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Introduction

If helical boundary conditions (Claerbout, 1998b) are imposed on a multi-dimensional system, an isomorphism exists between that system and an equivalent one-dimensional system. Previous authors, for example Claerbout (1998a), take advantage of this isomorphism to perform rapid multi-dimensional inverse filtering by recursion.

The Fourier analogue of convolution is multiplication: to convolve a 2-D signal with a 2-D filter, take their 2-D Fourier transforms, multiply them together and return to the original domain. The relationship between 1-D and 2-D convolution, FFT's and the helix is illustrated in Figure 1. With helical boundary conditions, we can take advantage of the isomorphism described above, and perform multi-dimensional convolutions by wrapping multi-dimensional signals and filters onto a helix, taking their 1-D FFT's, multiplying them together, and then returning to the original domain.

If we can use 1-D FFT's to do 2-D convolutions, the isomorphism due to the helical boundary conditions must extend into the Fourier domain. In this paper, we explore the relationship between 1-D and multi-dimensional FFT's in helical coordinate systems. Specifically we demonstrate the link between the wavenumber vector, ${\bf k}$, in a multi-dimensional system, and the wavenumber of a helical 1-D FFT, $k_h$.

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Figure 1.
Relationship between 1-D and 2-D convolution, FFT's and the helical boundary conditions.
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next up previous [pdf]

Next: Theory Up: Rickett & Guitton: Helical Previous: Rickett & Guitton: Helical

2013-03-03