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These questions are concerned with k-space.

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Symmetrical K-Space

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Why do we keep k-space symmetrical (N x N) such that N is a power of 2, even when we aren't acquiring all of k-space?

It enables us to perform a Fast Fourier Transform (FFT) instead of a regular (discrete) Fourier transform (DFT). The FFT is much faster. It actually reduces the number of calculations that are required. A regular FT on a symmetric matrix requires N2 calculations, whereas the FFT requires Nlog2N calculations. For a 256x256 matrix, this is a reduction of the number of calculations required by a factor of 32! DFT: 65536 calculations, FFT: 2048 calculations.

If we do not fill all of k-space in the acquisition, we can fill the rest of the data matrix with zeros (e.g. when we reduce the scan percentage), or simulate the data from other parts of k-space because of its conjugate symmetry (e.g. when using partial Fourier). In this way, the FFT can be used.

Further reading on this topic:
Books: MRI The Basics p153, Q&A in MRI p71-74

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