0.1 Two-dimensional Fourier transform imaging

A two-dimensional Fourier transform (2D-FT) is computed numerically or carried out in two stages, both involving ‘standard’, one-dimensional Fourier transforms. However, the second stage Fourier transform is not the inverse Fourier transform (which would result in the original function that was transformed at the first stage), but a Fourier transform in a second variable– which is ‘shifted’ in value– relative to that involved in the result of the first Fourier transform. Such 2D-FT analysis is a very powerful method for three-dimensional reconstruction of polymer and biopolymer structures by two-dimensional Nuclear Magnetic resonance (2D-FT NMR , [1]) of solutions for molecular weights (M_w) of the dissolved polymers up to about 50,000 M_w. For larger biopolymers or polymers, more complex methods have been developed to obtain the desired resolution needed for the 3D-reconstruction of higher molecular structures, e.g. for 900, 000M_w, methods that can also be utilized in vivo. The 2D-FT method is also widely utilized in optical spectroscopy, such as 2D-FT NIR Hyperspectral Imaging, or in MRI imaging for research and clinical, diagnostic applications in Medicine.

A more precise mathematical definition of the ‘double’ Fourier transform involved is specified next, and a precise example follows the definition.

Example 0.1 A 2D Fourier transformation and phase correction is applied to a set of 2D NMR (FID) signals s(t₁,t₂) yielding a real 2D-FT NMR ‘spectrum’ (collection of 1D FT-NMR spectra) represented by a matrix S whose elements are

where ν₁ and ν₂ denote the discrete indirect double-quantum and single-quantum(detection) axes, respectively, in the 2D NMR experiments. Next, the covariance matrix is calculated in the frequency domain according to the following equation:

with ν₂,ν₂′ taking all possible single-quantum frequency values and with the summation carried out over all discrete, double quantum frequencies ν₁.

0.2 Example 0.2

Atomic structure reconstruction by 2D-FT of STEM Images(obtained at Cornell University) reveals the electron distributions in a high-temperature cuprate superconductor ‘paracrystal’; both the domains (or ‘location’) and the local symmetry of the “pseudo-gap” are seen in the electron-pair correlation band responsible for the high–temperature superconductivity effect .

0.3 Remarks

So far there have been three Nobel prizes awarded for 2D-FT NMR/MRI during 1992-2003, and an additional, earlier Nobel prize for 2D-FT of X-ray data (‘CAT scans’); recently the advanced possibilities of 2D-FT techniques in Chemistry, Physiology and Medicine received very significant recognition.

References