Mathematical equivalence of zero-padding interpolation and circular sampling theorem interpolation with implications for direct Fourier image reconstruction

Patrick J. La Riviere; Xiaochuan Pan

doi:10.1117/12.310838

24 June 1998 Mathematical equivalence of zero-padding interpolation and circular sampling theorem interpolation with implications for direct Fourier image reconstruction

Patrick J. La Riviere, Xiaochuan Pan

Author Affiliations +

Proceedings Volume 3338, Medical Imaging 1998: Image Processing; (1998) https://doi.org/10.1117/12.310838
Event: Medical Imaging '98, 1998, San Diego, CA, United States

Abstract

The speed and accuracy of Direct Fourier image reconstruction methods have long been hampered by the need to interpolate between the polar grid of Fourier data that is obtained from the measured projection data and the Cartesian grid of Fourier data that is needed to recover an image using the 2D FFT. Fast but crude interpolation schemes such as bilinear interpolation often lead to unacceptable image artifacts, while more sophisticated but computationally intense techniques such as circular sampling theorem (CST) interpolation negate the speed advantages afforded by the use of the 2D FFT. One technique that has been found to yield high-quality images without much computational penalty is a hybrid one in which zero-padding interpolation is first used to increase the density of samples on the polar grid after which bilinear interpolation onto the Cartesian grid is performed. In this work, we attempt to account for the success of this approach relative to the CST approach in three ways. First and more importantly, we establish that zero-padding interpolation of periodic functions that are sampled in accordance with the Nyquist criterion--precisely the sort of function encountered in the angular dimension of the polar grid--is exact and equivalent to circular sampling theorem interpolation. Second, we point out that both approaches make comparable approximations in interpolating in the radial direction. Finally, we indicate that the error introduced by the bilinear interpolation step in the zero- padding approach can be minimized by choosing sufficiently large zero-padding factors.

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Patrick J. La Riviere and Xiaochuan Pan "Mathematical equivalence of zero-padding interpolation and circular sampling theorem interpolation with implications for direct Fourier image reconstruction", Proc. SPIE 3338, Medical Imaging 1998: Image Processing, (24 June 1998); https://doi.org/10.1117/12.310838

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