Fourier convolution theorem to Z-space

Ciulla, Carlo; Xhaferri, Ilva

doi:10.1007/s11760-023-02509-y

Cited by 2 publications

(1 citation statement)

References 17 publications

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“…Equation () is formulated by grouping the two Cartesian coordinates dx and dy into a single variable, scaling the variable by the number of pixels, and linking the variable to the cut‐off frequency ω 0 . This formulation is consistent with the one reported earlier for comparisons between k‐space filtering and Z‐space filtering [12].…”

Section: Methodssupporting

confidence: 90%

Sampling technique for Fourier convolution theorem based k‐space filtering

et al. 2023

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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.

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Section: Methodssupporting

confidence: 90%

Sampling technique for Fourier convolution theorem based k‐space filtering

et al. 2023

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Two-dimensional image zooming by adaptive Sinc-based technique

Ciulla,

Shabani,

Doda

et al. 2023

Innov. Emerg. Technol.

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This article reports a novel adaptive Sinc-based zooming technique. The theoretical basis of the zooming technique is the Whittaker–Shannon interpolation formula and the Nyquist–Shannon sampling theorem. The technique can be used to zoom out, zoom in, and reconstruct two-dimensional images with no zoom. The technique introduces a novel concept called pixel-local-scaled k-space, which is calculated as follows. The k-space magnitude and the bandwidth of the image to zoom are calculated. The bandwidth of the frequency components of the image to zoom is estimated by the difference between the maximum and the minimum numerical value of the k-space magnitude. The ratio between the standardised k-space magnitude of the image and the bandwidth is calculated pixel-by-pixel, and is called pixel-local-scaled k-space. The signal is reconstructed using a finite convolution between image pixel intensities and Sinc functions. The zooming factor is consequential to the numerical difference between the pixel-local-scaled k-space and the sampling rate. The novelty of this zooming technique is to be adaptive because the pixel-local-scaled k-space is a pixel-by-pixel map, that is, an image with the same number of pixels as the departing image. Adaptive Sinc-based zooming is validated through the comparison with non-adaptive Sinc-based zooming, and its smoothing effect is compared to the smoothing of a variety of image space and z-space filters. The adaptive Sinc-based zooming technique achieves image zooming reconstruction using sampling rates within an extended range versus the range allowed by non-adaptive Sinc-based zooming.

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Empirical evidence of the extension of the Fourier convolution theorem to Z-space

Cited by 2 publications

References 17 publications

Sampling technique for Fourier convolution theorem based k‐space filtering

Sampling technique for Fourier convolution theorem based k‐space filtering

Two-dimensional image zooming by adaptive Sinc-based technique

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